Math Background

Lesson: Using Inverse Operations to Solve Equations
Introducing the Concept

Although students have been introduced to the concept of using inverse operations to solve equations, they may not have understood that they are actually performing the same operation on both sides of the equation. By modeling this concept, students will move from solving equations through the use of related facts to solving by using inverse operations.

Materials: pan balance, 50 connecting cubes

Prerequisite Skills and Concepts: Students should know how to write and evaluate algebraic expressions and how to write and solve addition and subtraction equations using inverse operations. They should also know basic multiplication and division facts.

Begin by reviewing the difference between an expression and an equation.

  • Say: We've learned two ways to solve an equation. One way is by using a function table. The other way is by using inverse operations.
  • Ask: What is the inverse of multiplication? (division) What is the inverse of division? (multiplication) What makes multiplication and division inverse operations?
    Elicit that multiplication and division undo each another. Encourage students to provide examples in word or numerical form.
  • Say: That's right. Multiplication and division are opposites. When you multiply, you combine equal-sized groups. When you divide, you break apart equal-sized groups.
  • Ask: How would you use inverse operations to solve 6m = 18?

    Write the equation on the chalkboard. Students may say that since 6 times a number is 18, 18 ÷ 6 must be equal to that number. So, since 18 ÷ 6 = 3, m = 3.

  • Say: Let's model this equation by placing cubes on a pan balance. Think of the middle of the pan balance as the equals sign in the equation. Both sides of the pan balance must be level, just like both sides of an equation must be equal.

    On one side of the balance place 6 cube trains that are each 3 cubes long. On the other side, count out 18 cubes. Explain what you are doing as you place the cubes in the pans. Then point out that the pans are now balanced.

  • Say: Let's see what happens if I multiply this side of the equation by 2. Point to the side with the 6 cube trains that are each 3 cubes long.
  • Ask: Right now we have 6 groups of 3 on this side. If I multiply the number of groups by 2, how many groups of 3 will I have?
    Students should say 12. Add 6 more cube trains that are each 3 cubes long to that side of the balance.
  • Ask: The pans are no longer balanced. What do you think I can do to the other pan to get them to balance? Point to the pan containing the 18 cubes. Elicit that you will need to double the number of cubes in the second pan to get the pans to balance. Add 18 cubes to the second pan.
  • Say: We can see that the pans are once again balanced after we multiplied both sides by two. Let's make sure this works with the numerical version of the equation.
    Model multiplying both sides of the equation by two on the board and then solve for m.
6m x 2   =   18 x 2
12m   =   36 left arrow Think: 36 ÷ 12 = 3
m   =   3
  • Ask: What was the value of the variable in the first equation? (3) What was the value of the variable after both sides of the equation were multiplied by 2? (3)
  • Say: Now let's divide both sides of this new equation by 3 to see if we get the same value for m. We have 12 groups of 3 on this side of the equation.
  • Ask: If we divide the number of groups by 3, how many groups will we have left?
    Students may say 4. Remove 8 cube trains. Repeat with the other side and remove 24 cubes. Discuss the fact that the pans are once again balanced. On the chalkboard, divide both sides of 12m = 36 by 3. Then solve. Discuss the fact that the value of m is still 3. Repeat this procedure, first adding a number to each side of 4m = 12, and then subtracting a number from each side of the new equation to show that the value of the variable remains the same.
  • Ask: What happens to the value of a variable when you perform the same operation on both sides of an equation?
    Students may say that the value remains the same.
  • Say: That's right. When you perform the same operation on both sides of the equation, the value of the variable does not change. This is why we can use inverse operations to solve an equation. Look at our original equation, 6m = 18. To find the value of m, we can divide both sides of the equation by 6. Work through the problem on the board as you talk the students through it.
  • Say: Six divided by six is one, leaving one m on the left side of the equation. Eighteen divided by six is three. So m equals three.

    You may wish to have students work in pairs to further practice this concept using counters.

Houghton Mifflin Math Grade 5