Colorado Academic Standards Online

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Mathematics - 2019

Sixth Grade, Standard 2. Algebra and Functions

| Read the Colorado Essential Skills | Expand Set of GLEs Below

Prepared Graduates:

MP3. Construct viable arguments and critique the reasoning of others.
MP7. Look for and make use of structure.
MP8. Look for and express regularity in repeated reasoning.

Grade Level Expectation:

6.EE.A. Expressions & Equations: Apply and extend previous understandings of arithmetic to algebraic expressions.

Evidence Outcomes:

Students Can:

Write and evaluate numerical expressions involving whole-number exponents. (CCSS: 6.EE.A.1)
Write, read, and evaluate expressions in which letters stand for numbers. (CCSS: 6.EE.A.2)
1. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract $y$ from $5$ " as $5 - y$ . (CCSS: 6.EE.A.2.a)
2. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression $2 \left( 8 + 7 \right)$ as a product of two factors; view $\left( 8 + 7 \right)$ as both a single entity and a sum of two terms. (CCSS: 6.EE.A.2.b)
3. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas $V = s^3$ and $A = 6s^2$ to find the volume and surface area of a cube with sides of length $s = \frac{1}{2}$ . (CCSS: 6.EE.A.2.c)
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression $3 \left( 2 + x \right)$ to produce the equivalent expression $6 + 3x$ ; apply the distributive property to the expression $24x + 18y$ to produce the equivalent expression $6 \left( 4x + 3y \right)$ ; apply properties of operations to $y + y + y$ to produce the equivalent expression $3y$ . (CCSS: 6.EE.A.3)
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions $y + y + y$ and $3y$ are equivalent because they name the same number regardless of which number $y$ stands for. (CCSS: 6.EE.A.4)

Academic Contexts and Connections:

Colorado Essential Skills and Mathematical Practices:

Recognize that expressions can be written in multiple forms and describe cause-and-effect relationships and patterns. (Entrepreneurial Skills: Critical Thinking/Problem Solving and Inquiry/Analysis)
Communicate a justification of why expressions are equivalent using arguments about properties of operations and whole numbers. (MP3)
See the structure of an expression like $x + 2$ as a sum but also as a single factor in the product $3\left(x + 2\right)$ . (MP7)
Recognize equivalence in variable expressions with repeated addition (such as $y + y + y = 3y$ ) and repeated multiplication (such as $y \times y \times y = y^3$ ) and use arithmetic operations to justify the equivalence. (MP8)

Inquiry Questions:

How are algebraic expressions similar to and different from numerical expressions?
What does it mean for two variable expressions to be equivalent?
How might the application of the order of operations differ when using grouping symbols, such as parentheses, for numerical expressions as compared to algebraic expressions?

Coherence Connections:

This expectation represents major work of the grade.
In previous grades, students understand and apply properties of operations, relationships between inverse arithmetic operations, and write and interpret numerical expressions.
In Grade 6, this expectation connects to fluency with multi-digit numbers, finding common factors and multiples, and one-variable equations and inequalities.
In future grades, students work with radicals and integer exponents and interpret the structure of more complex algebraic expressions.

Prepared Graduates:

MP2. Reason abstractly and quantitatively.
MP6. Attend to precision.

Grade Level Expectation:

6.EE.B. Expressions & Equations: Reason about and solve one-variable equations and inequalities.

Evidence Outcomes:

Students Can:

Describe solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (CCSS: 6.EE.B.5)
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; recognize that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. (CCSS: 6.EE.B.6)
Solve real-world and mathematical problems by writing and solving equations of the form $x \pm p=q$ and $px=q$ for cases in which $p$ , $q$ and $x$ are all nonnegative rational numbers. (CCSS: 6.EE.B.7)
Write an inequality of the form $x \gt c$ , $x \geq c$ , $x \lt c$ , or $x \leq c$ to represent a constraint or condition in a real-world or mathematical problem. Show that inequalities of the form $x \gt c$ , $x \geq c$ , $x \lt c$ , or $x \leq c$ have infinitely many solutions; represent solutions of such inequalities on number line diagrams. (CCSS: 6.EE.B.8)

Academic Contexts and Connections:

Colorado Essential Skills and Mathematical Practices:

Investigate unknown values to form hypotheses, make observations, and draw conclusions. (Entrepreneurial Skills: Inquiry/Analysis)
Reason about the values and operations of an equation both within a real-world context and abstracted from it. (MP2)
State precisely the meaning of variables used when setting up equations. (MP6)

Inquiry Questions:

What are the different ways a variable can be used in an algebraic equation or inequality? For example, how are these uses of the variable $x$ different from each other? (a) $x + 5 = 8$ ; (b) $x = \frac{1}{2}$ ; (c) $x \gt 5$ .
How is the solution to an inequality different than a solution to an equation?

Coherence Connections:

This expectation represents major work of the grade.
In previous grades, students write simple expressions that record calculations with numbers, interpret numerical expressions without evaluating them, and generate ordered pairs from two numerical rules.
This expectation connects to several others in Grade 6: (a) understanding ratio concepts and use ratio reasoning to solve problems, (b) applying and extending previous understandings of multiplication and division to divide fractions by fractions, (c) applying and extending previous understandings of numbers to the system of rational numbers, (d) applying and extending previous understandings of arithmetic to algebraic expressions, and (e) representing and analyzing quantitative relationships between dependent and independent variables.
In Grade 7, students solve real-life and mathematical problems involving two-step equations and inequalities. In Grade 8, students work with radicals and integer exponents and solve linear equations and pairs of simultaneous linear equations.

Prepared Graduates:

MP2. Reason abstractly and quantitatively.
MP4. Model with mathematics.

Grade Level Expectation:

6.EE.C. Expressions & Equations: Represent and analyze quantitative relationships between dependent and independent variables.

Evidence Outcomes:

Students Can:

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation $d = 65t$ to represent the relationship between distance and time. (CCSS: 6.EE.C.9)

Academic Contexts and Connections:

Colorado Essential Skills and Mathematical Practices:

Analyze relationships between dependent and independent variables. (Entrepreneurial Skills: Inquiry/Analysis)
Reason about the operations that relate constant and variable quantities in equations with dependent and independent variables. (MP2)
Model with mathematics by describing real-world situations with equations and inequalities. (MP4)

Inquiry Questions:

How can you determine if a variable is the independent variable or the dependent variable?
What are the advantages of showing the relationship between an independent and dependent variable in multiple representations (table, graph, equation)?

Coherence Connections:

This expectation represents major work of the grade.
In Grade 5, students analyze numerical patterns and relationships, including generating and graphing ordered pairs in the first quadrant.
In Grade 6, this expectation connects with understanding ratio concepts and using ratio reasoning to solve problems.
In Grade 7, students decide if two quantities are in a proportional relationship and identify the unit rate in tables, graphs, equations, diagrams, and verbal descriptions.

Need Help? Submit questions or requests for assistance to bruno_j@cde.state.co.us