Subject:
Algebra, Geometry
Material Type:
Lesson Plan
Level:
Middle School
Grade:
7
Provider:
Pearson
Tags:
7th Grade Mathematics, Inequalities, Problem Solving, Triangles
License:
Creative Commons Attribution Non-Commercial
Language:
English
Media Formats:
Text/HTML

Education Standards

Analyzing The Properties Of An Inequality

Analyzing The Properties Of An Inequality

Overview

Students write and solve inequalities in order to solve two problems. One of the problems is a real-world problem that involves selling a house and paying the real estate agent a commission. The second problem involves the relationship of the lengths of the sides of a triangle.

Key Concepts

In this lesson, students again use algebraic inequalities to solve word problems, including real-world situations. Students represent a quantity with a variable, write an inequality to solve the problem, use the properties of inequality to solve the inequality, express the solution in words, and make sure that the solution makes sense.

Students explore the relationships of the lengths of the sides of a triangle. They apply the knowledge that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side to solve for the lengths of sides of a triangle using inequalities. They solve the inequality for the length of the third side.

Goals and Learning Objectives

  • Use an algebraic inequality to solve problems, including real-world problems.
  • Use the properties of inequalities to solve an inequality.

Math Mission

Lesson Guide

Discuss the Math Mission. Students will analyze and solve problems by writing and solving inequalities.

Opening

Analyze and solve problems by writing and solving inequalities.

Selling Prices

Lesson Guide

Have students work in pairs. Make sure that students understand the task.

You may want to go over what commission is and when commission is paid to a real estate agent on the sale of house. For example, if a house sells for $100,000 and the commission is 7%, the real estate agent is paid a commission of 7% of $100,000, or $7,000.

Preparing for Ways of Thinking

Look for students who have difficulty with the Selling Prices activity. Determine whether the difficulty is due to unfamiliarity with the concept of commission, difficulty setting up an inequality, or difficulty solving an inequality. Make sure that these topics are covered during the Ways of Thinking discussion.

Mathematical Practices

Mathematical Practice 2: Reason abstractly and quantitatively.

Students look at a real-world situation and reason quantitatively to translate that situation into an algebraic inequality. They then work abstractly to solve the inequality, but then go back and reason quantitatively to see if their solution to the inequality does in fact match the real-world situation.

Mathematical Practice 4: Model with mathematics.

In this lesson, students use an inequality to model a real-world situation.

Interventions

Student does not know how to get started.

  • What do you know?
  • Is the fee to the real estate agent added or subtracted from the selling price of a house?

For the Selling Price problem, student does not know whether to use ≥ or ≤.

  • What does the seller want to have after he pays the fee to the real estate agent?

Student has an incorrect solution.

  • Does your answer make sense?
  • Have you checked your work?

Possible Answers

  • Let x equal the sales price of the house:
    x − 0.07x ≥ 150,000
  • If x is the sales price of the house, then 0.07x is 7% of the sales price. In order for the homeowner to make at least $150,000, there must be at least $150,000 left after paying the real estate agent 7% of the sales price. So, x − 0.07x is the amount of money that will be left after paying the agent, which means x − 0.07x must be at least $150,000: x −0.07x ≥ 150,000.
  • Inequality solution and explanation of steps

    x −0.07x ≥ 150,000

    0.93x ≥ 150,000                         Subtract.

    0.93x ÷ 0.93 ≥ 150,000 ÷ 0.93   Division property of inequality

    x ≥ 161,290.32                            Divide.

  • The homeowner needs to sell the house for $161,290.32 in order to make at least $150,000 after paying the real estate agent.
  • Yes, the solution makes sense in terms of the problem situation because 7% of $161,290.32 is $11,290.32. If the homeowner is able to sell his home for $161,290.32, he will pay his realtor the service fees of $11,290.32, and then he will receive $150,000 for the sale of his house.

Work Time

Selling Prices

A homeowner is selling his house. He has to pay 7% of the selling price to the real estate agent for her services. He wants to make at least $150,000 on the sale after he pays the agent.

What price (to the nearest dollar) does the homeowner need to sell his house for in order to make at least $150,000?

  • Write an inequality that represents the problem.
  • Explain how your inequality represents the problem. Which quantity in the problem did you represent with a variable?
  • Use your inequality to solve the problem, and explain each step you used to solve the inequality.
  • Express the solution in words.
  • Does your solution makes sense in terms of the problem situation?

Hint:

Think back to what you learned about percents earlier this year. What quantity do you need to find?

Triangles

Lesson Guide

Have students work in pairs. Make sure that students understand the task.

Preparing for Ways of Thinking

Look for students who have difficulty with the relationships of the sides of a triangle. You may need to review the relationships of the sides of a triangle with students. Use the example from the image to point out the relationships. In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Students explored this property in Unit 3.

ELL: Provide ELLs with materials and/or manipulatives and give them ample work time on problems. The use of manipulatives helps strengthen the mathematical connections students make and also helps make sense of more difficult, abstract concepts.

Interventions

Student does not know how to get started.

  • What do you know?

Student has difficulty setting up the inequalities for the Triangles problem.

  • What do you know about the lengths of the sides of a triangle?
  • What do you know about the side whose length is x?

Student has an incorrect solution.

  • Does your answer make sense?
  • Have you checked your work?

Possible Answers

  • x < 3 + 4x < 7
  • 4 < 3 + x1
  • 3 < 4 + x − 1
  • The variable x cannot be a negative number or 0. The length of a side must be > 0.
  • Answers will vary although we know that x cannot be negative or 0. In addition, based on the equalities above we know that 1 < x < 7.

Work Time

Triangles

Consider this triangle with side lengths of 3, 4, and x.

  • Write an inequality for the relationship of the side length x in terms of side lengths 4 and 3. Solve for x.
  • Write an inequality for the relationship of the side length 4 in terms of side lengths 3 and x. Solve for x.
  • Write an inequality for the relationship of the side length 3 in terms of side lengths 4 and x. Solve for x.
  • Are there any values for x that would not make sense for this triangle? Explain.
  • Write three possible values for x.

Hint:

  • Remember that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
  • This rule applies not only to the variable side length, but also to the sides of length 3 and 4.

Prepare a Presentation

Preparing for Ways of Thinking

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

  • Students who solved the Selling Prices or Triangles problems without setting up an inequality (have these students share their work first)
  • Students who solved the problems as an equation and then checked for the inequality
  • Students who checked their work (make sure several of these students present)
  • Students who solved the Selling Prices problem by setting up an inequality and solving it  (then choose another student who did the same with the Triangles problem)

Possible Answers

Presentations will vary. Presentations should include students’ analyses of the Selling Prices and Triangles problems.

Challenge Problem

Answers

  • It will take Angela 3 to 4 hours to visit her friend in college.
  • 3 hours ≤ time driven ≤ 4 hours

Work Time

Prepare a Presentation

Describe your approach for writing and solving inequalities for the Selling Prices and Triangles problems.

Challenge Problem

Marcus’s older sister Angela decides to visit her friend in college. The college is 120 miles away.

  • If Angela drives at speeds ranging from 30 to 40 miles per hour, how long will it take her to reach her destination?
  • Use inequalities to express your answer.

Make Connections

Mathematics

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

  • How did you determine what inequality to write to represent the situation?
  • Could you have written a different inequality to represent the situation?
  • Did you need to change the direction of the inequality sign?
  • Can you explain how you used the properties of inequality to solve the inequality?
  • Does the solution make sense? Why or why not?
  • In the Triangles problem, what is the relationship between the inequalities that you wrote?
  • What is true about the length of a side of a triangle?
  • What did you determine about the length of side x of the triangle? Why?

SWD: Rephrase for some students:
Which presentation was easy to understand? Why do you think that is the case?
How did (presenters) answer the question? What did they do to figure out the problem?

ELL: Be cognizant that ELLs may encounter difficulties when they have to express themselves in a foreign language. If you hear that they say the right things but use the wrong grammar structure, show signs of agreement and softly rephrase using the correct grammar. Use the student's words as much as possible.

Performance Task

Ways of Thinking: Make Connections

Take notes about your classmates’ approaches to writing and solving inequalities.

Hint:

As your classmates present, ask questions such as:

  • How did you determine which inequality to write to represent the situation?
  • Why did you change the direction of the inequality sign?
  • How did the problem situation affect the way you interpreted your solution?
  • In the triangle situation, what is the relationship between the inequalities you wrote?

Use Inequalities to Solve Problems

Lesson Guide

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

A Possible Summary

In this lesson, you again use an algebraic inequality to solve a word problem. The word problem gives you the information needed in order to write the inequality. Reasoning about whether the answer to the word problem makes sense helps you recognize when you have made a mistake in solving an inequality. You use the properties of inequalities to solve the inequality.

You set up the inequality for the selling price of the house to ensure the homeowner would make at least $150,000. The selling price of the house, x, minus the payment to the real estate agent needed to be greater than or equal to $150,000.

You set up the inequalities for the triangles problem to determine the possible lengths for side x. You need to consider that the length of a side of a triangle must be greater than 0.

Formative Assessment

Summary of the Math: Use Inequalities to Solve Problems

Write a summary about using inequalities to solve problems.

Hint:

Check your summary:

  • Do you explain how to write an inequality?
  • Do you explain why you sometimes need to reverse the inequality symbol when solving an inequality?
  • Do you discuss the importance of interpreting your solution in terms of the problem situation?

Reflect On Your Work

Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to determine where students lack understanding regarding solving 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.

Something I still do not understand about solving inequalities is …