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.

Unit Summary

In Unit 3, seventh-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 (Standard for Mathematical Practice 7). Students also face authentic real-world, multi-step problems that require strategic use of rational numbers and estimation where appropriate. 

In sixth 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 sixth grade to those with rational numbers such as $${-(a+b)-\frac{3}{2}(a-b)}$$ in the seventh grade.

The next seventh-grade unit, Unit 4, Equations and Inequalities, will continue to engage students in working with expressions with rational numbers. In eighth 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)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 7th Grade Scope and Sequence Recommended Adjustments.

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This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep


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

Model Example
Rectangular array


Essential Understandings


  • The order of operations is a guide to understanding how an expression is organized and how it can be evaluated.
  • Expressions can be expanded, factored, added, and subtracted using the properties of operations and the rules of operating with rational numbers. 
  • Expressions can be rewritten into equivalent, often simpler forms that can be interpreted in new and different ways. 



order of operations

expand an expression

factor an expression

combine like terms


commutative property

distributive property

numerical expression

algebraic expression

greatest common factor (gcf)

constant term

To see all the vocabulary for this course, view our 7th Grade Vocabulary Glossary.

Lesson Map

Topic A: Evaluating Numerical and Algebraic Expressions




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




Write and evaluate expressions for mathematical and contextual situations.

Topic B: Generating Equivalent Expressions



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



Expand and factor expressions with negative rational numbers.



Add and simplify expressions by combining like terms.



Subtract and simplify expressions.



Write and interpret expressions in different ways to shed new meaning on a context.

Topic C: Solving Multi-Step Problems using Expressions




Solve multi-step real-world problems with rational numbers.




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

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards


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.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."

  • 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
  • 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


Expressions and Equations
  • 6.EE.A.2

  • 6.EE.A.2.C

  • 6.EE.A.3

  • 6.EE.A.4

The Number System
  • 7.NS.A.1

  • 7.NS.A.2

Future Standards


Expressions and Equations
  • 7.EE.B.4

  • 8.EE.C.7

  • 8.EE.C.8

  • 7.G.B.4

  • 7.G.B.5

  • 7.G.B.6

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.