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Operations with Rational Numbers
Students extend the operations of addition, subtraction, multiplication, and division to include positive and negative rational numbers, and build fluency with evaluating numerical expressions.
Math
Unit 2
7th Grade
Unit Summary
Please Note: In May 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 2, 7th grade students extend the operations of addition, subtraction, multiplication, and division to include positive and negative rational numbers. Standards 7.NS.1, 7.NS.2, and 7.NS.3 represent a culmination in students’ work involving all four operations with rational numbers. In this unit, students model addition and subtraction on the number line, and through repeated reasoning and application of properties of operations, they determine efficient rules for computing with rational numbers MP.8 Look for and express regularity in repeated reasoning. . Students gain the ability to model a greater scope of real-world contexts to include situations involving elevations, temperature changes, debts and credits, and proportional relationships with negative rates of change MP.4 Model with mathematics. . They also develop greater fluency with evaluating numerical expressions, using the properties of operations to increase their flexibility in approach.
Starting in 1st grade, students learn about the commutative and associative properties of addition, and the relationship between addition and subtraction. In 3rd grade, students extend their understanding of the properties of operations to include multiplication and the distributive property. Throughout the years, students have applied these properties and relationships between the operations to whole numbers, fractions, and decimals. In 7th grade, all of these skills and concepts come together as students now operate with all rational numbers, including negative numbers.
7th-grade students will rely on their increased number sense and ability to compute with rational numbers in future units, particularly in Unit 3 and Unit 4. By the time students enter 8th grade, students should have a strong grasp on operating with rational numbers, which will be an underlying skill in many algebraic concepts. In 8th grade, students are introduced to irrational numbers, rounding out their understanding of the real number system before learning about complex numbers in high school.
Included in the materials for this unit are some activities that aim to support and build students’ fluency with integer computations, especially mental math. See our A Guide to Procedural Skill and Fluency for additional information and strategy and activity suggestions.
Pacing: 22 instructional days (18 lessons, 3 flex days, 1 assessment day)
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Assessment
The following assessments accompany Unit 2.
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 10.
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 Progressions for the Common Core Standards in Mathematics, The Number System, 6-8 for the relevant standards within the Number System domain, pages 197-210.
Essential Understandings
- The properties of operations for addition, subtraction, multiplication, and division hold true for all rational numbers.
- In the equation $${p+q=r}$$, where $$p$$, $$q$$, and $$r$$ are rational numbers, $$\left | q \right |$$ represents the distance between $$p$$ and $$r$$, and is also represented as $$\left | r-p \right |$$.
- A rational number and its opposite have a sum of zero, which helps define subtraction as adding the opposite, or additive inverse.
- The quotient or product of two numbers with the same sign is positive, and the quotient or product of two numbers with different signs is negative.
- Every quotient of integers with a non-zero divisor is a rational number and can be represented as a fraction $$\frac{a}{b}$$ or a decimal that eventually terminates in 0s or repeats.
Materials
Calculators (1 per student)
Graph Paper (1 sheet per student)
Dry erase marker (1 per student)
Colored Pencils (1 set per student group)
Optional: Laminated number line (1 per student)
Optional: Game piece or token (1 per student)
Number cards (1 set per student) –These are used in Lesson 2 and require some preparation.
Optional: Standard deck of playing cards (1 per student or small group)
Integer chips –Or counters, or access to a digital integer chip manipulative
Optional: Painter's tape
Vocabulary and Models
Unit Vocabulary
additive inverse
multiplicative inverse
rational number
repeating decimal
terminating decimal
Foundational Vocabulary
absolute value
associative property
commutative property
distributive property
opposite
reciprocal
To see all the vocabulary for Unit 2, view our 7th Grade Vocabulary Glossary.
Access foundational vocabulary for Unit 2 in the same 7th Grade Vocabulary Glossary.
Models
| Model | Example |
| number line | The number lines below show the addition of the integers -7 and 4. |
| integer chips | The integer chips below show –6 + 4 = –2. You start with 6 red chips representing –6 and add in 4 yellow chips to represent the positive 4. Four zero pairs can be created, leaving 2 red chips remaining for a sum of –2. |
Lesson Map
Topic A: Adding and Subtracting Rational Numbers
Topic B: Multiplying and Dividing Rational Numbers
Topic C: Using all Four Operations with Rational Numbers
Common Core Standards
Key
Major Cluster
Supporting Cluster
Additional Cluster
Core Standards
The Number System
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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.
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7.NS.A.1.A — Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
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7.NS.A.1.B — Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
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7.NS.A.1.C — Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
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7.NS.A.1.D — Apply properties of operations as strategies to add and subtract rational numbers.
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7.NS.A.2 — Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
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7.NS.A.2.A — Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
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7.NS.A.2.B — Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
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7.NS.A.2.C — Apply properties of operations as strategies to multiply and divide rational numbers.
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7.NS.A.2.D — Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
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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
Number and Operations in Base Ten
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4.NBT.B.5
Number and Operations—Fractions
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4.NF.B.3
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5.NF.A.1
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5.NF.A.2
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5.NF.B.3
Operations and Algebraic Thinking
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1.OA.B.3
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3.OA.B.5
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3.OA.B.6
The Number System
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6.NS.A.1
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6.NS.B.2
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6.NS.B.3
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6.NS.C.5
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6.NS.C.6
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6.NS.C.7
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6.NS.C.7.C
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6.NS.C.8
Future Standards
High School — Number and Quantity
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N.RN.A.1
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N.RN.A.2
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N.RN.B.3
The Number System
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8.NS.A.1
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8.NS.A.2
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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