1.1 Students should understand numbers, ways of representing numbers, relationships among numbers, and number systems

By the end of grade 6 students will
1.1.1 Work flexibly with fractions, decimals, and percents to solve problems;
1.1.2 Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
1.1.4 Understand and use ratios and proportions to represent quantitative relationships;

1.2 Students should understand meanings of operations and how they relate to one another

By the end of grade 6 students will
1.2.1 Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;
1.2.2 Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals;
1.2.3 Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

1.3 Students should compute fluently and make reasonable estimates

By the end of grade 6 students will
1.3.1 Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;
1.3.2 Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;
1.3.3 Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results;
1.3.4 Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Standard 2: Algebra

2.1 Students should understand patterns, relations, and functions

By the end of grade 6 students will
2.1.1 Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;

2.2 Students should represent and analyze mathematical situations and structures using algebraic symbols

By the end of grade 6 students will
2.2.1 Develop an initial conceptual understanding of different uses of variables;
2.2.3 Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;
2.2.4 Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

2.3 Students should use mathematical models to represent and understand quantitative relationships

By the end of grade 6 students will
2.3.1 Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

Standard 3: Geometry

3.1 Students should be able to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

By the end of grade 6 students will
3.1.1 Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;

3.4 Students should use visualization, spatial reasoning, and geometric modeling to solve problems

By the end of grade 6 students will
3.4.5 Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Standard 4: Measurement

4.1 Students should understand measurable attributes of objects and the units, systems, and processes of measurement

By the end of grade 6 students will
4.1.1 Understand both metric and customary systems of measurement;
4.1.2 Understand relationships among units and convert from one unit to another within the same system;
4.1.3 Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

4.2 Students should apply appropriate techniques, tools, and formulas to determine measurements.

By the end of grade 6 students will
4.2.5 Solve problems involving scale factors, using ratio and proportion;
4.2.6 Solve simple problems involving rates and derived measurements for such attributes as velocity and density.
Instructional programs from KP through grade 12 should enable all students to demonstrate these process standards:

Standard 6: Problem Solving

6.1 Build new mathematical knowledge through problem solving
6.2 Solve problems that arise in mathematics and in other contexts
6.3 Apply and adapt a variety of appropriate strategies to solve problems
6.4 Monitor and reflect on the process of mathematical problem solving

Standard 7: Reasoning and Proof

7.1 Recognize reasoning and proof as fundamental aspects of mathematics
7.2 Make and investigate mathematical conjectures
7.3 Develop and evaluate mathematical arguments and proofs
7.4 Select and use various types of reasoning and methods of proof

Standard 8: Communication

8.1 Organize and consolidate their mathematical thinking through communication
8.2 Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
8.3 Analyze and evaluate the mathematical thinking and strategies of others;
8.4 Use the language of mathematics to express mathematical ideas precisely.

Standard 9: Connections

9.1 Recognize and use connections among mathematical ideas
9.2 Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
9.3 Recognize and apply mathematics in contexts outside of mathematics

Standard 10: Representations

10.1 Create and use representations to organize, record, and communicate mathematical ideas
10.2 Select, apply, and translate among mathematical representations to solve problems

## Standard 1: Number and Operations

1.1 Students should understand numbers, ways of representing numbers, relationships among numbers, and number systemsBy the end of grade 6 students will

1.1.1 Work flexibly with fractions, decimals, and percents to solve problems;

1.1.2 Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;

1.1.4 Understand and use ratios and proportions to represent quantitative relationships;

1.2 Students should understand meanings of operations and how they relate to one anotherBy the end of grade 6 students will

1.2.1 Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;

1.2.2 Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals;

1.2.3 Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

1.3 Students should compute fluently and make reasonable estimatesBy the end of grade 6 students will

1.3.1 Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;

1.3.2 Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;

1.3.3 Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results;

1.3.4 Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

## Standard 2: Algebra

2.1 Students should understand patterns, relations, and functionsBy the end of grade 6 students will

2.1.1 Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;

2.2 Students should represent and analyze mathematical situations andstructures using algebraic symbolsBy the end of grade 6 students will

2.2.1 Develop an initial conceptual understanding of different uses of variables;

2.2.3 Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;

2.2.4 Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

2.3 Students should use mathematical models to represent and understand quantitative relationshipsBy the end of grade 6 students will

2.3.1 Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

## Standard 3: Geometry

3.1 Students should be able to analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationshipsBy the end of grade 6 students will

3.1.1 Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;

3.4 Students should use visualization, spatial reasoning, and geometric modeling to solve problemsBy the end of grade 6 students will

3.4.5 Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

## Standard 4: Measurement

4.1 Students should understand measurable attributes of objects and the units, systems, and processes of measurementBy the end of grade 6 students will

4.1.1 Understand both metric and customary systems of measurement;

4.1.2 Understand relationships among units and convert from one unit to another within the same system;

4.1.3 Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

4.2 Students should apply appropriate techniques, tools, and formulas to determine measurements.By the end of grade 6 students will

4.2.5 Solve problems involving scale factors, using ratio and proportion;

4.2.6 Solve simple problems involving rates and derived measurements for such attributes as velocity and density.

Instructional programs from KP through grade 12 should enable all students to demonstrate these process standards:

## Standard 6: Problem Solving

6.1 Build new mathematical knowledge through problem solving

6.2 Solve problems that arise in mathematics and in other contexts

6.3 Apply and adapt a variety of appropriate strategies to solve problems

6.4 Monitor and reflect on the process of mathematical problem solving

## Standard 7: Reasoning and Proof

7.1 Recognize reasoning and proof as fundamental aspects of mathematics

7.2 Make and investigate mathematical conjectures

7.3 Develop and evaluate mathematical arguments and proofs

7.4 Select and use various types of reasoning and methods of proof

## Standard 8: Communication

8.1 Organize and consolidate their mathematical thinking through communication

8.2 Communicate their mathematical thinking coherently and clearly to peers, teachers, and others

8.3 Analyze and evaluate the mathematical thinking and strategies of others;

8.4 Use the language of mathematics to express mathematical ideas precisely.

## Standard 9: Connections

9.1 Recognize and use connections among mathematical ideas

9.2 Understand how mathematical ideas interconnect and build on one another to produce a coherent whole

9.3 Recognize and apply mathematics in contexts outside of mathematics

## Standard 10: Representations

10.1 Create and use representations to organize, record, and communicate mathematical ideas

10.2 Select, apply, and translate among mathematical representations to solve problems