Numerical and Algebraic Expressions

Students manipulate expressions into different equivalent forms as they expand, factor, add, and subtract numerical and algebraic expressions and face authentic real-world, multi-step problems.

Math

Unit 3

7th Grade

Unit Summary

In Unit 3, 7th grade students bring several prior skills together to manipulate expressions into different equivalent forms. In the preceding unit, students operated and reasoned with positive and negative rational numbers. In this unit, they use these new skills to expand, factor, add, and subtract numerical and algebraic expressions. Students pay particular attention to the structure of expressions in order to better understand what an expression means and how it can be manipulated MP.7 Look for and make use of structure. . Students also face authentic real-world, multi-step problems that require strategic thinking, use of rational numbers, and estimation where appropriate MP.1 Make sense of problems and persevere in solving them. .

In 6th grade, students learned how the same rules that govern arithmetic also apply to algebraic expressions. They learned to expand and factor expressions using the distributive property, and they combined terms where variables are the same. With new knowledge of the number system, students go from working with expressions like $${5(6x+3y)}$$ in 6th grade to those with rational numbers such as $${-(a+b)-\frac{3}{2}(a-b)}$$ in the 7th grade.

The next unit of the 7th grade course, Unit 4, will continue to engage students in working with expressions with rational numbers, with a shift in focus to writing and solving equations and inequalities. In 8th grade, students will work with expressions and equations in both one variable and two variables, solving single linear equations and systems of linear equations. Throughout all of their future work with expressions, students’ ability to look for and make use of the structure in expressions will be as important as their ability to work with them procedurally.

Pacing: 15 instructional days (11 lessons, 3 flex days, 1 assessment day)

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Assessment

The following assessments accompany Unit 3.

Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Mid-Unit

Have students complete the Mid-Unit Assessment after Lesson 6.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Unit Launch

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

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 through the Progressions for the Common Core State Standards in Mathematics, 6-8, Expressions and Equations for the relevant standards within the Expressions and Equations domain.

Essential Understandings

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

The structure of an expression helps determine how it can be simplified or manipulated.
The order of operations provides a consistent framework for simplifying and evaluating numerical and algebraic expressions. Expressions can be expanded, factored, added, and subtracted using the properties of operations and the rules of operating with rational numbers.
Rewriting an expression in a simpler or more useful form can help solve problems, uncover patterns, or clarify relationships between quantities.
Algebraic and numerical expressions can be used to represent and solve real-world and multi-step problems.

Materials

The materials, representations, and tools teachers and students will need for this unit

Colored Pencils (one set per student group) –or colored pens

Algebra tiles (one set per student group) –or access to digital manipulatives, such as Polypad

Graph Paper

Vocabulary and Models

Unit Vocabulary

constant term

expand an expression

factor an expression

Foundational Vocabulary

algebraic expression

approximate/approximation

coefficient

combine like terms

commutative property

distributive property

estimate

evaluate

greatest common factor (GCF)

like terms

numerical expression

term

variable

To see all the vocabulary for Unit 3, view our 7th Grade Vocabulary Glossary.

Access foundational vocabulary for Unit 3 in the same 7th Grade Vocabulary Glossary.

Models

Models and representations that students learn or use in the unit

Model	Example
rectangular array	$$\frac{1}{3}(9x-12y-18)=3x-4y-6$$
Algebra tiles

Model

Example

rectangular array

$$\frac{1}{3}(9x-12y-18)=3x-4y-6$$

Algebra tiles

Bar diagram showing two groups of bars, each representing algebraic expressions.

Lesson Map

Topic A: Evaluating Numerical and Algebraic Expressions

Lesson

Evaluate numerical expressions with rational numbers using the order of operations.

Standards

7.EE.A.17.NS.A.3

Lesson

Write and evaluate expressions for mathematical and contextual situations.

Standards

7.EE.A.17.NS.A.3

Topic B: Generating Equivalent Expressions

Lesson

Expand and factor expressions using the distributive property and the greatest common factor.

Standards

7.EE.A.1

Lesson

Expand and factor expressions with negative rational numbers.

Standards

7.EE.A.1

Lesson

Add and simplify expressions by combining like terms.

Standards

7.EE.A.1

Lesson

Subtract and simplify expressions.

Standards

7.EE.A.1

Lesson

Simplify expressions by combining like terms and using the properties of operations (Part 1).

Standards

7.EE.A.1

Lesson

Simplify expressions by combining like terms and using the properties of operations (Part 2).

Standards

7.EE.A.1

Lesson

Solve real-world problems, using equivalent expressions to shed new meaning on a context.

Standards

7.EE.A.27.EE.B.37.NS.A.3

Topic C: Solving Multi-Step Problems using Expressions

Lesson

Model and solve multi-step real-world problems involving rational numbers.

Standards

7.EE.B.37.NS.A.3

Lesson

Model real-world problems involving rational numbers using reasoned estimates.

Standards

7.EE.B.37.NS.A.3

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Expressions and Equations

Expressions and Equations

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

7.EE.A.2 — Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."
Expressions and Equations

7.EE.B.3 — Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

The Number System

The Number System

7.NS.A.3 — Solve real-world and mathematical problems involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

Foundational Standards

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

Expressions and Equations

Expressions and Equations

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

The Number System

The Number System

7.NS.A.1 — Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
The Number System

7.NS.A.2 — Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Future Standards

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

Expressions and Equations

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.
Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.
Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

Geometry

Geometry

7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Geometry

7.G.B.5 — Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
Geometry

7.G.B.6 — Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

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.

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