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.

Math

Unit 4

7th Grade

Unit Summary

In Unit 4, 7th grade students continue to build on the last two units by solving equations and inequalities with rational numbers. They use models such as tape diagrams as a way to visually model situations in the form $$px+q=r$$ and $$p(x+q)=r$$. These models offer a pathway to solving equations using arithmetic, which students then compare to a different approach of solving equations algebraically. They work with the full range of rational numbers across all operations, building on their work from the previous units in 7th grade. Students encounter word problems and real-world situations that can be modeled and solved using equations and inequalities MP.4 Model with mathematics. . 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 Reason abstractly and quantitatively. . Students also practice solving equations throughout the unit, ensuring they are working towards fluency which is an expectation in 7th grade.

In 6th 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 7th 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 8th 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 7th grade. In 8th grade, students will also investigate situations that result in solutions such as 5 = 5 or 5 = 8, and they will extend their understanding of solutions to include no solution and infinite solutions. 8th grade students work with writing, simplifying, and interpreting functions and equations in two variables that involve rates of change and constants. Finally, 8th grade students work with simplifying and solving systems of equations in two variables using various strategies.

Pacing: 16 instructional days (12 lessons, 3 flex days, 1 assessment day)

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Assessment

The following assessments accompany Unit 4.

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 4.

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 the Progressions for the Common Core State Standards in Mathematics, 6-8, starting on page 171 for relevant standards in the Expressions and Equations domain.

Essential Understandings

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

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.

Materials

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

Algebra tiles (one set per student group)

Graph Paper

Laminated number line

Calculators

Optional: Toothpicks

Optional: Poster paper

Optional: Markers

Optional: Plastic cup

Optional: Counters

Optional: Two-pan balances

Optional: Digital manipulatives –Students may access digital manipulatives on a computer or tablet.

Vocabulary and Models

Unit Vocabulary

inverse operations

properties of equality

Foundational Vocabulary

equation

inequality

reciprocal

solution

substitution

To see all the vocabulary for Unit 4, view our 7th Grade Vocabulary Glossary.

Access foundational vocabulary for Unit 4 in the same 7th Grade Vocabulary Glossary.

Models

Models and representations that students learn or use in the unit

Model	Example
tape diagram and equations	$$3(x+4)=45$$ $Rectangular diagram with three equal parts, each labeled $x + 4$, totaling 45.$ $$3x+4=45$$
Algebra tiles
number lines with inequalities

Lesson Map

Topic A: Solving and Modeling with Equations

Lesson

Solve one-step equations with rational numbers.

Standards

7.EE.B.4.A

Lesson

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

Standards

7.EE.B.4.A

Lesson

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

Standards

7.EE.B.37.EE.B.4.A

Lesson

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

Standards

7.EE.B.4.A

Lesson

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

Standards

7.EE.B.37.EE.B.4.A

Lesson

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

Standards

7.EE.B.37.EE.B.4.A

Lesson

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

Standards

7.EE.B.37.EE.B.4.A

Topic B: Solving and Modeling with Inequalities

Lesson

Solve and graph one-step inequalities.

Standards

7.EE.B.4.B

Lesson

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

Standards

7.EE.B.4.B

Lesson

Predict and check solutions for inequalities with negative coefficients.

Standards

7.EE.B.4.B

Lesson

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

Standards

7.EE.B.4.B

Lesson

Model with inequalities.

Standards

7.EE.B.37.EE.B.4.B

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.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.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.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.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

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

Expressions and Equations

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.
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.
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

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

Expressions and Equations

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.

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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