Colorado State Standards for Mathematics: Grade 5

CO.1. Number Sense, Properties, and Operations

1.1. The characteristics of numbers can be used to classify them in various ways. Students can:

1.1.a. Apply concepts of squares, primes, composites, factors, and multiples to solve problems

1.1.b. Use the identity, associative, commutative, and distributive properties to solve problems

1.1.c. Describe and use divisibility rules for two, three, four, five, six, nine, and 10 to solve problems

1.2. In the real number system, commonly used rational numbers have multiple equivalent representations. Students can:

1.2.a. Find equivalent forms of commonly used fractions, decimals, and percents using models, drawings, and computational strategies

1.2.b. Use common fractions and percents to calculate parts of whole numbers in problem situations including comparisons of savings rates at different financial institutions (PFL)

1.2.c. Model addition, subtraction, and multiplication of fractions, decimals, and percents

1.2.d. Compose and decompose multi-digit whole numbers and decimals based on place value

1.2.e. Represent numbers to 1,000,000 with expanded notation and exponents

1.3. Formulate, represent, and use algorithms to multiply and divide multi-digit whole numbers with flexibility, accuracy, and efficiency. Students can:

1.3.a. Use flexible methods of computing including standard algorithms to multiply and divide multi-digit numbers by two-digit factors or divisors

1.3.b. Model multiplication and division using area, linear, and grouping models

1.3.c. Interpret remainders and select the most useful form of the quotient in division problems

1.3.d. Select and use appropriate methods to estimate products and quotients or calculate them mentally depending on the context and numbers involved

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Number patterns and relationships can be described using a variety of tools. Students can:

2.1.a. Analyze and describe patterns and relationships using words, tables, graphs, symbols, and technology

2.1.b. Explain, extend, and use patterns and relationships in solving problems, including those involving saving and checking accounts such as understanding that spending more means saving less (PFL)

2.2. When a relationship exists between two quantities, a change in one results in a change in the other. Students can:

2.2.a. Express change relationships involving whole numbers with if/then statements, input/output boxes, function tables, and rule statements

2.2.b. Select, describe, and use symbols to express unknown quantities

2.2.c. Use patterns to solve problems including those involving saving and checking accounts such as the pattern created when saving $10 a month (PFL)

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays and summary statistics are used to describe and interpret data. Students can:

3.1.a. Formulate a question and hypothesis to design appropriate data collection and display methods

3.1.b. Select and create appropriate displays of data including double bar graphs, time plots, and line graphs

3.1.c. Interpret data using the concepts of shape of distribution, range, mode, median and mean

3.1.d. Draw conclusions, and make convincing arguments based on categorical and numerical data analysis

3.2. Mathematical models are used to determine probability, analyze and describe the outcomes of events. Students can:

3.2.a. Organize all possible outcomes of events in a list or chart

3.2.b. Use fractions, decimals, and percents to quantify the likelihood of events

3.2.c. Explain why a game involving chance devices such as number cubes or spinners is fair or unfair

3.2.d. Compare individual data to class data collected from chance devices to describe the differences in outcomes based on sample size

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Geometric figures in the plane and in space are described and analyzed by their attributes. Students can:

4.1.a. Relate two-dimensional shapes to three-dimensional shapes using faces, edges, and vertices

4.1.b. Predict and describe the results of transformations: translations, reflections, rotations

4.1.c. Classify and compare angles

4.1.d. Apply concepts of parallel, perpendicular, congruence and line symmetry

4.2. Linear measure, area, and volume are fundamentally different and require different units of measure. Students can:

4.2.a. Accurately measure length to the nearest 1/8 inch or millimeter

4.2.b. Determine the perimeter of polygons and area of rectangles

4.2.c. Distinguish between appropriate units for area and linear measures

4.2.d. Model volume using cubic units

4.2.e. Use, apply, and select appropriate scales on number lines, graphs, and maps

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