Equations and Inequalities

Students solve equations and inequalities with rational numbers, and encounter real-world situations that can be modeled and solved using equations and inequalities.

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Unit Summary

In Unit 4, seventh-grade students continue to build on the last two units by solving equations and inequalities with rational numbers. They use familiar tape diagrams as a way to visually model situations in the form $$px+q=r$$ and $$p(x+q)=r$$. These tape diagrams offer a pathway to solving equations using arithmetic, which students compare to a different approach of solving equations algebraically. Throughout the unit, students encounter word problems and real-world situations, covering the full range of rational numbers, that can be modeled and solved using equations and inequalities (MP.4). As they work with equations and inequalities, they build on their abilities to abstract information with symbols and to interpret those symbols in context (MP.2). Students also practice solving equations throughout the unit, ensuring they are working towards fluency which is an expectation in 7th grade.

In sixth grade, students understood solving equations and inequalities as a process of finding the values that made the equation or inequality true. They wrote and solved equations in the forms $$x+p=q$$ and $$px=q$$, using nonnegative rational numbers. In seventh grade, students reach back to recall these concepts and skills in order to solve one- and two-step equations and inequalities with rational numbers including negatives.

In eighth grade, students explore complex multi-step equations; however, they will discover that these multi-step equations can be simplified into forms that are familiar to what they’ve seen in seventh grade. Eighth-grade students will also investigate situations that result in solutions such as 5 = 5 or 5 = 8, and they will extend their understanding of solution to include no solution and infinite solutions.

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 7th Grade Scope and Sequence Recommended Adjustments.

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Assessment

This assessment accompanies Unit 4 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

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

Model

Example

Tape diagram and equations

$$3(x+4)=45$$

$$3x+4=45$$

Essential Understandings

Equations and inequalities are powerful tools that can be used to model and solve real-world situations with unknown quantities.
Equations can be solved by reasoning about the arithmetic needed to uncover the value of the unknown. Equations can also be solved algebraically by using properties of operations and equality.
Inequalities have infinite solutions, which can be represented graphically on a number line. In context, these solutions are sometimes constrained by what makes sense for the situation; for example, if solving for the maximum number of people who can fit onto a boat, the solution set would be limited to positive integers.

Vocabulary

tape diagram

equation

solution

substitution

inequality

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

Lesson Map

Topic A: Solving and Modeling with Equations

1

7.EE.B.4.A

Solve one-step equations with rational numbers.

2

7.EE.B.4.A

Represent equations in the forms $${px+q=r}$$ and $${p(x+q)=r}$$ using tape diagrams.

3

7.EE.B.3

7.EE.B.4.A

Solve equations in the forms $${px+q=r}$$ and $${p(x+q)=r}$$ using tape diagrams.

4

7.EE.B.4.A

Solve equations in the forms $${px+q=r }$$ and $${p(x+q)=r}$$ algebraically.

5

7.EE.B.3

7.EE.B.4.A

Solve word problems leading to equations in the forms $${px+q=r}$$ and $${p(x+q)=r}$$ (Part 1).

6

7.EE.B.3

7.EE.B.4.A

Solve word problems leading to equations in the forms $${px+q=r}$$ and $${p(x+q)=r }$$ (Part 2).

7

7.EE.B.3

7.EE.B.4.A

Model with equations in the form $${px+q=r}$$ and $${p(x+q)=r}$$.

Topic B: Solving and Modeling with Inequalities

8

7.EE.B.4.B

Solve and graph one-step inequalities.

9

7.EE.B.4.B

Write and solve inequalities in the forms $${px+q>r}$$ or $${px+q<r}$$ and $${p(x+q)>r }$$ or $${p(x+q)<r.}$$

10

7.EE.B.4.B

Solve inequalities with negative coefficients.

11

7.EE.B.4.B

Solve word problems leading to inequalities in the forms $${px+q>r}$$ or $${px+q<r}$$ and $${p(x+q)>r}$$ or $${p(x+q)<r}$$.

12

7.EE.B.3

7.EE.B.4.B

Model with inequalities.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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.

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.
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

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.
7.EE.B.4.A — Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

Expressions and Equations

7.EE.B.4.A — Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4.B — Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Expressions and Equations

7.EE.B.4.B — Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Foundational Standards

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.

Future Standards

Expressions and Equations

8.EE.C.7

Expressions and Equations

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

Expressions and Equations

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

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.

Equations and Inequalities

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Internalization of Standards via the Post-Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Vocabulary

Lesson Map

Topic A: Solving and Modeling with Equations

1

2

3

4

5

6

7

Topic B: Solving and Modeling with Inequalities

8

9

10

11

12

Common Core Standards

Core Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Foundational Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Future Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Standards for Mathematical Practice

Unit 3

Numerical and Algebraic Expressions

Unit 5

Percent and Scaling