- Author:
- Chris Adcock
- Subject:
- Algebra
- Material Type:
- Lesson Plan
- Level:
- Middle School
- Grade:
- 7
- Provider:
- Pearson
- Tags:

- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Text/HTML

# Earning at Least 200 Dollars

# Graphing an Inequality

# Strategies For Solving An Algebraic Inequality

## Overview

Students solve a problem about a salesperson's compensation. They solve the problem first by arithmetic and then by writing and solving an inequality.

# Key Concepts

In Lesson 11, students learned how to solve inequalities using the addition and multiplication properties of inequality. In this lesson, they solve word problems by writing and solving inequalities.

To help students make connections and see how problems can be solved in different ways, students first solve the same problem using arithmetic.

# Goals and Learning Objectives

- Write and solve an algebraic inequality to solve a word problem.

# How Many Sales?

# Lesson Guide

Have students work individually to read about Sophie’s mom. Then bring the class together to discuss the questions.

ELL: Allow and encourage the use of dictionaries or translation sites, if students need them to better understand the topic and word problems. Help ELLs to improve their use of academic vocabulary.

# Mathematics

Ask volunteers to discuss advantages and disadvantages of being paid a given amount and extra for each sale. Students may see that great salespersons may consider this an advantage, whereas salespeople who sell very little may consider this a disadvantage.

Have students share their solution strategy. Encourage solutions who use arithmetic at this time. The algebraic solution is discussed during Work Time. To calculate the number of sales Sophie’s mom needs to make to get exactly $200, subtract $100 from $200 and then divide the result by $5. With 20 sales, Sophie’s mom makes $200.

## Opening

# How Many Sales?

Sophie’s mom is a salesperson. She earns $100 per week, plus $5 for each sale that she makes.

- What are the advantages and disadvantages of this payment system? Do you think it is a fair system?
- Sophie’s mom wants to earn $200 this week. How can you determine the number of sales she needs to make?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will compare their solution to a problem solved with an equation and one solved with an inequality.

## Opening

Compare your solution to a problem solved with an equation and one solved with an inequality.

# Earn $200

# Lesson Guide

Have students work in pairs. Remind students to graph the solution using the interactive sketch. Also, make sure that students understand that they need to check their solution and determine if it makes sense.

# Mathematical Practices

**Mathematical Practice 4: Model with mathematics.**

Students model the same real-world problem in two different ways: first, by using arithmetic, and then by writing and solving an algebraic equation. Students model the answer to the problem using both an algebraic equation and a sentence. Students then model a different yet similar real-world situation that involves an inequality.

# Interventions

**Student does not know how to begin.**

- What is the problem asking you to find?
- What do you know?
- What does the variable represent?

**Student does not know how to begin writing a sentence to find the solution to the problem.**

- Should you set up an equation or an inequality? Why?

**Student has a solution.**

- Explain your strategy for solving the problem.
- How did you know how to graph the solution?
- Did you use an open circle or a closed circle? Why?

**Student has an incorrect solution.**

- Does your answer make sense?
- Have you checked your work?
- Do the points that you graphed make the equation or inequality true?

# Possible Answer

- 100 + 5
*x*= 200 Equation solution:

$\begin{array}{c}100+5x=200\\ 100-100+5x=200-100\\ 5x=100\\ \frac{1}{5}\cdot 5x=\frac{1}{5}\cdot 100\\ x=20\end{array}$- For Sophie's mom to earn $200. she must make 20 sales. This solution makes sense because 20 times $5 is $100, and if you add that to the $100 she makes for working that week, you get $200.

## Work Time

# Earn $200

Sophie’s mom earns $100 per week, plus $5 for each sale that she makes. She wants to earn $200 per week. How can you find out the number of sales she needs to make each week?

Let *x* = the number of sales.

- Write an equation that shows the number of sales Sophie’s mom needs to make in order to earn $200.
- Solve the equation.
- Graph the equation on the number line.
- Check that your solution makes sense in terms of the problem situation.

HANDOUT: Earning 200 Dollars

INTERACTIVE: Graphing an Inequality

# Earn at Least $200

# Lesson Guide

Have students work in pairs. Continue as you did with Task 3 using the same Interventions as needed.

# Interventions

**Student does not know how to begin.**

- What is the problem asking you to find?
- What do you know?
- What does the variable represent?

**Student does not know how to begin writing a sentence to find the solution to the problem.**

- Should you set up an equation or an inequality? Why?

**Student has a solution.**

- Explain your strategy for solving the problem.
- How did you know how to graph the solution?
- Did you use an open circle or a closed circle? Why?

**Student has an incorrect solution.**

- Does your answer make sense?
- Have you checked your work?
- Do the points that you graphed make the equation or inequality true?

# Possible Answer

- 100 + 5
*x*≥ 200 Inequality solution:

$\begin{array}{c}100+5x\ge 200\\ 100-100+5x\ge 200-100\\ 5x\ge 100\\ \frac{1}{5}\cdot 5x\ge \frac{1}{5}\cdot 100\\ x\ge 20\end{array}$- For Sophie's mom to make at least $200, she must make at least 20 sales; this solution makes sense because 20 or any number greater than 20, when multiplied by $5 and added to $100, will allow Sophie's mom to make at least $200 for this week of work.

## Work Time

# Earn at Least $200

Sophie’s mom earns $100 per week, plus $5 for each sale that she makes. She wants to earn *at least* $200 per week. How can you find out the number of sales she needs to make each week?

Let *x* = the number of sales.

- Write an inequality that shows the number of sales Sophie’s mom needs to make in order to earn
*at least*$200. - Solve the inequality.
- Graph the inequality on the number line.
- Check that your solution makes sense in terms of the problem situation.

HANDOUT: Earning at Least 200 Dollars

INTERACTIVE: Graphing an Inequality

## Hint:

- Should you use
*<_ or_>*in your inequality? Think about the question. - The number of sales Sophie’s mom needs to make in order to earn
*at least*$200 is not just one number. - Use the addition and multiplication properties of inequality to solve the inequality.

# Prepare a Presentation

# Preparing for Ways of Thinking

Look for these types of responses to share during the Ways of Thinking discussion:

- Student pairs with both correct and incorrect solutions
- Students who set up the problems correctly with the equality sign and the inequality sign
- Students who may use the wrong inequality sign or who may reverse the sign when it is unnecessary
- Students who solve the equality or inequality in different ways; for example, do they add −100 to both sides, or do they subtract 100 from both sides? Do they multiply each side by $\frac{1}{5}$, or do they divide both sides by 5?
- Students who attempt the Challenge Problem

SWD: Students with disabilities may use an incorrect operation. Ask the students to read the expression to you; see if they correct the misconception. Pay special attention to their use of the inequality sign.

# Challenge Problem

## Answer

- Let
*x*equal the number of sales per week so that Plan A is the better option.

200 + 6*x* > 250 + 5*x*

200 − 200 + 6*x* > 250 − 200 + 5*x*

6*x* > 50 + 5*x*

6*x *− 5*x* > 50 + 5*x *− 5*x*

*x* > 50

When Sophie’s mom completes more than 50 sales per week, Plan A is the better option.

- For 50 sales, the amount earned will be equal for both plans; for more than 50 sales, Plan A is the better option; and for fewer than 50 sales, Plan B is the better option.

250 + 5*x* > 200 + 6*x*

250 − 200 + 5*x* > 200 − 200 + 6*x*

50 + 5*x* > 6*x*

50 + 5*x* − 5*x* > 6*x *− 5*x*

50 > *x,* or *x* < 50

## Work Time

# Prepare a Presentation

Compare your solution about how Sophie’s mom could earn *at least* $200 in one week to your discussion at the start of the lesson about how Sophie’s mom could earn exactly $200.

- What is different about the solutions?
- Show your work for your solution to the inequality.

# Challenge Problem

Sophie’s mom is offered a raise. She can choose from two plans:

Plan A: $200 per week plus $6 per sale

Plan B: $250 per week plus $5 per sale

- For what number of sales per week would Plan A be the better option?
- For what number of sales per week would Plan B be the better option?

# Make Connections

# Mathematics

Facilitate the discussion to help students understand the mathematics of the lesson informally. Ask questions such as the following:

- Did you use >, <, ≥, or ≤ in your inequality? Why?
- How many values are there for
*x*so that Sophie's mom earns $200? - How many values are there for
*x*so that Sophie's mom earns at least $200? - How did you use the addition and multiplication properties of equality to solve the equation?
- How did you use the addition and multiplication properties of inequality to solve the inequality?
- In solving the inequality, did you reverse the inequality sign? Why?
- How do you graph the solution of the equation?
- How do you graph the solution of the inequality? Did you use a closed circle or an open circle in your graph? Why?
- What did you notice about the inequality used to solve the Challenge Problem?

SWD: When participating in a class discussion, Ways of Thinking can be intimidating for students with language-based learning vulnerabilities and/or learning challenges. Have students work on the speaking and listening skills implicit to this portion of the lesson. Supports for students during this portion of the lesson include:

- In small groups or with partners, give students a few minutes to discuss their ideas, the questions posed, and what has taken place during the lesson.
- Conference with individual students prior to the discussion to ascertain what they might be able to successfully contribute to the discussion. Students should rehearse their contribution and/or write notes for reference when they speak. This will support students with expressive language difficulties and/or students who are anxious or reluctant to participate in class discussions.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about the inequalities your classmates wrote and the solutions they found.

## Hint:

As your classmates present, ask questions such as:

- Why isn't the answer just “20 sales”?
- How did you decide whether to use the “greater than or equal to” symbol versus the “greater than” symbol?
- When you multiplied both sides of the inequality by 1.5, why didn't you reverse the inequality sign?
- Can you explain how you used the addition and multiplication properties of inequality to find the solution?
- How did you decide whether to fill in the circle on the number line?

# Inequalities and Real-World Problems

# Lesson Guide

Have each student write a summary of the math in this lesson, then write a class summary. When done, if you think the summary is helpful, share it with the class.

SWD: At this point in the course, students should be comfortable in presenting. Extend student responses by using prompts and targeted questions that are appropriate to the student's ability.

ELL: When writing the summary, provide ELLs access to a dictionary and give them time to discuss their summary with a partner before writing, to help them organize their thoughts. Allow ELLs who share the same primary language to discuss in their language of choice.

# A Possible Summary

You can solve a problem that is a real-world situation by using an equation or an inequality. Whether you use an equation or an inequality depends on the problem.

In the first problem, you need to find out how many sales Sophie's mom must make in order to earn $200. The steps were: set up the equation, solve the equation, graph the solution, and check the answer to make sure that the equation is true with this value for the variable.

In the second problem, you need to find out how many sales Sophie's mom must make in order to earn at least $200. This problem needed an inequality. The steps were: set up the inequality, solve the inequality using the addition and multiplication properties of inequality, graph the solution, and check the answer to make sure that it is reasonable. Many numbers are part of the solution and can be checked. Since the problem asked for Sophie's mom to earn at least $200, the graph of the solution used a closed circle.

## Formative Assessment

# Summary of the Math: Inequalities and Real-World Problems

Write a summary about using inequalities to solve real-world problems.

## Hint:

Check your summary:

- Do you explain why there is more than one solution for an inequality?
- Do you explain how to represent an inequality on a number line?
- Do you explain how to check your solutions?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to be aware of the strategies that students used for working with inequalities.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**A strategy that I found to be useful when working with inequalities is …**