New York State Standards for Mathematics: Grade 5

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).