Colorado State Standards for Mathematics: Grade 6

CO.1. Number Sense, Properties, and Operations

1.1. In the real number system, positive rational numbers are represented in multiple equivalent forms. Students can:

1.1.a. Read, write, compare, convert and order positive rational numbers in a variety of forms including proper and improper fractions, mixed numbers, decimals, and percents

1.1.b. Express whole numbers as products of prime factors with exponents and use prime factorization to find the greatest common factor and least common multiple of two numbers

1.1.c. Express the quotient and remainder of a whole number division problem (a/b or a / b) using fractions, terminating decimals, or repeating decimals

1.1.d. Locate positive fractions and decimals on a number line

1.2. Formulate, represent, and use algorithms with positive rational numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Model and compute the addition, subtraction, multiplication and division of positive fractions, decimals, and combinations of fractions and decimals

1.2.b. Solve multi-step word problems involving fractions, decimals and whole numbers

1.2.c. Estimate sums, differences, products and quotients of rational numbers using common fractions, common decimals, and whole numbers

1.2.d. Compare and round positive numbers from thousandths through millions

1.3. Quantities can be expressed and compared using ratios and rates. Students can:

1.3.a. Apply the multiplicative identity to create equivalent fractions and to reduce fractions to simplest form

1.3.b. Express the comparison of two whole number quantities using differences, part-to-part ratios, and part-to-whole ratios in real contexts, including investing and saving (PFL)

1.3.c. Compute unit rates in real-world situations involving mixtures, concentrations, and distance-time relationships

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Patterns can be described using words, tables, and graphs. Students can:

2.1.a. Extend the pattern and describe the rule for arithmetic and geometric sequences

2.1.b. Model linear situations using tables and graphs, and convert between these two representations

2.1.c. Given a linear equation, substitute non-negative input values to create a table and graph coordinate points in the first quadrant

2.2. Variables are used to represent unknown quantities. Students can:

2.2.a. Describe patterns by using words and variables with mathematical symbols

2.2.b. Evaluate expressions by substituting whole number values for variables

CO.3. Data Analysis, Statistics, and Probability

3.1. Questions can be answered by collecting and analyzing data and data displays. Students can:

3.1.a. Formulate questions for populations larger than the classroom

3.1.b. Recognize that a sample may not represent a population accurately

3.1.c. Recognize bias in surveys

3.1.d. Utilize appropriate techniques to design a random sample

3.1.e. Recognize the use of deceptive scales on a graph that make differences look much larger than they are, or the use of pictographs with areas that are proportioned incorrectly

3.2. Mathematical models are used to determine probability. Students can:

3.2.a. Determine probabilities through experiments or simulations

3.2.b. Express the probability of an event using fractions, decimals, and percents

3.2.c. Make a table, tree diagram or an organized list to determine possible outcomes of two or more compound events

3.2.d. Predict outcomes of events using experimental and theoretical probabilities

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Polygons can be described, classified, and analyzed by their attributes. Students can:

4.1.a. Develop and apply formulas and procedures for finding area of triangles, parallelograms, and trapezoids

4.1.b. Describe properties of polygons up to ten sides using accurate vocabulary and notation

4.1.c. Classify triangles and apply angle and side properties, including the sum of the interior angles

4.1.d. Use accurate geometric notation to describe angles, lines, and segments

4.2. Standard units provide common language for communicating measurements. Students can:

4.2.a. Connect metric prefixes to place value

4.2.b. Measure to the nearest sixteenth of an inch

4.2.c. Select and use appropriate units to accurately measure length, weight, capacity and time in problem-solving situations

4.2.d. Use a protractor to measure angles to the nearest degree

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions