Connecticut State Standards for Mathematics:

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Sort and classify objects by attributes including size, shape, color, texture, orientation, position and use, and explain the reason for each sort.

1.1.2. Describe and make comparisons of qualitative and quantitative changes of a given pattern using terms such as warmer, softer, more, one more, less, one less, bigger, smaller, longer and shorter.

1.1.3. Recognize, reproduce, extend and create repeating patterns using movement, sounds, color, shapes, numbers and textures.

1.1.4. Identify and extend visual, auditory and physical patterns to make predictions.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Represent quantities of up to 30 objects in a set.

2.1.2. Compare sets of up to 30 objects and use the terms more, less or the same to compare the two sets and identify a set with one more or one less than a given set.

2.1.3. Order sets of up to 30 objects from least to greatest.

2.1.4. Identify the ordinal position of objects: first, second, third, fourth, fifth and last.

2.1.5. Use a variety of models and familiar object to compare two parts of a whole and describe the parts as being closer to a whole or closer to very little.

2.1.6. Use a variety of models and familiar objects to:

2.1.6.1. Identify one whole and one half of an object.

2.1.6.2. Recognize a half and put two halves of an object together to make a whole.

2.1.6.3. Form a whole from two smaller sets that have equal amounts.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.7. Count by rote to at least 30.

2.2.8. Count and group up to 30 objects by tens.

2.2.9. Identify the numerals 1-30 and match each numeral to an appropriate set of objects.

2.2.10. Act out and solve addition and subtraction story problems that reflect real-world experiences and contextual problems using sets of up to 10 objects and describe the strategy or reasoning used to solve a problem. For example: Put two crayons together with four crayons; then count to determine the number of crayons needed for all students at a table.

2.2.11. Write the number sentences that correspond to story problems using addition, subtraction and equals symbols (+, -, =) correctly.

2.2.12. Estimate the amount of objects in a set using 10 as a benchmark and then count to determine if the amount is more or less than 10.

2.2.13. Identify and name pennies and dimes.

2.2.14. Count pennies and trade pennies for objects.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Identify and describe familiar shapes (triangles, squares, rectangles and circles) and solids (cubes, spheres, cylinders, cones and prisms) in the environment.

3.1.2. Compare and sort familiar shapes and solids in the environment and contextual situations.

3.1.3. Construct small sets of shapes and solids using a variety of materials.

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.4. Describe location, direction, and position of objects or parts of objects, using terms such as under/over, inside/outside, next to/near, top/bottom, in front of, first and last.

3.2.5. Complete simple shape and jigsaw puzzles and explain the reasoning used to complete the puzzle and solve the problem.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.6. Recognize events that reoccur (at specific times of the day or week).

3.3.7. Locate yesterday, today, and tomorrow on a calendar to sequence events and use terms such as before and after to compare events.

3.3.8. Use nonstandard units, physical referents (such as a finger) or everyday objects such as links, Unifix cubes or blocks to compare, estimate and order measures of length, area, capacity, weight and temperature and describe the reasoning and strategies used.

3.3.9. Describe and order small sets of familiar objects by size, length or area using comparative language such as more, bigger, longer, shorter and taller.

3.3.10. Use a balance scale to compare the weight of two objects and identify which is heavier.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Pose questions about objects and events in the environment that can be used to guide the collection of data.

4.1.2. Collect data, record and the results using real graphs and picture graphs.

4.1.3. Arrange information in a systematic way using counting, sorting, lists and graphic organizers.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.4. Describe data using the terms more, less and the same.

4.2.5. Identify and extend patterns from organized data to make predictions. For example: More boys than girls in our class watch television every day. We predict that the same will be true for another kindergarten class.

4.3. Understand and apply basic concepts of probability.

4.3.6. Describe the likelihood of the future occurrence of events based on patterns and personal experiences using terms such as likely, unlikely or certainly.

4.3.7. Engage in simple probability activities and discuss the results.

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Sort, classify and order numbers and objects by one and two attributes including size, shape, color, texture, orientation, position and use, and explain the reason or rule used.

1.1.2. Recognize, extend and create one- attribute and two-attribute patterns, e.g., size and shape, counting, e.g., by 5 or 10, and number patterns, e.g., n + 2. Describe the pattern and the rule used to make it.

1.1.3. Replicate a pattern using a different representation, e.g., from color to shape.

1.1.4. Develop and test generalizations based on observations of patterns and relationships.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.5. Model real-life situations that represent the result of counting, combining and separation of sets of objects (addition and subtraction of whole numbers) with objects, pictures, symbols and open sentences.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.6. Demonstrate understanding of equivalence or balance with objects, models, diagrams, operations or numbers, e.g., using a balance scale, or an arm balance showing the same amount on both sides.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Represent and identify whole numbers up to 100 as groups of tens and ones using models and number lines.

2.1.2. Compare and order quantities of up to 100 objects, including naming a number that is one or ten more or less than a given number.

2.1.3. Describe and estimate quantities using benchmark amounts such as zero, 10 and 100.

2.1.4. Identify ordinal numbers up to 10th with an ordered set of objects, e.g., point to the fifth crayon lined up on the table.

2.1.5. Use a variety of models and familiar objects to compare two parts of a whole object and describe the parts as being closer to very little, one half or one whole.

2.1.6. Use a variety of models and familiar objects to:

2.1.6.1. Make a whole of equal size parts of familiar objects.

2.1.6.2. Show and identify equal size pieces of a whole as halves, thirds or fourths

2.1.6.3. Identify pieces of a whole as not being halves, thirds or fourths.

2.1.7. Determine half of a whole set of up to 20 objects.

2.1.8. Describe ratios in terms of the patterns that develop in the relationships between quantities, e.g., if one cat has four legs, then two cats have eight legs.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.9. Count by rote to at least 100.

2.2.10. Count on from a given amount, orally and with models, and count back from 10.

2.2.11. Count and group at least 100 objects by tens.

2.2.12. Identify, read and write numerals to 100.

2.2.13. Create problems and write one- and two-digit number sentences that reflect contextual situations and real world experiences. Solve the problems using a variety of methods including models, pictures, pencil and paper, estimation and mental computation, and describe the reasoning or strategies used. For example: Tell a story or draw a picture for a problem that can be solved using the number sentence 10 + 6 = 16.

2.2.14. Solve contextual problems using all addition sums to 18 and subtraction differences from 10 with flexibility and fluency.

2.2.15. Estimate the amount of objects in a set using zero, 10 and 100 as benchmarks and then determine if the estimate was reasonable.

2.2.16. Identify and name pennies, nickels, dimes and quarters.

2.2.17. Identify pennies, nickels, dimes and quarters.

2.2.18. Determine and compare sets of pennies and dimes valued up to $1.00; trade sets of pennies for dimes and vice versa. For example: Jose has three dimes and eight pennies. Andrea has two dimes and 17 pennies. If they do not have the same amount of money, who has more or less? How much more or less?

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Identify and describe familiar two- dimensional shapes and three-dimensional solids in the environment and contextual situations.

3.1.2. Copy two- and three-dimensional designs from visual memory.

3.1.3. Compare and sort familiar shapes and solids and designs found in the environment and contextual situations

3.1.4. Construct shapes and solids using a variety of materials and create two-dimensional shapes and designs with a line of symmetry.

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.5. Describe location, direction and position of objects or parts of objects, using terms such as left, right and opposite.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.6. Know the days of the week in order and locate dates, days, weeks and months on a calendar. Use the information to solve problems involving the planning and sequencing of events.

3.3.7. Solve problems involving telling time to the nearest hour using digital and analog clocks. Estimate and compare the length of time needed to complete a task using comparative language such as longer, shorter, more or less.

3.3.8. Use nonstandard units or physical referents to estimate answers to measurement problems involving length, area, weight, temperature, volume and capacity, and then justify the reasonableness of the answers. Suggested materials include Unifix or locking cubes, paperclips, Popsicle sticks, square tiles, water and sand.

3.3.9. Use nonstandard units, references or direct comparison of objects (appearance), to order objects by length, area and capacity.

3.3.10. Explore using standard units of measure (inch and centimeter) to communicate measurement in a universal manner.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Pose questions that can be used to guide data collection, organization and representation.

4.1.2. Collect and systematically organize and represent the data that answers the questions using lists, charts and tables, tallies, glyphs (coded pictures), picture graphs and bar graphs.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.3. Describe data that have been organized and make comparisons using terms such as largest, smallest, most often or least often.

4.3. Understand and apply basic concepts of probability.

4.3.4. Describe and explain the likelihood of the occurrence of various events in the student's world using terms such as possible, impossible, likely, unlikely or certain.

4.3.5. Engage in simple probability activities and games including the use of number cubes and spinners; record, graph and describe the results of the activities and games.

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Sort, classify and order objects and numbers in more than one way and by one and two attributes and describe the rule used. Use attributes such as size, shape, color, texture, orientation, position and use; and characteristics such as symmetry and congruence.

1.1.2. Recognize, extend, and create repeating, growing, number; e.g., skip counting, odd/even, counting on by 10; and one and two attribute patterns. Describe the pattern and the rule used to make it.

1.1.3. Replicate the pattern using a different representation, e.g., letters to numbers.

1.1.4. Use patterns and the rules that describe the patterns to identify a missing object, objects with common or different attributes, and the complement of a set of objects.

1.1.5. Analyze and describe observable changes in patterns using language that describes number characteristics and qualitative characteristics such as attributes, orientation and position.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.6. Model real-life situations that represent the addition and subtraction of whole numbers with objects, pictures, symbols and open sentences.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.7. Demonstrate an understanding of equivalence or balance of sets using objects, models, diagrams, numbers whole number relationships (operations) and the equals sign, e.g., 2 + 3 = 5 is the same as 5 = 2 + 3 and the same as 4 + 1 = 5.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Locate, label, compare, and order whole numbers up to 1,000 using pictures, place value models, number lines, and benchmarks of 0, 10 and 100, including naming the number that is 10 or 100 more or less than a given number.

2.1.2. Represent whole numbers up to 1,000 by modeling and writing numbers in expanded forms, e.g., 37 = (3 x 10) + (7 x 1), and regrouped forms, e.g., (2 x 10) + (17 x 1) = 37, and use the forms to support computational strategies.

2.1.3. Represent multiplication and division (with factors of 1, 2, 5 and 10 ) using a variety of models and strategies such as arrays, pictures, skip counting, extending number patterns, and repeated addition and subtraction; describe the connection between multiplication and division.

2.1.4. Use a variety of models and familiar objects to compare, order and estimate parts of a whole using the unit fractions 1/2, 1/3, 1/4.

2.1.5. Use a variety of models to represent and describe parts of groups as unit fractions 1/2, through 1/10.

2.1.6. Estimate and determine 1/2, 1/3, 1/4 of a small group of up to 20 objects, such as finding 1/2, 1/3, 1/4 of 12 cookies.

2.1.7. Describe ratios in terms of the linear patterns that develop from the relationships between quantities, e.g., In a pattern of green, green, red blocks there are always two green blocks for one red block.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.8. Count whole numbers to 1,000 and beyond.

2.2.9. Count on by tens from a given amount, e.g., 17, 27, 37, etc.

2.2.10. Read and write numerals up to 1,000.

2.2.11. Skip count by twos, fives, tens and hundreds to 1,000 and beyond.

2.2.12. Determine whether a set of objects has an odd or even number of items by pairing objects and creating arrays.

2.2.13. Create word problems and write and solve two- and three-digit number sentences that reflect contextual situations and real-world experiences involving addition and subtraction. Construct and solve open sentences, e.g., __ + 5 = 11. Solve the problems using a variety of methods including models, pictures, pencil and paper, estimation and mental computation, and describe the reasoning or strategies used.

2.2.14. Solve problems using addition and subtraction facts involving sums and differences to 20 with flexibility and fluency

2.2.15. Add two-digit numbers with and without regrouping. Subtract two-digit numbers without regrouping and with regrouping using models.

2.2.16. Determine when an estimate for a problem involving two- and three-digit numbers is appropriate or when an exact answer is needed.

2.2.17. Use a variety of strategies to estimate solutions and to determine if a solution to a computation or word problem reflecting real-world experiences involving addition and subtraction of two- and three-digit whole numbers is reasonable.

2.2.18. Determine and compare the value of pennies, nickels, dimes, quarters and half dollars.

2.2.19. Count, compare and trade sets of pennies, dimes and dollars up to $10.00

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Identify, describe and draw polygons (triangles, quadrilaterals including trapezoids and rhombuses, pentagons and hexagons), solids, and other familiar two- and three- dimensional objects in the environment.

3.1.2. Compare and sort familiar polygons, solids, and other two- and three- dimensional objects in the environment.

3.1.3. Construct polygons, solids and other two- and three-dimensional objects using a variety of materials and create two-dimensional shapes and designs with one or more lines of reflective symmetry (lines that divide the shape or design into two congruent parts).

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.4. Investigate and predict the result of putting together and taking apart two- and three-dimensional shapes in the environment, e.g. use objects to find other shapes that can be made from three triangles or a rectangle and a triangle.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.5. Know the months of the year in order and locate dates, days, weeks and months on a calendar. Use the information to write and solve problems involving calendars.

3.3.6. Solve problems involving telling time, including estimating and measuring the length of time needed to complete a task, to the half-hour using analog and digital clocks.

3.3.7. Use measurement tools such as thermometers to measure temperature, basic rulers to measure length to the nearest half-inch or centimeter, and balance scales to measure weight /mass in grams.

3.3.8. Use nonstandard referents and standard benchmarks to estimate and measure the following:

3.3.8.1. Length(to the nearest inch, half-inch, foot, yard, centimeter or meter);

3.3.8.2. Area (in square inches);

3.3.8.3. Capacity (in liters and cups);

3.3.8.4. Weight (in grams);

3.3.8.5. Temperature; and

3.3.8.6. Volume (using water or sand).

3.3.9. Describe the strategy used to determine an estimate and determine if the estimate is reasonable.

3.3.10. Describe the relationships between and centimeter and meter among inch, foot and yard.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Pose questions that can be used to guide data collection, organization and representation.

4.1.2. Collect and systematically organize and represent the data that answer the questions using lists, charts and tables, tallies, glyphs (coded pictures), picture graphs and bar graphs.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.3. Describe data that have been organized and make comparisons using terms such as largest, smallest, most often or least often.

4.2.4. Determine patterns and make predictions from data displayed in tables and graphs.

4.3. Understand and apply basic concepts of probability.

4.3.5. Describe and explain the likelihood of the occurrence of various events. State possibilities, make predictions and test the predictions in practical situations.

4.3.6. Conduct simple probability investigations involving activities of chance and games with number cubes and spinners; record, graph and describe the results of the investigations.

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Sort, classify and order a group of objects and numbers in more than one way and explain the reason or describe the rule used.

1.1.2. Create and construct numerical and spatial patterns and sequences that repeat and grow.

1.1.3. Analyze, describe and extend repeating and growing patterns and sequences, including those found in real-world contexts, by constructing and using tables, graphs and charts.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.4. Describe mathematical relationships and situations involving computation of whole numbers (addition, subtraction, multiplication and division) using words, symbols, open number sentences and equations, e.g., 56 + __ = 100 and 3 x 5 = 9 + 6.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.5. .Demonstrate understanding of equivalence as a balanced relationship of quantities by using the equals sign to relate two quantities that are equivalent and the inequality symbols, < and >, to relate two quantities that are not equivalent. (23 x 5 > 23 x 2)

1.3.6. Solve problems and demonstrate an understanding of equivalence using the equals sign in number sentences that reflect the commutative and associative properties of addition and multiplication of whole numbers, e.g. 3 x 5 = 5 x 3.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Locate, label, compare and order whole numbers up to 10,000 using place value models, number lines and number patterns (including multiples of 100 and 1,000).

2.1.2. Identify the number that is 100 and 1,000 more or less than a given number up to 10,000 using place value models, pictures and number lines.

2.1.3. Round three- and four-digit numbers to the nearest hundred and thousand using place value models, number lines and number patterns.

2.1.4. Represent three- and four-digit numbers up to 10,000 in expanded forms, e.g., 5,472 = (5 x 1,000) + (4 x 100) + (7 x 10) + (2 x 1), and regrouped forms, e.g., 5,472 = (4 x 1,000) + (14 x 100) + (6 x 10) + (12 x 1). Use the forms to support computational strategies.

2.1.5. Represent fractions with like and unlike denominators of 2, 3, 4, 5, 6 and 8 using a variety of materials; label the fractional parts using words and fraction symbols.

2.1.6. Locate, label and estimate fractions with like and unlike denominators of 2, 3, 4, 5, 6 and 8 by constructing and using models, pictures and number lines.

2.1.7. Determine equivalence, compare and order fractions through the construction and use of models, pictures and number lines with like and unlike denominators of 2, 3, 4, 5, 6 and 8, including identifying a whole object or a whole set of objects as a fraction with the same numerator and denominator.

2.1.8. .Use models, number patterns and counting and grouping of objects, to find equal parts of a set of objects and identify amounts such as 2/3 of 12 is 8.

2.1.9. Describe quantitative relationships using ratios and identify patterns with equivalent ratios such as 3 out of 6 crayons are red or 4 out of 8 crayons are red and are the same as 1 out of 2 crayons is red.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.10. Recall the multiplication and division facts for 1, 2, 3, 4, 5 and 10.

2.2.11. Write multiplication and division story problems to match a given multiplication or division number sentence and vice versa; solve the problems and justify the solution.

2.2.12. Solve problems involving addition and subtraction of two- and three-digit whole numbers and money amounts up to $100.00 with and without regrouping, using a variety of strategies, including models.

2.2.13. Create and solve addition and subtraction word problems by using place value patterns and algebraic properties (commutative and associative for addition).

2.2.14. Solve problems involving the multiplication and division of two- and three-digit numbers by one digit (2, 3, 4, 5 or 10) with models, arrays and pictures of sets.

2.2.15. Determine when an estimate for a problem involving two- and three-digit numbers is appropriate or when an exact answer is needed.

2.2.16. Use a variety of estimation strategies to determine and justify the reasonableness of an answer to a computation or word problem involving addition and subtraction of two- and three-digit whole numbers and money amounts up to $100.00.

2.2.17. Determine when a strategy will result in an overestimate or an underestimate in problems involving two- and three-digit numbers.

2.2.18. Determine and compare the value of sets of coins and write the values using decimal notation, e.g., two quarters = 50 cents or $0.50 (50 of 100 cents in a dollar) and is less than two quarters, two dimes and a nickel or $0.75.

2.2.19. Determine, compare and write the value of money amounts up to $100.00 and identify equivalent ways to represent a given amount of money, including combinations of pennies, nickels, dimes, quarters and half dollars, e.g., $0.25 can be five nickels, two dimes and one nickel or one quarter.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Identify, describe, construct and draw two- dimensional shapes such as quadrilaterals (including parallelograms), pentagons and hexagons.

3.1.2. Identify, describe, construct and represent three-dimensional figures such as cubes, spheres, cylinders, cones, pyramids, prisms.

3.1.3. Compare and classify polygons and solids and determine congruence by using attributes such as the number and length of sides, faces and edges, and the number and kinds of angles (acute, right and obtuse).

3.1.4. Create two-dimensional figures with one or more lines of reflective symmetry.

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.5. Draw and interpret simple maps using shapes or pictures on a coordinate grid.

3.2.6. Investigate ways to tile or tessellate a shape or region using a variety of polygons.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.7. Use calendar and clocks to plan and sequence events and identify events and times as occurring in the a.m. and p.m.

3.3.8. Solve problems involving telling time to the nearest quarter hour, five minutes and minute using analog and digital clocks.

3.3.9. Develop an understanding and describe the relationships between appropriate units of measure through concrete experiences (ounces and pounds; gram and kilograms; inches, feet and yards; meters and kilometers; cups, pints and quarts; and milliliters and liters).

3.3.10. Estimate and measure using nonstandard units and appropriate customary and metric tools and units:

3.3.10.1. Length and perimeter to the nearest 1/4 inch or 1/2 centimeter;

3.3.10.2. Area in square inches or square centimeters;

3.3.10.3. Capacity in cups, pints, quarts, milliliters or liters,

3.3.10.4. Weight in ounces, pounds and grams (mass is weighed in grams);

3.3.10.5. Temperature to the nearest degree; and

3.3.10.6. Volume using inch cubes and centimeter cubes.

3.3.11. Describe and use estimation strategies that can identify a reasonable answer to a measurement problem when an estimate is appropriate.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Pose questions that can be used to guide data collection, organization, and representation.

4.1.2. Collect and organize the data that answer the questions using diagrams, charts, tables, lists, pictographs, bar graphs and line plots

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.3. Analyze data that have been collected and organized, to draw and defend conclusions based on the data.

4.2.4. Describe an event or element as typical based upon the range, median and mode of a set of data.

4.3. Understand and apply basic concepts of probability.

4.3.5. Experiment to test predictions and determine probability in practical situations such as investigating the fairness of games using a variety of spinners and dice.

4.3.6. Describe the probability of an outcome as __ out of __, e.g., 3 out of 5.

4.3.7. Investigate combinations using models.

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Extend and compare numerical and geometric sequences and classify patterns as growing or repeating, e.g. 2, 4, 8, _, _, grows and visual sequence.

1.1.2. Develop and test generalizations based on observable patterns and relationships and describe the rules for number patterns using equations, e.g., in this sequence 1, 6, 16, 36 ..., to get the next number the current number can be doubled and four added to the product.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.3. Describe mathematical relationships and situations, involving ratios and computation of whole numbers, in all four operations with using symbols, number sentences and equations. If one trapezoid = three triangles, then three trapezoids = __.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.4. Represent possible values by using symbols, e.g., variables, to represent quantities in expressions and number sentences. Use number sentences (equations) to model and solve word problems.

1.3.5. Solve problems and demonstrate an understanding of equivalence in mathematical situations that reflect the commutative and associative properties of addition and multiplication of whole numbers and the distributive property.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Locate, label, compare and order numbers up to 100,000 using place value models, number lines and number patterns (including multiples of 1,000 and 10,000).

2.1.2. Extend number patterns to determine 1,000 and 10,000 more and less than a given number in practical situations.

2.1.3. Round whole numbers up to 100,000 using number patterns, number lines, diagrams and place value models.

2.1.4. Write and describe equivalent representations of four- and five-digit whole numbers up to 100,000 and beyond, in expanded and regrouped forms. Use the forms to support computational strategies.

2.1.5. Relate multiplication and division to number patterns and models of groups and rectangular arrays.

2.1.6. Identify and define prime and composite numbers through the use of models including rectangular arrays, place value models and pictures.

2.1.7. Construct and use number lines, pictures and models, including rulers, to determine and identify equivalent ratios and fractions.

2.1.8. Locate, label and estimate (round) fractions with like and unlike denominators of 2, 3, 4, 5, 6, 8 and 10 by constructing and using models, pictures and number lines.

2.1.9. Construct and use models, pictures and number lines, including rulers to compare and order fractional parts of a whole and mixed numbers with like and unlike denominators of 2, 3, 4, 5, 6 and 8 and 10.

2.1.10. Construct and use models, pictures and number lines, including rulers, to identify wholes and parts of a whole (including a part of a group or groups) as simple fractions and mixed numbers.

2.1.11. Use models to represent tenths and hundredths and record the representations using equivalent ratio, fraction and decimal notation (1/10, 0.1)

2.1.12. Express a ratio or division problem as a fraction and describe the relationship between the divisor and the remainder written as a fraction. For example: When determining the number of groups of 3 in 14, we say 14 / 3 = 4 with a remainder of 2 or 4 2/3).

2.1.13. Solve practical problems involving simple ratios and proportions, e.g., determining distance on maps, by using models, pictures and number patterns

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.14. Develop and use a variety of computation strategies including place value concepts, number lines and the commutative and associative properties to add and subtract three- and four-digit numbers and money amounts up to $1,000.00.

2.2.15. Solve contextual problems involving addition and subtraction of whole numbers using a variety of methods, including writing appropriate number sentences (equations) and explaining the strategies used.

2.2.16. Create story problems to match a given number sentence (equation).

2.2.17. Recall the multiplication and division facts 1 through 10.

2.2.18. Write multiplication and division story problems involving basic facts and two- and three-digit by one-digit numbers to match a given number sentence and vice versa; solve the problems using strategies that include models and arrays and justify the solutions.

2.2.19. Determine and explain in writing when an estimate is appropriate and whether a particular estimation strategy is reasonable or will result in an overestimate or underestimate involving computation with three- and four- digit numbers and money amounts up to $1,000.

2.2.20. Use models and pictures to add and subtract fractions with like and unlike denominators of 2, 3, 4, 5, 6, 8 and 10 and match number sentences or equations to the problems.

2.2.21. Identify or write number sentences to solve simple problems involving fractions with like denominators, decimals (tenths) and mixed numbers.

2.2.22. Write contextual problems involving the addition and subtraction of fractions with like denominators, decimals (tenths) and mixed numbers; solve the problems and justify the solutions.

2.2.23. Estimate a reasonable answer to simple problems involving fractions, mixed numbers and decimals (tenths).

2.2.24. Write and solve multistep contextual problems, including problems with extraneous information and explain orally and in writing how the answers were determined.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Describe and represent polygons, solids, and other familiar two- and three-dimensional objects.

3.1.2. Compare and classify polygons based on relationships such as parallel or perpendicular lines, symmetry and congruence.

3.1.3. Make and test conjectures about polygons using geometric relationships such as symmetry and congruence.

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.4. Draw and interpret simple maps with ordered pairs of numbers and/or letters in quadrant one of an x, y coordinate system and find possible paths between two points.

3.2.5. Analyze geometric reflections (flips), rotations (turns), and translations (slides) of plane figures and describe the relationship to the original figure.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.6. Use calendars and clocks to solve problems and schedule events involving elapsed time.

3.3.7. Write and solve problems involving the conversion of simple measures of time, e.g., minutes to hours, hours to days and days to weeks and months.

3.3.8. Use customary and metric tools and units and non-standard units to estimate, measure and solve problems involving length and perimeter to the nearest quarter-inch or half-centimeter, area, capacity, weight, temperature and volume.

3.3.9. Use estimation strategies to predict reasonable answers to measurement problems and explain the reasoning used orally and in writing.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Pose questions and develop a plan to collect data using observations, surveys and experiments to answer the questions.

4.1.2. Collect, organize and represent the data that answer the questions using simple circle graphs and broken line graphs.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.3. Discuss, make predictions and write about patterns and trends in categorical and numerical data that have been represented in a variety of ways.

4.2.4. Determine the range, median, mode and mean of a set of data and describe characteristics of the data set as typical or average based on those determinations.

4.3. Understand and apply basic concepts of probability.

4.3.5. Conduct probability experiments and express the probability based on possible outcomes, e.g., 8 out of 10 tiles chosen were red.

4.3.6. Determine and describe possible combinations, where order does not matter, e.g., when there is a choice of vanilla (V), chocolate (C) or strawberry (S) ice cream for a two-scoop cone and two different scoops are desired, the possible combinations are CV, CS, or VS.

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Represent, extend and compare geometric and numeric patterns using words, tables, graphs and equations

1.1.2. Analyze patterns and data to make generalizations, make predictions and to identify trends.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.3. Represent and describe mathematical relationships using variables or symbols in expressions, equations and inequalities

1.2.4. Describe how a change in one variable relates to a change in a second variable in context. For example: If a recipe requires two cups of flour for eight servings, the flour must be doubled for 16 servings or increased by one-half for 12 servings.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.5. Replace variables or symbols in algebraic expressions with given values and evaluate or simplify the expression, e.g., If __ =5, find the value of 4 x __ +7.

1.3.6. Model, write and solve one-step equations by using appropriate concrete materials that model equivalence, e.g., If 4 x __ = 36, then __ equals 9.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Compare, order and round whole numbers to 1,000,000 using number patterns, number lines and diagrams.

2.1.2. Represent whole numbers up to 1,000,000 in expanded and regrouped forms and use the forms to support computation.

2.1.3. Construct and use models, number patterns and pictorial representations to extend place value concepts and patterns to decimals, e.g., 0.1 is one-tenth of one and 0.01 is one one-hundredth of one and one-tenth of 0.1.

2.1.4. Investigate negative integers (values less than zero) using place value models, diagrams and number lines and represent negative integers in practical applications, e.g. temperatures, money and locations below sea level.

2.1.5. Classify numbers as prime, composite or perfect squares and identify factor pairs using rectangular arrays.

2.1.6. Represent equivalent fractions, decimals, ratios and percents using models, pictures, number patterns and common factors.

2.1.7. Choose and use benchmarks to approximate locations, of fractions, mixed numbers and decimals, on number lines and coordinate grids.

2.1.8. Write division problems in fraction form and round the fraction form to estimate an answer to a division problem, e.g., 14/3 = 4 2/3 is approximately equal to 5.

2.1.9. Use models and pictures to identify and compare ratios and represent ratios in equivalent fraction and decimal forms.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.10. Solve practical problems involving 10, 100, 1,000 and 10,000 more or less than a number.

2.2.11. Estimate products and missing factors using multiples of 10, 100 and 1,000.

2.2.12. Develop and use strategies involving place value relationships, inverse operations and algebraic properties (commutative, associative and distributive) to simplify addition, subtraction and multiplication problems with three-, four- and five-digit numbers and money amounts and division by one-digit factors.

2.2.13. Multiply and divide decimals and money amounts by whole numbers.

2.2.14. Write and solve multistep problems for all four operations involving multidigit whole numbers and money amounts and explain how answers were determined, orally and in writing.

2.2.15. Find fractional parts of a set by using estimation, counting, grouping of objects, number patterns, equivalent ratios and division.

2.2.16. Add and subtract fractions, decimals and mixed numbers using a variety of strategies, e.g., models, mental math, equivalence and substitution: 1/2 + 3/4 can also be solved using 0.5 + 0.75.

2.2.17. Construct and use models and pictorial representations to multiply common fractions and mixed numbers by whole numbers.

2.2.18. Use ratios and proportions to solve practical problems, e.g., interpreting scale drawings and maps and determining the probability of an event.

2.2.19. Use estimation to predict results and to recognize when an answer is or is not reasonable, or will result in an overestimate or underestimate and explain the reasoning used orally and in writing.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Represent the surface of three-dimensional solids using two-dimensional nets.

3.1.2. Develop formulas for finding the perimeter and area of squares, rectangles and triangles and use them to solve problems.

3.1.3. Use the attributes of parallel sides, perpendicular sides, congruent sides/angles, number and length of sides or faces and number and kinds of angles (right, acute or obtuse) to describe, classify and sort polygons and solids (cube, prism, pyramid and sphere).

3.1.4. Make and test conjectures about polygons using geometric relationships

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.5. Use an x, y coordinate system to plot points, to estimate the distance between points and to determine the horizontal or vertical distance between two points.

3.2.6. Analyze and describe the effect that changing the dimensions (perimeter) of a polygon has on its area and vice versa.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.7. Use calendars and clocks to plan and sequence events and to solve problems involving the conversion of measures of time and elapsed time using days, hours, minutes and seconds.

3.3.8. Estimate and measure to solve a variety of problems that involve angles, length, area, weight, mass, temperature, capacity and volume in either metric or customary units explain the reasoning used orally and in writing.

3.3.9. Use cubic inch or cubic centimeter models to find the volume of rectangular solids.

3.3.10. Solve length problems involving conversions of measure within the customary (inches, feet, yards and miles) or metric systems (millimeters, centimeters, meters and kilometers).

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Represent sets of data using line plots, bar graphs, double bar graphs, pictographs, simple circle graphs, stem and leaf plots and scatter plots.

4.1.2. Compare different representations of the same data set and evaluate how well each kind of display represents the features of the data.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.3. Design and conduct surveys of a representative sample of a population and use the data collected to begin to make inferences about the general population.

4.2.4. Determine the mean, mode and median of a data set and explain in writing, how they are affected by a change in the data set.

4.3. Understand and apply basic concepts of probability.

4.3.5. Design and conduct probability experiments and simple games of chance to test predictions about outcomes and fairness.

4.3.6. Determine and describe possible outcomes and express the likelihood of events as a fraction.

4.3.7. Determine and describe possible outcomes using permutations, where order does matter, e.g., when there is a choice of vanilla (V), chocolate (C) or strawberry (S) ice cream for a three-scoop cone, there are two possible ways to have the chocolate scoop on top CVS or CSV.

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Analyze, describe in writing and extend a variety of patterns to justify predictions and identify trends.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.2. Create tables of values and scatterplots from mathematical relationships and equations and vice versa to solve problems.

1.2.3. Examine tables, graphs and equations to determine patterns of change in linear relationships.

1.2.4. Write expressions, formulas, equations or inequalities using symbols or variables to denote a pattern or represent a contextual situation.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.5. Evaluate algebraic expressions and formulas using substitution.

1.3.6. Write, model and solve one-step equations using mental math, tables, substitution and concrete models that demonstrate equivalence and justify the solution.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Locate and label whole numbers, fractions, decimals and positive and negative integers on number lines, scales, coordinate grids (all four quadrants) and measurement tools.

2.1.2. Compare and order whole numbers, fractions, decimals and positive and negative integers in context using number lines and scales.

2.1.3. Represent and compare whole numbers (to a billion) and decimals (to thousandths) in expanded notation, e.g., 75.654 = (7 x 10) + (5 x 1) + (6 x 0.1) + (5 x 0.01) + (4 x 0.001).

2.1.4. Represent chain multiplication, including powers of 10 in exponential and standard form, e.g., 5 x 5 x 5 = 5^3 = 125.

2.1.5. Factor composite numbers and express them as a product of primes using exponents.

2.1.6. Determine equivalent fraction, decimal, and percent representations and choose among these forms to solve problems.

2.1.7. Use ratios and rates (involving different units) to compare quantities.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.8. Understand place value and patterns in place value when multiplying and dividing decimals by powers of 10.

2.2.9. Develop, describe and use strategies for solving, simplifying and estimating multiplication and division problems involving large numbers, decimals and powers of 10, e.g., 4.25 x 100 = 425 and 365,000 / 6,000 = 365 / 6 ; 365 / 6 is approximately equal to 360 / 6 is approximately equal to 60.

2.2.10. Estimate and find percentages of a number in context using benchmarks and number patterns and ratios to 100.

2.2.11. Solve practical problems involving rates, ratios, percentages and proportionality.

2.2.12. Add, subtract, multiply and divide by fractions and decimals in context.

2.2.13. Describe situations in writing that connect multiplying fractions to determining the fractional part of a set.

2.2.14. Examine the relationships between multiplication by a unit fraction and dividing by the fraction's denominator, e.g., 1/2 of $6 is the same as $6 / 2, and use this to solve problems.

2.2.15. Use the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.

2.2.16. Understand and defend in writing the magnitude of the result of multiplication or division problems involving fractions or decimals.

2.2.17. Determine when an estimate is sufficient or when an exact answer is needed.

2.2.18. Estimate solutions to problems and justify the reasonableness of estimates in writing.

2.2.19. Write and solve multistep problems in context involving addition, subtraction, multiplication and division with whole numbers, fractions, decimals, money and simple percentages.

2.2.20. Understand and use divisibility rules, factors of composite numbers and powers of 10 to find products and quotients.

2.2.21. Apply the order of operations and algebraic properties; i.e., commutative, associative, distributive, inverse operations, and the additive and multiplicative identities; to compute and solve multistep problems and explain solutions in writing.

2.2.22. Use concrete models to develop strategies to add and subtract integers.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Classify sets and subsets of polygons using the relationship of the sides (length, parallel and perpendicular) and angles (types and measure).

3.1.2. Make and test conjectures about polygons and congruence using side and angle relationships and describe the results in writing.

3.1.3. Identify lines of symmetry and reflections, rotations and translations of geometric figures.

3.1.4. Use rectangles as basic shapes to model and develop formulas for finding the area of triangles, parallelograms and trapezoids.

3.1.5. Recognize the relationships among radius, diameter, circumference and area of circles and develop formulas for finding circumference and area based on these relationships.

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.6. Use and describe concrete strategies for finding the volume of rectangular solids and cylinders.

3.2.7. Use measurements to examine the ratios between corresponding side lengths of scale models and similar figures.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.8. Select and use appropriate strategies, tools and units to estimate and solve measurement problems involving length, perimeter, area, volume, capacity, mass and weight.

3.3.9. Use ratios to convert between customary units of length, mass, capacity and time.

3.3.10. Use ratios and powers of ten to convert between metric units.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Compare sets of data between two populations, e.g., heights of two classes of students, or within a population, e.g., height vs. arm length of sixth-grade students, using a variety of graphical representations.

4.1.2. Select, create and use appropriate graphical representations of data including, circle graphs, scatter plots, histograms and stem and leaf plots.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.3. Describe the shape of numerical data sets using measures of spread (range) and central tendency (mean, median, mode) and outliers.

4.2.4. Determine how the mean, median, mode and range change as a result of changes in the data set and describe in writing.

4.3. Understand and apply basic concepts of probability.

4.3.1. Investigate and describe the relationship between the number of trials in an experiment and the predicted outcomes.

4.3.2. Design and conduct probability experiments to test predictions about outcomes and fairness.

4.3.3. Express probabilities as fractions, ratios, decimals and percentages.

4.3.4. Find all possible outcomes by systematic listing and counting strategies to solve problems.

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Analyze a variety of patterns (physical phenomena, numeric and geometric patterns, arithmetic sequences) and generalize with algebraic expressions, formulas or equations.

1.1.2. Identify and describe in writing the independent and dependent variables in a mathematical situation, e.g. age vs. height of children.

1.1.3. Determine when mathematical situations are continuous (distance traveled over time) or discrete sets of points, e.g., weekly sales.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.4. Write expressions, formulas, equations or inequalities using variables to represent mathematical relationships and solve problems.

1.2.5. Represent and compare the characteristics of linear and nonlinear relationships using verbal descriptions, e.g., linear - ''increases $1 per month'' vs. nonlinear - ''doubles every month,'' tables, graphs, equations or inequalities (when possible).

1.2.6. Examine situations with constant or varying rates of change and know that a constant rate of change describes a linear relationship.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.7. Evaluate and simplify algebraic expressions, equations and formulas using algebraic properties (i.e. commutative, associative, distributive, inverse operations, and the additive and multiplicative identities) and the order of operations.

1.3.8. Solve real world problems using a variety of algebraic methods including tables, graphs, equations and inequalities.

1.3.9. Write, model and solve one- and two-step, e.g., 2x + 3 = 11, equations using a variety of methods such as tables, concrete models and the Properties of Equality and justify the solution.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Compare and order rational numbers, e.g., -2, 3/8, -3.15, 0.8, in context and locate them on number lines, scales and coordinate grids.

2.1.2. Represent rational numbers in equivalent fraction, decimal and percent forms.

2.1.3. Represent fractions as terminating, e.g., 1/2 = 0.5, or repeating, e.g., 1/3 = 0.333... decimals and determine when it is appropriate to round the decimal form in context.

2.1.4. Use patterns to compute with and write whole numbers and fractions as powers of whole numbers and vice versa, e.g., 2^2 = 4, 2^1 = 2, 2^0 = 1, 2^-1 = 1/2, 2^-2 = 1/4.

2.1.5. Understand the relationship between squares and square roots.

2.1.6. Read, write, compare and solve problems with whole numbers in scientific notation and vice versa.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.7. Estimate solutions to problems in context or computations with rational numbers and justify the reasonableness of the estimate in writing.

2.2.8. Apply the order of operations and algebraic properties; i.e. commutative, associative, distributive, inverse operations, and the additive and multiplicative identities; to write, simplify, e.g., 4 (3 1/2) = 4 (3) + 4 (1/2) = 12 + 2 = 14, and solve problems, including those with parentheses and exponents.

2.2.9. Apply a variety of strategies to write and solve problems involving addition, subtraction, multiplication and division of positive rational numbers, i.e., whole numbers, fractions and decimals.

2.2.10. Write ratios and proportions to solve problems in context involving rates, scale factors and percentages.

2.2.11. Find and/or estimate a percentage of a number, including percentages that are more than 100 percent and less than 1 percent using a variety of strategies, including:

2.2.11.1. Number patterns - e.g., find 20 percent of 50. Solution: 10 percent of 50 = 5, so 20 percent of 50 = 2 (5) = 10

2.2.11.2. Distributive Property - e.g., find 150 percent of 20. Solution: 150 percent of 20 = 100 percent of 20 + 50 percent of 20. 20 + 10 = 30

2.2.11.3. Proportions - e.g., 75 percent of 48. Solution: 75/100 = x/48; x = 36

2.2.11.4. Multiplication of decimal equivalent - e.g., 0.7 percent of 48. Solution: 0.007 (48) = 0.336

2.2.11.5. Estimation - e.g., 22 percent of $49.95. Estimate 22 percent of $49.95 is approximately equal to 20 percent of 50. 10 percent of 50 = 5, so 20 percent of 50 = 2 (5) = 10, therefore, 22 percent of $49.95 is approximately equal to $10

2.2.12. Solve percent problems in context including what percentage one number is of another, percentage increase and percentage decrease using a variety of strategies, e.g., proportions or equations.

2.2.13. Compare the magnitude of and compute with whole numbers expressed as positive powers of 10.

2.2.14. Develop and describe strategies for estimating and multiplying whole numbers expressed in scientific notation.

2.2.15. Estimate and solve problems containing whole numbers expressed is expanded notation, powers of 10 and scientific notation.

2.2.16. Develop and describe in writing strategies for addition, subtraction, multiplication and division and solve problems with positive and negative integers using models, number lines, coordinate grids and computational strategies.

2.2.17. Develop an understanding of absolute value using a number line while solving problems involving distance.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Classify two- and three-dimensional geometric figures based on their properties including relationships of sides and angles and symmetry (line and/or rotational) and apply this information to solve problems.

3.1.2. Identify polygons that have line and/or rotational symmetry.

3.1.3. Draw the result of transformations on polygons on coordinate planes including translations, rotations, reflections and dilations (reductions and enlargements).

3.1.4. Describe the effect of transformations; i.e., position and orientation from the original figure, size; on polygons that have line and/or rotational symmetry.

3.1.5. Compare and describe in writing the relationships (including congruence, equality, scale) between the angles, sides, perimeter and area of congruent and similar geometric shapes.

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.6. Identify and/or draw two-dimensional representations of three dimensional geometric solids using nets, cross-sections, front, side and top views to solve problems.

3.2.7. Use two-dimensional representations of rectangular prisms, pyramids and cylinders to determine surface area.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.8. Use formulas to solve problems involving perimeters and areas of polygons and circles.

3.3.9. Develop and use formulas to determine volumes of geometric solids (rectangular prisms and cylinders).

3.3.10. Use estimation and measurement strategies to solve problems involving area of irregular polygons and volumes of irregular solids and justify solutions in writing.

3.3.11. Write and solve problems in context involving conversions of customary or metric units and units of time.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Formulate questions and design studies; e.g., surveys, experiments, research using published sources and the internet; to collect and analyze data.

4.1.2. Organize and display data using appropriate graphical representation such as, tables and charts, line, bar and circle graphs, Venn diagrams, stem and leaf plots, scatter plots, histograms.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.3. Make and defend in writing predictions based on patterns and trends from the graphical representations.

4.2.4. Find, use and interpret measures of central tendency and spread, including mean, median, mode, range and outliers.

4.2.5. Compare two sets of data based on their spread and measures of central tendency.

4.3. Understand and apply basic concepts of probability.

4.3.6. Identifying all possible outcomes using models, tree diagrams, tables and/or organized lists to determine theoretical probabilities.

4.3.7. Perform experiments to determine experimental probabilities.

4.3.8. Compare and contrast experimental probability results to theoretical probabilities in writing.

4.3.9. Solve probability problems in familiar contexts including simple events (flipping a coin) and compound events (flipping a coin and rolling a number cube).

CT.1. Algebraic Reasoning: Patterns and Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Understand and describe patterns and functional relationships.

1.1.1. Generalize the relationships in patterns in a variety of ways including recursive and explicit descriptions; e.g., the pattern 1, 4, 7, 10... is represented as follows:

1.1.1.1. Recursively as ''add 3 to the previous number''

1.1.1.2. Explicitly as 3n + 1

1.1.2. Determine whether relationships are linear or nonlinear.

1.1.3. Write and solve problems involving proportional relationships (direct variation) using linear equations (y = mx).

1.1.4. Examine and make comparisons in writing between linear and non-linear mathematical relationships including y = mx, y = mx^2 and y = mx^3 using a variety of representations.

1.2. Represent and analyze quantitative relationships in a variety of ways.

1.2.5. Represent linear and nonlinear mathematical relationships with verbal descriptions, tables, graphs and equations (when possible).

1.2.6. Determine the constant rate of change in a linear relationship and recognize this as the slope of a line.

1.2.7. Compare and contrast the slopes and the graphs of lines that have a positive slope, negative slope, zero slope, undefined slope, slopes greater than one and slopes between zero and one.

1.2.8. Compare and contrast the slopes and the graphs of lines to classify lines as parallel, perpendicular or intersecting.

1.2.9. Interpret and describe slope and y-intercepts from contextual situations, graphs and linear equations.

1.3. Use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.10. Evaluate and simplify algebraic expressions, equations and formulas including those with powers using algebraic properties and the order of operations.

1.3.11. Examine systems of two linear equations in context that have a common solution, i.e. point of intersection, using tables, graphs and substitution and interpret the solution.

1.3.12. Write and solve multistep equations using various algebraic methods including the distributive property, e.g., 3 (x + 2) =10, combining like terms, e.g., 3x + 2x = 15, and properties of equality and justify the solutions.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Understand that a variety of numerical representations can be used to describe quantitative relationships.

2.1.1. Compare and order rational and common irrational numbers; e.g., -5, 1/16, -4 1/2, pi^2, pi; and locate them on number lines, scales and coordinate grids.

2.1.2. Identify perfect squares and their square roots; e.g., squares 1, 4, 9, 16...to corresponding roots 1, 2, 3, 4 ...; and use these relationships to estimate other square roots.

2.1.3. Read and represent whole numbers and those between zero and one in scientific notation (and vice versa) and compare their magnitudes.

2.1.4. Represent fractions, mixed numbers, decimals and percentages in equivalent forms.

2.2. Use numbers and their properties to compute flexibly and fluently and to reasonably estimate measures and quantities.

2.2.5. Compute (using addition, subtraction, multiplication and division) and solve problems with positive and negative rational numbers.

2.2.6. Calculate the square roots of positive rational numbers using technology.

2.2.7. Develop and use strategies for multiplying and dividing with numbers expressed in scientific notation using the commutative and associative property.

2.2.8. Estimate reasonable answers and solve problems in context involving rational and common irrational numbers, ratios and percentages (including percentage of increase and decrease) and justify solutions in writing.

2.2.9. Use proportional reasoning to write and solve problems in context.

2.2.10. Solve a variety of problems in context involving percents, including the following:

2.2.10.1. Percent of a number, e.g., If 65 percent of the 250 applicants will be accepted to the Arts Magnet School, how many students will be accepted?

2.2.10.2. The percentage one number is of another number, e.g., Find the percent of students who play soccer if 39 students play soccer out of a total of 387 students.

2.2.10.3. The percentage of a missing amount, e.g., 5 percent of the money from a fundraiser will be donated to a charity. If $25 is donated to the charity, how much money was made from the fundraiser?

2.2.10.4. Percentage increase/decrease, e.g., The number of music downloads have increased from 1,345 per minute to 1,567 per minute. What is the percent increase?

2.2.11. Use the rules for exponents to multiply and divide with powers of 10 and extend to other bases.

2.2.11.1. 10^2 x 10^3 = 10^5 - Add exponents

2.2.11.2. 2^5 / 2^7 = 2^-2 = 1/2^2 = 1/4 - Subtract exponents

2.2.12. Estimate answers to problems in context containing numbers expressed in scientific notation.

2.2.13. Solve problems in context that involve repetitive multiplication; e.g., compound interest, depreciation; using tables, spreadsheets and calculators to develop an understanding of exponential growth and decay.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.1. Determine the effect of scale factors (resulting in similar figures) on the perimeters and areas of two-dimensional shapes and the surface areas and volumes of three-dimensional solids.

3.1.2. Make and test conjectures about the angle and side relationships to determine that similar figures have congruent angles and corresponding sides proportional and congruent figures have congruent angles and sides.

3.1.3. Construct and/or examine right triangles and make and test conjectures about the relationships of the angles and sides and develop the Pythagorean theorem.

3.1.4. Apply side and angle relationships in geometric figures to solve problems including the Pythagorean theorem and similar figures.

3.2. Use spatial reasoning, location and geometric relationships to solve problems.

3.2.5. Use a coordinate plane to make and test conjectures about changes in the coordinates of the vertices of polygons as a result of a transformation (translation and/ or reflection) and describe the results in writing.

3.2.6. Develop and use formulas to determine the surface areas of rectangular prisms, cylinders and pyramids.

3.2.7. Develop formulas using measurement strategies and concrete models; and use formulas to determine the volumes of pyramids, cones and spheres.

3.3. Develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.8. Understand and describe in writing that measurement tools, measurements and estimates of measures are not precise and can affect the results of calculations.

3.3.9. Use estimation and measurement strategies, including formulas, to solve surface area and volume problems in context.

3.3.10. Solve customary or metric measurement problems in context using Dimensional Analysis (the Unit Factor Method) and justify the results in writing.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Collect, organize and display data using appropriate statistical and graphical methods.

4.1.1. Collect, organize and display data using an appropriate representation (including box-and-whisker plots, stem and leaf plots, scatter plots, histograms) based on the size and type of data set and purpose for its use.

4.1.2. Use appropriate representations to compare and analyze large data sets.

4.1.3. Identify where measures of central tendency and spread are found in graphical displays including box-and-whisker plots, stem and leaf plots, scatter plots and histograms.

4.2. Analyze data sets to form hypotheses and make predictions.

4.2.4. Use descriptive statistics, including range, mode, median, mean, quartiles and outliers to describe data and support conclusions in writing.

4.2.5. Make predictions from scatter plots by using or estimating a line-of-best-fit.

4.2.6. Make observations and inferences and evaluate hypotheses based on collected and/or experimental data.

4.2.7. Describe in writing the accuracy of statistical claims, e.g., 4 out of 5 dentists prefer Brand x toothpaste, by recognizing when a sample is biased or when data is misrepresented.

4.2.8. Explain the effects of sample size and sampling techniques (convenience sampling, voluntary response sampling, systematic sampling and random sampling) on statistical claims.

4.3. Understand and apply basic concepts of probability.

4.3.9. Determine when a situation is a permutation (changing the order results in a different outcome) or a combination (changing the order does not result in a different outcome.)

4.3.10. Use tree diagrams, lists, or the Counting Principle to determine all possible outcomes in permutations and combinations.

4.3.11. Apply permutations and combinations to predict possible outcomes and find probabilities to solve problems in a variety of contexts.

CT.1. Algebraic Reasoning: Patterns And Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Students should understand and describe patterns and functional relationships.

1.1.C.a. Core: Describe relationships and make generalizations about patterns and functions.

1.1.E.a. Extended: Model real-world situations and make generalizations about mathematical relationships using a variety of patterns and functions.

1.2. Students should represent and analyze quantitative relationships in a variety of ways.

1.2.C.a. Core: Represent and analyze linear and nonlinear functions and relations symbolically and with tables and graphs.

1.2.E.a. Extended: Relate the behavior of functions and relations to specific parameters and determine functions to model real-world situations.

1.3. Students should use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.C.a. Core: Manipulate equations, inequalities and functions to solve problems.

1.3.E.a. Extended: Use and extend algebraic concepts to include real and complex numbers, vectors and matrices.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Students should understand that a variety of numerical representations can be used to describe quantitative relation-ships.

2.1.C.a. Core: Extend the understanding of number to include integers, rational numbers and real numbers.

2.1.C.b. Core: Interpret and represent large sets of numbers with the aid of technologies.

2.1.E.a. Extended: Extend the understanding of number to include the set of complex numbers.

2.2. Students should use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities.

2.2.C.a. Core: Develop strategies for computation and estimation using properties of number systems to solve problems.

2.2.C.b. Core: Solve proportional reasoning problems.

2.2.E.a. Extended: Investigate mathematical properties and operations related to objects that are not numbers.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Students should use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.C.a. Core: Investigate relationships among plane and solid geometric figures using geometric models, constructions and tools.

3.1.C.b. Core: Develop and evaluate mathematical arguments using reasoning and proof.

3.1.E.a. Extended: Use methods of deductive and inductive reasoning to make, test and validate geometric conjectures.

3.1.E.b. Extended: Explore non-Euclidean geometries.

3.2. Students should use spatial reasoning, location and geometric relationships to solve problems.

3.2.C.a. Core: Verify geometric relationships using algebra, coordinate geometry, and transformations.

3.2.E.a. Extended: Use a variety of coordinate systems and transformations to solve geometric problems in 2 and 3 dimensions using appropriate tools and technologies.

3.3. Students should develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.C.a. Core: Solve a variety of problems involving 1-, 2- and 3-dimensional measurements using geometric relationships and trigonometric ratios.

3.3.E.a. Extended: Approximate measurements that cannot be directly determined with some degree of precision using appropriate tools, techniques and strategies.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Students should collect, organize and display data using appropriate statistical and graphical methods.

4.1.C.a. Core: Create the appropriate visual or graphical representation of real data.

4.1.E.a. Extended: Model real data graphically using appropriate tools, technologies and strategies.

4.2. Students should analyze data sets to form hypotheses and make predictions.

4.2.C.a. Core: Analyze real- world problems using statistical techniques.

4.2.E.a. Extended: Describe and analyze sets of data using statistical models.

4.3. Students should understand and apply basic concepts of probability.

4.3.C.a. Core: Understand and apply the principles of probability in a variety of situations.

4.3.E.a. Extended: Solve problems using the methods of discrete mathematics.

4.3.E.b. Extended: Make statistical inferences through the use of probability.

CT.1. Algebraic Reasoning: Patterns And Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Students should understand and describe patterns and functional relationships.

1.1.C.a. Core: Describe relationships and make generalizations about patterns and functions.

1.1.E.a. Extended: Model real-world situations and make generalizations about mathematical relationships using a variety of patterns and functions.

1.2. Students should represent and analyze quantitative relationships in a variety of ways.

1.2.C.a. Core: Represent and analyze linear and nonlinear functions and relations symbolically and with tables and graphs.

1.2.E.a. Extended: Relate the behavior of functions and relations to specific parameters and determine functions to model real-world situations.

1.3. Students should use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.C.a. Core: Manipulate equations, inequalities and functions to solve problems.

1.3.E.a. Extended: Use and extend algebraic concepts to include real and complex numbers, vectors and matrices.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Students should understand that a variety of numerical representations can be used to describe quantitative relation-ships.

2.1.C.a. Core: Extend the understanding of number to include integers, rational numbers and real numbers.

2.1.C.b. Core: Interpret and represent large sets of numbers with the aid of technologies.

2.1.E.a. Extended: Extend the understanding of number to include the set of complex numbers.

2.2. Students should use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities.

2.2.C.a. Core: Develop strategies for computation and estimation using properties of number systems to solve problems.

2.2.C.b. Core: Solve proportional reasoning problems.

2.2.E.a. Extended: Investigate mathematical properties and operations related to objects that are not numbers.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Students should use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.C.a. Core: Investigate relationships among plane and solid geometric figures using geometric models, constructions and tools.

3.1.C.b. Core: Develop and evaluate mathematical arguments using reasoning and proof.

3.1.E.a. Extended: Use methods of deductive and inductive reasoning to make, test and validate geometric conjectures.

3.1.E.b. Extended: Explore non-Euclidean geometries.

3.2. Students should use spatial reasoning, location and geometric relationships to solve problems.

3.2.C.a. Core: Verify geometric relationships using algebra, coordinate geometry, and transformations.

3.2.E.a. Extended: Use a variety of coordinate systems and transformations to solve geometric problems in 2 and 3 dimensions using appropriate tools and technologies.

3.3. Students should develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.C.a. Core: Solve a variety of problems involving 1-, 2- and 3-dimensional measurements using geometric relationships and trigonometric ratios.

3.3.E.a. Extended: Approximate measurements that cannot be directly determined with some degree of precision using appropriate tools, techniques and strategies.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Students should collect, organize and display data using appropriate statistical and graphical methods.

4.1.C.a. Core: Create the appropriate visual or graphical representation of real data.

4.1.E.a. Extended: Model real data graphically using appropriate tools, technologies and strategies.

4.2. Students should analyze data sets to form hypotheses and make predictions.

4.2.C.a. Core: Analyze real- world problems using statistical techniques.

4.2.E.a. Extended: Describe and analyze sets of data using statistical models.

4.3. Students should understand and apply basic concepts of probability.

4.3.C.a. Core: Understand and apply the principles of probability in a variety of situations.

4.3.E.a. Extended: Solve problems using the methods of discrete mathematics.

4.3.E.b. Extended: Make statistical inferences through the use of probability.

CT.1. Algebraic Reasoning: Patterns And Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Students should understand and describe patterns and functional relationships.

1.1.C.a. Core: Describe relationships and make generalizations about patterns and functions.

1.1.E.a. Extended: Model real-world situations and make generalizations about mathematical relationships using a variety of patterns and functions.

1.2. Students should represent and analyze quantitative relationships in a variety of ways.

1.2.C.a. Core: Represent and analyze linear and nonlinear functions and relations symbolically and with tables and graphs.

1.2.E.a. Extended: Relate the behavior of functions and relations to specific parameters and determine functions to model real-world situations.

1.3. Students should use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.C.a. Core: Manipulate equations, inequalities and functions to solve problems.

1.3.E.a. Extended: Use and extend algebraic concepts to include real and complex numbers, vectors and matrices.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Students should understand that a variety of numerical representations can be used to describe quantitative relation-ships.

2.1.C.a. Core: Extend the understanding of number to include integers, rational numbers and real numbers.

2.1.C.b. Core: Interpret and represent large sets of numbers with the aid of technologies.

2.1.E.a. Extended: Extend the understanding of number to include the set of complex numbers.

2.2. Students should use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities.

2.2.C.a. Core: Develop strategies for computation and estimation using properties of number systems to solve problems.

2.2.C.b. Core: Solve proportional reasoning problems.

2.2.E.a. Extended: Investigate mathematical properties and operations related to objects that are not numbers.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Students should use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.C.a. Core: Investigate relationships among plane and solid geometric figures using geometric models, constructions and tools.

3.1.C.b. Core: Develop and evaluate mathematical arguments using reasoning and proof.

3.1.E.a. Extended: Use methods of deductive and inductive reasoning to make, test and validate geometric conjectures.

3.1.E.b. Extended: Explore non-Euclidean geometries.

3.2. Students should use spatial reasoning, location and geometric relationships to solve problems.

3.2.C.a. Core: Verify geometric relationships using algebra, coordinate geometry, and transformations.

3.2.E.a. Extended: Use a variety of coordinate systems and transformations to solve geometric problems in 2 and 3 dimensions using appropriate tools and technologies.

3.3. Students should develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.C.a. Core: Solve a variety of problems involving 1-, 2- and 3-dimensional measurements using geometric relationships and trigonometric ratios.

3.3.E.a. Extended: Approximate measurements that cannot be directly determined with some degree of precision using appropriate tools, techniques and strategies.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Students should collect, organize and display data using appropriate statistical and graphical methods.

4.1.C.a. Core: Create the appropriate visual or graphical representation of real data.

4.1.E.a. Extended: Model real data graphically using appropriate tools, technologies and strategies.

4.2. Students should analyze data sets to form hypotheses and make predictions.

4.2.C.a. Core: Analyze real- world problems using statistical techniques.

4.2.E.a. Extended: Describe and analyze sets of data using statistical models.

4.3. Students should understand and apply basic concepts of probability.

4.3.C.a. Core: Understand and apply the principles of probability in a variety of situations.

4.3.E.a. Extended: Solve problems using the methods of discrete mathematics.

4.3.E.b. Extended: Make statistical inferences through the use of probability.

CT.1. Algebraic Reasoning: Patterns And Functions: Patterns and functional relationships can be represented and analyzed using a variety of strategies, tools and technologies.

1.1. Students should understand and describe patterns and functional relationships.

1.1.C.a. Core: Describe relationships and make generalizations about patterns and functions.

1.1.E.a. Extended: Model real-world situations and make generalizations about mathematical relationships using a variety of patterns and functions.

1.2. Students should represent and analyze quantitative relationships in a variety of ways.

1.2.C.a. Core: Represent and analyze linear and nonlinear functions and relations symbolically and with tables and graphs.

1.2.E.a. Extended: Relate the behavior of functions and relations to specific parameters and determine functions to model real-world situations.

1.3. Students should use operations, properties and algebraic symbols to determine equivalence and solve problems.

1.3.C.a. Core: Manipulate equations, inequalities and functions to solve problems.

1.3.E.a. Extended: Use and extend algebraic concepts to include real and complex numbers, vectors and matrices.

CT.2. Numerical and Proportional Reasoning: Quantitative relationships can be expressed numerically in multiple ways in order to make connections and simplify calculations using a variety of strategies, tools and technologies.

2.1. Students should understand that a variety of numerical representations can be used to describe quantitative relation-ships.

2.1.C.a. Core: Extend the understanding of number to include integers, rational numbers and real numbers.

2.1.C.b. Core: Interpret and represent large sets of numbers with the aid of technologies.

2.1.E.a. Extended: Extend the understanding of number to include the set of complex numbers.

2.2. Students should use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities.

2.2.C.a. Core: Develop strategies for computation and estimation using properties of number systems to solve problems.

2.2.C.b. Core: Solve proportional reasoning problems.

2.2.E.a. Extended: Investigate mathematical properties and operations related to objects that are not numbers.

CT.3. Geometry and Measurement: Shapes and structures can be analyzed, visualized, measured and transformed using a variety of strategies, tools and technologies.

3.1. Students should use properties and characteristics of two- and three-dimensional shapes and geometric theorems to describe relationships, communicate ideas and solve problems.

3.1.C.a. Core: Investigate relationships among plane and solid geometric figures using geometric models, constructions and tools.

3.1.C.b. Core: Develop and evaluate mathematical arguments using reasoning and proof.

3.1.E.a. Extended: Use methods of deductive and inductive reasoning to make, test and validate geometric conjectures.

3.1.E.b. Extended: Explore non-Euclidean geometries.

3.2. Students should use spatial reasoning, location and geometric relationships to solve problems.

3.2.C.a. Core: Verify geometric relationships using algebra, coordinate geometry, and transformations.

3.2.E.a. Extended: Use a variety of coordinate systems and transformations to solve geometric problems in 2 and 3 dimensions using appropriate tools and technologies.

3.3. Students should develop and apply units, systems, formulas and appropriate tools to estimate and measure.

3.3.C.a. Core: Solve a variety of problems involving 1-, 2- and 3-dimensional measurements using geometric relationships and trigonometric ratios.

3.3.E.a. Extended: Approximate measurements that cannot be directly determined with some degree of precision using appropriate tools, techniques and strategies.

CT.4. Working with Data: Probability and Statistics: Data can be analyzed to make informed decisions using a variety of strategies, tools and technologies.

4.1. Students should collect, organize and display data using appropriate statistical and graphical methods.

4.1.C.a. Core: Create the appropriate visual or graphical representation of real data.

4.1.E.a. Extended: Model real data graphically using appropriate tools, technologies and strategies.

4.2. Students should analyze data sets to form hypotheses and make predictions.

4.2.C.a. Core: Analyze real- world problems using statistical techniques.

4.2.E.a. Extended: Describe and analyze sets of data using statistical models.

4.3. Students should understand and apply basic concepts of probability.

4.3.C.a. Core: Understand and apply the principles of probability in a variety of situations.

4.3.E.a. Extended: Solve problems using the methods of discrete mathematics.

4.3.E.b. Extended: Make statistical inferences through the use of probability.