Math / 6th Grade / Unit 5: Numerical and Algebraic Expressions

Numerical and Algebraic Expressions

Students venture into the Expressions and Equations domain, using variables to represent unknown or changing quantities, and using properties of operations to investigate equivalent expressions.

Math

Unit 5

6th Grade

Unit Summary

In Unit 5, sixth graders venture into the Expressions and Equations domain for the first time, extending on their understanding of arithmetic to see how it applies to algebraic expressions. They start with numerical expressions with exponents, rewriting the expressions into simpler forms until a final value is determined. Students then use variables in expressions to represent quantities that are unknown or quantities that change. Using the properties of operations, students will investigate what makes expressions equivalent to others, a concept that is threaded throughout the middle school Expressions and Equations domain. Throughout this unit, students pay close attention to the structure of expressions, understanding the role of parentheses, the order of operations, and the way expressions are described verbally (MP.7). A note on fluency: Evaluating expressions provides a good opportunity for students to continue developing and to demonstrate fluency with decimal operations. Several problems throughout the unit include decimal values; include additional problems in practice for students as needed.

In elementary school, students used variables to represent unknown quantities, and they evaluated and described numerical expressions without exponents. They used the commutative property to enhance their understanding of multiplication and addition, and they used the distributive property when modeling partial areas. All of these concepts come together and support student understanding in this sixth-grade unit.

Immediately following this unit, sixth graders will start a unit on Equations and Inequalities, where they will use algebra to model and solve real-world problems. They will also revisit percentages using new skills with expressions and equations to efficiently solve percent problems. In seventh and eighth grades, students continue to simplify and solve more complex expressions and equations using the same tools learned in this unit.

Pacing: 16 instructional days (12 lessons, 3 flex days, 1 assessment day)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 6th Grade Scope and Sequence Recommended Adjustments.

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Assessment

This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Post-Unit Assessment

Take the Post-Unit Assessment. Annotate for:
- Standards that each question aligns to
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that Assessment points to

Internalization of Trajectory of Unit

Read and annotate the Unit Summary.
Notice the progression of concepts through the unit using the Lesson Map.
Do all Target Tasks. Annotate the Target Tasks for:
- Essential Understandings
- Connection to Post-Unit Assessment questions
Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Read the Progressions for the Common Core State Standards in Mathematics, 6-8, Expressions and Equations for standards relevant to this unit.
Read the following table that includes models throughout the unit.

Model

Example

Tape diagram

$$4-m$$

$$m-4$$

$$3x+2=x+x+x+2$$

Area model

$$2\times5^2$$

$$2+5^2$$

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

Numerical expressions can be evaluated using the order of operations. Paying close attention to the structure of an expression, such as what role the parentheses play or where the exponents are, can provide insight into how an expression is evaluated to a single number.
When two algebraic expressions are equivalent, they are equal for any value of the variable(s).
Equivalent expressions can be identified and generated by using properties of operations, specifically the distributive property and the commutative property.
Algebraic expressions can be used to represent real-world situations where variables represent unknown quantities or quantities that vary.

Vocabulary

Terms and notation that students learn or use in the unit

variable

numerical expression

equivalent

term

commutative property

algebraic expression

distributive property

order of operations

To see all the vocabulary for this course, view our 6th Grade Vocabulary Glossary.

Lesson Map

Topic A: Numerical Expressions with Exponents

Understand the meaning of exponents.

6.EE.A.1

Evaluate numerical expressions involving whole-number exponents.

6.EE.A.1

Topic B: Introduction to Algebraic Expressions

Use variables to write algebraic expressions.

6.EE.A.2 6.EE.A.2.C 6.EE.B.6

Evaluate algebraic expressions.

6.EE.A.2 6.EE.A.2.C

Write expressions for verbal statements and vice versa (Part 1).

6.EE.A.2.A 6.EE.A.2.B

Write expressions for verbal statements and vice versa (Part 2).

6.EE.A.2.A 6.EE.A.2.B

Topic C: Equivalent Expressions & Applications

Identify equivalent expressions (Part 1).

6.EE.A.3 6.EE.A.4

Identify equivalent expressions (Part 2).

6.EE.A.3 6.EE.A.4

Write equivalent expressions using the distributive property (Part 1).

6.EE.A.3 6.EE.A.4

Write equivalent expressions using the distributive property (Part 2).

6.EE.A.3 6.EE.A.4

Write algebraic expressions for application situations (Part 1).

6.EE.B.6

Write algebraic expressions for application situations (Part 2).

6.EE.B.6

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Expressions and Equations

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.

Expressions and Equations

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2 — Write, read, and evaluate expressions in which letters stand for numbers.

Expressions and Equations

6.EE.A.2 — Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.2.A — 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.

Expressions and Equations

6.EE.A.2.A — 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.
6.EE.A.2.B — 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 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

Expressions and Equations

6.EE.A.2.B — 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 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
6.EE.A.2.C — 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 = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.

Expressions and Equations

6.EE.A.2.C — 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 = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.
6.EE.A.3 — Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Expressions and Equations

6.EE.A.3 — Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.4 — 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.

Expressions and Equations

6.EE.A.4 — 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.
6.EE.B.6 — Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Expressions and Equations

6.EE.B.6 — Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Measurement and Data

5.MD.C.5.B

Measurement and Data

5.MD.C.5.B — Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Number and Operations in Base Ten

4.NBT.B.5

Number and Operations in Base Ten

4.NBT.B.5 — Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Operations and Algebraic Thinking

4.OA.A.2

Operations and Algebraic Thinking

4.OA.A.2 — Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
4.OA.A.3

Operations and Algebraic Thinking

4.OA.A.3 — Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
5.OA.A.1

Operations and Algebraic Thinking

5.OA.A.1 — Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5.OA.A.2

Operations and Algebraic Thinking

5.OA.A.2 — Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Future Standards

Standards in future grades or units that connect to the content in this unit

Expressions and Equations

6.EE.B.5

Expressions and Equations

6.EE.B.5 — Understand 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.
6.EE.B.7

Expressions and Equations

6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.B.8

Expressions and Equations

6.EE.B.8 — Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
6.EE.C.9

Expressions and Equations

6.EE.C.9 — 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.
7.EE.A.1

Expressions and Equations

7.EE.A.1 — Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.B.4

Expressions and Equations

7.EE.B.4 — Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Unit 4

Rational Numbers

Unit 6

Equations and Inequalities

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