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Dividing as fair sharing and as making groups

Numbers · level 2, lesson 5 of 7

A question to start

Here are two different jobs with 12 cookies.

  • Job A. Share the 12 cookies fairly between 3 friends. How many does each friend get?
  • Job B. Pack the 12 cookies into bags, with 3 cookies in each bag. How many bags do you need?

These are not the same job. In Job A you know the number of friends and want the number in each share. In Job B you know the number in each bag and want the number of bags. Yet when you do them, both answers come out as 4.

Is that just luck? Or is there a reason? This lesson is about dividing, and by the end you will see that the two jobs are really one picture, looked at in two ways.

The idea

Dividing as fair sharing

To share fairly means that everyone gets the same amount, and nothing is left in the middle (if possible).

The easiest way to share fairly is to deal, like dealing cards: give one cookie to each friend in turn, then go round again, and again, until the cookies run out.

friend 1: 4 friend 2: 4 friend 3: 4

Round 1: each friend gets 1 (3 cookies used). Round 2: each gets 1 more (6 used). Round 3: 9 used. Round 4: 12 used. The cookies have run out after 4 rounds, so each friend has 4.

Why is dealing always fair? In every round, every friend gets exactly one. So after any number of rounds, they all have the same number. Nobody can get ahead.

We write this as 12 ÷ 3 = 4 and read it "twelve divided by three is four". The sign ÷ means "divided by". Dividing is splitting into equal parts.

Dividing as making groups

Now Job B. Take cookies 3 at a time and put them in a bag. Keep going until the cookies run out. Then count the bags.

bag 1 bag 2 bag 3 bag 4

There are 4 bags. This is also 12 ÷ 3 = 4, but now it means "how many 3s fit into 12?"

You can think of it as taking away 3 again and again: 12 − 3 = 9, 9 − 3 = 6, 6 − 3 = 3, 3 − 3 = 0. You took 3 away 4 times before reaching zero. So 12 ÷ 3 = 4. Dividing is repeated taking away, just as multiplying was repeated adding.

Why the two jobs give the same answer

Put the 12 cookies in an array: 3 rows of 4.

rows: 3 friends, 4 each columns: 4 bags of 3
  • Each row (orange) is one friend's share. There are 3 rows, and each has 4. That is Job A: 12 shared by 3 is 4 each.
  • Each column (blue) has one cookie from each row: 3 cookies. That is a bag of 3! There are 4 columns. That is Job B: 12 in groups of 3 is 4 groups.

It is the same array. Sharing reads it across; grouping reads it down. And when you deal cookies one at a time, each round of dealing makes exactly one column: one cookie for each friend, which is 3 cookies. So the number of rounds (the number in each share) equals the number of 3s in 12 (the number of bags). That is why the answers always match.

Dividing undoes multiplying

The array of 3 rows of 4 also shows 3 × 4 = 12. So multiplying and dividing come in families, like adding and taking away did in lesson 3:

  • 3 × 4 = 12 and 4 × 3 = 12;
  • 12 ÷ 3 = 4 and 12 ÷ 4 = 3.

Dividing undoes multiplying. If you put 4 cookies on each of 3 plates (3 × 4 = 12) and then share the 12 back out among 3 plates (12 ÷ 3), you get 4 on each plate again, where you started.

This gives the best way to divide in your head: think of the multiplication. To find 20 ÷ 5, ask: "5 times what makes 20?" You know 5 × 4 = 20, so 20 ÷ 5 = 4.

Leftovers

What if the cookies do not share out evenly? Share 14 cookies among 3 friends.

4 each2 left over

Deal them out. After 4 rounds, 12 cookies are used and each friend has 4. There are 2 cookies left, but that is not enough for another round (another round needs 3). So each friend gets 4, and 2 are left over.

The amount left over is called the remainder. We can write 14 ÷ 3 = 4, remainder 2.

Check it: 3 friends × 4 cookies = 12, and 12 + 2 = 14. ✓ The remainder must always be smaller than the number you divide by. If 3 or more were left over, you could deal another round.

What happens to the leftovers depends on the story. You could keep them for later, cut them into pieces (that leads to fractions, in the next lesson), or, if you are packing bags, use one more bag that is not full. Level 3 has much more on remainders in stories.

Some special cases

  • Divide by 1. 7 ÷ 1: share 7 cookies with 1 person. They get all 7.
  • A number divided by itself. 7 ÷ 7: share 7 cookies among 7 people. Each gets 1.
  • Zero divided. 0 ÷ 5: share no cookies among 5 people. Each gets 0.
  • Dividing by zero. 6 ÷ 0 would mean "how many groups of 0 fit into 6?" You could put empty bags out forever and never use up any cookies. There is no answer. Mathematicians say dividing by zero is not allowed (it has no meaning).

Worked examples

1. Sharing. 15 stickers are shared fairly among 5 children. How many each?

15 ÷ 5. Think: "5 times what makes 15?" 5 × 3 = 15. So each child gets 3 stickers.

2. Grouping. 18 children make teams of 6. How many teams?

18 ÷ 6. How many 6s fit into 18? Count by sixes: 6, 12, 18. That is three 6s. 3 teams.

3. A fact family. Write the fact family for 2, 8 and 16.

2 × 8 = 16, 8 × 2 = 16, 16 ÷ 2 = 8, 16 ÷ 8 = 2.

4. With a remainder. 17 pencils are packed in boxes of 5. How many full boxes? How many pencils left over?

Count by fives: 5, 10, 15. That is 3 boxes (15 pencils). 17 − 15 = 2 left. So 3 full boxes, 2 pencils left over. Check: 3 × 5 + 2 = 17. ✓

5. Repeated taking away. Find 20 ÷ 4 by taking away fours.

20 − 4 = 16, 16 − 4 = 12, 12 − 4 = 8, 8 − 4 = 4, 4 − 4 = 0. Five times. So 20 ÷ 4 = 5.

Try it

  1. Share 10 grapes fairly between 2 people. How many each? Write the division.
  2. 24 eggs go into boxes of 6. How many boxes?
  3. Find 30 ÷ 10 and 30 ÷ 3.
  4. Write the whole fact family for 4, 5 and 20.
  5. 13 apples are shared among 4 children. How many does each get, and how many are left?
  6. A Mars day is called a sol. A rover team plans 21 sols of work in blocks of 7 sols. How many blocks?
  7. Is 9 ÷ 3 the same as 3 ÷ 9? Explain with sharing.
  8. Mo says "23 ÷ 5 = 3, remainder 8". What is wrong? What is the right answer?
Answers
  1. 10 ÷ 2 = 5 each. (2 × 5 = 10.)
  2. 24 ÷ 6. Count by sixes: 6, 12, 18, 24, that is four. 4 boxes.
  3. 30 ÷ 10 = 3 (10 × 3 = 30). 30 ÷ 3 = 10 (3 × 10 = 30).
  4. 4 × 5 = 20, 5 × 4 = 20, 20 ÷ 4 = 5, 20 ÷ 5 = 4.
  5. Count by fours: 4, 8, 12. That is 3 rounds (12 apples), and 13 − 12 = 1 left. 3 each, 1 left over. Check: 4 × 3 + 1 = 13. ✓
  6. 21 ÷ 7: count by sevens, 7, 14, 21. 3 blocks.
  7. No. 9 ÷ 3 is 9 cookies shared among 3 people: 3 each. 3 ÷ 9 is 3 cookies shared among 9 people: not even one each! (Each would get a piece of a cookie, which needs fractions.) Unlike multiplying, the order matters for dividing.
  8. A remainder of 8 is too big: 8 is more than 5, so another group of 5 fits. 5 × 4 = 20, and 23 − 20 = 3. The right answer is 4, remainder 3. Check: 5 × 4 + 3 = 23. ✓

Common mistake

A remainder that is too big. The remainder must be smaller than the number you are dividing by. If it is not, you stopped too early: deal one more round.

Swapping the numbers. 12 ÷ 3 and 3 ÷ 12 are different. The number being shared (the cookies) comes first; the number of shares, or the size of each group, comes second.

Where it leads