New York State Standards for Mathematics:
Currently Perma-Bound only has suggested titles for grades K-8 in the Science and Social Studies areas. We are working on expanding this.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
K.PS.1. Explore, examine, and make observations about a social problem or mathematical situation.
K.PS.2. Interpret information correctly, identify the problem, and generate possible solutions.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
K.PS.3. Act out or model with manipulatives activities involving mathematical content from literature and/or story telling.
K.PS.4. Formulate problems and solutions from everyday situations (e.g., counting the number of children in the class, using the calendar to teach counting).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
K.PS.5. Use informal counting strategies to find solutions.
K.PS.6. Experience teacher-directed questioning process to understand problems.
K.PS.7. Compare and discuss ideas for solving a problem with teacher and/or students to justify their thinking.
K.PS.8. Use manipulatives (e.g., tiles, blocks) to model the action in problems.
K.PS.9. Use drawings/pictures to model the action in problems.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
K.PS.10. Explain to others how a problem was solved, giving strategies.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
K.RP.1. Understand that mathematical statements can be true or false.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
K.RP.2. Investigate the use of knowledgeable guessing as a mathematical tool.
K.RP.3. Explore guesses, using a variety of objects and manipulatives.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
K.RP.4. Listen to claims other students make.
3.8. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
K.CM.1. Understand how to organize their thought processes with teacher guidance.
3.9. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
K.CM.2. Share mathematical ideas through the manipulation of objects, drawings, pictures, and verbal explanations.
K.CM.3. Listen to solutions shared by other students.
K.CM.4. Formulate mathematically relevant questions with teacher guidance.
3.10. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
K.CM.5. Use appropriate mathematical terms, vocabulary, and language.
3.11. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
K.CN.1. Recognize the presence of mathematics in their daily lives.
K.CN.2. Use counting strategies to solve problems in their daily lives.
K.CN.3. Recognize and apply mathematics to objects and pictures.
3.12. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
K.R.1. Use multiple representations, including verbal language, acting out or modeling a situation, and drawing pictures as representations.
K.R.2. Use standard and nonstandard representations.
3.13. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
K.R.3. Use objects to show and understand physical phenomena (e.g., guess the number of cookies in a package).
K.R.4. Use objects to show and understand social phenomena (e.g., count and represent sharing cookies between friends).
K.R.5. Use objects to show and understand mathematical phenomena (e.g., draw pictures to show a story problem, show number value using fingers on your hand).
3.14. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
K.N.1. Number Systems: Count the items in a collection and know the last counting word tells how many items are in the collection (1 to 10).
K.N.2. Number Systems: Count out (produce) a collection of a specified size 1 to 10.
K.N.3. Number Systems: Numerically label a data set of 1 to 5.
K.N.4. Number Systems: Verbally count by 1's to 20.
K.N.5. Number Systems: Verbally count backwards from 10.
K.N.6. Number Systems: Represent collections with a finger pattern up to 10.
K.N.7. Number Systems: Draw pictures or other informal symbols to represent a spoken number up to 10.
K.N.8. Number Systems: Draw pictures or other informal symbols to represent how many in a collection up to 10.
K.N.9. Number Systems: Write numbers 1-10 to represent a collection.
K.N.10. Number Systems: Visually determine how many more or less, and then using the verbal counting sequence, match and count 1-10.
K.N.11. Number Systems: Use and understand verbal ordinal terms, first to tenth.
3.15. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
K.N.12. Operations: Solve and create addition and subtraction verbal word problems (use counting-based strategies, such as counting on and to ten).
K.N.13. Operations: Determine sums and differences by various means.
3.16. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
K.A.1. Patterns, Relations, and Functions: Use a variety of manipulatives to create patterns using attributes of color, size, or shape.
K.A.2. Patterns, Relations, and Functions: Recognize, describe, extend, and create patterns that repeat (e.g., ABABAB or ABAABAAAB).
3.17. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
K.G.1. Shapes: Describe characteristics and relationships of geometric objects.
3.18. Geometry Strand: Students will identify and justify geometric relationships, formally and informally.
K.G.2. Geometric: Sort groups of objects by size and size order (increasing and Relationships decreasing).
3.19. Geometry Strand: Students will apply transformations and symmetry to analyze problem solving situations.
K.G.3. Transformational Geometry: Explore vertical and horizontal orientation of objects.
K.G.4. Transformational Geometry: Manipulate two- and three-dimensional shapes to explore symmetry.
3.20. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
K.G.5. Coordinate Geometry Understand and use ideas such as over, under, above, below, on, beside, next to, and between.
3.21. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
K.M.1. Units of Measurement Name, discuss, and compare attributes of length (longer than, shorter than).
K.M.2. Units of Measurement Compare the length of two objects by representing each length with string or a paper strip.
K.M.3. Units of Measurement Relate specific times such as morning, noon, afternoon, and evening to activities and absence or presence of daylight.
3.22. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
K.S.1. Collection of Data Gather data in response to questions posed by the teacher and students.
K.S.2. Organization and Display of Data: Help to make simple pictographs for quantities up to 10, where one picture represents 1.
K.S.3. Organization and Display of Data Sort and organize objects by two attributes (e.g., color, size, or shape).
K.S.4. Organization and Display of Data: Represent data using manipulatives.
K.S.5. Analysis of Data: Identify more, less, and same amounts from pictographs or concrete models.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
1.PS.1. Explore, examine, and make observations about a social problem or mathematical situation.
1.PS.2. Interpret information correctly, identify the problem, and generate possible solutions.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
1.PS.3. Act out or model with manipulatives activities involving mathematical content from literature and/or story telling.
1.PS.4. Formulate problems and solutions from everyday situations (e.g., counting the number of children in the class or using the calendar to teach counting).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
1.PS.5. Use informal counting strategies to find solutions.
1.PS.6. Experience teacher-directed questioning process to understand problems.
1.PS.7. Compare and discuss ideas for solving a problem with teacher and/or students to justify their thinking.
1.PS.8. Use manipulatives (e.g., tiles, blocks) to model the action in problems.
1.PS.9. Use drawings/pictures to model the action in problems.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
1.PS.10. Explain to others how a problem was solved, giving strategies and justifications.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
1.RP.1. Understand that mathematical statements can be true or false.
1.RP.2. Recognize that mathematical ideas need to be supported by evidence.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
1.RP.3. Investigate the use of knowledgeable guessing as a mathematical tool.
1.RP.4. Explore guesses, using a variety of objects and manipulatives.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
1.RP.5. Justify general claims, using manipulatives.
1.RP.6. Develop and explain an argument verbally or with objects.
1.RP.7. Listen to and discuss claims other students make.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
1.RP.8. Use trial and error strategies to verify claims.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
1.CM.1. Understand how to organize their thought processes with teacher guidance.
1.CM.2 Verbally support their reasoning and answer.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
1.CM.3. Share mathematical ideas through the manipulation of objects, drawings, pictures, charts, and symbols in both written and verbal explanations.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
1.CM.4. Listen to solutions shared by other students.
1.CM.5. Formulate mathematically relevant questions.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
1.CM.6. Use appropriate mathematical terms, vocabulary, and language.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
1.CN.1. Recognize the connections of patterns in their everyday experiences to mathematical ideas.
1.CN.2. Understand the connections between numbers and the quantities they represent.
1.CN.3. Compare the similarities and differences of mathematical ideas.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
1.CN.4. Understand how models of situations involving objects, pictures, and symbols relate to mathematical ideas.
1.CN.5. Understand meanings of operations and how they relate to one another.
1.CN.6 Understand how mathematical models represent quantitative relationships.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
1.CN.7. Recognize the presence of mathematics in their daily lives.
1.CN.8. Recognize and apply mathematics to solve problems.
1.CN.9. Recognize and apply mathematics to objects, pictures, and symbols.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
1.R.1. Use multiple representations including verbal and written language, acting out or modeling a situation, drawings, and/or symbols as representations.
1.R.2. Share mental images of mathematical ideas and understandings.
1.R.3. Use standard and nonstandard representations.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
1.R.4. Connect mathematical representations with problem solving.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
1.R.5. Use mathematics to show and understand physical phenomena (e.g., estimate and represent the number of apples in a tree).
1.R.6. Use mathematics to show and understand social phenomena (e.g., count and represent sharing cookies between friends).
1.R.7. Use mathematics to show and understand mathematical phenomena (e.g., draw pictures to show a story problem, show number value using fingers on your hand).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
1.N.1. Number Systems: Count the items in a collection and know the last counting word tells how many items are in the collection (1 to 100).
1.N.2. Number Systems: Count out (produce) a collection of a specified size (10 to 100 items), using groups of ten.
1.N.3. Number Systems: Quickly see and label with a number, collections of 1 to 10.
1.N.4. Number Systems: Count by 1's to 100.
1.N.5. Number Systems: Skip count by 10's to 100.
1.N.6. Number Systems: Skip count by 5's to 50.
1.N.7. Number Systems: Skip count by 2's to 20.
1.N.8. Number Systems: Verbally count from a number other than one by 1's.
1.N.9. Number Systems: Count backwards from 20 by 1's.
1.N.10. Number Systems: Draw pictures or other informal symbols to represent a spoken number up to 20.
1.N.11. Number Systems: Identify that spacing of the same number of objects does not affect the quantity (conservation).
1.N.12. Number Systems: Arrange objects in size order (increasing and decreasing).
1.N.13. Number Systems: Write numbers to 100.
1.N.14. Number Systems: Read the number words one, two, three...ten.
1.N.15. Number Systems: Explore and use place value.
1.N.16. Number Systems: Compare and order whole numbers up to 100.
1.N.17. Number Systems: Develop an initial understanding of the base ten system: 10 ones = 1 ten; 10 tens = 1 hundred.
1.N.18. Number Systems: Use a variety of strategies to compose and decompose one-digit numbers.
1.N.19. Number Systems: Understand the commutative property of addition.
1.N.20. Number Systems: Name the number before and the number after a given number, and name the number(s) between two given numbers up to 100 (with and without the use of a number line or a hundreds chart).
1.N.21. Number Systems: Use before, after, or between to order numbers to 100 (with or without the use of a number line).
1.N.22. Number Systems: Use the words higher, lower, greater, and less to compare two numbers.
1.N.23. Number Systems: Use and understand verbal ordinal terms, first to twentieth.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
1.N.24. Operations: Develop and use strategies to solve addition and subtraction word problems.
1.N.25. Operations: Represent addition and subtraction word problems and their solutions as number sentences.
1.N.26. Operations: Create problem situations that represent a given number sentence.
1.N.27. Operations: Use a variety of strategies to solve addition and subtraction problems with one- and two-digit numbers without regrouping.
1.N.28. Operations: Demonstrate fluency and apply addition and subtraction facts to and including 10.
1.N.29. Operations: Understand that different parts can be added to get the same whole.
3.21. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
1.N.30. Estimation: Estimate the number in a collection to 50 and then compare by counting the actual items in the collection.
3.22. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
1.A.1. Patterns, Relations, and Functions: Determine and discuss patterns in arithmetic (what comes next in a repeating pattern, using numbers or objects).
3.23. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
1.G.1. Shapes: Match shapes and parts of shapes to justify congruency.
1.G.2. Shapes: Recognize, name, describe, create, sort, and compare two-dimensional and three-dimensional shapes.
3.24. Geometry Strand: Students will apply transformations and symmetry to analyze problem solving situations.
1.G.3. Transformational Geometry: Experiment with slides, flips, and turns of two-dimensional shapes.
1.G.4. Transformational Geometry: Identify symmetry in two-dimensional shapes.
3.25. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
1.G.5. Coordinate Geometry: Recognize geometric shapes and structures in the environment.
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
1.M.1. Units of Measurement: Recognize length as an attribute that can be measured.
1.M.2. Units of Measurement: Use non-standard units (including finger lengths, paper clips, students' feet and paces) to measure both vertical and horizontal lengths.
1.M.3. Units of Measurement: Informally explore the standard unit of measure, inch.
3.27. Measurement Strand: Students will use units to give meaning to measurements.
1.M.4. Units: Know vocabulary and recognize coins (penny, nickel, dime, quarter).
1.M.5. Units: Recognize the cent notation as cents.
1.M.6. Units: Use different combinations of coins to make money amounts up to 25 cents.
1.M.7. Units: Recognize specific times (morning, noon, afternoon, evening).
1.M.8. Units: Tell time to the hour, using both digital and analog clocks.
1.M.9. Units: Know the days of the week and months of the year in sequence.
1.M.10. Units: Classify months and connect to seasons and other events.
3.28. Measurement Strand: Students will develop strategies for estimating measurements.
1.M.11. Estimation: Select and use non-standard units to estimate measurements.
3.29. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
1.S.1. Collection of Data: Pose questions about themselves and their surrounding.
1.S.2. Collection of Data: Collect and record data related to a question.
1.S.3. Organization and Display of Data: Display data in simple pictographs for quantities up to 20 with units of one.
1.S.4. Organization and Display of Data: Display data in bar graphs using concrete objects with intervals of one.
1.S.5. Organization and Display of Data: Use Venn diagrams to sort and describe data.
1.S.6. Analysis of Data: Interpret data in terms of the words: most, least, greater than, less than, or equal to.
1.S.7. Analysis of Data: Interpret Answer simple questions related to data displayed in pictographs (e.g., category with most, how many more in a category compared to another, how many all together in two categories).
3.30. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
1.S.8. Predictions from Data: Discuss conclusions and make predictions in terms of the words likely and unlikely.
1.S.9. Predictions from Data: Construct a question that can be answered by using information from a graph.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
2.PS.1. Explore, examine, and make observations about a social problem or mathematical situation.
2.PS.2. Interpret information correctly, identify the problem, and generate possible solutions.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
2.PS.3. Act out or model with manipulatives activities involving mathematical content from literature and/or story telling.
2.PS.4. Formulate problems and solutions from everyday situations (e.g., counting the number of children in the class, using the calendar to teach counting).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
2.PS.5. Use informal counting strategies to find solutions.
2.PS.6. Experience teacher-directed questioning process to understand problems.
2.PS.7. Compare and discuss ideas for solving a problem with teacher and/or students to justify their thinking.
2.PS.8. Use manipulatives (e.g., tiles, blocks) to model the action in problems.
2.PS.9. Use drawings/pictures to model the action in problems.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
2.PS.10. Explain to others how a problem was solved, giving strategies and justifications.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
2.RP.1. Understand that mathematical statements can be true or false.
2.RP.2. Recognize that mathematical ideas need to be supported by evidence.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
2.RP.3. Investigate the use of knowledgeable guessing as a mathematical tool.
2.RP.4. Explore guesses, using a variety of objects and manipulatives.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
2.RP.5. Justify general claims, using manipulatives.
2.RP.6. Develop and explain an argument verbally or with objects.
2.RP.7. Listen to and discuss claims other students make.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
2.RP.8. Use trial and error strategies to verify claims.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
2.CM.1. Understand how to organize their thought processes.
2.CM.2. Verbally support their reasoning and answer.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
2.CM.3. Share mathematical ideas through the manipulation of objects, drawings, pictures, charts, and symbols in both written and verbal explanations.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
2.CM.4. Listen to solutions shared by other students.
2.CM.5. Formulate mathematically relevant questions.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
2.CM.6. Use appropriate mathematical terms, vocabulary, and language.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
2.CN.1. Recognize the connections of patterns in their everyday experiences to mathematical ideas.
2.CN.2. Understand and use the connections between numbers and the quantities they represent to solve problems.
2.CN.3. Compare the similarities and differences of mathematical ideas.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
2.CN.4. Understand how models of situations involving objects, pictures, and symbols relate to mathematical ideas.
2.CN.5. Understand meanings of operations and how they relate to one another.
2.CN.6. Understand how mathematical models represent quantitative relationships.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
2.CN.7. Recognize the presence of mathematics in their daily lives.
2.CN.8. Recognize and apply mathematics to solve problems.
2.CN.9. Recognize and apply mathematics to objects, pictures and symbols.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
2.R.1. Use multiple representations, including verbal and written language, acting out or modeling a situation, drawings, and/or symbols as representations.
2.R.2. Share mental images of mathematical ideas and understandings.
2.R.3. Use standard and nonstandard representations.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
2.R.4. Connect mathematical representations with problem solving.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
2.R.5. Use mathematics to show and understand physical phenomena (e.g., estimate and represent the number of apples in a tree).
2.R.6. Use mathematics to show and understand social phenomena (e.g., count and represent sharing cookies between friends).
2.R.7. Use mathematics to show and understand mathematical phenomena (e.g., draw pictures to show a story problem or show number value using fingers on your hand).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
2.N.1. Number Systems Skip count to 100 by 2's, 5's, 10's.
2.N.2. Number Systems Count back from 100 by 1's, 5's, 10's using a number chart.
2.N.3. Number Systems Skip count by 3's to 36 for multiplication readiness.
2.N.4. Number Systems Skip count by 4's to 48 for multiplication readiness.
2.N.5. Number Systems Compare and order numbers to 100.
2.N.6. Number Systems Develop an understanding of the base ten system: 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand.
2.N.7. Number Systems Use a variety of strategies to compose and decompose two-digit numbers.
2.N.8. Number Systems Understand and use the commutative property of addition.
2.N.9. Number Systems Name the number before and the number after a given number, and name the number(s) between two given numbers up to 100 (with and without the use of a number line or a hundreds chart).
2.N.10. Number Systems Use and understand verbal ordinal terms.
2.N.11. Number Systems Read written ordinal terms (first through ninth) and use them to represent ordinal relations.
2.N.12. Number Systems Use zero as the identity element for addition.
2.N.13. Number Systems Recognize the meaning of zero in the place value system (0-100).
2.N.14. Number Theory: Use concrete materials to justify a number as odd or even.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
2.N.15. Operations: Determine sums and differences of number sentences by various means (e.g., families, related facts, inverse operations, addition doubles, and doubles plus one).
2.N.16. Operations: Use a variety of strategies to solve addition and subtraction problems using one- and two-digit numbers with and without regrouping.
2.N.17. Operations: Demonstrate fluency and apply addition and subtraction facts up to and including 18.
2.N.18. Operations: Use doubling to add 2-digit numbers.
2.N.19. Operations: Use compensation to add 2-digit numbers.
2.N.20. Operations: Develop readiness for multiplication by using repeated addition.
2.N.21. Operations: Develop readiness for division by using repeated subtraction, dividing objects into groups (fair share).
3.21. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
2.N.22. Estimation: Estimate the number in a collection to 100 and then compare by counting the actual items in the collection.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
2.A.1. Equations and Inequalities: Use the symbols <, >, = (with and without the use of a number line) to compare whole numbers up to 100.
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
2.A.2. Patterns, Relations, and Functions: Describe and extend increasing or decreasing (+,-) sequences and patterns (numbers or objects up to 100).
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
2.G.1. Shapes: Experiment with slides, flips, and turns to compare two dimensional shapes.
2.G.2. Shapes: Identify and appropriately name two-dimensional shapes: circle, square, rectangle, and triangle (both regular and irregular).
2.G.3. Shapes: Compose (put together) and decompose (break apart) two-dimensional shapes.
3.25. Geometry Strand: Students will identify and justify geometric relationships, formally and informally.
2.G.4. Geometric Relationships: Group objects by like properties.
3.26. Geometry Strand: Students will apply transformations and symmetry to analyze problem solving situations.
2.G.5. Transformational Geometry: Explore and predict the outcome of slides, flips, and turns of two-dimensional shapes.
2.G.6. Transformational Geometry: Explore line symmetry.
3.27. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
2.M.1. Units of Measurement: Use non-standard and standard units to measure both vertical and horizontal lengths.
2.M.2. Units of Measurement: Use a ruler to measure standard units (including whole inches and whole feet).
2.M.3. Units of Measurement: Compare and order objects according to the attribute of length.
2.M.4. Units of Measurement: Recognize mass as a qualitative measure (e.g., Which is heavier? Which is lighter?).
2.M.5. Units of Measurement: Compare and order objects, using lighter than and heavier than.
3.28. Measurement Strand: Students will use units to give meaning to measurements.
2.M.6. Units: Know and recognize coins (penny, nickel, dime, quarter) and bills ($1, $5, $10, and $20).
2.M.7. Units: Recognize the whole dollar notation as $1, etc.
2.M.8. Units: Identify equivalent combinations to make one dollar.
2.M.9. Units: Tell time to the half hour and five minutes using both digital and analog clocks.
3.29. Measurement Strand: Students will develop strategies for estimating measurements.
2.M.10. Estimation: Select and use standard (customary) and non-standard units to estimate measurements.
3.30. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
2.S.1. Collection of Data: Formulate questions about themselves and their surroundings.
2.S.2. Collection of Data: Collect and record data (using tallies) related to the question.
2.S.3. Organization and Display of Data: Display data in pictographs and bar graphs using concrete objects or a representation of the object.
2.S.4. Analysis of Data Compare and interpret data in terms of describing quantity (similarity or differences).
3.31. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
2.S.5. Predictions from Data: Discuss conclusions and make predictions from graphs.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
3.PS.1. Explore, examine, and make observations about a social problem or mathematical situation.
3.PS.2. Understand that some ways of representing a problem are more helpful than others.
3.PS.3. Interpret information correctly, identify the problem, and generate possible solutions.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
3.PS.4. Act out or model with manipulatives activities involving mathematical content from literature.
3.PS.5. Formulate problems and solutions from everyday situations.
3.PS.6. Translate from a picture/diagram to a numeric expression.
3.PS.7. Represent problem situations in oral, written, concrete, pictorial, and graphical forms.
3.PS.8. Select an appropriate representation of a problem.
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
3.PS.9. Use trial and error to solve problems.
3.PS.10. Use process of elimination to solve problems.
3.PS.11. Make pictures/diagrams of problems.
3.PS.12. Use physical objects to model problems.
3.PS.13. Work in collaboration with others to solve problems.
3.PS.14. Make organized lists to solve numerical problems.
3.PS.15. Make charts to solve numerical problems.
3.PS.16. Analyze problems by identifying relationships.
3.PS.17. Analyze problems by identifying relevant versus irrelevant information.
3.PS.18. Analyze problems by observing patterns.
3.PS.19. The student will state a problem in their own words.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
3.PS.20. Determine what information is needed to solve a problem.
3.PS.21. Discuss with peers to understand a problem situation.
3.PS.22. Discuss the efficiency of different representations of a problem.
3.PS.23. Verify results of a problem.
3.PS.24. Recognize invalid approaches.
3.PS.25. Determine whether a solution is reasonable in the context of the original problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
3.RP.1. Use representations to support mathematical ideas.
3.RP.2. Determine whether a mathematical statement is true or false and explain why.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
3.RP.3. Investigate the use of knowledgeable guessing by generalizing mathematical ideas.
3.RP.4. Make conjectures from a variety of representations.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
3.RP.5. Justify general claims or conjectures, using manipulatives, models, and expressions.
3.RP.6. Develop and explain an argument using oral, written, concrete, pictorial, and/or graphical forms.
3.RP.7. Discuss, listen, and make comments that support or reject claims made by other students.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
3.RP.8. Support an argument by trying many cases.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
3.CM.1. Understand and explain how to organize their thought process.
3.CM.2. Verbally explain their rationale for strategy selection.
3.CM.3. Provide reasoning both in written and verbal form.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
3.CM.4. Organize and accurately label work.
3.CM.5. Share organized mathematical ideas through the manipulation of objects, drawings, pictures, charts, graphs, tables, diagrams, models, symbols, and expressions in written and verbal form.
3.CM.6. Answer clarifying questions from others.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
3.CM.7. Listen for understanding of mathematical solutions shared by other students.
3.CM.8. Consider strategies used and solutions found in relation to their own work.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
3.CM.9. Increase their use of mathematical vocabulary and language when communicating with others.
3.CM.10. Describe objects, relationships, solutions and rationale using appropriate vocabulary.
3.CM.11. Decode and comprehend mathematical visuals and symbols to construct meaning.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
3.CN.1. Recognize, understand, and make connections in their everyday experiences to mathematical ideas.
3.CN.2. Compare and contrast mathematical ideas.
3.CN.3. Connect and apply mathematical information to solve problems.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
3.CN.4. Understand multiple representations and how they are related.
3.CN.5. Model situations with objects and representations and be able to make observations.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
3.CN.6. Recognize the presence of mathematics in their daily lives.
3.CN.7. Apply mathematics to solve problems that develop outside of mathematics.
3.CN.8. Recognize and apply mathematics to other disciplines.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
3.R.1. Use verbal and written language, physical models, drawing charts, graphs, tables, symbols, and equations as representations.
3.R.2. Share mental images of mathematical ideas and understandings.
3.R.3. Recognize and use external mathematical representations.
3.R.4. Use standard and nonstandard representations with accuracy and detail.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
3.R.5. Understand similarities and differences in representations.
3.R.6. Connect mathematical representations with problem solving.
3.R.7. Construct effective representations to solve problems.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
3.R.8. Use mathematics to show and understand physical phenomena (e.g., estimate and represent the number of apples in a tree).
3.R.9. Use mathematics to show and understand social phenomena (e.g., determine the number of buses required for a field trip).
3.R.10. Use mathematics to show and understand mathematical phenomena (e.g., use a multiplication grid to solve odd and even number problems).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
3.N.1. Number Systems: Skip count by 25's, 50's, 100's to 1,000.
3.N.2. Number Systems: Read and write whole numbers to 1,000.
3.N.3. Number Systems: Compare and order numbers to 1,000.
3.N.4. Number Systems: Understand the place value structure of the base ten number system: 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand.
3.N.5. Number Systems: Use a variety of strategies to compose and decompose three-digit numbers.
3.N.6. Number Systems: Use and explain the commutative property of addition and multiplication.
3.N.7. Number Systems: Use 1 as the identity element for multiplication.
3.N.8. Number Systems: Use the zero property of multiplication.
3.N.9. Number Systems: Understand and use the associative property of addition.
3.N.10. Number Systems: Develop an understanding of fractions as part of a whole unit and as parts of a collection.
3.N.11. Number Systems: Use manipulatives, visual models, and illustrations to name and represent unit fractions or a set of objects.
3.N.12. Number Systems: Understand and recognize the meaning of numerator and denominator in the symbolic form of a fraction.
3.N.13. Number Systems: Recognize fractional numbers as equal parts of a whole.
3.N.14. Number Systems: Explore equivalent fractions (1/2, 1/3, 1/4).
3.N.15. Number Systems: Compare and order unit fractions (1/2, 1/3, 1/4) and find their approximate locations on a number line.
3.N.16. Number Theory: Identify odd and even numbers.
3.N.17. Number Theory: Develop an understanding of the properties of odd/even numbers as a result of addition or subtraction.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
3.N.18. Operations: Use a variety of strategies to add and subtract 3-digit numbers (with and without regrouping).
3.N.19. Operations: Develop fluency with single-digit multiplication facts.
3.N.20. Operations: Use a variety of strategies to solve multiplication problems with factors up to 12 x 12.
3.N.21. Operations: Use the area model, tables, patterns, arrays, and doubling to provide meaning for multiplication.
3.N.22. Operations: Demonstrate fluency and apply single-digit division facts.
3.N.23. Operations: Use tables, patterns, halving, and manipulatives to provide meaning for division.
3.N.24. Operations: Develop strategies for selecting the appropriate computational and operational method in problem solving situations.
3.21. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
3.N.25. Estimation: Estimate numbers up to 500.
3.N.26. Estimation: Recognize real world situations in which an estimate (rounding) is more appropriate.
3.N.27. Estimation: Check reasonableness of an answer by using estimation.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
3.A.1. Equations and Inequalities: Use the symbols <, >, = (with and without the use of a number line) to compare whole numbers and unit fractions.
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
3.A.2. Patterns, Relations, and Functions: Describe and extend numeric (+, -) and geometric patterns.
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
3.G.1. Shapes Define and use correct terminology when referring to shapes (circle, triangle, square, rectangle, rhombus, trapezoid, and hexagon).
3.G.2. Shapes Identify congruent and similar figures.
3.G.3. Shapes Name, describe, compare, and sort three-dimensional shapes: cube, cylinder, sphere, prism, and cone.
3.G.4. Shapes Identify the faces on a three-dimensional shape as two-dimensional shapes.
3.25. Geometry Strand: Students will apply transformations and symmetry to analyze problem solving situations.
3.G.5. Transformational Geometry: Identify and construct lines of symmetry.
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
3.M.1. Units of Measurement: Select tools and units (customary) appropriate for the length measured.
3.M.2. Units of Measurement: Use a ruler/yardstick to measure to the nearest standard unit (whole and 1/2 inches, whole feet, and whole yards).
3.M.3. Units of Measurement: Measure objects, using ounces and pounds.
3.M.4. Units of Measurement: Recognize capacity as an attribute that can be measured.
3.M.5. Units of Measurement: Compare capacities (e.g., Which contains more? Which contains less?).
3.M.6. Units of Measurement: Measure capacity, using cups, pints, quarts, and gallons.
3.27. Measurement Strand: Students will use units to give meaning to measurements.
3.M.7. Units: Count and represent combined coins and dollars, using currency symbols ($0.00).
3.M.8. Units: Relate unit fractions to the face of the clock: Whole = 60 minutes; 1/2 = 30 minutes; 1/4 = 15 minutes.
3.28. Measurement Strand: Students will develop strategies for estimating measurements.
3.M.9. Estimation: Tell time to the minute, using digital and analog clocks.
3.M.10. Estimation: Select and use standard (customary) and non-standard units to estimate measurements.
3.29. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
3.S.1. Collection of Data: Formulate questions about themselves and their surroundings.
3.S.2. Collection of Data: Collect data using observation and surveys, and record appropriately.
3.S.3. Organization and Display of Data: Construct a frequency table to represent a collection of data.
3.S.4. Organization and Display of Data: Identify the parts of pictographs and bar graphs.
3.S.5. Organization and Display of Data: Display data in pictographs and bar graphs.
3.S.6. Organization and Display of Data: State the relationships between pictographs and bar graphs.
3.S.7. Analysis of Data: Read and interpret data in bar graphs and pictographs.
3.30. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
3.S.8. Predictions from Data: Formulate conclusions and make predictions from graphs.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
4.PS.1. Explore, examine, and make observations about a social problem or mathematical situation.
4.PS.2. Understand that some ways of representing a problem are more helpful than others.
4.PS.3. Interpret information correctly, identify the problem, and generate possible solutions.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
4.PS.4. Act out or model with manipulatives activities involving mathematical content from literature.
4.PS.5. Formulate problems and solutions from everyday situations.
4.PS.6. Translate from a picture/diagram to a numeric expression.
4.PS.7. Represent problem situations in oral, written, concrete, pictorial, and graphical forms.
4.PS.8. Select an appropriate representation of a problem.
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
4.PS.9. Use trial and error to solve problems.
4.PS.10. Use process of elimination to solve problems.
4.PS.11. Make pictures/diagrams of problems.
4.PS.12. Use physical objects to model problems.
4.PS.13. Work in collaboration with others to solve problems.
4.PS.14. Make organized lists to solve numerical problems.
4.PS.15. Make charts to solve numerical problems.
4.PS.16. Analyze problems by identifying relationships.
4.PS.17. Analyze problems by identifying relevant versus irrelevant information.
4.PS.18. Analyze problems by observing patterns.
4.PS.19. State a problem in their own words.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
4.PS.20. Determine what information is needed to solve a problem.
4.PS.21. Discuss with peers to understand a problem situation.
4.PS.22. Discuss the efficiency of different representations of a problem.
4.PS.23. Verify results of a problem.
4.PS.24. Recognize invalid approaches.
4.PS.25. Determine whether a solution is reasonable in the context of the original problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
4.RP.1. Use representations to support mathematical ideas.
4.RP.2. Determine whether a mathematical statement is true or false and explain why.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
4.RP.3. Investigate the use of knowledgeable guessing by generalizing mathematical ideas.
4.RP.4. Make conjectures from a variety of representations.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
4.RP.5. Justify general claims or conjectures, using manipulatives, models, and expressions.
4.RP.6. Develop and explain an argument using oral, written, concrete, pictorial, and/or graphical forms.
4.RP.7. Discuss, listen, and make comments that support or reject claims made by other students.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
4.RP.8. Support an argument by trying many cases.
4.RP.9. Disprove an argument by finding counterexamples.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
4.CM.1. Understand and explain how to organize their thought process.
4.CM.2. Verbally explain their rationale for strategy selection.
4.CM.3. Provide reasoning both in written and verbal form.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
4.CM.4. Organize and accurately label work.
4.CM.5. Share organized mathematical ideas through the manipulation of objects, drawing, pictures, charts, graphs, tables, diagrams, models, symbols, and expressions in written and verbal form.
4.CM.6. Answer clarifying questions from others.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
4.CM.7. Restate mathematical solutions shared by other students.
4.CM.8. Consider strategies used and solutions found in relation to their own work.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
4.CM.9. Increase their use of mathematical vocabulary and language when communicating with others.
4.CM.10. Describe objects, relationships, solutions, and rationale using appropriate vocabulary.
4.CM.11. Decode and comprehend mathematical visuals and symbols to construct meaning.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
4.CN.1. Recognize, understand, and make connections in their everyday experiences to mathematical ideas.
4.CN.2. Compare and contrast mathematical ideas.
4.CN.3. Connect and apply mathematical information to solve problems.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
4.CN.4. Understand multiple representations and how they are related.
4.CN.5. Model situations with objects and representations and be able to make observations.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
4.CN.6. Recognize the presence of mathematics in their daily lives.
4.CN.7. Apply mathematics to solve problems that develop outside of mathematics.
4.CN.8. Recognize and apply mathematics to other disciplines.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
4.R.1. Use verbal and written language, physical models, drawing charts, graphs, tables, symbols, and equations as representations.
4.R.2. Share mental images of mathematical ideas and understandings.
4.R.3. Recognize and use external mathematical representations.
4.R.4. Use standard and nonstandard representations with accuracy and detail.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
4.R.5. Understand similarities and differences in representations.
4.R.6. Connect mathematical representations with problem solving.
4.R.7. Construct effective representations to solve problems.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
4.R.8. Use mathematics to show and understand physical phenomena (e.g., estimate and represent the number of apples in a tree).
4.R.9. Use mathematics to show and understand social phenomena (e.g., determine the number of buses required for a field trip).
4.R.10. Use mathematics to show and understand mathematical phenomena (e.g., use a multiplication grid to solve odd and even number problems).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
4.N.1. Number Systems: Skip count by 1,000's.
4.N.2. Number Systems: Read and write whole numbers to 10,000.
4.N.3. Number Systems: Compare and order numbers to 10,000.
4.N.4. Number Systems: Understand the place value structure of the base ten number system: 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand; 10 thousands = 1 ten thousand.
4.N.5. Number Systems: Recognize equivalent representations for numbers up to four digits and generate them by decomposing and composing numbers.
4.N.6. Number Systems: Understand, use, and explain the associative property of multiplication.
4.N.7. Number Systems: Develop an understanding of fractions as locations on number lines and as divisions of whole numbers.
4.N.8. Number Systems: Recognize and generate equivalent fractions (halves, fourths, thirds, fifths, sixths, and tenths) using manipulatives, visual models, and illustrations.
4.N.9. Number Systems: Use concrete materials and visual models to compare and order unit fractions or fractions with the same denominator (with and without the use of a number line).
4.N.10. Number Systems: Develop an understanding of decimals as part of a whole.
4.N.11. Number Systems: Read and write decimals to hundredths, using money as a context.
4.N.12. Number Systems: Use concrete materials and visual models to compare and order decimals (less than 1) to the hundredths place in the context of money.
4.N.13. Number Theory: Develop an understanding of the properties of odd/even numbers as a result of multiplication.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
4.N.14. Operations: Use a variety of strategies to add and subtract numbers up to 10000.
4.N.15. Operations: Select appropriate computational and operational methods to solve problems.
4.N.16. Operations: Understand various meanings of multiplication and division.
4.N.17. Operations: Use multiplication and division as inverse operations to solve problems.
4.N.18. Operations: Use a variety of strategies to multiply two-digit numbers by one digit numbers (with and without regrouping).
4.N.19. Operations: Use a variety of strategies to multiply two-digit numbers by two digit numbers (with and without regrouping).
4.N.20. Operations: Develop fluency in multiplying and dividing multiples of 10 and 100 up to 1,000.
4.N.21. Operations: Use a variety of strategies to divide two-digit dividends by one digit divisors (with and without remainders).
4.N.22. Operations: Interpret the meaning of remainders.
4.N.23. Operations: Add and subtract proper fractions with common denominators.
4.N.24. Operations: Express decimals as an equivalent form of fractions to tenths and hundredths.
4.N.25. Operations: Add and subtract decimals to tenths and hundredths using a hundreds chart.
3.21. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
4.N.26. Estimation: Round numbers less than 1,000 to the nearest tens and hundreds.
4.N.27. Estimation: Check reasonableness of an answer by using estimation.
3.22. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
4.A.1. Variables and Expressions: Evaluate and express relationships using open sentences with one operation.
3.23. Algebra Strand: Students will perform algebraic procedures accurately.
4.A.2. Equations and Inequalities: Use the symbols <, >, =, and not equal to (with and without the use of a number line) to compare whole numbers and unit fractions and decimals (up to hundredths).
4.A.3. Equations and Inequalities: Find the value or values that will make an open sentence true, if it contains < or >.
3.24. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
4.A.4. Patterns, Relations, and Functions: Describe, extend, and make generalizations about numeric (addition, subtraction, multiplication, division) and geometric patterns.
4.A.5. Patterns, Relations, and Functions: Analyze a pattern or a whole-number function and state the rule, given a table or an input/output box.
3.25. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
4.G.1. Shapes: Identify and name polygons, recognizing that their names are related to the number of sides and angles (triangle, quadrilateral, pentagon, hexagon, and octagon).
4.G.2. Shapes: Identify points and line segments when drawing a plane figure.
4.G.3. Shapes: Find perimeter of polygons by adding sides.
4.G.4. Shapes: Find the area of a rectangle by counting the number of squares needed to cover the rectangle.
4.G.5. Shapes: Define and identify vertices, faces, and edges of three-dimensional shapes.
3.26. Geometry Strand: Students will identify and justify geometric relationships, formally and informally.
4.G.6. Geometric Relationships: Draw and identify intersecting, perpendicular, and parallel lines.
4.G.7. Geometric Relationships: Identify points and rays when drawing angles.
4.G.8. Geometric Relationships: Classify angles as acute, obtuse, right, and straight.
3.27. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
4.M.1. Units of Measurement: Select tools and units (customary and metric) appropriate for the length being measured.
4.M.2. Units of Measurement: Use a ruler to measure to the nearest standard unit (whole, 1/2 and 1/4 inches, whole feet, whole yards, whole centimeters, and whole meters).
4.M.3. Units of Measurement: Know and understand equivalent standard units of length: 12 inches = 1 foot; 3 feet = 1 yard.
4.M.4. Units of Measurement: Select tools and units appropriate to the mass of the object being measured (grams and kilograms).
4.M.5. Units of Measurement: Measure mass, using grams.
4.M.6. Units of Measurement: Select tools and units appropriate to the capacity being measured (milliliters and liters).
4.M.7. Units of Measurement: Measure capacity, using milliliters and liters.
3.28. Measurement Strand: Students will use units to give meaning to measurements.
4.M.8. Units: Make change, using combined coins and dollar amounts.
4.M.9. Units: Calculate elapsed time in hours and half hours, not crossing A.M./P.M.
4.M.10. Units: Calculate elapsed time in days and weeks, using a calendar.
3.29. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
4.S.1. Collection of Data: Design investigations to address a question from given data.
4.S.2. Collection of Data: Collect data using observations, surveys, and experiments and record appropriately.
4.S.3. Organization and Display of Data: Represent data using tables, bar graphs, and pictographs.
4.S.4. Analysis of Data: Read and interpret line graphs.
3.30. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
4.S.5. Predictions from Data: Develop and make predictions that are based on data.
4.S.6. Predictions from Data: Formulate conclusions and make predictions from graphs.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
5.PS.1. Know the difference between relevant and irrelevant information when solving problems.
5.PS.2. Understand that some ways of representing a problem are more efficient than others.
5.PS.3. Interpret information correctly, identify the problem, and generate possible strategies and solutions.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
5.PS.4. Act out or model with manipulatives activities involving mathematical content from literature.
5.PS.5. Formulate problems and solutions from everyday situations.
5.PS.6. Translate from a picture/diagram to a numeric expression.
5.PS.7. Represent problem situations verbally, numerically, algebraically, and/or graphically.
5.PS.8. Select an appropriate representation of a problem.
5.PS.9. Understand the basic language of logic in mathematical situations (and, or, not).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
5.PS.10. Work in collaboration with others to solve problems.
5.PS.11. Translate from a picture/diagram to a number or symbolic expression.
5.PS.12. Use trial and error and the process of elimination to solve problems.
5.PS.13. Model problems with pictures/diagrams or physical objects.
5.PS.14. Analyze problems by observing patterns.
5.PS.15. Make organized lists or charts to solve numerical problems.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
5.PS.16. Discuss with peers to understand a problem situation.
5.PS.17. Determine what information is needed to solve problem.
5.PS.18. Determine the efficiency of different representations of a problem.
5.PS.19. Differentiate between valid and invalid approaches.
5.PS.20. Understand valid counterexamples.
5.PS.21. Explain the methods and reasoning behind the problem solving strategies used.
5.PS.22. Discuss whether a solution is reasonable in the context of the original problem.
5.PS.23. Verify results of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
5.RP.1. Recognize that mathematical ideas can be supported using a variety of strategies.
5.RP.2. Understand that mathematical statements can be supported, using models, facts, and relationships to explain their thinking.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
5.RP.3. Investigate conjectures, using arguments and appropriate mathematical terms.
5.RP.4. Make and evaluate conjectures, using a variety of strategies.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
5.RP.5. Justify general claims or conjectures, using manipulatives, models, expressions, and mathematical relationships.
5.RP.6. Develop and explain an argument verbally, numerically, and/or graphically.
5.RP.7. Verify claims other students make, using examples and counterexamples when appropriate.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
5.RP.8. Support an argument through examples/counterexamples and special cases.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
5.CM.1. Provide an organized thought process that is correct, complete, coherent, and clear.
5.CM.2. Explain a rationale for strategy selection.
5.CM.3. Organize and accurately label work.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
5.CM.4. Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form.
5.CM.5. Answer clarifying questions from others.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
5.CM.6. Understand mathematical solutions shared by other students.
5.CM.7. Raise questions that elicit, extend, or challenge others' thinking.
5.CM.8. Consider strategies used and solutions found by others in relation to their own work.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
5.CM.9. Increase their use of mathematical vocabulary and language when communicating with others.
5.CM.10. Use appropriate vocabulary when describing objects, relationships, mathematical solutions, and rationale.
5.CM.11. Decode and comprehend mathematical visuals and symbols to construct meaning.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
5.CN.1. Understand and make connections and conjectures in their everyday experiences to mathematical ideas.
5.CN.2. Explore and explain the relationship between mathematical ideas.
5.CN.3. Connect and apply mathematical information to solve problems.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
5.CN.4. Understand multiple representations and how they are related.
5.CN.5. Model situations with objects and representations and be able to draw conclusions.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
5.CN.6. Recognize and provide examples of the presence of mathematics in their daily lives.
5.CN.7. Apply mathematics to problem situations that develop outside of mathematics.
5.CN.8. Investigate the presence of mathematics in careers and areas of interest.
5.CN.9. Recognize and apply mathematics to other disciplines and areas of interest.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
5.R.1. Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations.
5.R.2. Explain, describe, and defend mathematical ideas using representations.
5.R.3. Read, interpret, and extend external models.
5.R.4. Use standard and nonstandard representations with accuracy and detail.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
5.R.5. Use representations to explore problem situations.
5.R.6. Investigate relationships between different representations and their impact on a given problem.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
5.R.7. Use mathematics to show and understand physical phenomena (e.g., determine the perimeter of a bulletin board).
5.R.8. Use mathematics to show and understand social phenomena (e.g., construct tables to organize data showing book sales).
5.R.9. Use mathematics to show and understand mathematical phenomena (e.g., find the missing value that makes the equation true: (3 + 4) + 5 = 3 + (4 + ___ ).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
5.N.1. Number Systems: Read and write whole numbers to millions.
5.N.2. Number Systems: Compare and order numbers to millions.
5.N.3. Number Systems: Understand the place value structure of the base ten number system: 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand; 10 thousands = 1 ten thousand; 10 ten thousands = 1 hundred thousand; 10 hundred thousands = 1 million.
5.N.4. Number Systems: Create equivalent fractions, given a fraction.
5.N.5. Number Systems: Compare and order fractions including unlike denominators (with and without the use of a number line) Note: Commonly used fractions such as those that might be indicated on ruler, measuring cup, etc.
5.N.6. Number Systems: Understand the concept of ratio.
5.N.7. Number Systems: Express ratios in different forms.
5.N.8. Number Systems: Read, write, and order decimals to thousandths.
5.N.9. Number Systems: Compare fractions using <, >, or =.
5.N.10. Number Systems: Compare decimals using <, >, or =.
5.N.11. Number Systems: Understand that percent means part of 100, and write percents as fractions and decimals.
5.N.12. Number Theory: Recognize that some numbers are only divisible by one and themselves (prime) and others have multiple divisors (composite).
5.N.13. Number Theory: Calculate multiples of a whole number and the least common multiple of two numbers.
5.N.14. Number Theory: Identify the factors of a given number.
5.N.15. Number Theory: Find the common factors and the greatest common factor of two numbers.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
5.N.16. Operations: Use a variety of strategies to multiply three-digit by three-digit numbers Note: Multiplication by anything greater than a three digit multiplier/ multiplicand should be done using technology.
5.N.17. Operations: Use a variety of strategies to divide three-digit numbers by one and two-digit numbers Note: Division by anything greater than a two-digit divisor should be done using technology.
5.N.18. Operations: Evaluate an arithmetic expression using order of operations including multiplication, division, addition, subtraction and parentheses.
5.N.19. Operations: Simplify fractions to lowest terms.
5.N.20. Operations: Convert improper fractions to mixed numbers, and mixed numbers to improper fractions.
5.N.21. Operations: Use a variety of strategies to add and subtract fractions with like denominators.
5.N.22. Operations: Add and subtract mixed numbers with like denominators.
5.N.23. Operations: Use a variety of strategies to add, subtract, multiply, and divide decimals to thousandths.
3.21. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
5.N.24. Estimation: Round numbers to the nearest hundredth and up to 10,000.
5.N.25. Estimation: Estimate sums and differences of fractions with like denominators.
5.N.26. Estimation: Estimate sums, differences, products, and quotients of decimals.
5.N.27. Estimation: Justify the reasonableness of answers using estimation.
3.22. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
5.A.1. Variables and Expressions: Define and use appropriate terminology when referring to constants, variables, and algebraic expressions.
5.A.2. Variables and Expressions: Translate simple verbal expressions into algebraic expressions.
3.23. Algebra Strand: Students will perform algebraic procedures accurately.
5.A.3. Variables and Expressions: Substitute assigned values into variable expressions and evaluate using order of operations.
5.A.4. Equations and Inequalities: Solve simple one-step equations using basic whole-number facts.
5.A.5. Equations and Inequalities: Solve and explain simple one-step equations using inverse operations involving whole numbers.
5.A.6. Equations and Inequalities: Evaluate the perimeter formula for given input values.
3.24. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
5.A.7. Patterns, Relations, and Functions: Create and explain patterns and algebraic relationships (e.g., 2,4,6,8...) algebraically: 2n (doubling).
5.A.8. Patterns, Relations, and Functions: Create algebraic or geometric patterns using concrete objects or visual drawings (e.g., rotate and shade geometric shapes).
3.25. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
5.G.1. Shapes: Calculate the perimeter of regular and irregular polygons.
3.26. Geometry Strand: Students will identify and justify geometric relationships, formally and informally.
5.G.2. Geometric Relationships: Identify pairs of similar triangles.
5.G.3. Geometric Relationships: Identify the ratio of corresponding sides of similar triangles.
5.G.4. Geometric Relationships: Classify quadrilaterals by properties of their angles and sides.
5.G.5. Geometric Relationships: Know that the sum of the interior angles of a quadrilateral is 360 degrees.
5.G.6. Geometric Relationships: Classify triangles by properties of their angles and sides.
5.G.7. Geometric Relationships: Know that the sum of the interior angles of a triangle is 180 degrees.
5.G.8. Geometric Relationships: Find a missing angle when given two angles of a triangle.
5.G.9. Geometric Relationships: Identify pairs of congruent triangles.
5.G.10. Geometric Relationships: Identify corresponding parts of congruent triangles.
3.27. Geometry Strand: Students will apply transformations and symmetry to analyze problem solving situations.
5.G.11. Transformational Geometry: Identify and draw lines of symmetry of basic geometric shapes.
3.28. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
5.G.12. Coordinate Geometry: Identify and plot points in the first quadrant.
5.G.13. Coordinate Geometry: Plot points to form basic geometric shapes (identify and classify).
5.G.14. Coordinate Geometry: Calculate perimeter of basic geometric shapes drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths and parallel to the axes).
3.29. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
5.M.1. Units of Measurement: Use a ruler to measure to the nearest inch.
5.M.2. Units of Measurement: Identify customary equivalent units of length.
5.M.3. Units of Measurement: Measure to the nearest centimeter.
5.M.4. Units of Measurement: Identify equivalent metric units of length.
5.M.5. Units of Measurement: Convert measurement within a given system.
5.M.6. Tools and Methods: Determine the tool and technique to measure with an appropriate level of precision: lengths and angles.
3.30. Measurement Strand: Students will use units to give meaning to measurements.
5.M.7. Units: Calculate elapsed time in hours and minutes.
5.M.8. Units: Measure and draw angles using a protractor.
3.31. Measurement Strand: Students will develop strategies for estimating measurements.
5.M.9. Estimation: Determine personal references for customary units of length (e.g., your pace is approximately 3 feet, your height is approximately 5 feet, etc.).
5.M.10. Estimation: Determine personal references for metric units of length.
5.M.11. Estimation: Justify the reasonableness of estimates.
3.32. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
5.S.1. Collection of Data: Collect and record data from a variety of sources (e.g., newspapers, magazines, polls, charts, and surveys).
5.S.2. Organization and Display of Data: Display data in a line graph to show an increase or decrease over time.
5.S.3. Analysis of Data: Calculate the mean for a given set of data and use to describe a set of data.
3.33. Statistics and Probability Strand Students will make predictions that are based upon data analysis.
5.S.4. Predictions from Data: Formulate conclusions and make predictions from graphs.
3.34. Statistics and Probability Strand Students will understand and apply concepts of probability.
5.S.5. Probability: List the possible outcomes for a single-event experiment.
5.S.6. Probability: Record experiment results using fractions/ratios.
5.S.7. Probability: Create a sample space and determine the probability of a single event, given a simple experiment (e.g., rolling a number cube).
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
6.PS.1. Know the difference between relevant and irrelevant information when solving problems.
6.PS.2. Understand that some ways of representing a problem are more efficient than others.
6.PS.3. Interpret information correctly, identify the problem, and generate possible strategies and solutions.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
6.PS.4. Act out or model with manipulatives activities involving mathematical content from literature.
6.PS.5. Formulate problems and solutions from everyday situations.
6.PS.6. Translate from a picture/diagram to a numeric expression.
6.PS.7. Represent problem situations verbally, numerically, algebraically, and/or graphically.
6.PS.8. Select an appropriate representation of a problem.
6.PS.9. Understand the basic language of logic in mathematical situations (and, or, and not).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
6.PS.10. Work in collaboration with others to solve problems.
6.PS.11. Translate from a picture/diagram to a number or symbolic expression.
6.PS.12. Use trial and error and the process of elimination to solve problems.
6.PS.13. Model problems with pictures/diagrams or physical objects.
6.PS.14. Analyze problems by observing patterns.
6.PS.15. Make organized lists or charts to solve numerical problems.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
6.PS.16. Discuss with peers to understand a problem situation.
6.PS.17. Determine what information is needed to solve problem.
6.PS.18. Determine the efficiency of different representations of a problem.
6.PS.19. Differentiate between valid and invalid approaches.
6.PS.20. Understand valid counterexamples.
6.PS.21. Explain the methods and reasoning behind the problem solving strategies used.
6.PS.22. Discuss whether a solution is reasonable in the context of the original problem.
6.PS.23. Verify results of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
6.RP.1. Recognize that mathematical ideas can be supported using a variety of strategies.
6.RP.2. Understand that mathematical statements can be supported, using models, facts, and relationships to explain their thinking.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
6.RP.3. Investigate conjectures, using arguments and appropriate mathematical terms.
6.RP.4. Make and evaluate conjectures, using a variety of strategies.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
6.RP.5. Justify general claims or conjectures, using manipulatives, models, expressions, and mathematical relationships.
6.RP.6. Develop and explain an argument verbally, numerically, algebraically, and/or graphically.
6.RP.7. Verify claims other students make, using examples and counterexamples when appropriate.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
6.RP.8. Support an argument through examples/counterexamples and special cases.
6.RP.9. Devise ways to verify results.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
6.CM.1. Provide an organized thought process that is correct, complete, coherent, and clear.
6.CM.2. Explain a rationale for strategy selection.
6.CM.3. Organize and accurately label work.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
6.CM.4. Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form.
6.CM.5. Answer clarifying questions from others.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
6.CM.6. Understand mathematical solutions shared by other students.
6.CM.7. Raise questions that elicit, extend, or challenge others' thinking.
6.CM.8. Consider strategies used and solutions found by others in relation to their own work.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
6.CM.9. Increase their use of mathematical vocabulary and language when communicating with others.
6.CM.10. Use appropriate vocabulary when describing objects, relationships, mathematical solutions, and rationale.
6.CM.11. Decode and comprehend mathematical visuals and symbols to construct meaning.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
6.CN.1. Understand and make connections and conjectures in their everyday experiences to mathematical ideas.
6.CN.2. Explore and explain the relationship between mathematical ideas.
6.CN.3. Connect and apply mathematical information to solve problems.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
6.CN.4. Understand multiple representations and how they are related.
6.CN.5. Model situations with objects and representations and be able to draw conclusions.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
6.CN.6. Recognize and provide examples of the presence of mathematics in their daily lives.
6.CN.7. Apply mathematics to problem situations that develop outside of mathematics.
6.CN.8. Investigate the presence of mathematics in careers and areas of interest.
6.CN.9. Recognize and apply mathematics to other disciplines and areas of interest.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
6.R.1. Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations.
6.R.2. Explain, describe, and defend mathematical ideas using representations.
6.R.3. Read, interpret, and extend external models.
6.R.4. Use standard and nonstandard representations with accuracy and detail.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
6.R.5. Use representations to explore problem situations.
6.R.6. Investigate relationships between different representations and their impact on a given problem.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
6.R.7. Use mathematics to show and understand physical phenomena (e.g., determine the perimeter of a bulletin board).
6.R.8. Use mathematics to show and understand social phenomena (e.g., construct tables to organize data showing book sales).
6.R.9. Use mathematics to show and understand mathematical phenomena (e.g., Find the missing value: (3 + 4) + 5 = 3 + (4 + ___ ).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
6.N.1. Number Systems: Read and write whole numbers to trillions.
6.N.2. Number Systems: Define and identify the commutative and associative properties of addition and multiplication.
6.N.3. Number Systems: Define and identify the distributive property of multiplication over addition.
6.N.4. Number Systems: Define and identify the identity and inverse properties of addition and multiplication.
6.N.5. Number Systems: Define and identify the zero property of multiplication.
6.N.6. Number Systems: Understand the concept of rate.
6.N.7. Number Systems: Express equivalent ratios as a proportion.
6.N.8. Number Systems: Distinguish the difference between rate and ratio.
6.N.9. Number Systems: Solve proportions using equivalent fractions.
6.N.10. Number Systems: Verify the proportionality using the product of the means equals the product of the extremes.
6.N.11. Number Systems: Read, write, and identify percents of a whole (0 percent to 100 percent).
6.N.12. Number Systems: Solve percent problems involving percent, rate, and base.
6.N.13. Number Systems: Define absolute value and determine the absolute value of rational numbers (including positive and negative).
6.N.14. Number Systems: Locate rational numbers on a number line (including positive and negative).
6.N.15. Number Systems: Order rational numbers (including positive and negative).
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
6.N.16. Operations: Add and subtract fractions with unlike denominators.
6.N.17. Operations: Multiply and divide fractions with unlike denominators.
6.N.18. Operations: Add, subtract, multiply, and divide mixed numbers with unlike denominators.
6.N.19. Operations: Identify the multiplicative inverse (reciprocal) of a number.
6.N.20. Operations: Represent fractions as terminating or repeating decimals.
6.N.21. Operations: Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100).
6.N.22. Operations: Evaluate numerical expressions using order of operations (may include exponents of two and three).
6.N.23. Operations: Represent repeated multiplication in exponential form.
6.N.24. Operations: Represent exponential form as repeated multiplication.
6.N.25. Operations: Evaluate expressions having exponents where the power is an exponent of one, two, or three.
3.21. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
6.N.26. Estimation: Estimate a percent of quantity (0 percent to 100 percent).
6.N.27. Estimation: Justify the reasonableness of answers using estimation (including rounding).
3.22. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
6.A.1. Variables and Expressions: Translate two-step verbal expressions into algebraic expressions.
3.23. Algebra Strand: Students will perform algebraic procedures accurately.
6.A.2. Variables and Expressions: Use substitution to evaluate algebraic expressions (may include exponents of one, two and three).
6.A.3. Equations and Inequalities: Translate two-step verbal sentences into algebraic equations.
6.A.4. Equations and Inequalities: Solve and explain two-step equations involving whole numbers using inverse operations.
6.A.5. Equations and Inequalities: Solve simple proportions within context.
6.A.6. Equations and Inequalities: Evaluate formulas for given input values (circumference, area, volume, distance, temperature, interest, etc.).
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
6.G.1. Shapes: Calculate the length of corresponding sides of similar triangles, using proportional reasoning.
6.G.2. Shapes: Determine the area of triangles and quadrilaterals (squares, rectangles, rhombi, and trapezoids) and develop formulas.
6.G.3. Shapes: Use a variety of strategies to find the area of regular and irregular polygons.
6.G.4. Shapes: Determine the volume of rectangular prisms by counting cubes and develop the formula.
6.G.5. Shapes: Identify radius, diameter, chords and central angles of a circle.
6.G.6. Shapes: Understand the relationship between the diameter and radius of a circle.
6.G.7. Shapes: Determine the area and circumference of a circle, using the appropriate formula.
6.G.8. Shapes: Calculate the area of a sector of a circle, given the measure of a central angle and the radius of the circle.
6.G.9. Shapes: Understand the relationship between the circumference and the diameter of a circle.
3.25. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
6.G.10. Coordinate Geometry: Identify and plot points in all four quadrants.
6.G.11. Coordinate Geometry: Calculate the area of basic polygons drawn on a coordinate plane (rectangles and shapes composed of rectangles having sides with integer lengths).
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
6.M.1. Units of Measurement: Measure capacity and calculate volume of a rectangular prism.
6.M.2. Units of Measurement: Identify customary units of capacity (cups, pints, quarts, and gallons).
6.M.3. Units of Measurement: Identify equivalent customary units of capacity (cups to pints, pints to quarts, and quarts to gallons).
6.M.4. Units of Measurement: Identify metric units of capacity (liter and milliliter).
6.M.5. Units of Measurement: Identify equivalent metric units of capacity (milliliter to liter and liter to milliliter).
6.M.6. Tools and Methods: Determine the tool and technique to measure with an appropriate level of precision: capacity.
3.27. Measurement Strand: Students will develop strategies for estimating measurements.
6.M.7. Estimation: Estimate volume, area, and circumference (see figures identified in geometry strand).
6.M.8. Estimation: Justify the reasonableness of estimates.
6.M.9. Estimation: Determine personal references for capacity.
3.28. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
6.S.1. Collection of Data: Develop the concept of sampling when collecting data from a population and decide the best method to collect data for a particular question.
6.S.2. Organization and Display of Data: Record data in a frequency table.
6.S.3. Organization and Display of Data: Construct Venn diagrams to sort data.
6.S.4. Organization and Display of Data: Determine and justify the most appropriate graph to display a given set of data (pictograph, bar graph, line graph, histogram, or circle graph).
6.S.5. Analysis of Data: Determine the mean, mode and median for a given set of data.
6.S.6. Analysis of Data: Determine the range for a given set of data.
6.S.7. Analysis of Data: Read and interpret graphs.
3.29. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
6.S.8. Predictions from Data: Justify predictions made from data.
3.30. Statistics and Probability Strand: Students will understand and apply concepts of probability.
6.S.9. Probability: List possible outcomes for compound events.
6.S.10. Probability: Determine the probability of dependent events.
6.S.11. Probability: Determine the number of possible outcomes for a compound event by using the fundamental counting principle and use this to determine the probabilities of events when the outcomes have equal probability.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
7.PS.1. Use a variety of strategies to understand new mathematical content and to develop more efficient methods.
7.PS.2. Construct appropriate extensions to problem situations.
7.PS.3. Understand and demonstrate how written symbols represent mathematical ideas.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
7.PS.4. Observe patterns and formulate generalizations.
7.PS.5. Make conjectures from generalizations.
7.PS.6. Represent problem situations verbally, numerically, algebraically, and graphically.
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
7.PS.7. Understand that there is no one right way to solve mathematical problems but that different methods have advantages and disadvantages.
7.PS.8. Understand how to break a complex problem into simpler parts or use a similar problem type to solve a problem.
7.PS.9. Work backwards from a solution.
7.PS.10. Use proportionality to model problems.
7.PS.11. Work in collaboration with others to solve problems.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
7.PS.12. Interpret solutions within the given constraints of a problem.
7.PS.13. Set expectations and limits for possible solutions.
7.PS.14. Determine information required to solve the problem.
7.PS.15. Choose methods for obtaining required information.
7.PS.16. Justify solution methods through logical argument.
7.PS.17. Evaluate the efficiency of different representations of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
7.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
7.RP.2. Use mathematical strategies to reach a conclusion.
7.RP.3. Evaluate conjectures by distinguishing relevant from irrelevant information to reach a conclusion or make appropriate estimates.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
7.RP.4. Provide supportive arguments for conjectures.
7.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
7.RP.6. Support an argument by using a systematic approach to test more than one case.
7.RP.7. Devise ways to verify results or use counterexamples to refute incorrect statements.
7.RP.8. Apply inductive reasoning in making and supporting mathematical conjectures.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
7.CM.1. Provide a correct, complete, coherent, and clear rationale for thought process used in problem solving.
7.CM.2. Provide an organized argument which explains rationale for strategy selection.
7.CM.3. Organize and accurately label work.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
7.CM.4. Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form.
7.CM.5. Answer clarifying questions from others.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
7.CM.6. Analyze mathematical solutions shared by others.
7.CM.7. Compare strategies used and solutions found by others in relation to their own work.
7.CM.8. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
7.CM.9. Increase their use of mathematical vocabulary and language when communicating with others.
7.CM.10. Use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
7.CM.11. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
7.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
7.CN.2. Recognize connections between subsets of mathematical ideas.
7.CN.3. Connect and apply a variety of strategies to solve problems.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
7.CN.4. Model situations mathematically, using representations to draw conclusions and formulate new situations.
7.CN.5. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
7.CN.6. Recognize and provide examples of the presence of mathematics in their daily lives.
7.CN.7. Apply mathematical ideas to problem situations that develop outside of mathematics.
7.CN.8. Investigate the presence of mathematics in careers and areas of interest.
7.CN.9. Recognize and apply mathematics to other disciplines, areas of interest, and societal issues.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
7.R.1. Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations.
7.R.2. Explain, describe, and defend mathematical ideas using representations.
7.R.3. Recognize, compare, and use an array of representational forms.
7.R.4. Explain how different representations express the same relationship.
7.R.5. Use standard and non-standard representations with accuracy and detail.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
7.R.6. Use representations to explore problem situations.
7.R.7. Investigate relationships between different representations and their impact on a given problem.
7.R.8. Use representation as a tool for exploring and understanding mathematical ideas.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
7.R.9. Use mathematics to show and understand physical phenomena (e.g., make and interpret scale drawings of figures or scale models of objects).
7.R.10. Use mathematics to show and understand social phenomena (e.g., determine profit from sale of yearbooks).
7.R.11. Use mathematics to show and understand mathematical phenomena (e.g., use tables, graphs, and equations to show a pattern underlying a function).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
7.N.1. Number Systems: Distinguish between the various subsets of real numbers (counting/natural numbers, whole numbers, integers, rational numbers, and irrational numbers).
7.N.2. Number Systems: Recognize the difference between rational and irrational numbers (e.g., explore different approximations of pi).
7.N.3. Number Systems: Place rational and irrational numbers (approximations) on a number line and justify the placement of the numbers.
7.N.4. Number Systems: Develop the laws of exponents for multiplication and division.
7.N.5. Number Systems: Write numbers in scientific notation.
7.N.6. Number Systems: Translate numbers from scientific notation into standard form.
7.N.7. Number Systems: Compare numbers written in scientific notation.
7.N.8. Number Theory: Find the common factors and greatest common factor of two or more numbers.
7.N.9. Number Theory: Determine multiples and least common multiple of two or more numbers.
7.N.10. Number Theory: Determine the prime factorization of a given number and write in exponential form.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
7.N.11. Operations: Simplify expressions using order of operations Note: Expressions may include absolute value and/or integral exponents greater than 0.
7.N.12. Operations: Add, subtract, multiply, and divide integers.
7.N.13. Operations: Add and subtract two integers (with and without the use of a number line).
7.N.14. Operations: Develop a conceptual understanding of negative and zero exponents with a base of ten and relate to fractions and decimals (e.g., 10^-2 =.01 = 1/100).
7.N.15. Operations: Recognize and state the value of the square root of a perfect square (up to 225).
7.N.16. Operations: Determine the square root of non-perfect squares using a calculator.
7.N.17. Operations: Classify irrational numbers as non-repeating/non-terminating decimals.
3.21. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
7.N.18. Estimation: Identify the two consecutive whole numbers between which the square root of a non-perfect square whole number less than 225 lies (with and without the use of a number line).
7.N.19. Estimation: Justify the reasonableness of answers using estimation.
3.22. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
7.A.1. Variables and Expressions: Translate two-step verbal expressions into algebraic expressions.
3.23. Algebra Strand: Students will perform algebraic procedures accurately.
7.A.2. Variables and Expressions: Add and subtract monomials with exponents of one.
7.A.3. Variables and Expressions: Identify a polynomial as an algebraic expression containing one or more terms.
7.A.4. Equations and Inequalities: Solve multi-step equations by combining like terms, using the distributive property, or moving variables to one side of the equation.
7.A.5. Equations and Inequalities: Solve one-step inequalities (positive coefficients only) (See 7.G.10).
7.A.6. Equations and Inequalities: Evaluate formulas for given input values (surface area, rate, and density problems).
3.24. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
7.A.7. Patterns, Relations, and Functions: Draw the graphic representation of a pattern from an equation or from a table of data.
7.A.8. Patterns, Relations, and Functions: Create algebraic patterns using charts/tables, graphs, equations, and expressions.
7.A.9. Patterns, Relations, and Functions: Build a pattern to develop a rule for determining the sum of the interior angles of polygons.
7.A.10. Patterns, Relations, and Functions: Write an equation to represent a function from a table of values.
3.25. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
7.G.1. Shapes: Calculate the radius or diameter, given the circumference or area of a circle.
7.G.2. Shapes: Calculate the volume of prisms and cylinders, using a given formula and a calculator.
7.G.3. Shapes: Identify the two-dimensional shapes that make up the faces and bases of three-dimensional shapes (prisms, cylinders, cones, and pyramids).
7.G.4. Shapes: Determine the surface area of prisms and cylinders, using a calculator and a variety of methods.
3.26. Geometry Strand: Students will identify and justify geometric relationships, formally and informally.
7.G.5. Geometric Relationships Identify the right angle, hypotenuse, and legs of a right triangle.
7.G.6. Geometric Relationships Explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem.
7.G.7. Geometric Relationships Find a missing angle when given angles of a quadrilateral.
7.G.8. Geometric Relationships Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle.
7.G.9. Geometric Relationships Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator.
3.27. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
7.G.10. Coordinate Geometry: Graph the solution set of an inequality (positive coefficients only) on a number line (See 7.A.5).
3.28. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
7.M.1. Units of Measurement: Calculate distance using a map scale.
7.M.2. Units of Measurement: Convert capacities and volumes within a given system.
7.M.3. Units of Measurement: Identify customary and metric units of mass.
7.M.4. Units of Measurement: Convert mass within a given system.
7.M.5. Units of Measurement: Calculate unit price using proportions.
7.M.6. Units of Measurement: Compare unit prices.
7.M.7. Units of Measurement: Convert money between different currencies with the use of an exchange rate table and a calculator.
7.M.8. Units of Measurement: Draw central angles in a given circle using a protractor (circle graphs).
7.M.9. Tools and Methods: Determine the tool and technique to measure with an appropriate level of precision: mass.
3.29. Measurement Strand: Students will develop strategies for estimating measurements.
7.M.10. Estimation: Identify the relationships between relative error and magnitude when dealing with large numbers (e.g., money, population).
7.M.11. Estimation: Estimate surface area.
7.M.12. Estimation: Determine personal references for customary /metric units of mass.
7.M.13. Estimation: Justify the reasonableness of the mass of an object.
3.30. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
7.S.1. Collection of Data: Identify and collect data using a variety of methods.
7.S.2. Organization and Display of Data: Display data in a circle graph.
7.S.3. Organization and Display of Data: Convert raw data into double bar graphs and double line graphs.
7.S.4. Analysis of Data: Calculate the range for a given set of data.
7.S.5. Analysis of Data: Select the appropriate measure of central tendency.
7.S.6. Analysis of Data: Read and interpret data represented graphically (pictograph, bar graph, histogram, line graph, double line/bar graphs or circle graph).
3.31. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
7.S.7. Predictions from Data: Identify and explain misleading statistics and graphs.
3.32. Statistics and Probability Strand: Students will understand and apply concepts of probability.
7.S.8. Probability: Interpret data to provide the basis for predictions and to establish experimental probabilities.
7.S.9. Probability: Determine the validity of sampling methods to predict outcomes.
7.S.10. Probability: Predict the outcome of an experiment.
7.S.11. Probability: Design and conduct an experiment to test predictions.
7.S.12. Probability: Compare actual results to predicted results.
NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
8.PS.1. Use a variety of strategies to understand new mathematical content and to develop more efficient methods.
8.PS.2. Construct appropriate extensions to problem situations.
8.PS.3. Understand and demonstrate how written symbols represent mathematical ideas.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
8.PS.4. Observe patterns and formulate generalizations.
8.PS.5. Make conjectures from generalizations.
8.PS.6. Represent problem situations verbally, numerically, algebraically, and graphically.
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
8.PS.7. Understand that there is no one right way to solve mathematical problems but that different methods have advantages and disadvantages.
8.PS.8. Understand how to break a complex problem into simpler parts or use a similar problem type to solve a problem.
8.PS.9. Work backwards from a solution.
8.PS.10. Use proportionality to model problems.
8.PS.11. Work in collaboration with others to solve problems.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
8.PS.12. Interpret solutions within the given constraints of a problem.
8.PS.13. Set expectations and limits for possible solutions.
8.PS.14. Determine information required to solve the problem.
8.PS.15. Choose methods for obtaining required information.
8.PS.16. Justify solution methods through logical argument.
8.PS.17. Evaluate the efficiency of different representations of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
8.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
8.RP.2. Use mathematical strategies to reach a conclusion.
8.RP.3. Evaluate conjectures by distinguishing relevant from irrelevant information to reach a conclusion or make appropriate estimates.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
8.RP.4. Provide supportive arguments for conjectures.
8.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
8.RP.6. Support an argument by using a systematic approach to test more than one case.
8.RP.7. Devise ways to verify results or use counterexamples to refute incorrect statements.
8.RP.8. Apply inductive reasoning in making and supporting mathematical conjectures.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
8.CM.1. Provide a correct, complete, coherent, and clear rationale for thought process used in problem solving.
8.CM.2. Provide an organized argument which explains rationale for strategy selection.
8.CM.3. Organize and accurately label work.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
8.CM.4. Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form.
8.CM.5. Answer clarifying questions from others.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
8.CM.6. Analyze mathematical solutions shared by others.
8.CM.7. Compare strategies used and solutions found by others in relation to their own work.
8.CM.8. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
8.CM.9. Increase their use of mathematical vocabulary and language when communicating with others.
8.CM.10. Use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
8.CM.11. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
8.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
8.CN.2. Recognize connections between subsets of mathematical ideas.
8.CN.3. Connect and apply a variety of strategies to solve problems.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
8.CN.4. Model situations mathematically, using representations to draw conclusions and formulate new situations.
8.CN.5. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
8.CN.6. Recognize and provide examples of the presence of mathematics in their daily lives.
8.CN.7. Apply mathematical ideas to problem situations that develop outside of mathematics.
8.CN.8. Investigate the presence of mathematics in careers and areas of interest.
8.CN.9. Recognize and apply mathematics to other disciplines, areas of interest, and societal issues.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
8.R.1. Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations.
8.R.2. Explain, describe, and defend mathematical ideas using representations.
8.R.3. Recognize, compare, and use an array of representational forms.
8.R.4. Explain how different representations express the same relationship.
8.R.5. Use standard and non-standard representations with accuracy and detail.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
8.R.6. Use representations to explore problem situations.
8.R.7. Investigate relationships between different representations and their impact on a given problem.
8.R.8. Use representation as a tool for exploring and understanding mathematical ideas.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
8.R.9. Use mathematics to show and understand physical phenomena (e.g., make and interpret scale drawings of figures or scale models of objects).
8.R.10. Use mathematics to show and understand social phenomena (e.g., determine profit from sale of yearbooks).
8.R.11. Use mathematics to show and understand mathematical phenomena (e.g., use tables, graphs, and equations to show a pattern underlying a function).
3.19. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
8.N.1. Operations: Develop and apply the laws of exponents for multiplication and division.
8.N.2. Operations: Evaluate expressions with integral exponents.
8.N.3. Operations: Read, write, and identify percents less than 1 percent and greater than 100 percent.
8.N.4. Operations: Apply percents to: Tax; Percent increase/decrease; Simple interest; Sale price; Commission; Interest rates; Gratuities.
3.20. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.
8.N.5. Estimation: Estimate a percent of quantity, given an application.
8.N.6. Estimation: Justify the reasonableness of answers using estimation.
3.21. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
8.A.1. Variables and Expressions: Translate verbal sentences into algebraic inequalities.
8.A.2. Variables and Expressions: Write verbal expressions that match given mathematical expressions.
8.A.3. Variables and Expressions: Describe a situation involving relationships that matches a given graph.
8.A.4. Variables and Expressions: Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship.
8.A.5. Variables and Expressions: Use physical models to perform operations with polynomials.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
8.A.6. Variables and Expressions: Multiply and divide monomials.
8.A.7. Variables and Expressions: Add and subtract polynomials (integer coefficients).
8.A.8. Variables and Expressions: Multiply a binomial by a monomial or a binomial (integer coefficients).
8.A.9. Variables and Expressions: Divide a polynomial by a monomial (integer coefficients). Note: The degree of the denominator is less than or equal to the degree of the numerator for all variables.
8.A.10. Variables and Expressions: Factor algebraic expressions using the GCF.
8.A.11. Variables and Expressions: Factor a trinomial in the form ax to the power of 2 + bx + c; a=1 and c having no more than three sets of factors.
8.A.12. Equations and Inequalities: Apply algebra to determine the measure of angles formed by or contained in parallel lines cut by a transversal and by intersecting lines.
8.A.13. Equations and Inequalities: Solve multi-step inequalities and graph the solution set on a number line.
8.A.14. Equations and Inequalities: Solve linear inequalities by combining like terms, using the distributive property, or moving variables to one side of the inequality (include multiplication or division of inequalities by a negative number).
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
8.A.15. Patterns, Relations, and Functions: Understand that numerical information can be represented in multiple ways: arithmetically, algebraically, and graphically.
8.A.16. Patterns, Relations, and Functions: Find a set of ordered pairs to satisfy a given linear numerical pattern (expressed algebraically); then plot the ordered pairs and draw the line.
8.A.17. Patterns, Relations, and Functions: Define and use correct terminology when referring to function (domain and range).
8.A.18. Patterns, Relations, and Functions: Determine if a relation is a function.
8.A.19. Patterns, Relations, and Functions: Interpret multiple representations using equation, table of values, and graph.
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
8.G.0. Constructions: Construct the following, using a straight edge and compass: Segment congruent to a segment; Angle congruent to an angle; Perpendicular bisector; Angle bisector.
3.25. Algebra Strand: Students will identify and justify geometric relationships, formally and informally.
8.G.1. Geometric Relationships: Identify pairs of vertical angles as congruent.
8.G.2. Geometric Relationships: Identify pairs of supplementary and complementary angles.
8.G.3. Geometric Relationships: Calculate the missing angle in a supplementary or complementary pair.
8.G.4. Geometric Relationships: Determine angle pair relationships when given two parallel lines cut by a transversal.
8.G.5. Geometric Relationships: Calculate the missing angle measurements when given two parallel lines cut by a transversal.
8.G.6. Geometric Relationships: Calculate the missing angle measurements when given two intersecting lines and an angle.
3.26. Algebra Strand: Students will apply transformations and symmetry to analyze problem solving situations.
8.G.7. Transformational Geometry: Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations).
8.G.8. Transformational Geometry: Draw the image of a figure under rotations of 90 and 180 degrees.
8.G.9. Transformational Geometry: Draw the image of a figure under a reflection over a given line.
8.G.10. Transformational Geometry: Draw the image of a figure under a translation.
8.G.11. Transformational Geometry: Draw the image of a figure under a dilation.
8.G.12. Transformational Geometry: Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation.
3.27. Algebra Strand: Students will apply coordinate geometry to analyze problem solving situations.
8.G.13. Coordinate Geometry: Determine the slope of a line from a graph and explain the meaning of slope as a constant rate of change.
8.G.14. Coordinate Geometry: Determine the y-intercept of a line from a graph and be able to explain the y-intercept.
8.G.15. Coordinate Geometry: Graph a line using a table of values.
8.G.16. Coordinate Geometry: Determine the equation of a line given the slope and the y-intercept.
8.G.17. Coordinate Geometry: Graph a line from an equation in slope-intercept form (y=mx+b).
8.G.18. Coordinate Geometry: Solve systems of equations graphically (only linear, integral solutions, y=mx+b format, no vertical/horizontal lines).
8.G.19. Coordinate Geometry: Graph the solution set of an inequality on a number line.
8.G.20. Coordinate Geometry: Distinguish between linear and nonlinear equations ax to the power of 2 + bx + c; a=1 (only graphically).
8.G.21. Coordinate Geometry: Recognize the characteristics of quadratics in tables, graphs, equations, and situations.
3.28. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
8.M.1. Units of Measurement: Solve equations/proportions to convert to equivalent measurements within metric and customary measurement systems. Note: Also allow Fahrenheit to Celsius and vice versa.
NY.3. Integrated Algebra: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
A.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
A.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
A.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A.PS.6. Use a variety of strategies to extend solution methods to other problems.
A.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
A.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A.PS.9. Interpret solutions within the given constraints of a problem.
A.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
A.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
A.RP.2. Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture.
A.RP.3. Recognize when an approximation is more appropriate than an exact answer.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
A.RP.4. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A.RP.5. Construct logical arguments that verify claims or counterexamples that refute them.
A.RP.6. Present correct mathematical arguments in a variety of forms.
A.RP.7. Evaluate written arguments for validity.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
A.RP.8. Support an argument by using a systematic approach to test more than one case.
A.RP.9. Devise ways to verify results or use counterexamples to refute incorrect statements.
A.RP.10. Extend specific results to more general cases.
A.RP.11. Use a Venn diagram to support a logical argument.
A.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
A.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
A.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A.CM.4. Explain relationships among different representations of a problem.
A.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
A.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A.CM.8. Reflect on strategies of others in relation to one's own strategy.
A.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
A.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A.CM.11. Represent word problems using standard mathematical notation.
A.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
A.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
A.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A.CN.5. Understand how quantitative models connect to various physical models and representations.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
A.CN.6. Recognize and apply mathematics to situations in the outside world.
A.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A.CN.8. Develop an appreciation for the historical development of mathematics.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
A.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A.R.2. Recognize, compare, and use an array of representational forms.
A.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
A.R.4. Select appropriate representations to solve problem situations.
A.R.5. Investigate relationships between different representations and their impact on a given problem.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
A.R.6. Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground).
A.R.7. Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales).
A.R.8. Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations y = x to the power of 2 and y = -x to the power of 2).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
A.N.1. Number Theory: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
A.N.2. Operations: Simplify radical terms (no variable in the radicand).
A.N.3. Operations: Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form.
A.N.4. Operations: Understand and use scientific notation to compute products and quotients of numbers.
A.N.5. Operations: Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation.
A.N.6. Operations: Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s).
A.N.7. Operations: Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting.
A.N.8. Operations: Determine the number of possible arrangements (permutations) of a list of items.
3.21. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
A.A.1. Variables and Expressions: Translate a quantitative verbal phrase into an algebraic expression.
A.A.2. Variables and Expressions: Write a verbal expression that matches a given mathematical expression.
A.A.3. Equations and Inequalities: Distinguish the difference between an algebraic expression and an algebraic equation.
A.A.4. Equations and Inequalities: Translate verbal sentences into mathematical equations or inequalities.
A.A.5. Equations and Inequalities: Write algebraic equations or inequalities that represent a situation.
A.A.6. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable.
A.A.7. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables.
A.A.8. Equations and Inequalities: Analyze and solve verbal problems that involve quadratic equations.
A.A.9. Equations and Inequalities: Analyze and solve verbal problems that involve exponential growth and decay.
A.A.10. Equations and Inequalities: Solve systems of two linear equations in two variables algebraically (See A.G.7).
A.A.11. Equations and Inequalities: Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
A.A.12. Variables and Expressions: Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only.
A.A.13. Variables and Expressions: Add, subtract, and multiply monomials and polynomials.
A.A.14. Variables and Expressions: Divide a polynomial by a monomial or binomial, where the quotient has no remainder.
A.A.15. Variables and Expressions: Find values of a variable for which an algebraic fraction is undefined.
A.A.16. Variables and Expressions: Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms.
A.A.17. Variables and Expressions: Add or subtract fractional expressions with monomial or like binomial denominators.
A.A.18. Variables and Expressions: Multiply and divide algebraic fractions and express the product or quotient in simplest form.
A.A.19. Variables and Expressions: Identify and factor the difference of two perfect squares.
A.A.20. Variables and Expressions: Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF).
A.A.21. Equations and Inequalities: Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable.
A.A.22. Equations and Inequalities: Solve all types of linear equations in one variable.
A.A.23. Equations and Inequalities: Solve literal equations for a given variable.
A.A.24. Equations and Inequalities: Solve linear inequalities in one variable.
A.A.25. Equations and Inequalities: Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable.
A.A.26. Equations and Inequalities: Solve algebraic proportions in one variable which result in linear or quadratic equations.
A.A.27. Equations and Inequalities: Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots.
A.A.28. Equations and Inequalities: Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression.
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
A.A.29. Patterns, Relations, and Functions: Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form.
A.A.30. Patterns, Relations, and Functions: Find the complement of a subset of a given set, within a given universe.
A.A.31. Patterns, Relations, and Functions: Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets).
A.A.32. Coordinate Geometry: Explain slope as a rate of change between dependent and independent variables.
A.A.33. Coordinate Geometry: Determine the slope of a line, given the coordinates of two points on the line.
A.A.34. Coordinate Geometry: Write the equation of a line, given its slope and the coordinates of a point on the line.
A.A.35. Coordinate Geometry: Write the equation of a line, given the coordinates of two points on the line.
A.A.36. Coordinate Geometry: Write the equation of a line parallel to the x- or y-axis.
A.A.37. Coordinate Geometry: Determine the slope of a line, given its equation in any form.
A.A.38. Coordinate Geometry: Determine if two lines are parallel, given their equations in any form.
A.A.39. Coordinate Geometry: Determine whether a given point is on a line, given the equation of the line.
A.A.40. Coordinate Geometry: Determine whether a given point is in the solution set of a system of linear inequalities.
A.A.41. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 ).
A.A.42. Trigonometric Functions: Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides.
A.A.43. Trigonometric Functions: Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle.
A.A.44. Trigonometric Functions: Find the measure of a side of a right triangle, given an acute angle and the length of another side.
A.A.45. Trigonometric Functions: Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides.
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
A.G.1. Shapes: Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
A.G.2. Shapes: Use formulas to calculate volume and surface area of rectangular solids and cylinders.
3.25. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
A.G.3. Coordinate Geometry: Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations.
A.G.4. Coordinate Geometry: Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions.
A.G.5. Coordinate Geometry: Investigate and generalize how changing the coefficients of a function affects its graph.
A.G.6. Coordinate Geometry: Graph linear inequalities.
A.G.7. Coordinate Geometry: Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10).
A.G.8. Coordinate Geometry: Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions.
A.G.9. Coordinate Geometry: Solve systems of linear and quadratic equations graphically. Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.
A.G.10. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41 ) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
A.M.1. Units of Measurement: Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail).
A.M.2. Units of Measurement: Solve problems involving conversions within measurement systems, given the relationship between the units.
3.27. Measurement Strand: Understand that all measurement contains error and be able to determine its significance.
A.M.3. Error and Magnitude: Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure.
3.28. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
A.S.1. Organization and Display of Data: Categorize data as qualitative or quantitative.
A.S.2. Organization and Display of Data: Determine whether the data to be analyzed is univariate or bivariate.
A.S.3. Organization and Display of Data: Determine when collected data or display of data may be biased.
A.S.4. Organization and Display of Data: Compare and contrast the appropriateness of different measures of central tendency for a given data set.
A.S.5. Organization and Display of Data: Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data.
A.S.6. Organization and Display of Data: Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box and- whisker plot.
A.S.7. Organization and Display of Data: Create a scatter plot of bivariate data.
A.S.8. Organization and Display of Data: Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line.
A.S.9. Analysis of Data: Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot.
A.S.10. Analysis of Data: Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions.
A.S.11. Analysis of Data: Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles.
A.S.12. Analysis of Data: Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none).
A.S.13. Analysis of Data: Understand the difference between correlation and causation.
A.S.14. Analysis of Data: Identify variables that might have a correlation but not a causal relationship.
3.29. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
A.S.15. Predictions from Data: Identify and describe sources of bias and its effect, drawing conclusions from data.
A.S.16. Predictions from Data: Recognize how linear transformations of one-variable data affect the data's mean, median, mode, and range.
A.S.17. Predictions from Data: Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation.
3.30. Statistics and Probability Strand: Students will understand and apply concepts of probability.
A.S.18. Probability: Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
A.S.19. Probability: Determine the number of elements in a sample space and the number of favorable events.
A.S.20. Probability: Calculate the probability of an event and its complement.
A.S.21. Probability: Determine empirical probabilities based on specific sample data.
A.S.22. Probability: Determine, based on calculated probability of a set of events, if some or all are equally likely to occur; one is more likely to occur than another; whether or not an event is certain to happen or not to happen.
A.S.23. Probability: Calculate the probability of a series of independent events; a series of dependent events; two mutually exclusive events; two events that are not mutually exclusive.
G.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
G.PS.2. Observe and explain patterns to formulate generalizations and conjectures.
G.PS.3. Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations).
G.PS.4. Construct various types of reasoning, arguments, justifications and methods of proof for problems.
G.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
G.PS.6. Use a variety of strategies to extend solution methods to other problems.
G.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
G.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
G.PS.9. Interpret solutions within the given constraints of a problem.
G.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
G.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
G.RP.2. Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies.
G.RP.3. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
G.RP.4. Provide correct mathematical arguments in response to other students' conjectures, reasoning, and arguments.
G.RP.5. Present correct mathematical arguments in a variety of forms.
G.RP.6. Evaluate written arguments for validity.
G.RP.7. Construct a proof using a variety of methods (e.g., deductive, analytic, transformational).
G.RP.8. Devise ways to verify results or use counterexamples to refute incorrect statements.
G.RP.9. Apply inductive reasoning in making and supporting mathematical conjectures.
G.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
G.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
G.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
G.CM.4. Explain relationships among different representations of a problem.
G.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
G.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
G.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
G.CM.8. Reflect on strategies of others in relation to one's own strategy.
G.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
G.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
G.CM.11. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams.
G.CM.12. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
G.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
G.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
G.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
G.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
G.CN.5. Understand how quantitative models connect to various physical models and representations.
G.CN.6. Recognize and apply mathematics to situations in the outside world.
G.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
G.CN.8. Develop an appreciation for the historical development of mathematics.
G.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
G.R.2. Recognize, compare, and use an array of representational forms.
G.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
G.R.4. Select appropriate representations to solve problem situations.
G.R.5. Investigate relationships between different representations and their impact on a given problem.
G.R.6. Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank).
G.R.7. Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person's argument have a logical foundation).
G.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent).
G.G.1. Geometric Relationships: Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
G.G.2. Geometric Relationships: Know and apply that through a given point there passes one and only one plane perpendicular to a given line.
G.G.3. Geometric Relationships: Know and apply that through a given point there passes one and only one line perpendicular to a given plane.
G.G.4. Geometric Relationships: Know and apply that two lines perpendicular to the same plane are coplanar.
G.G.5. Geometric Relationships: Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.
G.G.6. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.
G.G.7. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
G.G.8. Geometric Relationships: Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines.
G.G.9. Geometric Relationships: Know and apply that if two planes are perpendicular to the same line, they are parallel.
G.G.10. Geometric Relationships: Know and apply that the lateral edges of a prism are congruent and parallel.
G.G.11. Geometric Relationships: Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal.
G.G.12. Geometric Relationships: Know and apply that the volume of a prism is the product of the area of the base and the altitude.
G.G.13. Geometric Relationships: Apply the properties of a regular pyramid, including lateral edges are congruent; lateral faces are congruent isosceles triangles; volume of a pyramid equals one-third the product of the area of the base and the altitude.
G.G.14. Geometric Relationships: Apply the properties of a cylinder, including bases are congruent; volume equals the product of the area of the base and the altitude; lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base.
G.G.15. Geometric Relationships: Apply the properties of a right circular cone, including lateral area equals one-half the product of the slant height and the circumference of its base volume is one-third the product of the area of its base and its altitude.
G.G.16. Geometric Relationships: Apply the properties of a sphere, including the intersection of a plane and a sphere is a circle; a great circle is the largest circle that can be drawn on a sphere; two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles; surface area is 4 pi r to the power of 2; volume is 4/3 pi r to the power of 3.
G.G.17. Constructions: Construct a bisector of a given angle, using a straightedge and compass, and justify the construction.
G.G.18. Constructions: Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction.
G.G.19. Constructions: Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction.
G.G.20. Constructions: Construct an equilateral triangle, using a straightedge and compass, and justify the construction.
G.G.21. Locus: Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles.
G.G.22. Locus: Solve problems using compound loci.
G.G.23. Locus: Graph and solve compound loci in the coordinate plane.
G.G.24. Informal and Formal Proofs: Determine the negation of a statement and establish its truth value.
G.G.25. Informal and Formal Proofs: Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true.
G.G.26. Informal and Formal Proofs: Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences.
G.G.27. Informal and Formal Proofs: Write a proof arguing from a given hypothesis to a given conclusion.
G.G.28. Informal and Formal Proofs: Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles.
G.G.29. Informal and Formal Proofs: Identify corresponding parts of congruent triangles.
G.G.30. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle.
G.G.31. Informal and Formal Proofs: Investigate, justify, and apply the isosceles triangle theorem and its converse.
G.G.32. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem.
G.G.33. Informal and Formal Proofs: Investigate, justify, and apply the triangle inequality theorem.
G.G.34. Informal and Formal Proofs: Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle.
G.G.35. Informal and Formal Proofs: Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines.
G.G.36. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons.
G.G.37. Informal and Formal Proofs: Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons.
G.G.38. Informal and Formal Proofs: Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals.
G.G.39. Informal and Formal Proofs: Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals.
G.G.40. Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals.
G.G.41. Informal and Formal Proofs: Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids.
G.G.42. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle.
G.G.43. Informal and Formal Proofs: Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1.
G.G.44. Informal and Formal Proofs: Establish similarity of triangles, using the following theorems: AA, SAS, and SSS.
G.G.45. Informal and Formal Proofs: Investigate, justify, and apply theorems about similar triangles.
G.G.46. Informal and Formal Proofs: Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle.
G.G.47. Informal and Formal Proofs: Investigate, justify, and apply theorems about mean proportionality: the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse; the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg.
G.G.48. Informal and Formal Proofs: Investigate, justify, and apply the Pythagorean theorem and its converse.
G.G.49. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding chords of a circle: perpendicular bisectors of chords the relative lengths of chords as compared to their distance from the center of the circle.
G.G.50. Informal and Formal Proofs: Investigate, justify, and apply theorems about tangent lines to a circle: a perpendicular to the tangent at the point of tangency; two tangents to a circle from the same external point; common tangents of two non-intersecting or tangent circles.
G.G.51. Informal and Formal Proofs: Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is inside the circle (two chords); on the circle (tangent and chord); outside the circle (two tangents, two secants, or tangent and secant).
G.G.52. Informal and Formal Proofs: Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines.
G.G.53. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding segments intersected by a circle: along two tangents from the same external point; along two secants from the same external point; along a tangent and a secant from the same external point; along two intersecting chords of a given circle.
G.G.54. Transformational Geometry: Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
G.G.55. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections.
G.G.56. Transformational Geometry: Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism.
G.G.57. Transformational Geometry: Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections).
G.G.58. Transformational Geometry: Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries).
G.G.59. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under similarities.
G.G.60. Transformational Geometry: Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism.
G.G.61. Transformational Geometry: Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90 degrees and 180 degrees, reflections over the lines x = 0, y = 0, and y = x, and dilations centered at the origin.
G.G.62. Coordinate Geometry: Find the slope of a perpendicular line, given the equation of a line.
G.G.63. Coordinate Geometry: Determine whether two lines are parallel, perpendicular, or neither, given their equations.
G.G.64. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line.
G.G.65. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line.
G.G.66. Coordinate Geometry: Find the midpoint of a line segment, given its endpoints.
G.G.67. Coordinate Geometry: Find the length of a line segment, given its endpoints.
G.G.68. Coordinate Geometry: Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment.
G.G.69. Coordinate Geometry: Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas.
G.G.70. Coordinate Geometry: Solve systems of equations involving one linear equation and one quadratic equation graphically.
G.G.71. Coordinate Geometry: Write the equation of a circle, given its center and radius or given the endpoints of a diameter.
G.G.72. Coordinate Geometry: Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer.
G.G.73. Coordinate Geometry: Find the center and radius of a circle, given the equation of the circle in center-radius form.
G.G.74. Coordinate Geometry: Graph circles of the form (x - h) to the power of 2 + (j - k) to the power of 2 = r to the power of 2.
A2.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A2.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
A2.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A2.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
A2.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A2.PS.6. Use a variety of strategies to extend solution methods to other problems.
A2.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
A2.PS.8. Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A2.PS.9. Interpret solutions within the given constraints of a problem.
A2.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
A2.RP.1. Support mathematical ideas using a variety of strategies.
A2.RP.2. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
A2.RP.3. Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer.
A2.RP.4. Recognize when an approximation is more appropriate than an exact answer.
A2.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A2.RP.6. Construct logical arguments that verify claims or counterexamples that refute claims.
A2.RP.7. Present correct mathematical arguments in a variety of forms.
A2.RP.8. Evaluate written arguments for validity.
A2.RP.9. Support an argument by using a systematic approach to test more than one case.
A2.RP.10. Devise ways to verify results, using counterexamples and informal indirect proof.
A2.RP.11. Extend specific results to more general cases.
A2.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
A2.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A2.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A2.CM.4. Explain relationships among different representations of a problem.
A2.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A2.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
A2.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A2.CM.8. Reflect on strategies of others in relation to one's own strategy.
A2.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
A2.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A2.CM.11. Represent word problems using standard mathematical notation.
A2.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A2.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
A2.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A2.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
A2.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A2.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A2.CN.5. Understand how quantitative models connect to various physical models and representations.
A2.CN.6. Recognize and apply mathematics to situations in the outside world.
A2.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A2.CN.8. Develop an appreciation for the historical development of mathematics.
A2.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A2.R.2. Recognize, compare, and use an array of representational forms.
A2.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
A2.R.4. Select appropriate representations to solve problem situations.
A2.R.5. Investigate relationships among different representations and their impact on a given problem.
A2.R.6. Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions).
A2.R.7. Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll).
A2.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss).
A2.N.1. Operations: Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers).
A2.N.2. Operations: Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form.
A2.N.3. Operations: Perform arithmetic operations with polynomial expressions containing rational coefficients.
A2.N.4. Operations: Perform arithmetic operations on irrational expressions.
A2.N.5. Operations: Rationalize a denominator containing a radical expression.
A2.N.6. Operations: Write square roots of negative numbers in terms of i.
A2.N.7. Operations: Simplify powers of i.
A2.N.8. Operations: Determine the conjugate of a complex number.
A2.N.9. Operations: Perform arithmetic operations on complex numbers and write the answer in the form a + bi. Note: This includes simplifying expressions with complex denominators.
A2.N.10. Operations: Know and apply sigma notation.
A2.A.1. Equations and Inequalities: Solve absolute value equations and inequalities involving linear expressions in one variable.
A2.A.2. Equations and Inequalities: Use the discriminant to determine the nature of the roots of a quadratic equation.
A2.A.3. Equations and Inequalities: Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots.
A2.A.4. Equations and Inequalities: Solve quadratic inequalities in one and two variables, algebraically and graphically.
A2.A.5. Equations and Inequalities: Use direct and inverse variation to solve for unknown values.
A2.A.6. Equations and Inequalities: Solve an application which results in an exponential function.
A2.A.7. Variables and Expressions: Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials.
A2.A.8. Variables and Expressions: Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents.
A2.A.9. Variables and Expressions: Rewrite algebraic expressions that contain negative exponents using only positive exponents.
A2.A.10. Variables and Expressions: Rewrite algebraic expressions with fractional exponents as radical expressions.
A2.A.11. Variables and Expressions: Rewrite algebraic expressions in radical form as expressions with fractional exponents.
A2.A.12. Variables and Expressions: Evaluate exponential expressions, including those with base e.
A2.A.13. Variables and Expressions: Simplify radical expressions.
A2.A.14. Variables and Expressions: Perform addition, subtraction, multiplication and division of radical expressions.
A2.A.15. Variables and Expressions: Rationalize denominators involving algebraic radical expressions.
A2.A.16. Variables and Expressions: Perform arithmetic operations with rational expressions and rename to lowest terms.
A2.A.17. Variables and Expressions: Simplify complex fractional expressions.
A2.A.18. Variables and Expressions: Evaluate logarithmic expressions in any base.
A2.A.19. Variables and Expressions: Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms.
A2.A.20. Equations and Inequalities: Determine the sum and product of the roots of a quadratic equation by examining its coefficients.
A2.A.21. Equations and Inequalities: Determine the quadratic equation, given the sum and product of its roots.
A2.A.22. Equations and Inequalities: Solve radical equations.
A2.A.23. Equations and Inequalities: Solve rational equations and inequalities.
A2.A.24. Equations and Inequalities: Know and apply the technique of completing the square.
A2.A.25. Equations and Inequalities: Solve quadratic equations, using the quadratic formula.
A2.A.26. Equations and Inequalities: Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula.
A2.A.27. Equations and Inequalities: Solve exponential equations with and without common bases.
A2.A.28. Equations and Inequalities: Solve a logarithmic equation by rewriting as an exponential equation.
A2.A.29. Patterns, Relations, and Functions: Identify an arithmetic or geometric sequence and find the formula for its nth term.
A2.A.30. Patterns, Relations, and Functions: Determine the common difference in an arithmetic sequence.
A2.A.31. Patterns, Relations, and Functions: Determine the common ratio in a geometric sequence.
A2.A.32. Patterns, Relations, and Functions: Determine a specified term of an arithmetic or geometric sequence.
A2.A.33. Patterns, Relations, and Functions: Specify terms of a sequence, given its recursive definition.
A2.A.34. Patterns, Relations, and Functions: Represent the sum of a series, using sigma notation.
A2.A.35. Patterns, Relations, and Functions: Determine the sum of the first n terms of an arithmetic or geometric series.
A2.A.36. Patterns, Relations, and Functions: Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion.
A2.A.37. Patterns, Relations, and Functions: Define a relation and function.
A2.A.38. Patterns, Relations, and Functions: Determine when a relation is a function.
A2.A.39. Patterns, Relations, and Functions: Determine the domain and range of a function from its equation.
A2.A.40. Patterns, Relations, and Functions: Write functions in functional notation.
A2.A.41. Patterns, Relations, and Functions: Use functional notation to evaluate functions for given values in the domain.
A2.A.42. Patterns, Relations, and Functions: Find the composition of functions.
A2.A.43. Patterns, Relations, and Functions: Determine if a function is one-to-one, onto, or both.
A2.A.44. Patterns, Relations, and Functions: Define the inverse of a function.
A2.A.45. Patterns, Relations, and Functions: Determine the inverse of a function and use composition to justify the result.
A2.A.46. Patterns, Relations, and Functions: Perform transformations with functions and relations: f(x + a), f(x) + a, f(-x), -f(x), af(x).
A2.A.47. Coordinate Geometry: Determine the center-radius form for the equation of circle in standard form.
A2.A.48. Coordinate Geometry: Write the equation of a circle, given its center and a point on the circle.
A2.A.49. Coordinate Geometry: Write the equation of a circle from its graph.
A2.A.50. Coordinate Geometry: Approximate the solution to polynomial equations of higher degree by inspecting the graph.
A2.A.51. Coordinate Geometry: Determine the domain and range of a function from its graph.
A2.A.52. Coordinate Geometry: Identify relations and functions, using graphs.
A2.A.53. Coordinate Geometry: Graph exponential functions of the form y = b to the power of x for positive values of b, including b = e.
A2.A.54. Graph logarithmic functions, using the inverse of the related exponential function.
A2.A.55. Trigonometric Functions: Express and apply the six trigonometric functions as ratios of the sides of a right triangle.
A2.A.56. Trigonometric Functions: Know the exact and approximate values of the sine, cosine, and tangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.57. Trigonometric Functions: Sketch and use the reference angle for angles in standard position.
A2.A.58. Trigonometric Functions: Know and apply the co-function and reciprocal relationships between trigonometric ratios.
A2.A.59. Trigonometric Functions: Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.60. Trigonometric Functions: Sketch the unit circle and represent angles in standard position.
A2.A.61. Trigonometric Functions: Determine the length of an arc of a circle, given its radius and the measure of its central angle.
A2.A.62. Trigonometric Functions: Find the value of trigonometric functions, if given a point on the terminal side of angle theta.
A2.A.63. Trigonometric Functions: Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function.
A2.A.64. Trigonometric Functions: Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent.
A2.A.65. Trigonometric Functions: Sketch the graph of the inverses of the sine, cosine, and tangent functions.
A2.A.66. Trigonometric Functions: Determine the trigonometric functions of any angle, using technology.
A2.A.67. Trigonometric Functions: Justify the Pythagorean identities.
A2.A.68. Trigonometric Functions: Solve trigonometric equations for all values of the variable from 0 degrees to 360 degrees.
A2.A.69. Trigonometric Functions: Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function.
A2.A.70. Trigonometric Functions: Sketch and recognize one cycle of a function of the form y = Asin Bx or y = Acos Bx.
A2.A.71. Trigonometric Functions: Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x).
A2.A.72. Trigonometric Functions: Write the trigonometric function that is represented by a given periodic graph.
A2.A.73. Trigonometric Functions: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines.
A2.A.74. Trigonometric Functions: Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle.
A2.A.75. Trigonometric Functions: Determine the solution(s) from the SSA situation (ambiguous case).
A2.A.76. Trigonometric Functions: Apply the angle sum and difference formulas for trigonometric functions.
A2.A.77. Trigonometric Functions: Apply the double-angle and half-angle formulas for trigonometric functions.
A2.M.1. Units of Measurement: Define radian measure.
A2.M.2. Units of Measurement: Convert between radian and degree measures.
A2.S.1. Collection of Data: Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment).
A2.S.2. Collection of Data: Determine factors which may affect the outcome of a survey.
A2.S.3. Organization and Display of Data: Calculate measures of central tendency with group frequency distributions.
A2.S.4. Organization and Display of Data: Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations.
A2.S.5. Organization and Display of Data: Know and apply the characteristics of the normal distribution.
A2.S.6. Predictions from Data: Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate.
A2.S.7. Predictions from Data: Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data.
A2.S.8. Predictions from Data: Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship.
A2.S.9. Probability: Differentiate between situations requiring permutations and those requiring combinations.
A2.S.10. Probability: Calculate the number of possible permutations (nPr) of n items taken r at a time.
A2.S.11. Probability: Calculate the number of possible combinations (nCr) of n items taken r at a time.
A2.S.12. Probability: Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event).
A2.S.13. Probability: Calculate theoretical probabilities, including geometric applications.
A2.S.14. Probability: Calculate empirical probabilities.
A2.S.15. Probability: Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most.
A2.S.16. Probability: Use the normal distribution as an approximation for binomial probabilities.
NY.3. Integrated Algebra: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
A.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
A.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
A.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A.PS.6. Use a variety of strategies to extend solution methods to other problems.
A.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
A.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A.PS.9. Interpret solutions within the given constraints of a problem.
A.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
A.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
A.RP.2. Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture.
A.RP.3. Recognize when an approximation is more appropriate than an exact answer.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
A.RP.4. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A.RP.5. Construct logical arguments that verify claims or counterexamples that refute them.
A.RP.6. Present correct mathematical arguments in a variety of forms.
A.RP.7. Evaluate written arguments for validity.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
A.RP.8. Support an argument by using a systematic approach to test more than one case.
A.RP.9. Devise ways to verify results or use counterexamples to refute incorrect statements.
A.RP.10. Extend specific results to more general cases.
A.RP.11. Use a Venn diagram to support a logical argument.
A.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
A.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
A.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A.CM.4. Explain relationships among different representations of a problem.
A.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
A.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A.CM.8. Reflect on strategies of others in relation to one's own strategy.
A.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
A.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A.CM.11. Represent word problems using standard mathematical notation.
A.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
A.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
A.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A.CN.5. Understand how quantitative models connect to various physical models and representations.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
A.CN.6. Recognize and apply mathematics to situations in the outside world.
A.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A.CN.8. Develop an appreciation for the historical development of mathematics.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
A.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A.R.2. Recognize, compare, and use an array of representational forms.
A.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
A.R.4. Select appropriate representations to solve problem situations.
A.R.5. Investigate relationships between different representations and their impact on a given problem.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
A.R.6. Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground).
A.R.7. Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales).
A.R.8. Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations y = x to the power of 2 and y = -x to the power of 2).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
A.N.1. Number Theory: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
A.N.2. Operations: Simplify radical terms (no variable in the radicand).
A.N.3. Operations: Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form.
A.N.4. Operations: Understand and use scientific notation to compute products and quotients of numbers.
A.N.5. Operations: Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation.
A.N.6. Operations: Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s).
A.N.7. Operations: Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting.
A.N.8. Operations: Determine the number of possible arrangements (permutations) of a list of items.
3.21. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
A.A.1. Variables and Expressions: Translate a quantitative verbal phrase into an algebraic expression.
A.A.2. Variables and Expressions: Write a verbal expression that matches a given mathematical expression.
A.A.3. Equations and Inequalities: Distinguish the difference between an algebraic expression and an algebraic equation.
A.A.4. Equations and Inequalities: Translate verbal sentences into mathematical equations or inequalities.
A.A.5. Equations and Inequalities: Write algebraic equations or inequalities that represent a situation.
A.A.6. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable.
A.A.7. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables.
A.A.8. Equations and Inequalities: Analyze and solve verbal problems that involve quadratic equations.
A.A.9. Equations and Inequalities: Analyze and solve verbal problems that involve exponential growth and decay.
A.A.10. Equations and Inequalities: Solve systems of two linear equations in two variables algebraically (See A.G.7).
A.A.11. Equations and Inequalities: Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
A.A.12. Variables and Expressions: Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only.
A.A.13. Variables and Expressions: Add, subtract, and multiply monomials and polynomials.
A.A.14. Variables and Expressions: Divide a polynomial by a monomial or binomial, where the quotient has no remainder.
A.A.15. Variables and Expressions: Find values of a variable for which an algebraic fraction is undefined.
A.A.16. Variables and Expressions: Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms.
A.A.17. Variables and Expressions: Add or subtract fractional expressions with monomial or like binomial denominators.
A.A.18. Variables and Expressions: Multiply and divide algebraic fractions and express the product or quotient in simplest form.
A.A.19. Variables and Expressions: Identify and factor the difference of two perfect squares.
A.A.20. Variables and Expressions: Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF).
A.A.21. Equations and Inequalities: Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable.
A.A.22. Equations and Inequalities: Solve all types of linear equations in one variable.
A.A.23. Equations and Inequalities: Solve literal equations for a given variable.
A.A.24. Equations and Inequalities: Solve linear inequalities in one variable.
A.A.25. Equations and Inequalities: Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable.
A.A.26. Equations and Inequalities: Solve algebraic proportions in one variable which result in linear or quadratic equations.
A.A.27. Equations and Inequalities: Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots.
A.A.28. Equations and Inequalities: Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression.
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
A.A.29. Patterns, Relations, and Functions: Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form.
A.A.30. Patterns, Relations, and Functions: Find the complement of a subset of a given set, within a given universe.
A.A.31. Patterns, Relations, and Functions: Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets).
A.A.32. Coordinate Geometry: Explain slope as a rate of change between dependent and independent variables.
A.A.33. Coordinate Geometry: Determine the slope of a line, given the coordinates of two points on the line.
A.A.34. Coordinate Geometry: Write the equation of a line, given its slope and the coordinates of a point on the line.
A.A.35. Coordinate Geometry: Write the equation of a line, given the coordinates of two points on the line.
A.A.36. Coordinate Geometry: Write the equation of a line parallel to the x- or y-axis.
A.A.37. Coordinate Geometry: Determine the slope of a line, given its equation in any form.
A.A.38. Coordinate Geometry: Determine if two lines are parallel, given their equations in any form.
A.A.39. Coordinate Geometry: Determine whether a given point is on a line, given the equation of the line.
A.A.40. Coordinate Geometry: Determine whether a given point is in the solution set of a system of linear inequalities.
A.A.41. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 ).
A.A.42. Trigonometric Functions: Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides.
A.A.43. Trigonometric Functions: Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle.
A.A.44. Trigonometric Functions: Find the measure of a side of a right triangle, given an acute angle and the length of another side.
A.A.45. Trigonometric Functions: Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides.
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
A.G.1. Shapes: Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
A.G.2. Shapes: Use formulas to calculate volume and surface area of rectangular solids and cylinders.
3.25. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
A.G.3. Coordinate Geometry: Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations.
A.G.4. Coordinate Geometry: Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions.
A.G.5. Coordinate Geometry: Investigate and generalize how changing the coefficients of a function affects its graph.
A.G.6. Coordinate Geometry: Graph linear inequalities.
A.G.7. Coordinate Geometry: Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10).
A.G.8. Coordinate Geometry: Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions.
A.G.9. Coordinate Geometry: Solve systems of linear and quadratic equations graphically. Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.
A.G.10. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41 ) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
A.M.1. Units of Measurement: Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail).
A.M.2. Units of Measurement: Solve problems involving conversions within measurement systems, given the relationship between the units.
3.27. Measurement Strand: Understand that all measurement contains error and be able to determine its significance.
A.M.3. Error and Magnitude: Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure.
3.28. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
A.S.1. Organization and Display of Data: Categorize data as qualitative or quantitative.
A.S.2. Organization and Display of Data: Determine whether the data to be analyzed is univariate or bivariate.
A.S.3. Organization and Display of Data: Determine when collected data or display of data may be biased.
A.S.4. Organization and Display of Data: Compare and contrast the appropriateness of different measures of central tendency for a given data set.
A.S.5. Organization and Display of Data: Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data.
A.S.6. Organization and Display of Data: Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box and- whisker plot.
A.S.7. Organization and Display of Data: Create a scatter plot of bivariate data.
A.S.8. Organization and Display of Data: Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line.
A.S.9. Analysis of Data: Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot.
A.S.10. Analysis of Data: Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions.
A.S.11. Analysis of Data: Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles.
A.S.12. Analysis of Data: Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none).
A.S.13. Analysis of Data: Understand the difference between correlation and causation.
A.S.14. Analysis of Data: Identify variables that might have a correlation but not a causal relationship.
3.29. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
A.S.15. Predictions from Data: Identify and describe sources of bias and its effect, drawing conclusions from data.
A.S.16. Predictions from Data: Recognize how linear transformations of one-variable data affect the data's mean, median, mode, and range.
A.S.17. Predictions from Data: Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation.
3.30. Statistics and Probability Strand: Students will understand and apply concepts of probability.
A.S.18. Probability: Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
A.S.19. Probability: Determine the number of elements in a sample space and the number of favorable events.
A.S.20. Probability: Calculate the probability of an event and its complement.
A.S.21. Probability: Determine empirical probabilities based on specific sample data.
A.S.22. Probability: Determine, based on calculated probability of a set of events, if some or all are equally likely to occur; one is more likely to occur than another; whether or not an event is certain to happen or not to happen.
A.S.23. Probability: Calculate the probability of a series of independent events; a series of dependent events; two mutually exclusive events; two events that are not mutually exclusive.
G.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
G.PS.2. Observe and explain patterns to formulate generalizations and conjectures.
G.PS.3. Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations).
G.PS.4. Construct various types of reasoning, arguments, justifications and methods of proof for problems.
G.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
G.PS.6. Use a variety of strategies to extend solution methods to other problems.
G.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
G.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
G.PS.9. Interpret solutions within the given constraints of a problem.
G.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
G.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
G.RP.2. Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies.
G.RP.3. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
G.RP.4. Provide correct mathematical arguments in response to other students' conjectures, reasoning, and arguments.
G.RP.5. Present correct mathematical arguments in a variety of forms.
G.RP.6. Evaluate written arguments for validity.
G.RP.7. Construct a proof using a variety of methods (e.g., deductive, analytic, transformational).
G.RP.8. Devise ways to verify results or use counterexamples to refute incorrect statements.
G.RP.9. Apply inductive reasoning in making and supporting mathematical conjectures.
G.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
G.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
G.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
G.CM.4. Explain relationships among different representations of a problem.
G.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
G.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
G.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
G.CM.8. Reflect on strategies of others in relation to one's own strategy.
G.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
G.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
G.CM.11. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams.
G.CM.12. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
G.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
G.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
G.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
G.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
G.CN.5. Understand how quantitative models connect to various physical models and representations.
G.CN.6. Recognize and apply mathematics to situations in the outside world.
G.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
G.CN.8. Develop an appreciation for the historical development of mathematics.
G.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
G.R.2. Recognize, compare, and use an array of representational forms.
G.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
G.R.4. Select appropriate representations to solve problem situations.
G.R.5. Investigate relationships between different representations and their impact on a given problem.
G.R.6. Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank).
G.R.7. Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person's argument have a logical foundation).
G.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent).
G.G.1. Geometric Relationships: Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
G.G.2. Geometric Relationships: Know and apply that through a given point there passes one and only one plane perpendicular to a given line.
G.G.3. Geometric Relationships: Know and apply that through a given point there passes one and only one line perpendicular to a given plane.
G.G.4. Geometric Relationships: Know and apply that two lines perpendicular to the same plane are coplanar.
G.G.5. Geometric Relationships: Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.
G.G.6. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.
G.G.7. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
G.G.8. Geometric Relationships: Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines.
G.G.9. Geometric Relationships: Know and apply that if two planes are perpendicular to the same line, they are parallel.
G.G.10. Geometric Relationships: Know and apply that the lateral edges of a prism are congruent and parallel.
G.G.11. Geometric Relationships: Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal.
G.G.12. Geometric Relationships: Know and apply that the volume of a prism is the product of the area of the base and the altitude.
G.G.13. Geometric Relationships: Apply the properties of a regular pyramid, including lateral edges are congruent; lateral faces are congruent isosceles triangles; volume of a pyramid equals one-third the product of the area of the base and the altitude.
G.G.14. Geometric Relationships: Apply the properties of a cylinder, including bases are congruent; volume equals the product of the area of the base and the altitude; lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base.
G.G.15. Geometric Relationships: Apply the properties of a right circular cone, including lateral area equals one-half the product of the slant height and the circumference of its base volume is one-third the product of the area of its base and its altitude.
G.G.16. Geometric Relationships: Apply the properties of a sphere, including the intersection of a plane and a sphere is a circle; a great circle is the largest circle that can be drawn on a sphere; two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles; surface area is 4 pi r to the power of 2; volume is 4/3 pi r to the power of 3.
G.G.17. Constructions: Construct a bisector of a given angle, using a straightedge and compass, and justify the construction.
G.G.18. Constructions: Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction.
G.G.19. Constructions: Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction.
G.G.20. Constructions: Construct an equilateral triangle, using a straightedge and compass, and justify the construction.
G.G.21. Locus: Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles.
G.G.22. Locus: Solve problems using compound loci.
G.G.23. Locus: Graph and solve compound loci in the coordinate plane.
G.G.24. Informal and Formal Proofs: Determine the negation of a statement and establish its truth value.
G.G.25. Informal and Formal Proofs: Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true.
G.G.26. Informal and Formal Proofs: Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences.
G.G.27. Informal and Formal Proofs: Write a proof arguing from a given hypothesis to a given conclusion.
G.G.28. Informal and Formal Proofs: Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles.
G.G.29. Informal and Formal Proofs: Identify corresponding parts of congruent triangles.
G.G.30. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle.
G.G.31. Informal and Formal Proofs: Investigate, justify, and apply the isosceles triangle theorem and its converse.
G.G.32. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem.
G.G.33. Informal and Formal Proofs: Investigate, justify, and apply the triangle inequality theorem.
G.G.34. Informal and Formal Proofs: Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle.
G.G.35. Informal and Formal Proofs: Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines.
G.G.36. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons.
G.G.37. Informal and Formal Proofs: Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons.
G.G.38. Informal and Formal Proofs: Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals.
G.G.39. Informal and Formal Proofs: Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals.
G.G.40. Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals.
G.G.41. Informal and Formal Proofs: Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids.
G.G.42. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle.
G.G.43. Informal and Formal Proofs: Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1.
G.G.44. Informal and Formal Proofs: Establish similarity of triangles, using the following theorems: AA, SAS, and SSS.
G.G.45. Informal and Formal Proofs: Investigate, justify, and apply theorems about similar triangles.
G.G.46. Informal and Formal Proofs: Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle.
G.G.47. Informal and Formal Proofs: Investigate, justify, and apply theorems about mean proportionality: the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse; the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg.
G.G.48. Informal and Formal Proofs: Investigate, justify, and apply the Pythagorean theorem and its converse.
G.G.49. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding chords of a circle: perpendicular bisectors of chords the relative lengths of chords as compared to their distance from the center of the circle.
G.G.50. Informal and Formal Proofs: Investigate, justify, and apply theorems about tangent lines to a circle: a perpendicular to the tangent at the point of tangency; two tangents to a circle from the same external point; common tangents of two non-intersecting or tangent circles.
G.G.51. Informal and Formal Proofs: Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is inside the circle (two chords); on the circle (tangent and chord); outside the circle (two tangents, two secants, or tangent and secant).
G.G.52. Informal and Formal Proofs: Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines.
G.G.53. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding segments intersected by a circle: along two tangents from the same external point; along two secants from the same external point; along a tangent and a secant from the same external point; along two intersecting chords of a given circle.
G.G.54. Transformational Geometry: Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
G.G.55. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections.
G.G.56. Transformational Geometry: Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism.
G.G.57. Transformational Geometry: Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections).
G.G.58. Transformational Geometry: Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries).
G.G.59. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under similarities.
G.G.60. Transformational Geometry: Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism.
G.G.61. Transformational Geometry: Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90 degrees and 180 degrees, reflections over the lines x = 0, y = 0, and y = x, and dilations centered at the origin.
G.G.62. Coordinate Geometry: Find the slope of a perpendicular line, given the equation of a line.
G.G.63. Coordinate Geometry: Determine whether two lines are parallel, perpendicular, or neither, given their equations.
G.G.64. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line.
G.G.65. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line.
G.G.66. Coordinate Geometry: Find the midpoint of a line segment, given its endpoints.
G.G.67. Coordinate Geometry: Find the length of a line segment, given its endpoints.
G.G.68. Coordinate Geometry: Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment.
G.G.69. Coordinate Geometry: Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas.
G.G.70. Coordinate Geometry: Solve systems of equations involving one linear equation and one quadratic equation graphically.
G.G.71. Coordinate Geometry: Write the equation of a circle, given its center and radius or given the endpoints of a diameter.
G.G.72. Coordinate Geometry: Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer.
G.G.73. Coordinate Geometry: Find the center and radius of a circle, given the equation of the circle in center-radius form.
G.G.74. Coordinate Geometry: Graph circles of the form (x - h) to the power of 2 + (j - k) to the power of 2 = r to the power of 2.
A2.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A2.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
A2.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A2.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
A2.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A2.PS.6. Use a variety of strategies to extend solution methods to other problems.
A2.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
A2.PS.8. Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A2.PS.9. Interpret solutions within the given constraints of a problem.
A2.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
A2.RP.1. Support mathematical ideas using a variety of strategies.
A2.RP.2. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
A2.RP.3. Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer.
A2.RP.4. Recognize when an approximation is more appropriate than an exact answer.
A2.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A2.RP.6. Construct logical arguments that verify claims or counterexamples that refute claims.
A2.RP.7. Present correct mathematical arguments in a variety of forms.
A2.RP.8. Evaluate written arguments for validity.
A2.RP.9. Support an argument by using a systematic approach to test more than one case.
A2.RP.10. Devise ways to verify results, using counterexamples and informal indirect proof.
A2.RP.11. Extend specific results to more general cases.
A2.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
A2.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A2.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A2.CM.4. Explain relationships among different representations of a problem.
A2.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A2.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
A2.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A2.CM.8. Reflect on strategies of others in relation to one's own strategy.
A2.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
A2.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A2.CM.11. Represent word problems using standard mathematical notation.
A2.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A2.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
A2.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A2.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
A2.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A2.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A2.CN.5. Understand how quantitative models connect to various physical models and representations.
A2.CN.6. Recognize and apply mathematics to situations in the outside world.
A2.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A2.CN.8. Develop an appreciation for the historical development of mathematics.
A2.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A2.R.2. Recognize, compare, and use an array of representational forms.
A2.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
A2.R.4. Select appropriate representations to solve problem situations.
A2.R.5. Investigate relationships among different representations and their impact on a given problem.
A2.R.6. Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions).
A2.R.7. Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll).
A2.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss).
A2.N.1. Operations: Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers).
A2.N.2. Operations: Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form.
A2.N.3. Operations: Perform arithmetic operations with polynomial expressions containing rational coefficients.
A2.N.4. Operations: Perform arithmetic operations on irrational expressions.
A2.N.5. Operations: Rationalize a denominator containing a radical expression.
A2.N.6. Operations: Write square roots of negative numbers in terms of i.
A2.N.7. Operations: Simplify powers of i.
A2.N.8. Operations: Determine the conjugate of a complex number.
A2.N.9. Operations: Perform arithmetic operations on complex numbers and write the answer in the form a + bi. Note: This includes simplifying expressions with complex denominators.
A2.N.10. Operations: Know and apply sigma notation.
A2.A.1. Equations and Inequalities: Solve absolute value equations and inequalities involving linear expressions in one variable.
A2.A.2. Equations and Inequalities: Use the discriminant to determine the nature of the roots of a quadratic equation.
A2.A.3. Equations and Inequalities: Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots.
A2.A.4. Equations and Inequalities: Solve quadratic inequalities in one and two variables, algebraically and graphically.
A2.A.5. Equations and Inequalities: Use direct and inverse variation to solve for unknown values.
A2.A.6. Equations and Inequalities: Solve an application which results in an exponential function.
A2.A.7. Variables and Expressions: Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials.
A2.A.8. Variables and Expressions: Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents.
A2.A.9. Variables and Expressions: Rewrite algebraic expressions that contain negative exponents using only positive exponents.
A2.A.10. Variables and Expressions: Rewrite algebraic expressions with fractional exponents as radical expressions.
A2.A.11. Variables and Expressions: Rewrite algebraic expressions in radical form as expressions with fractional exponents.
A2.A.12. Variables and Expressions: Evaluate exponential expressions, including those with base e.
A2.A.13. Variables and Expressions: Simplify radical expressions.
A2.A.14. Variables and Expressions: Perform addition, subtraction, multiplication and division of radical expressions.
A2.A.15. Variables and Expressions: Rationalize denominators involving algebraic radical expressions.
A2.A.16. Variables and Expressions: Perform arithmetic operations with rational expressions and rename to lowest terms.
A2.A.17. Variables and Expressions: Simplify complex fractional expressions.
A2.A.18. Variables and Expressions: Evaluate logarithmic expressions in any base.
A2.A.19. Variables and Expressions: Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms.
A2.A.20. Equations and Inequalities: Determine the sum and product of the roots of a quadratic equation by examining its coefficients.
A2.A.21. Equations and Inequalities: Determine the quadratic equation, given the sum and product of its roots.
A2.A.22. Equations and Inequalities: Solve radical equations.
A2.A.23. Equations and Inequalities: Solve rational equations and inequalities.
A2.A.24. Equations and Inequalities: Know and apply the technique of completing the square.
A2.A.25. Equations and Inequalities: Solve quadratic equations, using the quadratic formula.
A2.A.26. Equations and Inequalities: Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula.
A2.A.27. Equations and Inequalities: Solve exponential equations with and without common bases.
A2.A.28. Equations and Inequalities: Solve a logarithmic equation by rewriting as an exponential equation.
A2.A.29. Patterns, Relations, and Functions: Identify an arithmetic or geometric sequence and find the formula for its nth term.
A2.A.30. Patterns, Relations, and Functions: Determine the common difference in an arithmetic sequence.
A2.A.31. Patterns, Relations, and Functions: Determine the common ratio in a geometric sequence.
A2.A.32. Patterns, Relations, and Functions: Determine a specified term of an arithmetic or geometric sequence.
A2.A.33. Patterns, Relations, and Functions: Specify terms of a sequence, given its recursive definition.
A2.A.34. Patterns, Relations, and Functions: Represent the sum of a series, using sigma notation.
A2.A.35. Patterns, Relations, and Functions: Determine the sum of the first n terms of an arithmetic or geometric series.
A2.A.36. Patterns, Relations, and Functions: Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion.
A2.A.37. Patterns, Relations, and Functions: Define a relation and function.
A2.A.38. Patterns, Relations, and Functions: Determine when a relation is a function.
A2.A.39. Patterns, Relations, and Functions: Determine the domain and range of a function from its equation.
A2.A.40. Patterns, Relations, and Functions: Write functions in functional notation.
A2.A.41. Patterns, Relations, and Functions: Use functional notation to evaluate functions for given values in the domain.
A2.A.42. Patterns, Relations, and Functions: Find the composition of functions.
A2.A.43. Patterns, Relations, and Functions: Determine if a function is one-to-one, onto, or both.
A2.A.44. Patterns, Relations, and Functions: Define the inverse of a function.
A2.A.45. Patterns, Relations, and Functions: Determine the inverse of a function and use composition to justify the result.
A2.A.46. Patterns, Relations, and Functions: Perform transformations with functions and relations: f(x + a), f(x) + a, f(-x), -f(x), af(x).
A2.A.47. Coordinate Geometry: Determine the center-radius form for the equation of circle in standard form.
A2.A.48. Coordinate Geometry: Write the equation of a circle, given its center and a point on the circle.
A2.A.49. Coordinate Geometry: Write the equation of a circle from its graph.
A2.A.50. Coordinate Geometry: Approximate the solution to polynomial equations of higher degree by inspecting the graph.
A2.A.51. Coordinate Geometry: Determine the domain and range of a function from its graph.
A2.A.52. Coordinate Geometry: Identify relations and functions, using graphs.
A2.A.53. Coordinate Geometry: Graph exponential functions of the form y = b to the power of x for positive values of b, including b = e.
A2.A.54. Graph logarithmic functions, using the inverse of the related exponential function.
A2.A.55. Trigonometric Functions: Express and apply the six trigonometric functions as ratios of the sides of a right triangle.
A2.A.56. Trigonometric Functions: Know the exact and approximate values of the sine, cosine, and tangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.57. Trigonometric Functions: Sketch and use the reference angle for angles in standard position.
A2.A.58. Trigonometric Functions: Know and apply the co-function and reciprocal relationships between trigonometric ratios.
A2.A.59. Trigonometric Functions: Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.60. Trigonometric Functions: Sketch the unit circle and represent angles in standard position.
A2.A.61. Trigonometric Functions: Determine the length of an arc of a circle, given its radius and the measure of its central angle.
A2.A.62. Trigonometric Functions: Find the value of trigonometric functions, if given a point on the terminal side of angle theta.
A2.A.63. Trigonometric Functions: Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function.
A2.A.64. Trigonometric Functions: Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent.
A2.A.65. Trigonometric Functions: Sketch the graph of the inverses of the sine, cosine, and tangent functions.
A2.A.66. Trigonometric Functions: Determine the trigonometric functions of any angle, using technology.
A2.A.67. Trigonometric Functions: Justify the Pythagorean identities.
A2.A.68. Trigonometric Functions: Solve trigonometric equations for all values of the variable from 0 degrees to 360 degrees.
A2.A.69. Trigonometric Functions: Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function.
A2.A.70. Trigonometric Functions: Sketch and recognize one cycle of a function of the form y = Asin Bx or y = Acos Bx.
A2.A.71. Trigonometric Functions: Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x).
A2.A.72. Trigonometric Functions: Write the trigonometric function that is represented by a given periodic graph.
A2.A.73. Trigonometric Functions: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines.
A2.A.74. Trigonometric Functions: Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle.
A2.A.75. Trigonometric Functions: Determine the solution(s) from the SSA situation (ambiguous case).
A2.A.76. Trigonometric Functions: Apply the angle sum and difference formulas for trigonometric functions.
A2.A.77. Trigonometric Functions: Apply the double-angle and half-angle formulas for trigonometric functions.
A2.M.1. Units of Measurement: Define radian measure.
A2.M.2. Units of Measurement: Convert between radian and degree measures.
A2.S.1. Collection of Data: Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment).
A2.S.2. Collection of Data: Determine factors which may affect the outcome of a survey.
A2.S.3. Organization and Display of Data: Calculate measures of central tendency with group frequency distributions.
A2.S.4. Organization and Display of Data: Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations.
A2.S.5. Organization and Display of Data: Know and apply the characteristics of the normal distribution.
A2.S.6. Predictions from Data: Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate.
A2.S.7. Predictions from Data: Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data.
A2.S.8. Predictions from Data: Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship.
A2.S.9. Probability: Differentiate between situations requiring permutations and those requiring combinations.
A2.S.10. Probability: Calculate the number of possible permutations (nPr) of n items taken r at a time.
A2.S.11. Probability: Calculate the number of possible combinations (nCr) of n items taken r at a time.
A2.S.12. Probability: Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event).
A2.S.13. Probability: Calculate theoretical probabilities, including geometric applications.
A2.S.14. Probability: Calculate empirical probabilities.
A2.S.15. Probability: Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most.
A2.S.16. Probability: Use the normal distribution as an approximation for binomial probabilities.
NY.3. Integrated Algebra: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
A.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
A.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
A.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A.PS.6. Use a variety of strategies to extend solution methods to other problems.
A.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
A.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A.PS.9. Interpret solutions within the given constraints of a problem.
A.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
A.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
A.RP.2. Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture.
A.RP.3. Recognize when an approximation is more appropriate than an exact answer.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
A.RP.4. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A.RP.5. Construct logical arguments that verify claims or counterexamples that refute them.
A.RP.6. Present correct mathematical arguments in a variety of forms.
A.RP.7. Evaluate written arguments for validity.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
A.RP.8. Support an argument by using a systematic approach to test more than one case.
A.RP.9. Devise ways to verify results or use counterexamples to refute incorrect statements.
A.RP.10. Extend specific results to more general cases.
A.RP.11. Use a Venn diagram to support a logical argument.
A.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
A.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
A.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A.CM.4. Explain relationships among different representations of a problem.
A.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
A.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A.CM.8. Reflect on strategies of others in relation to one's own strategy.
A.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
A.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A.CM.11. Represent word problems using standard mathematical notation.
A.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
A.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
A.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A.CN.5. Understand how quantitative models connect to various physical models and representations.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
A.CN.6. Recognize and apply mathematics to situations in the outside world.
A.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A.CN.8. Develop an appreciation for the historical development of mathematics.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
A.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A.R.2. Recognize, compare, and use an array of representational forms.
A.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
A.R.4. Select appropriate representations to solve problem situations.
A.R.5. Investigate relationships between different representations and their impact on a given problem.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
A.R.6. Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground).
A.R.7. Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales).
A.R.8. Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations y = x to the power of 2 and y = -x to the power of 2).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
A.N.1. Number Theory: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
A.N.2. Operations: Simplify radical terms (no variable in the radicand).
A.N.3. Operations: Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form.
A.N.4. Operations: Understand and use scientific notation to compute products and quotients of numbers.
A.N.5. Operations: Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation.
A.N.6. Operations: Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s).
A.N.7. Operations: Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting.
A.N.8. Operations: Determine the number of possible arrangements (permutations) of a list of items.
3.21. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
A.A.1. Variables and Expressions: Translate a quantitative verbal phrase into an algebraic expression.
A.A.2. Variables and Expressions: Write a verbal expression that matches a given mathematical expression.
A.A.3. Equations and Inequalities: Distinguish the difference between an algebraic expression and an algebraic equation.
A.A.4. Equations and Inequalities: Translate verbal sentences into mathematical equations or inequalities.
A.A.5. Equations and Inequalities: Write algebraic equations or inequalities that represent a situation.
A.A.6. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable.
A.A.7. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables.
A.A.8. Equations and Inequalities: Analyze and solve verbal problems that involve quadratic equations.
A.A.9. Equations and Inequalities: Analyze and solve verbal problems that involve exponential growth and decay.
A.A.10. Equations and Inequalities: Solve systems of two linear equations in two variables algebraically (See A.G.7).
A.A.11. Equations and Inequalities: Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
A.A.12. Variables and Expressions: Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only.
A.A.13. Variables and Expressions: Add, subtract, and multiply monomials and polynomials.
A.A.14. Variables and Expressions: Divide a polynomial by a monomial or binomial, where the quotient has no remainder.
A.A.15. Variables and Expressions: Find values of a variable for which an algebraic fraction is undefined.
A.A.16. Variables and Expressions: Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms.
A.A.17. Variables and Expressions: Add or subtract fractional expressions with monomial or like binomial denominators.
A.A.18. Variables and Expressions: Multiply and divide algebraic fractions and express the product or quotient in simplest form.
A.A.19. Variables and Expressions: Identify and factor the difference of two perfect squares.
A.A.20. Variables and Expressions: Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF).
A.A.21. Equations and Inequalities: Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable.
A.A.22. Equations and Inequalities: Solve all types of linear equations in one variable.
A.A.23. Equations and Inequalities: Solve literal equations for a given variable.
A.A.24. Equations and Inequalities: Solve linear inequalities in one variable.
A.A.25. Equations and Inequalities: Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable.
A.A.26. Equations and Inequalities: Solve algebraic proportions in one variable which result in linear or quadratic equations.
A.A.27. Equations and Inequalities: Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots.
A.A.28. Equations and Inequalities: Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression.
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
A.A.29. Patterns, Relations, and Functions: Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form.
A.A.30. Patterns, Relations, and Functions: Find the complement of a subset of a given set, within a given universe.
A.A.31. Patterns, Relations, and Functions: Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets).
A.A.32. Coordinate Geometry: Explain slope as a rate of change between dependent and independent variables.
A.A.33. Coordinate Geometry: Determine the slope of a line, given the coordinates of two points on the line.
A.A.34. Coordinate Geometry: Write the equation of a line, given its slope and the coordinates of a point on the line.
A.A.35. Coordinate Geometry: Write the equation of a line, given the coordinates of two points on the line.
A.A.36. Coordinate Geometry: Write the equation of a line parallel to the x- or y-axis.
A.A.37. Coordinate Geometry: Determine the slope of a line, given its equation in any form.
A.A.38. Coordinate Geometry: Determine if two lines are parallel, given their equations in any form.
A.A.39. Coordinate Geometry: Determine whether a given point is on a line, given the equation of the line.
A.A.40. Coordinate Geometry: Determine whether a given point is in the solution set of a system of linear inequalities.
A.A.41. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 ).
A.A.42. Trigonometric Functions: Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides.
A.A.43. Trigonometric Functions: Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle.
A.A.44. Trigonometric Functions: Find the measure of a side of a right triangle, given an acute angle and the length of another side.
A.A.45. Trigonometric Functions: Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides.
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
A.G.1. Shapes: Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
A.G.2. Shapes: Use formulas to calculate volume and surface area of rectangular solids and cylinders.
3.25. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
A.G.3. Coordinate Geometry: Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations.
A.G.4. Coordinate Geometry: Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions.
A.G.5. Coordinate Geometry: Investigate and generalize how changing the coefficients of a function affects its graph.
A.G.6. Coordinate Geometry: Graph linear inequalities.
A.G.7. Coordinate Geometry: Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10).
A.G.8. Coordinate Geometry: Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions.
A.G.9. Coordinate Geometry: Solve systems of linear and quadratic equations graphically. Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.
A.G.10. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41 ) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
A.M.1. Units of Measurement: Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail).
A.M.2. Units of Measurement: Solve problems involving conversions within measurement systems, given the relationship between the units.
3.27. Measurement Strand: Understand that all measurement contains error and be able to determine its significance.
A.M.3. Error and Magnitude: Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure.
3.28. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
A.S.1. Organization and Display of Data: Categorize data as qualitative or quantitative.
A.S.2. Organization and Display of Data: Determine whether the data to be analyzed is univariate or bivariate.
A.S.3. Organization and Display of Data: Determine when collected data or display of data may be biased.
A.S.4. Organization and Display of Data: Compare and contrast the appropriateness of different measures of central tendency for a given data set.
A.S.5. Organization and Display of Data: Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data.
A.S.6. Organization and Display of Data: Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box and- whisker plot.
A.S.7. Organization and Display of Data: Create a scatter plot of bivariate data.
A.S.8. Organization and Display of Data: Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line.
A.S.9. Analysis of Data: Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot.
A.S.10. Analysis of Data: Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions.
A.S.11. Analysis of Data: Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles.
A.S.12. Analysis of Data: Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none).
A.S.13. Analysis of Data: Understand the difference between correlation and causation.
A.S.14. Analysis of Data: Identify variables that might have a correlation but not a causal relationship.
3.29. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
A.S.15. Predictions from Data: Identify and describe sources of bias and its effect, drawing conclusions from data.
A.S.16. Predictions from Data: Recognize how linear transformations of one-variable data affect the data's mean, median, mode, and range.
A.S.17. Predictions from Data: Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation.
3.30. Statistics and Probability Strand: Students will understand and apply concepts of probability.
A.S.18. Probability: Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
A.S.19. Probability: Determine the number of elements in a sample space and the number of favorable events.
A.S.20. Probability: Calculate the probability of an event and its complement.
A.S.21. Probability: Determine empirical probabilities based on specific sample data.
A.S.22. Probability: Determine, based on calculated probability of a set of events, if some or all are equally likely to occur; one is more likely to occur than another; whether or not an event is certain to happen or not to happen.
A.S.23. Probability: Calculate the probability of a series of independent events; a series of dependent events; two mutually exclusive events; two events that are not mutually exclusive.
G.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
G.PS.2. Observe and explain patterns to formulate generalizations and conjectures.
G.PS.3. Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations).
G.PS.4. Construct various types of reasoning, arguments, justifications and methods of proof for problems.
G.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
G.PS.6. Use a variety of strategies to extend solution methods to other problems.
G.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
G.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
G.PS.9. Interpret solutions within the given constraints of a problem.
G.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
G.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
G.RP.2. Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies.
G.RP.3. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
G.RP.4. Provide correct mathematical arguments in response to other students' conjectures, reasoning, and arguments.
G.RP.5. Present correct mathematical arguments in a variety of forms.
G.RP.6. Evaluate written arguments for validity.
G.RP.7. Construct a proof using a variety of methods (e.g., deductive, analytic, transformational).
G.RP.8. Devise ways to verify results or use counterexamples to refute incorrect statements.
G.RP.9. Apply inductive reasoning in making and supporting mathematical conjectures.
G.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
G.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
G.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
G.CM.4. Explain relationships among different representations of a problem.
G.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
G.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
G.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
G.CM.8. Reflect on strategies of others in relation to one's own strategy.
G.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
G.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
G.CM.11. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams.
G.CM.12. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
G.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
G.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
G.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
G.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
G.CN.5. Understand how quantitative models connect to various physical models and representations.
G.CN.6. Recognize and apply mathematics to situations in the outside world.
G.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
G.CN.8. Develop an appreciation for the historical development of mathematics.
G.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
G.R.2. Recognize, compare, and use an array of representational forms.
G.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
G.R.4. Select appropriate representations to solve problem situations.
G.R.5. Investigate relationships between different representations and their impact on a given problem.
G.R.6. Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank).
G.R.7. Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person's argument have a logical foundation).
G.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent).
G.G.1. Geometric Relationships: Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
G.G.2. Geometric Relationships: Know and apply that through a given point there passes one and only one plane perpendicular to a given line.
G.G.3. Geometric Relationships: Know and apply that through a given point there passes one and only one line perpendicular to a given plane.
G.G.4. Geometric Relationships: Know and apply that two lines perpendicular to the same plane are coplanar.
G.G.5. Geometric Relationships: Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.
G.G.6. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.
G.G.7. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
G.G.8. Geometric Relationships: Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines.
G.G.9. Geometric Relationships: Know and apply that if two planes are perpendicular to the same line, they are parallel.
G.G.10. Geometric Relationships: Know and apply that the lateral edges of a prism are congruent and parallel.
G.G.11. Geometric Relationships: Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal.
G.G.12. Geometric Relationships: Know and apply that the volume of a prism is the product of the area of the base and the altitude.
G.G.13. Geometric Relationships: Apply the properties of a regular pyramid, including lateral edges are congruent; lateral faces are congruent isosceles triangles; volume of a pyramid equals one-third the product of the area of the base and the altitude.
G.G.14. Geometric Relationships: Apply the properties of a cylinder, including bases are congruent; volume equals the product of the area of the base and the altitude; lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base.
G.G.15. Geometric Relationships: Apply the properties of a right circular cone, including lateral area equals one-half the product of the slant height and the circumference of its base volume is one-third the product of the area of its base and its altitude.
G.G.16. Geometric Relationships: Apply the properties of a sphere, including the intersection of a plane and a sphere is a circle; a great circle is the largest circle that can be drawn on a sphere; two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles; surface area is 4 pi r to the power of 2; volume is 4/3 pi r to the power of 3.
G.G.17. Constructions: Construct a bisector of a given angle, using a straightedge and compass, and justify the construction.
G.G.18. Constructions: Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction.
G.G.19. Constructions: Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction.
G.G.20. Constructions: Construct an equilateral triangle, using a straightedge and compass, and justify the construction.
G.G.21. Locus: Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles.
G.G.22. Locus: Solve problems using compound loci.
G.G.23. Locus: Graph and solve compound loci in the coordinate plane.
G.G.24. Informal and Formal Proofs: Determine the negation of a statement and establish its truth value.
G.G.25. Informal and Formal Proofs: Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true.
G.G.26. Informal and Formal Proofs: Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences.
G.G.27. Informal and Formal Proofs: Write a proof arguing from a given hypothesis to a given conclusion.
G.G.28. Informal and Formal Proofs: Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles.
G.G.29. Informal and Formal Proofs: Identify corresponding parts of congruent triangles.
G.G.30. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle.
G.G.31. Informal and Formal Proofs: Investigate, justify, and apply the isosceles triangle theorem and its converse.
G.G.32. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem.
G.G.33. Informal and Formal Proofs: Investigate, justify, and apply the triangle inequality theorem.
G.G.34. Informal and Formal Proofs: Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle.
G.G.35. Informal and Formal Proofs: Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines.
G.G.36. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons.
G.G.37. Informal and Formal Proofs: Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons.
G.G.38. Informal and Formal Proofs: Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals.
G.G.39. Informal and Formal Proofs: Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals.
G.G.40. Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals.
G.G.41. Informal and Formal Proofs: Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids.
G.G.42. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle.
G.G.43. Informal and Formal Proofs: Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1.
G.G.44. Informal and Formal Proofs: Establish similarity of triangles, using the following theorems: AA, SAS, and SSS.
G.G.45. Informal and Formal Proofs: Investigate, justify, and apply theorems about similar triangles.
G.G.46. Informal and Formal Proofs: Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle.
G.G.47. Informal and Formal Proofs: Investigate, justify, and apply theorems about mean proportionality: the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse; the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg.
G.G.48. Informal and Formal Proofs: Investigate, justify, and apply the Pythagorean theorem and its converse.
G.G.49. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding chords of a circle: perpendicular bisectors of chords the relative lengths of chords as compared to their distance from the center of the circle.
G.G.50. Informal and Formal Proofs: Investigate, justify, and apply theorems about tangent lines to a circle: a perpendicular to the tangent at the point of tangency; two tangents to a circle from the same external point; common tangents of two non-intersecting or tangent circles.
G.G.51. Informal and Formal Proofs: Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is inside the circle (two chords); on the circle (tangent and chord); outside the circle (two tangents, two secants, or tangent and secant).
G.G.52. Informal and Formal Proofs: Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines.
G.G.53. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding segments intersected by a circle: along two tangents from the same external point; along two secants from the same external point; along a tangent and a secant from the same external point; along two intersecting chords of a given circle.
G.G.54. Transformational Geometry: Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
G.G.55. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections.
G.G.56. Transformational Geometry: Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism.
G.G.57. Transformational Geometry: Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections).
G.G.58. Transformational Geometry: Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries).
G.G.59. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under similarities.
G.G.60. Transformational Geometry: Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism.
G.G.61. Transformational Geometry: Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90 degrees and 180 degrees, reflections over the lines x = 0, y = 0, and y = x, and dilations centered at the origin.
G.G.62. Coordinate Geometry: Find the slope of a perpendicular line, given the equation of a line.
G.G.63. Coordinate Geometry: Determine whether two lines are parallel, perpendicular, or neither, given their equations.
G.G.64. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line.
G.G.65. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line.
G.G.66. Coordinate Geometry: Find the midpoint of a line segment, given its endpoints.
G.G.67. Coordinate Geometry: Find the length of a line segment, given its endpoints.
G.G.68. Coordinate Geometry: Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment.
G.G.69. Coordinate Geometry: Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas.
G.G.70. Coordinate Geometry: Solve systems of equations involving one linear equation and one quadratic equation graphically.
G.G.71. Coordinate Geometry: Write the equation of a circle, given its center and radius or given the endpoints of a diameter.
G.G.72. Coordinate Geometry: Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer.
G.G.73. Coordinate Geometry: Find the center and radius of a circle, given the equation of the circle in center-radius form.
G.G.74. Coordinate Geometry: Graph circles of the form (x - h) to the power of 2 + (j - k) to the power of 2 = r to the power of 2.
A2.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A2.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
A2.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A2.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
A2.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A2.PS.6. Use a variety of strategies to extend solution methods to other problems.
A2.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
A2.PS.8. Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A2.PS.9. Interpret solutions within the given constraints of a problem.
A2.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
A2.RP.1. Support mathematical ideas using a variety of strategies.
A2.RP.2. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
A2.RP.3. Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer.
A2.RP.4. Recognize when an approximation is more appropriate than an exact answer.
A2.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A2.RP.6. Construct logical arguments that verify claims or counterexamples that refute claims.
A2.RP.7. Present correct mathematical arguments in a variety of forms.
A2.RP.8. Evaluate written arguments for validity.
A2.RP.9. Support an argument by using a systematic approach to test more than one case.
A2.RP.10. Devise ways to verify results, using counterexamples and informal indirect proof.
A2.RP.11. Extend specific results to more general cases.
A2.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
A2.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A2.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A2.CM.4. Explain relationships among different representations of a problem.
A2.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A2.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
A2.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A2.CM.8. Reflect on strategies of others in relation to one's own strategy.
A2.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
A2.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A2.CM.11. Represent word problems using standard mathematical notation.
A2.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A2.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
A2.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A2.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
A2.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A2.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A2.CN.5. Understand how quantitative models connect to various physical models and representations.
A2.CN.6. Recognize and apply mathematics to situations in the outside world.
A2.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A2.CN.8. Develop an appreciation for the historical development of mathematics.
A2.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A2.R.2. Recognize, compare, and use an array of representational forms.
A2.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
A2.R.4. Select appropriate representations to solve problem situations.
A2.R.5. Investigate relationships among different representations and their impact on a given problem.
A2.R.6. Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions).
A2.R.7. Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll).
A2.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss).
A2.N.1. Operations: Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers).
A2.N.2. Operations: Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form.
A2.N.3. Operations: Perform arithmetic operations with polynomial expressions containing rational coefficients.
A2.N.4. Operations: Perform arithmetic operations on irrational expressions.
A2.N.5. Operations: Rationalize a denominator containing a radical expression.
A2.N.6. Operations: Write square roots of negative numbers in terms of i.
A2.N.7. Operations: Simplify powers of i.
A2.N.8. Operations: Determine the conjugate of a complex number.
A2.N.9. Operations: Perform arithmetic operations on complex numbers and write the answer in the form a + bi. Note: This includes simplifying expressions with complex denominators.
A2.N.10. Operations: Know and apply sigma notation.
A2.A.1. Equations and Inequalities: Solve absolute value equations and inequalities involving linear expressions in one variable.
A2.A.2. Equations and Inequalities: Use the discriminant to determine the nature of the roots of a quadratic equation.
A2.A.3. Equations and Inequalities: Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots.
A2.A.4. Equations and Inequalities: Solve quadratic inequalities in one and two variables, algebraically and graphically.
A2.A.5. Equations and Inequalities: Use direct and inverse variation to solve for unknown values.
A2.A.6. Equations and Inequalities: Solve an application which results in an exponential function.
A2.A.7. Variables and Expressions: Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials.
A2.A.8. Variables and Expressions: Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents.
A2.A.9. Variables and Expressions: Rewrite algebraic expressions that contain negative exponents using only positive exponents.
A2.A.10. Variables and Expressions: Rewrite algebraic expressions with fractional exponents as radical expressions.
A2.A.11. Variables and Expressions: Rewrite algebraic expressions in radical form as expressions with fractional exponents.
A2.A.12. Variables and Expressions: Evaluate exponential expressions, including those with base e.
A2.A.13. Variables and Expressions: Simplify radical expressions.
A2.A.14. Variables and Expressions: Perform addition, subtraction, multiplication and division of radical expressions.
A2.A.15. Variables and Expressions: Rationalize denominators involving algebraic radical expressions.
A2.A.16. Variables and Expressions: Perform arithmetic operations with rational expressions and rename to lowest terms.
A2.A.17. Variables and Expressions: Simplify complex fractional expressions.
A2.A.18. Variables and Expressions: Evaluate logarithmic expressions in any base.
A2.A.19. Variables and Expressions: Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms.
A2.A.20. Equations and Inequalities: Determine the sum and product of the roots of a quadratic equation by examining its coefficients.
A2.A.21. Equations and Inequalities: Determine the quadratic equation, given the sum and product of its roots.
A2.A.22. Equations and Inequalities: Solve radical equations.
A2.A.23. Equations and Inequalities: Solve rational equations and inequalities.
A2.A.24. Equations and Inequalities: Know and apply the technique of completing the square.
A2.A.25. Equations and Inequalities: Solve quadratic equations, using the quadratic formula.
A2.A.26. Equations and Inequalities: Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula.
A2.A.27. Equations and Inequalities: Solve exponential equations with and without common bases.
A2.A.28. Equations and Inequalities: Solve a logarithmic equation by rewriting as an exponential equation.
A2.A.29. Patterns, Relations, and Functions: Identify an arithmetic or geometric sequence and find the formula for its nth term.
A2.A.30. Patterns, Relations, and Functions: Determine the common difference in an arithmetic sequence.
A2.A.31. Patterns, Relations, and Functions: Determine the common ratio in a geometric sequence.
A2.A.32. Patterns, Relations, and Functions: Determine a specified term of an arithmetic or geometric sequence.
A2.A.33. Patterns, Relations, and Functions: Specify terms of a sequence, given its recursive definition.
A2.A.34. Patterns, Relations, and Functions: Represent the sum of a series, using sigma notation.
A2.A.35. Patterns, Relations, and Functions: Determine the sum of the first n terms of an arithmetic or geometric series.
A2.A.36. Patterns, Relations, and Functions: Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion.
A2.A.37. Patterns, Relations, and Functions: Define a relation and function.
A2.A.38. Patterns, Relations, and Functions: Determine when a relation is a function.
A2.A.39. Patterns, Relations, and Functions: Determine the domain and range of a function from its equation.
A2.A.40. Patterns, Relations, and Functions: Write functions in functional notation.
A2.A.41. Patterns, Relations, and Functions: Use functional notation to evaluate functions for given values in the domain.
A2.A.42. Patterns, Relations, and Functions: Find the composition of functions.
A2.A.43. Patterns, Relations, and Functions: Determine if a function is one-to-one, onto, or both.
A2.A.44. Patterns, Relations, and Functions: Define the inverse of a function.
A2.A.45. Patterns, Relations, and Functions: Determine the inverse of a function and use composition to justify the result.
A2.A.46. Patterns, Relations, and Functions: Perform transformations with functions and relations: f(x + a), f(x) + a, f(-x), -f(x), af(x).
A2.A.47. Coordinate Geometry: Determine the center-radius form for the equation of circle in standard form.
A2.A.48. Coordinate Geometry: Write the equation of a circle, given its center and a point on the circle.
A2.A.49. Coordinate Geometry: Write the equation of a circle from its graph.
A2.A.50. Coordinate Geometry: Approximate the solution to polynomial equations of higher degree by inspecting the graph.
A2.A.51. Coordinate Geometry: Determine the domain and range of a function from its graph.
A2.A.52. Coordinate Geometry: Identify relations and functions, using graphs.
A2.A.53. Coordinate Geometry: Graph exponential functions of the form y = b to the power of x for positive values of b, including b = e.
A2.A.54. Graph logarithmic functions, using the inverse of the related exponential function.
A2.A.55. Trigonometric Functions: Express and apply the six trigonometric functions as ratios of the sides of a right triangle.
A2.A.56. Trigonometric Functions: Know the exact and approximate values of the sine, cosine, and tangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.57. Trigonometric Functions: Sketch and use the reference angle for angles in standard position.
A2.A.58. Trigonometric Functions: Know and apply the co-function and reciprocal relationships between trigonometric ratios.
A2.A.59. Trigonometric Functions: Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.60. Trigonometric Functions: Sketch the unit circle and represent angles in standard position.
A2.A.61. Trigonometric Functions: Determine the length of an arc of a circle, given its radius and the measure of its central angle.
A2.A.62. Trigonometric Functions: Find the value of trigonometric functions, if given a point on the terminal side of angle theta.
A2.A.63. Trigonometric Functions: Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function.
A2.A.64. Trigonometric Functions: Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent.
A2.A.65. Trigonometric Functions: Sketch the graph of the inverses of the sine, cosine, and tangent functions.
A2.A.66. Trigonometric Functions: Determine the trigonometric functions of any angle, using technology.
A2.A.67. Trigonometric Functions: Justify the Pythagorean identities.
A2.A.68. Trigonometric Functions: Solve trigonometric equations for all values of the variable from 0 degrees to 360 degrees.
A2.A.69. Trigonometric Functions: Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function.
A2.A.70. Trigonometric Functions: Sketch and recognize one cycle of a function of the form y = Asin Bx or y = Acos Bx.
A2.A.71. Trigonometric Functions: Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x).
A2.A.72. Trigonometric Functions: Write the trigonometric function that is represented by a given periodic graph.
A2.A.73. Trigonometric Functions: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines.
A2.A.74. Trigonometric Functions: Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle.
A2.A.75. Trigonometric Functions: Determine the solution(s) from the SSA situation (ambiguous case).
A2.A.76. Trigonometric Functions: Apply the angle sum and difference formulas for trigonometric functions.
A2.A.77. Trigonometric Functions: Apply the double-angle and half-angle formulas for trigonometric functions.
A2.M.1. Units of Measurement: Define radian measure.
A2.M.2. Units of Measurement: Convert between radian and degree measures.
A2.S.1. Collection of Data: Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment).
A2.S.2. Collection of Data: Determine factors which may affect the outcome of a survey.
A2.S.3. Organization and Display of Data: Calculate measures of central tendency with group frequency distributions.
A2.S.4. Organization and Display of Data: Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations.
A2.S.5. Organization and Display of Data: Know and apply the characteristics of the normal distribution.
A2.S.6. Predictions from Data: Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate.
A2.S.7. Predictions from Data: Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data.
A2.S.8. Predictions from Data: Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship.
A2.S.9. Probability: Differentiate between situations requiring permutations and those requiring combinations.
A2.S.10. Probability: Calculate the number of possible permutations (nPr) of n items taken r at a time.
A2.S.11. Probability: Calculate the number of possible combinations (nCr) of n items taken r at a time.
A2.S.12. Probability: Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event).
A2.S.13. Probability: Calculate theoretical probabilities, including geometric applications.
A2.S.14. Probability: Calculate empirical probabilities.
A2.S.15. Probability: Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most.
A2.S.16. Probability: Use the normal distribution as an approximation for binomial probabilities.
NY.3. Integrated Algebra: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.
3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.
A.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.
A.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.
A.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A.PS.6. Use a variety of strategies to extend solution methods to other problems.
A.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.
A.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A.PS.9. Interpret solutions within the given constraints of a problem.
A.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.
A.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.
A.RP.2. Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture.
A.RP.3. Recognize when an approximation is more appropriate than an exact answer.
3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.
A.RP.4. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A.RP.5. Construct logical arguments that verify claims or counterexamples that refute them.
A.RP.6. Present correct mathematical arguments in a variety of forms.
A.RP.7. Evaluate written arguments for validity.
3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.
A.RP.8. Support an argument by using a systematic approach to test more than one case.
A.RP.9. Devise ways to verify results or use counterexamples to refute incorrect statements.
A.RP.10. Extend specific results to more general cases.
A.RP.11. Use a Venn diagram to support a logical argument.
A.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.
A.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams.
3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
A.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A.CM.4. Explain relationships among different representations of a problem.
A.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.
A.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A.CM.8. Reflect on strategies of others in relation to one's own strategy.
A.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.
A.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A.CM.11. Represent word problems using standard mathematical notation.
A.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.
A.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
A.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A.CN.5. Understand how quantitative models connect to various physical models and representations.
3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.
A.CN.6. Recognize and apply mathematics to situations in the outside world.
A.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A.CN.8. Develop an appreciation for the historical development of mathematics.
3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.
A.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A.R.2. Recognize, compare, and use an array of representational forms.
A.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.
A.R.4. Select appropriate representations to solve problem situations.
A.R.5. Investigate relationships between different representations and their impact on a given problem.
3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.
A.R.6. Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground).
A.R.7. Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales).
A.R.8. Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations y = x to the power of 2 and y = -x to the power of 2).
3.19. Number Sense and Operations Strand: Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems.
A.N.1. Number Theory: Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas.
3.20. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.
A.N.2. Operations: Simplify radical terms (no variable in the radicand).
A.N.3. Operations: Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form.
A.N.4. Operations: Understand and use scientific notation to compute products and quotients of numbers.
A.N.5. Operations: Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation.
A.N.6. Operations: Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s).
A.N.7. Operations: Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting.
A.N.8. Operations: Determine the number of possible arrangements (permutations) of a list of items.
3.21. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.
A.A.1. Variables and Expressions: Translate a quantitative verbal phrase into an algebraic expression.
A.A.2. Variables and Expressions: Write a verbal expression that matches a given mathematical expression.
A.A.3. Equations and Inequalities: Distinguish the difference between an algebraic expression and an algebraic equation.
A.A.4. Equations and Inequalities: Translate verbal sentences into mathematical equations or inequalities.
A.A.5. Equations and Inequalities: Write algebraic equations or inequalities that represent a situation.
A.A.6. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable.
A.A.7. Equations and Inequalities: Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables.
A.A.8. Equations and Inequalities: Analyze and solve verbal problems that involve quadratic equations.
A.A.9. Equations and Inequalities: Analyze and solve verbal problems that involve exponential growth and decay.
A.A.10. Equations and Inequalities: Solve systems of two linear equations in two variables algebraically (See A.G.7).
A.A.11. Equations and Inequalities: Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers.
3.22. Algebra Strand: Students will perform algebraic procedures accurately.
A.A.12. Variables and Expressions: Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only.
A.A.13. Variables and Expressions: Add, subtract, and multiply monomials and polynomials.
A.A.14. Variables and Expressions: Divide a polynomial by a monomial or binomial, where the quotient has no remainder.
A.A.15. Variables and Expressions: Find values of a variable for which an algebraic fraction is undefined.
A.A.16. Variables and Expressions: Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms.
A.A.17. Variables and Expressions: Add or subtract fractional expressions with monomial or like binomial denominators.
A.A.18. Variables and Expressions: Multiply and divide algebraic fractions and express the product or quotient in simplest form.
A.A.19. Variables and Expressions: Identify and factor the difference of two perfect squares.
A.A.20. Variables and Expressions: Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF).
A.A.21. Equations and Inequalities: Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable.
A.A.22. Equations and Inequalities: Solve all types of linear equations in one variable.
A.A.23. Equations and Inequalities: Solve literal equations for a given variable.
A.A.24. Equations and Inequalities: Solve linear inequalities in one variable.
A.A.25. Equations and Inequalities: Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable.
A.A.26. Equations and Inequalities: Solve algebraic proportions in one variable which result in linear or quadratic equations.
A.A.27. Equations and Inequalities: Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots.
A.A.28. Equations and Inequalities: Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression.
3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.
A.A.29. Patterns, Relations, and Functions: Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form.
A.A.30. Patterns, Relations, and Functions: Find the complement of a subset of a given set, within a given universe.
A.A.31. Patterns, Relations, and Functions: Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets).
A.A.32. Coordinate Geometry: Explain slope as a rate of change between dependent and independent variables.
A.A.33. Coordinate Geometry: Determine the slope of a line, given the coordinates of two points on the line.
A.A.34. Coordinate Geometry: Write the equation of a line, given its slope and the coordinates of a point on the line.
A.A.35. Coordinate Geometry: Write the equation of a line, given the coordinates of two points on the line.
A.A.36. Coordinate Geometry: Write the equation of a line parallel to the x- or y-axis.
A.A.37. Coordinate Geometry: Determine the slope of a line, given its equation in any form.
A.A.38. Coordinate Geometry: Determine if two lines are parallel, given their equations in any form.
A.A.39. Coordinate Geometry: Determine whether a given point is on a line, given the equation of the line.
A.A.40. Coordinate Geometry: Determine whether a given point is in the solution set of a system of linear inequalities.
A.A.41. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 ).
A.A.42. Trigonometric Functions: Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides.
A.A.43. Trigonometric Functions: Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle.
A.A.44. Trigonometric Functions: Find the measure of a side of a right triangle, given an acute angle and the length of another side.
A.A.45. Trigonometric Functions: Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides.
3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.
A.G.1. Shapes: Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
A.G.2. Shapes: Use formulas to calculate volume and surface area of rectangular solids and cylinders.
3.25. Geometry Strand: Students will apply coordinate geometry to analyze problem solving situations.
A.G.3. Coordinate Geometry: Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations.
A.G.4. Coordinate Geometry: Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions.
A.G.5. Coordinate Geometry: Investigate and generalize how changing the coefficients of a function affects its graph.
A.G.6. Coordinate Geometry: Graph linear inequalities.
A.G.7. Coordinate Geometry: Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10).
A.G.8. Coordinate Geometry: Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions.
A.G.9. Coordinate Geometry: Solve systems of linear and quadratic equations graphically. Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers.
A.G.10. Coordinate Geometry: Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41 ) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value.
3.26. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.
A.M.1. Units of Measurement: Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail).
A.M.2. Units of Measurement: Solve problems involving conversions within measurement systems, given the relationship between the units.
3.27. Measurement Strand: Understand that all measurement contains error and be able to determine its significance.
A.M.3. Error and Magnitude: Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure.
3.28. Statistics and Probability Strand: Students will collect, organize, display, and analyze data.
A.S.1. Organization and Display of Data: Categorize data as qualitative or quantitative.
A.S.2. Organization and Display of Data: Determine whether the data to be analyzed is univariate or bivariate.
A.S.3. Organization and Display of Data: Determine when collected data or display of data may be biased.
A.S.4. Organization and Display of Data: Compare and contrast the appropriateness of different measures of central tendency for a given data set.
A.S.5. Organization and Display of Data: Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data.
A.S.6. Organization and Display of Data: Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box and- whisker plot.
A.S.7. Organization and Display of Data: Create a scatter plot of bivariate data.
A.S.8. Organization and Display of Data: Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line.
A.S.9. Analysis of Data: Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot.
A.S.10. Analysis of Data: Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions.
A.S.11. Analysis of Data: Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles.
A.S.12. Analysis of Data: Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none).
A.S.13. Analysis of Data: Understand the difference between correlation and causation.
A.S.14. Analysis of Data: Identify variables that might have a correlation but not a causal relationship.
3.29. Statistics and Probability Strand: Students will make predictions that are based upon data analysis.
A.S.15. Predictions from Data: Identify and describe sources of bias and its effect, drawing conclusions from data.
A.S.16. Predictions from Data: Recognize how linear transformations of one-variable data affect the data's mean, median, mode, and range.
A.S.17. Predictions from Data: Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation.
3.30. Statistics and Probability Strand: Students will understand and apply concepts of probability.
A.S.18. Probability: Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
A.S.19. Probability: Determine the number of elements in a sample space and the number of favorable events.
A.S.20. Probability: Calculate the probability of an event and its complement.
A.S.21. Probability: Determine empirical probabilities based on specific sample data.
A.S.22. Probability: Determine, based on calculated probability of a set of events, if some or all are equally likely to occur; one is more likely to occur than another; whether or not an event is certain to happen or not to happen.
A.S.23. Probability: Calculate the probability of a series of independent events; a series of dependent events; two mutually exclusive events; two events that are not mutually exclusive.
G.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
G.PS.2. Observe and explain patterns to formulate generalizations and conjectures.
G.PS.3. Use multiple representations to represent and explain problem situations (e.g., spatial, geometric, verbal, numeric, algebraic, and graphical representations).
G.PS.4. Construct various types of reasoning, arguments, justifications and methods of proof for problems.
G.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
G.PS.6. Use a variety of strategies to extend solution methods to other problems.
G.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
G.PS.8. Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
G.PS.9. Interpret solutions within the given constraints of a problem.
G.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
G.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.
G.RP.2. Recognize and verify, where appropriate, geometric relationships of perpendicularity, parallelism, congruence, and similarity, using algebraic strategies.
G.RP.3. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
G.RP.4. Provide correct mathematical arguments in response to other students' conjectures, reasoning, and arguments.
G.RP.5. Present correct mathematical arguments in a variety of forms.
G.RP.6. Evaluate written arguments for validity.
G.RP.7. Construct a proof using a variety of methods (e.g., deductive, analytic, transformational).
G.RP.8. Devise ways to verify results or use counterexamples to refute incorrect statements.
G.RP.9. Apply inductive reasoning in making and supporting mathematical conjectures.
G.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
G.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
G.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
G.CM.4. Explain relationships among different representations of a problem.
G.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
G.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
G.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
G.CM.8. Reflect on strategies of others in relation to one's own strategy.
G.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
G.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
G.CM.11. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams.
G.CM.12. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
G.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
G.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
G.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
G.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
G.CN.5. Understand how quantitative models connect to various physical models and representations.
G.CN.6. Recognize and apply mathematics to situations in the outside world.
G.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
G.CN.8. Develop an appreciation for the historical development of mathematics.
G.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
G.R.2. Recognize, compare, and use an array of representational forms.
G.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
G.R.4. Select appropriate representations to solve problem situations.
G.R.5. Investigate relationships between different representations and their impact on a given problem.
G.R.6. Use mathematics to show and understand physical phenomena (e.g., determine the number of gallons of water in a fish tank).
G.R.7. Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person's argument have a logical foundation).
G.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent).
G.G.1. Geometric Relationships: Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
G.G.2. Geometric Relationships: Know and apply that through a given point there passes one and only one plane perpendicular to a given line.
G.G.3. Geometric Relationships: Know and apply that through a given point there passes one and only one line perpendicular to a given plane.
G.G.4. Geometric Relationships: Know and apply that two lines perpendicular to the same plane are coplanar.
G.G.5. Geometric Relationships: Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.
G.G.6. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.
G.G.7. Geometric Relationships: Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
G.G.8. Geometric Relationships: Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines.
G.G.9. Geometric Relationships: Know and apply that if two planes are perpendicular to the same line, they are parallel.
G.G.10. Geometric Relationships: Know and apply that the lateral edges of a prism are congruent and parallel.
G.G.11. Geometric Relationships: Know and apply that two prisms have equal volumes if their bases have equal areas and their altitudes are equal.
G.G.12. Geometric Relationships: Know and apply that the volume of a prism is the product of the area of the base and the altitude.
G.G.13. Geometric Relationships: Apply the properties of a regular pyramid, including lateral edges are congruent; lateral faces are congruent isosceles triangles; volume of a pyramid equals one-third the product of the area of the base and the altitude.
G.G.14. Geometric Relationships: Apply the properties of a cylinder, including bases are congruent; volume equals the product of the area of the base and the altitude; lateral area of a right circular cylinder equals the product of an altitude and the circumference of the base.
G.G.15. Geometric Relationships: Apply the properties of a right circular cone, including lateral area equals one-half the product of the slant height and the circumference of its base volume is one-third the product of the area of its base and its altitude.
G.G.16. Geometric Relationships: Apply the properties of a sphere, including the intersection of a plane and a sphere is a circle; a great circle is the largest circle that can be drawn on a sphere; two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles; surface area is 4 pi r to the power of 2; volume is 4/3 pi r to the power of 3.
G.G.17. Constructions: Construct a bisector of a given angle, using a straightedge and compass, and justify the construction.
G.G.18. Constructions: Construct the perpendicular bisector of a given segment, using a straightedge and compass, and justify the construction.
G.G.19. Constructions: Construct lines parallel (or perpendicular) to a given line through a given point, using a straightedge and compass, and justify the construction.
G.G.20. Constructions: Construct an equilateral triangle, using a straightedge and compass, and justify the construction.
G.G.21. Locus: Investigate and apply the concurrence of medians, altitudes, angle bisectors, and perpendicular bisectors of triangles.
G.G.22. Locus: Solve problems using compound loci.
G.G.23. Locus: Graph and solve compound loci in the coordinate plane.
G.G.24. Informal and Formal Proofs: Determine the negation of a statement and establish its truth value.
G.G.25. Informal and Formal Proofs: Know and apply the conditions under which a compound statement (conjunction, disjunction, conditional, biconditional) is true.
G.G.26. Informal and Formal Proofs: Identify and write the inverse, converse, and contrapositive of a given conditional statement and note the logical equivalences.
G.G.27. Informal and Formal Proofs: Write a proof arguing from a given hypothesis to a given conclusion.
G.G.28. Informal and Formal Proofs: Determine the congruence of two triangles by using one of the five congruence techniques (SSS, SAS, ASA, AAS, HL), given sufficient information about the sides and/or angles of two congruent triangles.
G.G.29. Informal and Formal Proofs: Identify corresponding parts of congruent triangles.
G.G.30. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the angles of a triangle.
G.G.31. Informal and Formal Proofs: Investigate, justify, and apply the isosceles triangle theorem and its converse.
G.G.32. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric inequalities, using the exterior angle theorem.
G.G.33. Informal and Formal Proofs: Investigate, justify, and apply the triangle inequality theorem.
G.G.34. Informal and Formal Proofs: Determine either the longest side of a triangle given the three angle measures or the largest angle given the lengths of three sides of a triangle.
G.G.35. Informal and Formal Proofs: Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines.
G.G.36. Informal and Formal Proofs: Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons.
G.G.37. Informal and Formal Proofs: Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons.
G.G.38. Informal and Formal Proofs: Investigate, justify, and apply theorems about parallelograms involving their angles, sides, and diagonals.
G.G.39. Informal and Formal Proofs: Investigate, justify, and apply theorems about special parallelograms (rectangles, rhombuses, squares) involving their angles, sides, and diagonals.
G.G.40. Investigate, justify, and apply theorems about trapezoids (including isosceles trapezoids) involving their angles, sides, medians, and diagonals.
G.G.41. Informal and Formal Proofs: Justify that some quadrilaterals are parallelograms, rhombuses, rectangles, squares, or trapezoids.
G.G.42. Informal and Formal Proofs: Investigate, justify, and apply theorems about geometric relationships, based on the properties of the line segment joining the midpoints of two sides of the triangle.
G.G.43. Informal and Formal Proofs: Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1.
G.G.44. Informal and Formal Proofs: Establish similarity of triangles, using the following theorems: AA, SAS, and SSS.
G.G.45. Informal and Formal Proofs: Investigate, justify, and apply theorems about similar triangles.
G.G.46. Informal and Formal Proofs: Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle.
G.G.47. Informal and Formal Proofs: Investigate, justify, and apply theorems about mean proportionality: the altitude to the hypotenuse of a right triangle is the mean proportional between the two segments along the hypotenuse; the altitude to the hypotenuse of a right triangle divides the hypotenuse so that either leg of the right triangle is the mean proportional between the hypotenuse and segment of the hypotenuse adjacent to that leg.
G.G.48. Informal and Formal Proofs: Investigate, justify, and apply the Pythagorean theorem and its converse.
G.G.49. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding chords of a circle: perpendicular bisectors of chords the relative lengths of chords as compared to their distance from the center of the circle.
G.G.50. Informal and Formal Proofs: Investigate, justify, and apply theorems about tangent lines to a circle: a perpendicular to the tangent at the point of tangency; two tangents to a circle from the same external point; common tangents of two non-intersecting or tangent circles.
G.G.51. Informal and Formal Proofs: Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle when the vertex is inside the circle (two chords); on the circle (tangent and chord); outside the circle (two tangents, two secants, or tangent and secant).
G.G.52. Informal and Formal Proofs: Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines.
G.G.53. Informal and Formal Proofs: Investigate, justify, and apply theorems regarding segments intersected by a circle: along two tangents from the same external point; along two secants from the same external point; along a tangent and a secant from the same external point; along two intersecting chords of a given circle.
G.G.54. Transformational Geometry: Define, investigate, justify, and apply isometries in the plane (rotations, reflections, translations, glide reflections) Note: Use proper function notation.
G.G.55. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under translations, rotations, reflections, and glide reflections.
G.G.56. Transformational Geometry: Identify specific isometries by observing orientation, numbers of invariant points, and/or parallelism.
G.G.57. Transformational Geometry: Justify geometric relationships (perpendicularity, parallelism, congruence) using transformational techniques (translations, rotations, reflections).
G.G.58. Transformational Geometry: Define, investigate, justify, and apply similarities (dilations and the composition of dilations and isometries).
G.G.59. Transformational Geometry: Investigate, justify, and apply the properties that remain invariant under similarities.
G.G.60. Transformational Geometry: Identify specific similarities by observing orientation, numbers of invariant points, and/or parallelism.
G.G.61. Transformational Geometry: Investigate, justify, and apply the analytical representations for translations, rotations about the origin of 90 degrees and 180 degrees, reflections over the lines x = 0, y = 0, and y = x, and dilations centered at the origin.
G.G.62. Coordinate Geometry: Find the slope of a perpendicular line, given the equation of a line.
G.G.63. Coordinate Geometry: Determine whether two lines are parallel, perpendicular, or neither, given their equations.
G.G.64. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line perpendicular to the given line.
G.G.65. Coordinate Geometry: Find the equation of a line, given a point on the line and the equation of a line parallel to the desired line.
G.G.66. Coordinate Geometry: Find the midpoint of a line segment, given its endpoints.
G.G.67. Coordinate Geometry: Find the length of a line segment, given its endpoints.
G.G.68. Coordinate Geometry: Find the equation of a line that is the perpendicular bisector of a line segment, given the endpoints of the line segment.
G.G.69. Coordinate Geometry: Investigate, justify, and apply the properties of triangles and quadrilaterals in the coordinate plane, using the distance, midpoint, and slope formulas.
G.G.70. Coordinate Geometry: Solve systems of equations involving one linear equation and one quadratic equation graphically.
G.G.71. Coordinate Geometry: Write the equation of a circle, given its center and radius or given the endpoints of a diameter.
G.G.72. Coordinate Geometry: Write the equation of a circle, given its graph Note: The center is an ordered pair of integers and the radius is an integer.
G.G.73. Coordinate Geometry: Find the center and radius of a circle, given the equation of the circle in center-radius form.
G.G.74. Coordinate Geometry: Graph circles of the form (x - h) to the power of 2 + (j - k) to the power of 2 = r to the power of 2.
A2.PS.1. Use a variety of problem solving strategies to understand new mathematical content.
A2.PS.2. Recognize and understand equivalent representations of a problem situation or a mathematical concept.
A2.PS.3. Observe and explain patterns to formulate generalizations and conjectures.
A2.PS.4. Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically).
A2.PS.5. Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic).
A2.PS.6. Use a variety of strategies to extend solution methods to other problems.
A2.PS.7. Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving.
A2.PS.8. Determine information required to solve the problem, choose methods for obtaining the information, and define parameters for acceptable solutions.
A2.PS.9. Interpret solutions within the given constraints of a problem.
A2.PS.10. Evaluate the relative efficiency of different representations and solution methods of a problem.
A2.RP.1. Support mathematical ideas using a variety of strategies.
A2.RP.2. Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion.
A2.RP.3. Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer.
A2.RP.4. Recognize when an approximation is more appropriate than an exact answer.
A2.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
A2.RP.6. Construct logical arguments that verify claims or counterexamples that refute claims.
A2.RP.7. Present correct mathematical arguments in a variety of forms.
A2.RP.8. Evaluate written arguments for validity.
A2.RP.9. Support an argument by using a systematic approach to test more than one case.
A2.RP.10. Devise ways to verify results, using counterexamples and informal indirect proof.
A2.RP.11. Extend specific results to more general cases.
A2.RP.12. Apply inductive reasoning in making and supporting mathematical conjectures.
A2.CM.1. Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem.
A2.CM.2. Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
A2.CM.3. Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form.
A2.CM.4. Explain relationships among different representations of a problem.
A2.CM.5. Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
A2.CM.6. Support or reject arguments or questions raised by others about the correctness of mathematical work.
A2.CM.7. Read and listen for logical understanding of mathematical thinking shared by other students.
A2.CM.8. Reflect on strategies of others in relation to one's own strategy.
A2.CM.9. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.
A2.CM.10. Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures.
A2.CM.11. Represent word problems using standard mathematical notation.
A2.CM.12. Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.
A2.CM.13. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.
A2.CN.1. Understand and make connections among multiple representations of the same mathematical idea.
A2.CN.2. Understand the corresponding procedures for similar problems or mathematical concepts.
A2.CN.3. Model situations mathematically, using representations to draw conclusions and formulate new situations.
A2.CN.4. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.
A2.CN.5. Understand how quantitative models connect to various physical models and representations.
A2.CN.6. Recognize and apply mathematics to situations in the outside world.
A2.CN.7. Recognize and apply mathematical ideas to problem situations that develop outside of mathematics.
A2.CN.8. Develop an appreciation for the historical development of mathematics.
A2.R.1. Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts.
A2.R.2. Recognize, compare, and use an array of representational forms.
A2.R.3. Use representation as a tool for exploring and understanding mathematical ideas.
A2.R.4. Select appropriate representations to solve problem situations.
A2.R.5. Investigate relationships among different representations and their impact on a given problem.
A2.R.6. Use mathematics to show and understand physical phenomena (e.g., investigate sound waves using the sine and cosine functions).
A2.R.7. Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll).
A2.R.8. Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss).
A2.N.1. Operations: Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers).
A2.N.2. Operations: Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form.
A2.N.3. Operations: Perform arithmetic operations with polynomial expressions containing rational coefficients.
A2.N.4. Operations: Perform arithmetic operations on irrational expressions.
A2.N.5. Operations: Rationalize a denominator containing a radical expression.
A2.N.6. Operations: Write square roots of negative numbers in terms of i.
A2.N.7. Operations: Simplify powers of i.
A2.N.8. Operations: Determine the conjugate of a complex number.
A2.N.9. Operations: Perform arithmetic operations on complex numbers and write the answer in the form a + bi. Note: This includes simplifying expressions with complex denominators.
A2.N.10. Operations: Know and apply sigma notation.
A2.A.1. Equations and Inequalities: Solve absolute value equations and inequalities involving linear expressions in one variable.
A2.A.2. Equations and Inequalities: Use the discriminant to determine the nature of the roots of a quadratic equation.
A2.A.3. Equations and Inequalities: Solve systems of equations involving one linear equation and one quadratic equation algebraically Note: This includes rational equations that result in linear equations with extraneous roots.
A2.A.4. Equations and Inequalities: Solve quadratic inequalities in one and two variables, algebraically and graphically.
A2.A.5. Equations and Inequalities: Use direct and inverse variation to solve for unknown values.
A2.A.6. Equations and Inequalities: Solve an application which results in an exponential function.
A2.A.7. Variables and Expressions: Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials.
A2.A.8. Variables and Expressions: Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents.
A2.A.9. Variables and Expressions: Rewrite algebraic expressions that contain negative exponents using only positive exponents.
A2.A.10. Variables and Expressions: Rewrite algebraic expressions with fractional exponents as radical expressions.
A2.A.11. Variables and Expressions: Rewrite algebraic expressions in radical form as expressions with fractional exponents.
A2.A.12. Variables and Expressions: Evaluate exponential expressions, including those with base e.
A2.A.13. Variables and Expressions: Simplify radical expressions.
A2.A.14. Variables and Expressions: Perform addition, subtraction, multiplication and division of radical expressions.
A2.A.15. Variables and Expressions: Rationalize denominators involving algebraic radical expressions.
A2.A.16. Variables and Expressions: Perform arithmetic operations with rational expressions and rename to lowest terms.
A2.A.17. Variables and Expressions: Simplify complex fractional expressions.
A2.A.18. Variables and Expressions: Evaluate logarithmic expressions in any base.
A2.A.19. Variables and Expressions: Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms.
A2.A.20. Equations and Inequalities: Determine the sum and product of the roots of a quadratic equation by examining its coefficients.
A2.A.21. Equations and Inequalities: Determine the quadratic equation, given the sum and product of its roots.
A2.A.22. Equations and Inequalities: Solve radical equations.
A2.A.23. Equations and Inequalities: Solve rational equations and inequalities.
A2.A.24. Equations and Inequalities: Know and apply the technique of completing the square.
A2.A.25. Equations and Inequalities: Solve quadratic equations, using the quadratic formula.
A2.A.26. Equations and Inequalities: Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula.
A2.A.27. Equations and Inequalities: Solve exponential equations with and without common bases.
A2.A.28. Equations and Inequalities: Solve a logarithmic equation by rewriting as an exponential equation.
A2.A.29. Patterns, Relations, and Functions: Identify an arithmetic or geometric sequence and find the formula for its nth term.
A2.A.30. Patterns, Relations, and Functions: Determine the common difference in an arithmetic sequence.
A2.A.31. Patterns, Relations, and Functions: Determine the common ratio in a geometric sequence.
A2.A.32. Patterns, Relations, and Functions: Determine a specified term of an arithmetic or geometric sequence.
A2.A.33. Patterns, Relations, and Functions: Specify terms of a sequence, given its recursive definition.
A2.A.34. Patterns, Relations, and Functions: Represent the sum of a series, using sigma notation.
A2.A.35. Patterns, Relations, and Functions: Determine the sum of the first n terms of an arithmetic or geometric series.
A2.A.36. Patterns, Relations, and Functions: Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion.
A2.A.37. Patterns, Relations, and Functions: Define a relation and function.
A2.A.38. Patterns, Relations, and Functions: Determine when a relation is a function.
A2.A.39. Patterns, Relations, and Functions: Determine the domain and range of a function from its equation.
A2.A.40. Patterns, Relations, and Functions: Write functions in functional notation.
A2.A.41. Patterns, Relations, and Functions: Use functional notation to evaluate functions for given values in the domain.
A2.A.42. Patterns, Relations, and Functions: Find the composition of functions.
A2.A.43. Patterns, Relations, and Functions: Determine if a function is one-to-one, onto, or both.
A2.A.44. Patterns, Relations, and Functions: Define the inverse of a function.
A2.A.45. Patterns, Relations, and Functions: Determine the inverse of a function and use composition to justify the result.
A2.A.46. Patterns, Relations, and Functions: Perform transformations with functions and relations: f(x + a), f(x) + a, f(-x), -f(x), af(x).
A2.A.47. Coordinate Geometry: Determine the center-radius form for the equation of circle in standard form.
A2.A.48. Coordinate Geometry: Write the equation of a circle, given its center and a point on the circle.
A2.A.49. Coordinate Geometry: Write the equation of a circle from its graph.
A2.A.50. Coordinate Geometry: Approximate the solution to polynomial equations of higher degree by inspecting the graph.
A2.A.51. Coordinate Geometry: Determine the domain and range of a function from its graph.
A2.A.52. Coordinate Geometry: Identify relations and functions, using graphs.
A2.A.53. Coordinate Geometry: Graph exponential functions of the form y = b to the power of x for positive values of b, including b = e.
A2.A.54. Graph logarithmic functions, using the inverse of the related exponential function.
A2.A.55. Trigonometric Functions: Express and apply the six trigonometric functions as ratios of the sides of a right triangle.
A2.A.56. Trigonometric Functions: Know the exact and approximate values of the sine, cosine, and tangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.57. Trigonometric Functions: Sketch and use the reference angle for angles in standard position.
A2.A.58. Trigonometric Functions: Know and apply the co-function and reciprocal relationships between trigonometric ratios.
A2.A.59. Trigonometric Functions: Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0 degree, 30 degree, 45 degree, 60 degree, 90 degree, 180 degree, and 270 degree angles.
A2.A.60. Trigonometric Functions: Sketch the unit circle and represent angles in standard position.
A2.A.61. Trigonometric Functions: Determine the length of an arc of a circle, given its radius and the measure of its central angle.
A2.A.62. Trigonometric Functions: Find the value of trigonometric functions, if given a point on the terminal side of angle theta.
A2.A.63. Trigonometric Functions: Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function.
A2.A.64. Trigonometric Functions: Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent.
A2.A.65. Trigonometric Functions: Sketch the graph of the inverses of the sine, cosine, and tangent functions.
A2.A.66. Trigonometric Functions: Determine the trigonometric functions of any angle, using technology.
A2.A.67. Trigonometric Functions: Justify the Pythagorean identities.
A2.A.68. Trigonometric Functions: Solve trigonometric equations for all values of the variable from 0 degrees to 360 degrees.
A2.A.69. Trigonometric Functions: Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function.
A2.A.70. Trigonometric Functions: Sketch and recognize one cycle of a function of the form y = Asin Bx or y = Acos Bx.
A2.A.71. Trigonometric Functions: Sketch and recognize the graphs of the functions y = sec(x), y = csc(x), y = tan(x), and y = cot(x).
A2.A.72. Trigonometric Functions: Write the trigonometric function that is represented by a given periodic graph.
A2.A.73. Trigonometric Functions: Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines.
A2.A.74. Trigonometric Functions: Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle.
A2.A.75. Trigonometric Functions: Determine the solution(s) from the SSA situation (ambiguous case).
A2.A.76. Trigonometric Functions: Apply the angle sum and difference formulas for trigonometric functions.
A2.A.77. Trigonometric Functions: Apply the double-angle and half-angle formulas for trigonometric functions.
A2.M.1. Units of Measurement: Define radian measure.
A2.M.2. Units of Measurement: Convert between radian and degree measures.
A2.S.1. Collection of Data: Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment).
A2.S.2. Collection of Data: Determine factors which may affect the outcome of a survey.
A2.S.3. Organization and Display of Data: Calculate measures of central tendency with group frequency distributions.
A2.S.4. Organization and Display of Data: Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations.
A2.S.5. Organization and Display of Data: Know and apply the characteristics of the normal distribution.
A2.S.6. Predictions from Data: Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate.
A2.S.7. Predictions from Data: Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data.
A2.S.8. Predictions from Data: Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship.
A2.S.9. Probability: Differentiate between situations requiring permutations and those requiring combinations.
A2.S.10. Probability: Calculate the number of possible permutations (nPr) of n items taken r at a time.
A2.S.11. Probability: Calculate the number of possible combinations (nCr) of n items taken r at a time.
A2.S.12. Probability: Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event).
A2.S.13. Probability: Calculate theoretical probabilities, including geometric applications.
A2.S.14. Probability: Calculate empirical probabilities.
A2.S.15. Probability: Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most.
A2.S.16. Probability: Use the normal distribution as an approximation for binomial probabilities.