When Division is Not Exact

This is a complete lesson with teaching and exercises about the concept of remainder in division (when division is not exact), meant for third grade. It starts out by explaining the idea of a remainder using the example of sharing bananas. First, students work with visual examples, writing division sentences with remainders. Then the lesson explains how to find the remainder by multiplying and subtracting to find the difference (without a visual help).

If you divide 13 bananas evenly between Joe and Sally,
how many does each one get?

13 ÷ 2 = ?

Joe and Sally each get 6 bananas and one is left over.
We write this as:

13 ÷ 2 = 6, R1

The leftover banana is called the remainder, and is indicated after the letter R.

(If we didn't want any leftovers, then both could get 6 1/2 bananas.)

1. Fill in the blanks.

a. 14 bananas divided among 3 people
    gives 4 bananas to each and
    2 bananas that cannot be divided
    evenly.

14 ÷ 3 = 4,

remainder 2

b. 14 carrots divided among 5 people
gives 2 carrots to each and 4 carrots
that cannot be divided evenly.

14 ÷ 5 = 2,

remainder 4

c. 8 scissors divided among 5 people
gives 1 pair of scissors to each and
3 pairs that cannot be divided evenly.

8 ÷ 5 = ____,

remainder _____

d. 3 apples divided among 5 people
     means we cannot share them equally.
     So, no one gets any apples.
   All 3 are left over.

3 ÷ 5 = 0,

remainder ____

e. ___ rams divided among 6 people
gives ___ rams to each and ___ rams
that cannot be divided evenly.

____ ÷ 6 = ___,

remainder _____.

f. ___ camels divided between 2 people
gives ___ camels to each person,
and ___ camel left over.

___ ÷ 2 = ____,

remainder _____.

Here's another way of looking at division and remainder.
How many groups of 2 can we make out of 13 apples?

We can make six groups. One apple is left over.

13 ÷ 2 = 6 R1

2. Divide the dots into groups and write a division sentence.

a. Divide into
groups of 3.

20 ÷ 3 = _____

remainder ______

b. Divide into
groups of 4.

21 ÷ 4 = _____

remainder _____

c. Divide into
groups of 6.

_____ ÷ 6 = ____

remainder _____

d. Divide into
groups of 5.

_____ ÷ 5 = ____

remainder _____

e. Divide into
groups of 7.

_____ ÷ 7 = ____

remainder _____

f. Divide into
groups of 9.

_____ ÷ 9 = ____

remainder _____

g. Divide into
groups of 3.

_____ ÷ 3 = ____

remainder _____

h. Divide into
groups of 5.

_____ ÷ 5 = ____

remainder _____

4 ÷ 5 = ?

How many groups of 5 can we make out of 4 apples?

No groups. All four apples are left over.

4 ÷ 5 = 0 R4

3. Divide and indicate the remainders.

a. 7 ÷ 2 = _____, R ____

1 ÷ 2 = _____, R ____

b. 3 ÷ 4 = _____, R ____

11 ÷ 2 = _____, R ____

c. 18 ÷ 5 = _____, R ____

7 ÷ 6 = _____, R ____

20 ÷ 3 = ?

Think: How many groups of 3 are there in 20?
Or: How many times does 3 fit into 20?

6 × 3 = 18 and 7 × 3 = 21 (too much).

So 3 goes into 20 six times. Since 6 × 3 = 18
and 18 is 2 less than 20, the remainder is 2.

You will find the remainder by finding the difference between 20 and 6 × 3 = 18.

Example. 42 ÷ 8 = ?

Think: How many times does 8 fit into 42? 5 × 8 = 40 and 6 × 8 = 48.
So, 8 goes into 42 five times. And, 5 × 8 is 40. The remainder is the difference
between 40 and 42, or 2. So, 42 ÷ 8 = 5 R2

4. Practice some more!

a. 13 ÷ 5 = ____, R ____

14 ÷ 5 = ____, R ____

b. 5 ÷ 8 = ____, R _____

25 ÷ 8 = ____, R _____

c. 47 ÷ 6 = ____, R _____

50 ÷ 6 = ____, R _____

d. 13 ÷ 2 = ____, R ____

13 ÷ 5 = ____, R ____

e. 54 ÷ 8 = ____, R ____

67 ÷ 8 = ____, R ____

f. 57 ÷ 7 = ____, R _____

39 ÷ 9 = ____, R _____

5. Divide. What patterns do you notice?

21 ÷ 2 = _____, R ____

22 ÷ 2 = _____, R ____

23 ÷ 2 = _____, R ____

24 ÷ 2 = _____, R ____

25 ÷ 2 = _____, R ____

26 ÷ 2 = _____, R ____

27 ÷ 2 = _____, R ____

28 ÷ 2 = _____, R ____

29 ÷ 2 = _____, R ____

30 ÷ 2 = _____, R ____

21 ÷ 3 = _____, R ____

22 ÷ 3 = _____, R ____

23 ÷ 3 = _____, R ____

24 ÷ 3 = _____, R ____

25 ÷ 3 = _____, R ____

26 ÷ 3 = _____, R ____

27 ÷ 3 = _____, R ____

28 ÷ 3 = _____, R ____

29 ÷ 3 = _____, R ____

30 ÷ 3 = _____, R ____

21 ÷ 4 = _____, R ____

22 ÷ 4 = _____, R ____

23 ÷ 4 = _____, R ____

24 ÷ 4 = _____, R ____

25 ÷ 4 = _____, R ____

26 ÷ 4 = _____, R ____

27 ÷ 4 = _____, R ____

28 ÷ 4 = _____, R ____

29 ÷ 4 = _____, R ____

30 ÷ 4 = _____, R ____

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.

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