Author:
Chris Adcock
Subject:
Algebra
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Provider:
Pearson
Tags:
6th Grade Mathematics, Equations, Inequalities
License:
Creative Commons Attribution Non-Commercial
Language:
English
Media Formats:
Interactive, Text/HTML

Education Standards

Substituting A Given Value For A Variable

Substituting A Given Value For A Variable

Overview

Lesson Overview

Using a balance scale, students decide whether a certain value of a variable makes a given equation or inequality true. Then students extend what they learned using the balance scale to substituting a given value for a variable into an equation or inequality to decide if that value makes the equation or inequality true or false.

Key Concepts

Students will extend what they know about substituting a value for a variable into an expression to evaluate that expression.

Equations and inequalities may contain variables. These equations or inequalities are neither true nor false. When a value is substituted for a variable, the equation or inequality then becomes true or false. If the equation or inequality is true for that value of the variable, that value is considered a solution to the equation or inequality.

Goals and Learning Objectives

  • Understand what solving an equation or inequality means.
  • Use substitution to determine whether a given number makes an equation or inequality true.

Variables

Lesson Guide

Present the situation and explain the task as needed.

Mathematics

If necessary, explain how the left side of the balance scale represents + 5 and how the right side represents 2x + 3.

Students should understand that they remove each x from the scale and replace it with a
5-weight.

Ask students: Is it possible for more than one of the statements to be true when = 5?

  • The equation + 5 = 2+ 3 is false.
  • The inequality + 5 > 2x + 3 is false.
  • Statement + 5 < 2+ 3 is true.

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 them make sense of more difficult, abstract concepts. Make sure you make connections between the balanced/unbalanced scales and the equations/inequalities.

Opening

Variables

Equations and inequalities may contain one or more variables. When you substitute a value for a variable in an equation or inequality, the equation or inequality becomes either true or false.

Look at the balance scale in the interactive. It shows the expression x + 5 on the left and the expression 2x + 3 on the right.

  • Substitute 5 for each x in the expressions. What happens when = 5?
    • Is the equation + 5 = 2+ 3 true?
    • Is the inequality + 5 > 2+ 3 true?
    • Is the inequality + 5 < 2+ 3 true?

INTERACTIVE: Balance Scale X

Math Mission

Lesson Guide

Discuss the Math Mission. Students will determine whether a given number is a solution to an equation or to an inequality.

Opening

Determine whether a given number is a solution to an equation or to an inequality.

Solution or Not?

Lesson Guide

Have students work in pairs on the problems in Tasks 3, 4, and 5, and the presentation in Task 6. Check that all problems will be represented during the presentations. Assign pairs problems for the presentations if needed.

NOTE: Tell students to ignore the instructions in the interactive and follow the steps in each problem.

ELL: When forming partnerships, consider the task at hand. Since the work is less language dependent and more computational, consider pairing students together based on their math comprehension, rather than their language skills.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Look for students who understand how the situation on the balance scale shows an equation or inequality.

Mathematical Practice 6: Attend to precision.

Listen for students who use precise mathematical language: evaluate an expression for a value of the variable, a solution to the equation, a solution to the inequality.

Interventions

Student does not see the connection between the balance scale and the equation or inequality.

  • What is shown on the left side of the balance?
  • What is shown on the right side of the balance?
  • Is the scale balanced or unbalanced?
  • Which side of the balance is heavier?
  • Which side of the equation is greater?

Student replaces only one x with the value.

  • Are there any other instances of x on the scale?
  • What should you replace that x with?

Answers

  • Yes, 2 is a solution to 2x + 5 = x + 7.
  • The scale is balanced.

Work Time

Solution or Not?

When a given number is substituted for a variable in an equation or an inequality, and the resulting equation or inequality is true, that number is a solution to the equation or inequality.

NOTE: Ignore the instructions on the interactive and follow the steps below.

  • Is 2 a solution to 2x + 5 = x + 7?
  • Check your answer by substituting 2 for each x on the balance scale. Does the scale become balanced?

INTERACTIVE: Balance Scale X

Match the Solution

Lesson Guide

Have students work in pairs on the problems in Tasks 3, 4, and 5, and the presentation in Task 6. Check that all problems will be represented during the presentations. Assign pairs problems for the presentations if needed.

NOTE: Tell students to ignore the instructions in the interactive and follow the steps for each problem.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Look for students who understand how the situation on the balance scale shows an equation or inequality.

Mathematical Practice 6: Attend to precision.

Listen for students who use precise mathematical language: evaluate an expression for a value of the variable, a solution to the equation, a solution to the inequality.

Mathematical Practice 7: Look for and make use of structure.

Identify students who recognize in the second problem that only one of the statements can be true for a given value of the variable.

Interventions

[common error] For the second problem, student represents x + x + x as x + 3 on the balance scale.

  • What is another way to write x + x + x?
  • How do you show x + x + x on the scale?

Student does not see the connection between the balance scale and the equation or inequality.

  • What is shown on the left side of the balance?
  • What is shown on the right side of the balance?
  • Is the scale balanced or unbalanced?
  • Which side of the balance is heavier?
  • Which side of the equation is greater?

Student does not understand that when an inequality is true, the sides of the balance will not be equal.

  • Is the inequality 7 > 3 true?
  • Are the sides balanced?

Student replaces only one x with the value.

  • Are there any other instances of x on the scale?
  • What should you replace that x with?

Answers

  • 4 is a solution to the inequality + 5 < 3+ 1.
  • When you substitute 4 for x:

     + 5 < 3 + 1
    4 + 5 <  3(4) + 1
    9 < 13

 

 

Work Time

Match the Solution

  • For which equation or inequality is x = 4 a solution?

x + 5 = 3x + 1

+ 5 > 3+ 1

+ 5 < 3+ 1

  • Check your answer by using the Balance Scale X interactive. (Ignore the instructions on the interactive and follow these steps.) Represent the expression x + 5 on the left side and the expression 3+ 1 on the right side. Then subsitute 4 for each x on the scale.

INTERACTIVE: Balance Scale X

Solution of 2

Lesson Guide

Have students work in pairs on the problems in Tasks 3, 4, and 5, and the presentation in Task 6. Check that all problems will be represented during the presentations. Assign pairs problems for the presentations if needed.

NOTE: Tell students to ignore the instructions in the interactive and follow the steps in the problems.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Look for students who understand how the situation on the balance scale shows an equation or inequality.

Mathematical Practice 6: Attend to precision.

Listen for students who use precise mathematical language: evaluate an expression for a value of the variable, a solution to the equation, a solution to the inequality.

Interventions

Student does not see the connection between the balance scale and the equation or inequality.

  • What is shown on the left side of the balance?
  • What is shown on the right side of the balance?
  • Is the scale balanced or unbalanced?
  • Which side of the balance is heavier?
  • Which side of the equation is greater?

Student does not understand that when an inequality is true, the sides of the balance will not be equal.

  • Is the inequality 7 > 3 true?
  • Are the sides balanced?

Student replaces only one x with the value.

  • Are there any other instances of  on the scale?
  • What should you replace that  with?

Answers

  • Answers will vary. Possible answer: 4 =  + 6
  • When you substitute 2 for x:

    4 =  + 6
    4(2) = 2 + 6
    8 = 8
     
  • The scale representing the above combination would have four weights weighing 2 on the left of the scale. And on the right of the scale there would one weight weighing 2 and one weight weighing 6.

 

Work Time

Solution of 2

  • Make an equation that contains five instances of and has the solution 2. Set up the equation so the instances of are not all on the same side.
  • Check your answer by substituting 2 for each in your equation.
  • Check your answer by setting up a balance scale that represents your equation and substituting 2 for each x

INTERACTIVE: Balance Scale X

Prepare a Presentation

Preparing for Ways of Thinking

As students work, look for students who:

  • Understand that a solution to an equation means the scale is balanced.
  • Understand that a solution to an inequality means that the inequality statement is true for the given value of the variable and is represented with an unbalanced scale.

  • [common error] Think that a solution to an inequality means that the scale is balanced.

Answers

  • Presentations will vary.

Challenge Problem

Answers

  • Answers will vary.
  • Any number greater than 6 is a solution.

Work Time

Prepare a Presentation

  • Select one of the Work Time problems.
  • Be prepared to justify the solution to the equation or inequality.

Challenge Problem

  • Use the balance scale to find as many solutions as you can to the inequality x + 7 <  2x + 1.
  • What can you conclude about the solutions to this inequality?

INTERACTIVE: Balance Scale X

Make Connections

Lesson Guide

Have student pairs give their presentations. In Work Time, students used the balance scale to replace the x-weights with a given weight, and then used substitution to check their work. For each problem, make sure you review the steps used to evaluate an expression using substitution as outlined below.

Task 3: Substitute 2 for each x.

2x+5=x+722+5=2+74+5=2+79=9

As students show this process, check that they are evaluating each expression correctly and using precise language to describe what they are doing.

Since the two sides are equal, the equation 2x + 5 = x + 7 is true when = 2. It is important that students understand that this does not mean that the equation 2+ 5 = + 7 is always true. It means that it is true when = 2.

 

Task 4: Substitute 4 for each x.

x+5<3x+14+5<34+19<12+19<13

Since 9 < 13, students should know that this means that the inequality + 5 < 3+ 1 is true when = 4. (Another way to say this is that 4 is a solution to the inequality + 5 < 3+ 1.) Students should reason that if + 5 < 3+ 1 is true when = 4, then the equation + 5 = 3x+ 1 and the inequality + 5 > 3+ 1 will be false when = 4. It is important that students understand that this does not mean that the inequality + 5 < 3+ 1 is always true. It means that it is true when = 4.

Conduct a similar discussion for Task 5. Students' problems will vary. Here is a sample:

Task 5: Substitute 2 for each x.

4x=x+642=2+68=8

Again, as students show this process, check that they are evaluating each expression correctly and using precise language to describe what they are doing. Students will now be ready to use a similar process to decide which numbers are solutions to the equations and inequalities in the Apply the Learning section.

SWD: Some students may struggle to determine the salient information from the presentations. Provide students with summaries to reinforce the key ideas from the presentations and to model how to create strong summaries of mathematical information.

 

Performance Task

Ways of Thinking: Make Connections

Take notes about the ways that your classmates justified the solution to the equation or inequality they chose.

Find Solutions

Lesson Guide

Have students who struggle with rational numbers work with students who have mastered these concepts.

Mathematics

Encourage students to share any shortcuts they may have used. For example:

  • For + 2 = 7, students may see immediately that 5 is the only solution.
  • For + 6 = 6 + x, students may see that because of the commutative property of addition, any number will be a solution to this equation.

Answers

  1. 5
  2. 8
  3. 23, 5, 8
  4. 3, 6.2, 512
  5. 12

Work Time

Find Solutions

Identify the solution(s) for each equation or inequality from the list provided. There may be more than one solution.

  1. + 2 = 7

    • 2          
    • 212
    • 5
  2. > 5

    • 4          
    • 8          
    • 4.5
  3. + 6 = 6 + x

    • 23          
    • 5          
    • 8
  4. 2x

    • 3          
    • 6.2          
    • 512
  5. + 4 + 5x < 8 + 2x

    • 5          
    • 9          
    • 12

Solving With Substitution

A Possible Summary

To find out if an equation or inequality is true or false for a given value of a variable, you substitute the given value for the variable to get an equation or inequality with just numbers, and decide whether it is true or false.

ELL: Focus on and teach mathematical language. It is important that all students make the connection between the mathematics and the language. Point out that the word solution can have a different meaning in a science or English class.

SWD: Some students may struggle with the task of writing a summary of the math from the lesson. Possible supports:

  • Prior to writing the summary, have students discuss their ideas with a partner or adult and rehearse what they might write.
  • Allow students to map out their ideas in outline form or in a concept web.

Additional Discussion Points

When an equation or inequality has only numbers and no variables, it is either true or false: 3 = 2 + 1 is true and 7 = 4 + 1 is false; 4 > 3 is true and 3 > 4 is false. But when an equation or inequality contains a variable, it is true or false depending on the value of the variable. So, the equation + 6 = 15 is true when = 9, but false when = 5. The inequality + 6 < 10 is true when = 3, but false when = 5.

A value that makes an equation or inequality true is called a solution to the equation or inequality.

Formative Assessment

Summary of the Math: Solving With Substitution

Write a summary about solutions to equations and inequalities.

Reflect On Your Work

Lesson Guide

Have each student write a quick reflection before the end of the class. Review the reflections to find out if students understand that an equation with a variable can be either true or false depending on what value is substituted for the variable.

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.

An equation can either be true or false because …