Inverse Operations and Fact Families - Third Grade Mathematics

Multiplication and division are inverse operations, meaning they are mathematical opposites that undo each other. If you multiply two numbers to reach a total, division immediately takes that total and breaks it back down into the original numbers. In this chapter, you will discover how multiplication and division are joined at the hip, and how every set of three related numbers forms a complete fact family.

What Are Inverse Operations?

An inverse operation is an operation that reverses the effect of another operation:
- Addition and subtraction are inverse operations: 5 + 3 = 8, and 8 - 3 = 5.

- Multiplication and division are inverse operations: 4 x 6 = 24, and 24 ÷ 6 = 4.


Think of tying and untying your shoes: Multiplication puts the groups together into a knot; division unties the knot and separates the strands!

Multiply by 5:      4  ------->  20  (4 x 5 = 20)
Divide by 5:       20  ------->   4  (20 ÷ 5 = 4)

Because they undo each other, you can use multiplication to solve any division problem! When you see: 28 ÷ 4 = ? Think to yourself: "What number times 4 equals 28?" ( ? x 4 = 28 ). Since 7 x 4 = 28, you know immediately that 28 ÷ 4 = 7!

The Anatomy of a Fact Family

A fact family is a collection of four related mathematical equations made from the exact same three numbers (two factors and one product/dividend).

The Fact Family for 3, 8, and 24:
Multiplication Facts:
3 x 8 = 24
8 x 3 = 24

Division Facts:
24 ÷ 8 = 3
24 ÷ 3 = 8

Notice the key patterns in every fact family:
- In both multiplication sentences, the largest number (24) is the product on the right.

- In both division sentences, the largest number (24) is the dividend at the front.

- The two smaller numbers (3 and 8) swap positions using the commutative property.


Special Case: Square Numbers in Fact Families

When a number is multiplied by itself (a square number), the two factors are identical! Consider the numbers 6, 6, and 36:
- 6 x 6 = 36

- 36 ÷ 6 = 6
This fact family has only two unique equations instead of four because swapping the factors produces the exact same equation.

Using Fact Triangles

A fact triangle is a handy visual tool that organizes the three numbers in a fact family:

          /         / 24 \       <- Product / Dividend at the peak
        /------       / 3  x 8 \     <- Factors at the base
      +----------+

How to read the fact triangle:
- Cover the top number: 3 x 8 = 24.

- Cover the left number: 24 ÷ 8 = 3.

- Cover the right number: 24 ÷ 3 = 8.


Chapter Practice Exercises

  1. Write the complete fact family (4 equations) for each set of numbers: a. 4, 5, 20 b. 6, 7, 42 c. 8, 9, 72 d. 3, 9, 27
  2. Write the special fact family (2 equations) for the square number 7 x 7 = 49.
  3. Use a related multiplication fact to solve each division problem: a. 36 ÷ 4 = ___ (Think: ___ x 4 = 36) b. 54 ÷ 6 = ___ (Think: ___ x 6 = 54) c. 63 ÷ 7 = ___ (Think: ___ x 7 = 63) d. 48 ÷ 8 = ___ (Think: ___ x 8 = 48)
  4. Which equation does NOT belong to the fact family for 5, 6, and 30? a. 5 x 6 = 30 b. 30 ÷ 5 = 6 c. 30 - 6 = 24 d. 30 ÷ 6 = 5
  5. An array has 35 total tiles arranged in 5 equal rows. How many tiles are in each row? Write both a multiplication equation and a division equation that model this problem.

Solutions and Step-by-Step Answers

  1. Complete fact families: a. 4, 5, 20: 4 x 5 = 20; 5 x 4 = 20; 20 ÷ 5 = 4; 20 ÷ 4 = 5. b. 6, 7, 42: 6 x 7 = 42; 7 x 6 = 42; 42 ÷ 7 = 6; 42 ÷ 6 = 7. c. 8, 9, 72: 8 x 9 = 72; 9 x 8 = 72; 72 ÷ 9 = 8; 72 ÷ 8 = 9. d. 3, 9, 27: 3 x 9 = 27; 9 x 3 = 27; 27 ÷ 9 = 3; 27 ÷ 3 = 9.
  2. For 7, 7, 49: 7 x 7 = 49; 49 ÷ 7 = 7.
  3. Solving via related multiplication facts: a. 36 ÷ 4 = 9 because 9 x 4 = 36. b. 54 ÷ 6 = 9 because 9 x 6 = 54. c. 63 ÷ 7 = 9 because 9 x 7 = 63. d. 48 ÷ 8 = 6 because 6 x 8 = 48.
  4. Equation c (30 - 6 = 24) does not belong. It is a subtraction equation, whereas fact families for multiplication and division consist strictly of multiplication and division facts.
  5. Multiplication: 5 x ? = 35 -> 5 x 7 = 35. Division: 35 ÷ 5 = 7. There are 7 tiles in each row.