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

Education Standards

Additional Properties of Equality

Additional Properties of Equality

Overview

Lesson Overview

Students apply the addition property of equality to solve equations. They are introduced to this property using a balance scale.

Key Concepts

Up until this lesson, students have been solving equations informally. They used guess and check and reasoned about the quantities on either side of the equation in order to solve the equation.

In this lesson, students are introduced to the addition property of equality. As equations become more and more complicated, students will need to rely on formal methods for solving them. This property states that the same quantity can be added to both sides of an equation and the new equation will be equivalent to the original equation. That means the new equation will have the same solutions as the original equation.

To solve an equation such as + 6 = 15, –6 can be added to both sides to get the resulting equation x = 9. However, since adding a negative number has not been introduced yet, students will consider both adding and subtracting a number (which is the equivalent of adding a negative number) from both sides to be an application of the addition property of equality.

Students will apply the addition property of equality to an equation with the goal of getting the variable alone on one side of the equation and a number on the other.

Goals and Learning Objectives

  • Use the addition property of equality to keep a scale balanced.
  • Use the addition property of equality to solve equations of the form + = q for cases in which p, q, and x are all non-negative rational numbers.

Keep Balanced

Lesson Guide

Have students work with a partner to answer the two questions using the Balance Scale interactive.

 

Opening

Keep Balanced

  • Look at the balance scale in the Balance Scale interactive. 
  • Can you add a number to each side of the scale but still keep the scale balanced? Try it.
  • Based on your new scale, can you subtract a number from each side but still keep the scale balanced? Try it.

INTERACTIVE: Balance Scale

Find the Solution

Lesson Guide

Have pairs work on writing and solving the equation x + 3 = 5 using the Balance Scale X interactive. NOTE: On the board, show students how to set up this equation on the balance scale in the interactive beforehand. However, do not tell them what the equation is. Have them ignore any instructions in the interactive.

Remind students to follow the directions in the order they are given. Watch for students who move the weights on the scale before writing the equation to represent the scale. 

Mathematics

The whole-group discussion that follows should result in students' understanding that the scale will remain balanced if the same number (any number) is added to both sides of the equation or the same number (any number) is subtracted from both sides of the equation. This is known as the addition property of equality.

If 3 is subtracted from both sides of the scale, then x remains on the left and 2 remains on the right and the scale is still balanced. So, + 3 = 5 is equivalent to = 2 and the solution to both equations is 2.

Opening

Find the Solution

Set up the Balance Scale X interactive as your teacher instructs you. Ignore any instructions in the interactive. Then complete the following.

  • Write an equation that represents the balance scale.
  • Can you add or subtract something from both sides of the balance scale to find the value of x that makes the equation true? Experiment with the Balance Scale X interactive to see if you can find the solution using this method.
  • What is the solution?
  • Substitute your solution for x in your equation to check your answer.

INTERACTIVE: Balance Scale X

Math Mission

Lesson Guide

Discuss the Math Mission. Students will use the addition property of equality to solve equations.

Opening

Use the addition property of equality to solve equations.

Apply the Property

Lesson Guide

NOTE: On the board, show students how to set up each equation on the balance scale in the interactive beforehand. However, do not tell them what the equation is. Have them ignore any instructions in the interactive.

  • Problem 1: + 4 = 5 + 2
  • Problem 2: 6 = x + 1 + 3
  • Problem 3: x + 2 + 3 = 5 + 4

Have students work in pairs on the problems and the presentation. Make sure a variety of problems are chosen for the student presentations. Tell students to write down the steps they used for each problem.

Remind students to follow the directions in the order they are given. Watch for students who move the weights on the scale before writing the equation to represent the scale. 

SWD: As you confer with struggling pairs of students, try to move them toward efficiency. Scaffold their learning by starting with their strategy and another more efficient strategy. Help them to find the relationship between the guess and check method and using the addition property of equality.

Mathematical Practices

Mathematical Practice 2: Reason abstractly and quantitatively.

Listen for students who verbalize that as soon as x is by itself on one side and a number on the other side, they have the solution to the equation.

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

Look for students who understand how addition and subtraction undo one another; students may talk about undoing adding 3 by subtracting 3.

Mathematical Practice 8: Look for and express regularity in repeated reasoning.

Identify students who apply what they learned from solving one equation to solving the next.

Interventions

Student does not understand how to write an equation to represent the balance scale.

  • The two pans on the balance scale represent the two sides of the equation.
  • The equal sign separates the two sides of the equation.

Student does not know how to find the solution using the addition property of equality.

  • What does the addition property of equality mean?
  • As you move the weights around, the scale will tip one way or the other unless what is true? (Answer: Unless there is equal weight on each side.)
  • If one side goes up when you add or remove weight from the scale, what do you need to do to keep it balanced?
  • Document what is on each side of the scale at each point that it is balanced.

Student subtracts weights from only one side.

  • Was the scale balanced before you took off any weights?
  • What did you do that caused the scale to unbalance?
  • What can you do to make the scale balance again?

Student subtracts different numbers from each side.

  • What did you do to the left side?
  • What did you do to the right side?
  • Did you do the same thing to both sides?

Student is not sure he or she found the correct solution.

  • Go back to the original equation that the balance showed.
  • Substitute your solution for x. Do you get a true equation?

Answers

  • Problem 1: + 4 = 5 + 2

+ 4 = 7
+ 4 − 4 = 7 − 4
= 3

  • Problem 2: 6 = x + 1 + 3

6 = + 4
6 – 4 = + 4 – 4
2 = x

  • Problem 3: x + 2 + 3 = 5 + 4

+ 5 = 9
+ 5 – 5 = 9 – 5
= 4

Work Time

Apply the Property

Addition property of equality: If you add the same number to both sides of an algebraic equation or subtract the same number from both sides of an algebraic equation, the equations will be equivalent and the equations will have the same solution.

Look at the board to set up each problem in the Balance Scale X interactive. Ignore any instructions in the interactive. For each problem:

  • Write an equation that represents the balance scale.
  • Find the solution using the addition property of equality. Write down your solution steps.
  • Check your solution.

INTERACTIVE: Balance Scale X

Hint:

  • Try to get x by itself on one side of the balance scale.
  • How can you use substitution to check your solution?

Prepare a Presentation

Preparing for Ways of Thinking

Look for students who are able to write an equation to represent the balance scale and who use the addition property of equality to solve the equations. Watch for students who fail to get x alone on one side of the scale.

Mathematical Practices

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

Look for students who understand how addition and subtraction undo one another. Students may talk about undoing adding 4 by subtracting 4.

Answer

  • Presentations will vary.

Challenge Problem

Answer

  • Subtract 6.7 from both sides.

+ 6.7 = 9.45

+ 6.7 − 6.7 = 9.45 − 6.7

= 2.75

Work Time

Prepare a Presentation

Select one of the problems from Task 4.

  • Explain how you wrote an equation to represent the balance scale.
  • Explain how you used the addition property of equality to find the solution.
  • Explain how you checked your solution.

Challenge Problem

Use the addition property of equality to solve the following equation. Show each step of your work.

  • + 6.7 = 9.45

Make Connections

Lesson Guide

As students make their presentations and discuss their solution processes, be sure the following points are covered:

  • You have the solution to the equation when x is on one side and a number is on the other side. It does not matter whether the x is on the right or the left side.
  • If a number has been added to x, you can “undo” that by using the inverse operation. Addition and subtraction are inverse operations.
  • Even though the original equation and the new equation look different, they are equivalent. That means that they have the same solution.
  • Whenever you add the same number to both sides of an equation or subtract the same number from both sides of the equation, the new equation will be equivalent to the original equation. This is known as the addition property of equality.

After the discussion, students should be ready to move on to solving these kinds of equations without using the balance scale.

SWD: Go over the mathematical language used throughout the unit. Make sure students use that language when discussing the problems.

  • equation
  • inequality
  • equivalent
  • balanced/unbalanced
  • substitution
  • addition property of equality

Teacher Demonstration

Explain to students that they have just learned about using the addition property of equality to solve an equation like + 6 = 10. They will now learn how to record the steps in the process. Write the equation on the board: + 6 = 10.

  • What can we do to both sides of the equation to keep it balanced? (Answer: Subtract 6.)
  • Show students how to record the subtraction:

+ 6 − 6 = 10 − 6

= 4

Some students may question why they need to show the subtraction when they know just by looking at the equation that the solution is x = 4. Explain that they are learning a process; many equations aren't easy to solve using mental calculations, so they need to be able to record the steps. As an example, write this equation on the board:

+ 2+ 1 + = + 3x − + 5

  • Stress that the equation = 4 is equivalent to the equation + 6 = 10. Both have the same solution.
  • Have students substitute 4 into the original equation to show that it is the correct solution: + 6 = 10; 4 + 6 = 10
  • Explain the convention for writing the solution with the variable on the left, such as = 4 versus
    4 = x.

Performance Task

Ways of Thinking: Make Connections

  • Take notes about your classmates’ methods for solving equations using the addition property of equality.

Hint:

As your classmates present, ask questions such as:

  • How did you decide what to add or remove from each side of the equation?
  • Why do you need to do the same thing on both sides of an equation?
  • What could you do if you checked your solution and found that your solution was incorrect?

Solve the Equations

Lesson Guide

Pair up students to solve the equations. Have students who are struggling with fraction and decimal operations work with students who have mastered these skills. Remind students to think about inverse operations when solving the equations and to record their steps.

ELL: Encourage students to verbalize their explanations. Allow students to speak in small groups to gain confidence.

Answers

  • + 29 = 67

x + 29 − 29 = 67 − 29
= 38
Check: 38 + 29 = 67

  • x + 312 = 814    

    x + 312 – 312 = 814 – 312    
    x = 434
    Check: 434 + 312 = 814  
  •  x + 7.5 = 8.3

    + 7.5 – 7.5 = 8.3 – 7.5
    x = 0.8
    Check: 0.8 + 7.5 = 8.3
     
  • 219 = x34

    219 – 34 = x34 – 34
    x = 11316
    Check: 219 = 11316 + 34

Work Time

Apply the Learning: Solve the Equations

Solve the equations using the addition property of equality.

Show each step of your work, and check your solutions.

  • + 29 = 67
  • x + 312 = 814          
  • + 7.5 = 8.3
  • 219 = x34

The Addition Property of Equality

A Possible Summary

Solving an equation can be understood as a process of writing a series of equivalent equations by rebalancing the equation each time you change the value of one side. Adding or subtracting the same value from both sides of the equation will result in an equivalent equation. This is called the addition property of equality. For example, 8 = 8. If you add 2 to both sides, you get another true equation, 10 = 10. If you begin with 8 = 8, and subtract 2 from both sides, you get the true equation, 6 = 6.

To solve x + 8 = 19, you subtract 8 from both sides: + 8 − 8 = 19 − 8, or = 11.

Formative Assessment

Summary of the Math: The Addition Property of Equality

Write a summary about the addition property of equality and give an example.

Hint:

Check your summary:

  • Do you define the addition property of equality and provide a simple example of how it works? (Use an equation such as 8=8.)
  • Do you explain how to use the addition property of equality to solve equations that are similar to the ones you solved today (such as x+8=19)?

Reflect On Your Work

Lesson Guide

Have each student write a quick reflection before the end of the class. Review the reflections to learn one thing that still confuses students about solving equations.

ELL: When writing the reflection, allow some additional time for ELLs to discuss with a partner before writing, to help them organize their thoughts. Allow ELLs who share the same primary language to discuss in their preferred language and to use a dictionary.

Work Time

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

One thing that confuses me about solving equations is …