# The Connection Between Division and Multiplication

This is a complete lesson with teaching and exercises about the relationship between multiplication and division, meant for third grade. They are opposite operations, and both have to do with groups of equal size. Students write multiplication and division sentences from the same picture. In the later exercises, they write a division fact that matches the given multiplication, and vice versa. Lastly, students use their knowledge of multiplication facts to solve division problems.

We get both
a multiplication fact
and a division fact
from the same picture:
Three groups of 4 makes 12. 3 × 4 = 12

12 divided into groups of 4 is three groups.

12 ÷ 4 = 3
Multiplication and division are very closely related. They are opposite operations.
You could say division is “backwards” multiplication.

1. Fill in the blanks.

a.  Two groups of 6 is 12.

2 × 6 = 12

12 divided into groups of 6 is two groups.

12 ÷ 6 = 2

b.  Five groups of 2 is _____.

____ × 2 = ____

____ divided into groups of 2
is ___ groups.

_____ ÷ 2 = ____

c.  One group of 4 is 4.

____ × 4 = ____

4 divided into groups of 4 is one group.

_____ ÷ 4 = ____

d.  ____ groups of 3 is _____.

____ × ____ = ____

___ divided into groups of 3
is ___ groups.

_____ ÷____ = ____

e.  Five groups of 1 is 5.

____ × 1 = ____

5 divided into groups of 1
is ____ groups.

_____ ÷ 1 = ____

f.   ____ groups of ____ is _____.

____ × ____ = ____

___ divided into groups of 2
is ___ groups.

_____ ÷____ = ____

2. Make groups. Then write the division and multiplication facts that the pictures illustrate.

 a.  Make groups of four. ____ × 4 =  8 8 ÷ 4 = ____
 b.  Make groups of two.____ × 2 = _____ _____ ÷ 2 = ____
 c.  Make groups of four.     ____ × 4 = _____     _____ ÷ 4 = ____
 d.  Make groups of six. ____ × 6 =  _____ _____ ÷ 6 = ____
 e. ____ × 4 = _____ _____ ÷ 4 = ____
 f. ____ × 7 = _____ _____ ÷ 7 = ____
 g. ____ × 6 = ____ ____ ÷ 6 = ____
 h. ____ × 2 = ____ ____ ÷ 2 = ____
 i. ____ × 5 = ____ ____ ÷ 5 = ____

3. Now draw sticks or circles and make a picture yourself. Write the division
and multiplication sentences.

 a.  Draw 15 sticks.      Make groups of 5. _____ ×  5  = _______ _______ ÷  5  =  _____
 b.  Draw 24 sticks.      Make groups of 8. _____ × ____ = _______ ______ ÷ ____ =  _____
 c.  Draw 30 sticks.      Make groups of 5. _____ × ____ = _______ ______ ÷ ____ =  _____
 d.  Draw 27 sticks.      Make groups of 9. _____ × ____ = _______ _______ ÷ ____ =  _____
 c.  Draw 32 sticks.      Make groups of 16. _____ × ____ = _______ _______ ÷ ____ =  _____
 f.  Draw 16 sticks.      Make groups of 2. _____ × ____ = _______ _______ ÷ ____ =  _____
 g.  Draw 8 sticks.      Make a group of 8. _____ × ____ = _______ _______ ÷ ____ =  _____
 h.  Draw 18 sticks.      Make groups of 9. _____ × ____ = _______ _______ ÷ ____ =  _____
 i.  Draw 20 sticks.      Make groups of 5. _____ × ____ = _______ _______ ÷ ____ =  _____

4. For each multiplication fact, write also a division fact. Think about the groups!

 a.  7 × 2 = _____ ______ ÷ 2 = _____ b.  12 × 2 = ___________ ÷ 2 = _____ c.  8 × 5 = ___________ ÷ 5 = _____ d.  6 × 7 = _____ ______ ÷ ____ = _____ e.  7 × 7 = ___________ ÷ ____ = _____ f.  11 × 3 = ___________ ÷ ____ = _____ g.  9 × 8 = _____ ______ ÷ ____ = _____ h.  1 × 5 = ___________ ÷ ____ = _____ i. 7 × 9 = ___________ ÷ ____ = _____

You can solve a division problem by thinking of the matching multiplication.

 30 ÷ 6 = ___ Think: what times 6 is 30? ___  × 6 = 30

So, since you already know the multiplication tables, division will be easy!

5. For each division, think of the corresponding multiplication and solve.

 a.  14 ÷ 2 = ______     ____ × 2 = 14 b.  18 ÷ 2 = ______    ____ × 2 = ______ c.  21 ÷ 7 = ______    ____ × 7 = ______ d.  54 ÷ 6 = ______    ____ × ____ = ______ e.  24 ÷ 4 = ______  ____ × ____ = ______ f.  30 ÷ 3 = ______  ____ × ____ = ______ g.  32 ÷ 4 = ______    ____ × ____ = ______ h.  56 ÷ 7 = ______   ____ × ____ = ______ i.  55 ÷ 5 = ______   ____ × ____ = ______

6. Divide. Again think of the multiplication.

 a. b. c. d. 24 ÷ 4 = ______ 16 ÷ 2 = ______20 ÷ 2 = ______36 ÷ 9 = ______ 15 ÷ 5 = ______35 ÷ 5 = ______49 ÷ 7 = ______54 ÷ 9 = ______ 32 ÷ 8 = ______40 ÷ 8 = ______50 ÷ 5 = ______42 ÷ 6 = ______ 48 ÷ 6 = ______56 ÷ 8 = ______81 ÷ 9 = ______100 ÷ 10 = _____

Think of multiplication, and solve.

 a.  1,000 ÷ 100 = ______ b.  400 ÷ 50 = ______ c.  200 ÷ 4 = ______ d.  1,000 ÷ 500 = ______ e.  800 ÷ 800 = ______ f.  200 ÷ 40 = ______

This lesson is taken from Maria Miller's book Math Mammoth Division 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.

#### Math Mammoth Division 1

A self-teaching worktext for 3rd grade that covers division concept, division & multiplication fact families, word problems, division facts, remainder, zero and one in division, and more.Download (\$3.70). Also available as a printed copy.