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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
Please Note: In June 2025, this unit and its lesson plans received a round of enhancements. Teachers should pay close attention as they intellectually prepare to account for the updated pacing and content.
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.
Unit Prep
Intellectual Prep
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 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
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
order of operations
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
| Model | Example |
| rectangular array | $$\frac{1}{3}(9x-12y-18)=3x-4y-6$$
|
| Algebra tiles | |
Lesson Map
Topic A: Evaluating Numerical and Algebraic Expressions
Topic B: Generating Equivalent Expressions
Topic C: Solving Multi-Step Problems using Expressions
Common Core Standards
Key
Major Cluster
Supporting Cluster
Additional Cluster
Core Standards
Expressions and Equations
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7.EE.A.1 — Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
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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."
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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
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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
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6.EE.A.2
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6.EE.A.2.C
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6.EE.A.3
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6.EE.A.4
The Number System
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7.NS.A.1
-
7.NS.A.2
Future Standards
Expressions and Equations
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7.EE.B.4
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8.EE.C.7
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8.EE.C.8
Geometry
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7.G.B.4
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7.G.B.5
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7.G.B.6
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.
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