What division means
Twenty-eight beads are shared equally among 4 friends, and thirty buttons are sewn onto cards of 5. One story asks how many in each group, the other how many groups. Why is the same operation the answer to both?
Where you are 16 in Multiplication and division
Why this exists
Multiplication puts equal groups together. Four friends each holding 7 beads have 4 \times 7 = 28 beads in all, as What multiplication means shows. There we knew the groups and asked for the total.
Real problems often run the other way. The total is in your hands, and something about the groups is missing. At the craft club there are 28 beads to share fairly among 4 friends: how many does each friend get? Or there are 30 buttons to sew onto cards, 5 to a card: how many cards can be filled? Neither question is answered by putting groups together. Both are answered by taking a total apart into equal groups.
That is what division does. It starts from the total and finds the missing part of an equal-groups picture. The surprise of this lesson is that the two questions above, “how many in each group?” and “how many groups?”, are answered by the same operation, and the reason is a picture you can draw.
Definition
Division splits a total into equal groups. The division 28 \div 4 (read “twenty-eight divided by four”) has two meanings, and they always give the same number:
- Sharing. Split 28 into 4 equal groups. 28 \div 4 is how many are in each group.
- Grouping. Split 28 into groups of 4. 28 \div 4 is how many groups there are.
The numbers have names. In 28 \div 4 = 7, the total 28 is the dividend, the 4 we divide by is the divisor, and the answer 7 is the quotient.
Both meanings come from one statement about multiplication. For whole numbers a, b and c, with b not zero,
So 28 \div 4 = 7 because 4 \times 7 = 28. Every division in this lesson comes out even: the total splits into equal groups with nothing left over.
A third way to see a division, repeated subtraction, counts how many times the divisor can be taken away from the dividend before nothing is left. It is grouping written with minus signs, and Part 3 shows why.
Intuition
The lesson runs over four days. Part 1 is sharing, Part 2 is grouping, Part 3 is repeated subtraction, and Part 4 puts all three beside multiplication with a box for the unknown number.
Part 1: sharing, how many in each group
The craft club has 28 beads and 4 friends who should each get the same number. Nobody needs to know the answer ahead of time. Hand out the beads the way cards are dealt: one to the first friend, one to the second, one to the third, one to the fourth, and round again.
Each round of dealing hands out 4 beads, one to every friend. The dealing stops when the beads run out, and since every friend got one bead per round, every friend ends with the same number. Counting the rounds counts the beads in one share. Here the beads last for 7 rounds, so each friend gets 7, and we write
In a sharing story, the number of groups is known (the 4 friends) and the size of each group is what we want. The divisor counts the groups.
Dealing one at a time always works, but it is slow for a big total. Multiplication gives a faster way to think. “Four equal groups of how many make 28?” is the question 4 \times \square = 28, and a known fact, 4 \times 7 = 28, answers it.
Part 2: grouping, how many groups
Now the other question. There are 30 buttons, and each card holds 5. Put 5 buttons on a card, then 5 on the next card, and keep going until the buttons run out.
Six cards are filled, so
In a grouping story, the size of each group is known (the 5 buttons on a card) and the number of groups is what we want. The divisor is the group size. As a multiplication question, it asks “how many fives make 30?”, which is \square \times 5 = 30, and 6 \times 5 = 30 answers it.
So the sign \div covers two different questions. Look at what the divisor stands for:
| Story | What is known | What is missing | Equation |
|---|---|---|---|
| Sharing | how many groups | how many in each group | \text{groups} \times \square = \text{total} |
| Grouping | how many in each group | how many groups | \square \times \text{size} = \text{total} |
The reason one operation can answer both is shown in Part 4 and proved in “Why it is true”.
Part 3: repeated subtraction
Grouping can be done with nothing but subtraction. Hallvard has 18 felt squares, and each puppet needs 6. Every time he makes a puppet, 6 squares leave the pile:
The pile is empty after three subtractions, so he makes 3 puppets and 18 \div 6 = 3. On a number line, each subtraction is a jump of 6 to the left, and three jumps carry 18 back to 0.
This is grouping in disguise. Each subtraction removes one group of 6, so counting the subtractions counts the groups. It mirrors something already known: multiplication can be done as repeated addition, 6 + 6 + 6 = 18, and repeated subtraction undoes it step by step, landing back at 0.
Repeated subtraction answers the grouping question directly. It also answers a sharing question, because the two meanings give the same number, but the steps then stand for rounds of dealing rather than for groups.
Part 4: one picture, both meanings
Go back to the dealing in Part 1 and set each round of beads down in a column, one bead per friend. After 7 rounds the beads stand in 4 rows of 7.
This is an array, as in Arrays. Read across, each row is one friend’s share, so it answers the sharing question: 28 split into 4 equal groups has 7 in each. Read down, each column is a group of 4, one bead per friend, so it answers the grouping question: 28 makes 7 groups of 4. The same picture gives
and both come from the one multiplication fact 4 \times 7 = 28. Turning the array a quarter turn swaps rows and columns without changing the count, which is the commutative property of multiplication, 4 \times 7 = 7 \times 4.
That is why every division can be checked by multiplying, and why a box for the unknown can sit in any place of a multiplication equation:
- 4 \times \square = 28 (how many in each of 4 groups) is solved by 28 \div 4.
- \square \times 4 = 28 (how many groups of 4) is also solved by 28 \div 4.
- \square \div 4 = 7 (the total is missing) is solved by multiplying, 4 \times 7.
Results
Two meanings, one number. For whole numbers a and b, with b not zero and a splitting evenly into groups of b, the number of things in each of b equal groups and the number of groups of b are the same number, a \div b.
Division and multiplication. a \div b = c exactly when b \times c = a. So b \times \square = a and \square \times b = a are both solved by \square = a \div b, and \square \div b = c is solved by \square = b \times c.
Repeated subtraction. a \div b is the number of times b can be subtracted from a, one after another, to reach exactly 0.
Checking. A quotient is right when quotient times divisor gives back the dividend.
Why it is true
Sharing and grouping give the same number. Share a things among b groups by dealing: each round gives one thing to every group, so each round uses up b things. Suppose the dealing ends after c rounds with nothing left. Every group received one thing per round, so every group holds c things: the sharing answer is c. The things handed out in each round form a set of b, and there were c rounds, so a was split into c groups of b: the grouping answer is also c. Laid out with one column per round, the things make an array of b rows and c columns, and b \times c = a.
Repeated subtraction counts groups. Each subtraction of b takes one group of b away from what is left. Reaching 0 after c subtractions means a was used up by c groups of b, which says c \times b = a. By the commutative property, c \times b = b \times c, so c is a \div b.
Why one multiplication fact gives two divisions. In the array of b rows and c columns, the rows are b groups of c and the columns are c groups of b. So a \div b = c and a \div c = b both hold whenever b \times c = a.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.
Part 1: sharing beads at the craft club
At the craft club, 32 beads are shared equally among 4 friends. How many beads does each friend get?
SolutionThe number of groups is known: 4 friends. The size of each share is missing, so this is sharing.
Deal the beads one at a time, one to each friend per round. Each round hands out 4 beads, so count the beads used by fours: 4, 8, 12, 16, 20, 24, 28, 32. That is 8 rounds, and the beads are gone. Each round gave every friend one bead, so each friend has 8 beads. The picture is the first figure in Part 1 with one more bead in every box.
In symbols, 32 \div 4 = 8. Check by putting the shares back together: 4 groups of 8 is 4 \times 8 = 32, the beads we started with.
Part 1: buttons from the button jar
Rewi takes 45 buttons out of the button jar and puts the same number into each of 5 small bags. How many buttons go into each bag?
SolutionAgain the number of groups is known (5 bags) and the size of each group is missing: sharing.
Dealing would work, but think of it as a multiplication with a box: five equal groups of how many make 45? That is
Count by fives until 45: 5, 10, 15, 20, 25, 30, 35, 40, 45. That is nine fives, so 5 \times 9 = 45 and \square = 9. Each bag gets 9 buttons, and 45 \div 5 = 9.
The counting by fives is the dealing, done round by round: each five is one round that drops a button into every bag.
Part 2: sewing buttons onto cards
There are 28 buttons, and each card gets 7 buttons sewn onto it. How many cards are filled?
SolutionThis time the size of each group is known (7 buttons on a card) and the number of groups is missing: grouping.
Fill one card at a time. After each card, the buttons used so far are 7, 14, 21, 28. The buttons run out after the fourth card, so 4 cards are filled.
In symbols, 28 \div 7 = 4, which answers \square \times 7 = 28. Check: 4 \times 7 = 28.
Compare this with Example 2. There the divisor 5 counted bags, the groups. Here the divisor 7 is how many buttons go on each card, the size of a group. The sign \div is the same in both.
Part 2: same numbers, two stories
(a) Hinewai has 42 beads and threads 6 beads on each bracelet. How many bracelets can she make?
(b) Hinewai has 42 beads and threads them equally onto 6 bracelets. How many beads go on each bracelet?
Solution(a) The size of each group is known (6 beads), so this is grouping. Count by sixes: 6, 12, 18, 24, 30, 36, 42. That is 7 sixes, so she makes 7 bracelets: 42 \div 6 = 7.
(b) The number of groups is known (6 bracelets), so this is sharing. Six equal groups of how many make 42? That is 6 \times \square = 42, and 6 \times 7 = 42, so each bracelet gets 7 beads: 42 \div 6 = 7.
The equation is the same, but the 7 means different things. In (a) it counts bracelets; in (b) it counts beads on one bracelet. The pictures differ too: (a) is 7 groups of 6, and (b) is 6 groups of 7. Both hold 42 beads, because 7 \times 6 = 6 \times 7 by the commutative property.
Part 3: felt squares for puppets
Hallvard has 30 felt squares, and each puppet uses 6. Use repeated subtraction to find how many puppets he can make.
SolutionEach puppet takes 6 squares away from the pile, so subtract 6 until nothing is left:
There are five subtractions, one for each puppet, so he makes 5 puppets and 30 \div 6 = 5. On a number line like the one in Part 3, these are five jumps of 6 from 30 back to 0.
Check: five groups of 6 put back together is 6 + 6 + 6 + 6 + 6 = 30, or 5 \times 6 = 30.
Part 3: emptying the button jar
The button jar holds 56 buttons. Rewi takes them out 8 at a time. How many handfuls empty the jar?
SolutionEach handful takes away 8. Subtract 8 until the jar is empty:
That is 7 subtractions, so 7 handfuls and 56 \div 8 = 7. Check: 7 \times 8 = 56.
There is a quicker road. Five handfuls are 5 \times 8 = 40 buttons, so take those away at once: 56 - 40 = 16. Then 16 - 8 = 8 and 8 - 8 = 0 are two more handfuls. In all, 5 + 2 = 7 handfuls. Taking away several groups in one step is the idea that Partial quotients later grows into a way to divide large numbers.
Part 4: a box in different places
(a) Hinewai makes bracelets with 9 beads on each and uses 63 beads. How many bracelets did she make? Write an equation with a box, then solve it.
(b) Rewi shares some buttons equally among 3 cards, and each card gets 8. How many buttons did he share? Write an equation with a box, then solve it.
Solution(a) The group size is 9 and the number of groups is missing:
Count by nines: 9, 18, 27, 36, 45, 54, 63 is seven nines, so 7 \times 9 = 63 and \square = 7. She made 7 bracelets, and 63 \div 9 = 7.
(b) This time the total is the missing number. Sharing the total among 3 cards gave 8 each, so
The total is 3 groups of 8 put back together, so this one is solved by multiplying: \square = 3 \times 8 = 24. Rewi shared 24 buttons. Check: 24 \div 3 = 8, since 3 \times 8 = 24.
A division equation does not always call for dividing. What decides is which number is missing.
Part 4: more friends, smaller shares
The craft club has 36 beads. How many does each friend get if they are shared equally among 2 friends? Among 3, 4, 6 or 9 friends?
SolutionEach is a sharing question, 36 \div \text{friends}. Find the share that makes the multiplication come out to 36:
| Friends | Each gets | Check |
|---|---|---|
| 2 | 18 | 2 \times 18 = 36 |
| 3 | 12 | 3 \times 12 = 36 |
| 4 | 9 | 4 \times 9 = 36 |
| 6 | 6 | 6 \times 6 = 36 |
| 9 | 4 | 9 \times 4 = 36 |
For 2 friends, 18 + 18 = 36; for 3 friends, 12 + 12 + 12 = 36. The rest are known facts.
The pattern: the more friends, the fewer beads each. The total stays 36, so splitting it into more groups must make each group smaller. Notice too that 36 \div 4 = 9 and 36 \div 9 = 4 both appear in the table, as Part 4 promised: one fact, 4 \times 9 = 36, gives both.
Boundaries
No dividing by zero. 6 \div 0 has no answer. Sharing 6 beads among 0 friends makes no sense, since there is nobody to hold them. Grouping fails too: groups of 0 beads never use any beads up, so there is no number of such groups that makes 6. In the language of multiplication, 0 \times \square = 6 has no solution, because 0 times any number is 0.
Only totals that split evenly, for now. Every division in this lesson comes out even. Some do not: 36 beads cannot be shared equally among 5 friends with whole beads, because 5 \times 7 = 35 is too few and 5 \times 8 = 40 is too many. What to do with what is left is the subject of Remainders and what they mean.
The groups must be equal. Splitting 28 beads into piles of 10, 10 and 8 is not division. Division always means equal groups, the same requirement that makes equal shares name a fraction in Equal shares and naming the whole.
Order matters. 28 \div 4 and 4 \div 28 are different questions. The dividend is the total being split, and the divisor is either the number of groups or the group size. Swapping them asks to split 4 beads into 28 equal groups, which cannot be done with whole beads. Multiplication is commutative; division is not.
Where this goes next
The two meanings and the link to multiplication are the base for Division facts, where the quotients that go with each known multiplication fact are learned until they come quickly, and for Multiplication and division undo each other, which shows in general that dividing by a number undoes multiplying by it. Stories that call for multiplying or dividing are gathered in Equal group problems.
Sharing is the idea under Mean as fair share: computing the mean, where amounts that are not equal are pooled and dealt back out evenly, and under A fraction of a set, where a fraction of a group of objects means one of several equal shares. Grouping returns with area: a rectangle of known area and one known side has its other side found by asking how many rows fit, as in Finding a missing side from the area. Repeated subtraction, done with several groups at a time, becomes Partial quotients. Much later the same two meanings come back for fractions, in sharing a fraction into equal parts and asking how many of one fraction fit into another.
Exercises
Part 1: sharing
- There are 20 beads, shared equally among 5 friends. How many beads does each friend get? Write the division.
- Rewi puts 35 buttons equally into 7 cups. How many buttons are in each cup?
- The craft club shares 48 sequins equally among 6 bags. How many sequins go in each bag?
- Describe how dealing would share 27 beads among 3 friends. How many rounds does it take, and how many beads does each friend get?
- Rewi shares 16 buttons equally between 2 friends. Write the division, and name its dividend, divisor and quotient.
Part 2: grouping
- There are 40 buttons to sew onto cards, 8 to a card. How many cards are filled?
- Hinewai threads 3 beads on each key ring and has 21 beads. How many key rings can she make?
- There are 64 pipe cleaners, put into bundles of 8. How many bundles are there?
- There are 50 felt squares. (a) They are packed 5 to a pack. How many packs? (b) They are shared equally among 5 tables. How many does each table get? Say which question is sharing and which is grouping.
- Write a grouping story that 54 \div 6 answers, and find the answer.
Part 3: repeated subtraction
- Use repeated subtraction to find 24 \div 4. Write each subtraction.
- Hallvard has 49 felt squares and uses 7 for each puppet. Use repeated subtraction to find how many puppets he can make.
- On a number line, jumps of 10 start at 70 and go back to 0. How many jumps are there, and what division do they show?
- Write the chain of subtractions that shows 14 \div 2 = 7.
- The button jar holds 72 buttons, taken out 8 at a time. How many handfuls empty it? Use repeated subtraction, then check by multiplying.
Part 4: equations and pictures
- Find the number in the box: 9 \times \square = 81. Which division does it answer?
- Find the number in the box: \square \div 2 = 9.
- An array has 3 rows of 8 buttons. Write two division equations that the array shows, and say what each one means in the picture.
- Find the number in the box: 40 \div \square = 4.
- Rewi puts the same number of buttons on each of 3 cards and uses 15 buttons. Write an equation with a box and solve it.
- Hinewai threads 5 beads on each strand and uses 25 beads. Write an equation with a box and solve it.
Explaining and reasoning
- Draw or describe one picture of 35 beads that shows both 35 \div 5 = 7 and 35 \div 7 = 5, and explain why one picture can show both.
- Share 12 buttons equally among 2, 3, 4 and 6 friends. How many does each friend get? Explain why more friends means fewer buttons each.
- Is 10 \div 2 the same as 2 \div 10? Explain using sharing.
- Explain why repeated subtraction of 4 from 20 and grouping 20 buttons onto cards of 4 must give the same answer.
Answers
- 4 beads each, 20 \div 5 = 4. Five groups of 4 make 20: 5 \times 4 = 20.
- 5 buttons in each cup. 7 \times 5 = 35, so 35 \div 7 = 5.
- 8 sequins in each bag. 6 \times 8 = 48, so 48 \div 6 = 8.
- Each round gives one bead to each of the 3 friends, using 3 beads. Counting by threes, 3, 6, 9, 12, 15, 18, 21, 24, 27 is 9 rounds, so each friend gets 9 beads: 27 \div 3 = 9.
- 16 \div 2 = 8, since 2 \times 8 = 16. The dividend is 16, the divisor is 2 and the quotient is 8.
- 5 cards. Counting by eights, 8, 16, 24, 32, 40 is five eights, so 40 \div 8 = 5.
- 7 key rings. 7 \times 3 = 21, so 21 \div 3 = 7.
- 8 bundles. 8 \times 8 = 64, so 64 \div 8 = 8.
- (a) 10 packs; this is grouping, since the size of each pack is known. (b) 10 each; this is sharing, since the number of tables is known. Both are 50 \div 5 = 10, because 5 \times 10 = 50.
- One story: 54 beads are threaded 6 to a bracelet; how many bracelets? The answer is 9 bracelets, since 9 \times 6 = 54, so 54 \div 6 = 9.
- 24 - 4 = 20, 20 - 4 = 16, 16 - 4 = 12, 12 - 4 = 8, 8 - 4 = 4, 4 - 4 = 0. Six subtractions, so 24 \div 4 = 6.
- 7 puppets. 49 - 7 = 42, 42 - 7 = 35, 35 - 7 = 28, 28 - 7 = 21, 21 - 7 = 14, 14 - 7 = 7, 7 - 7 = 0: seven subtractions, so 49 \div 7 = 7.
- 7 jumps: 70, 60, 50, 40, 30, 20, 10, 0. They show 70 \div 10 = 7.
- 14 - 2 = 12, 12 - 2 = 10, 10 - 2 = 8, 8 - 2 = 6, 6 - 2 = 4, 4 - 2 = 2, 2 - 2 = 0. Seven subtractions, so 14 \div 2 = 7.
- 9 handfuls. 72 - 8 = 64, then 56, 48, 40, 32, 24, 16, 8, 0: nine subtractions in all. Check: 9 \times 8 = 72.
- \square = 9, because 9 \times 9 = 81. It answers 81 \div 9 = 9.
- \square = 18. The total is 2 groups of 9, so \square = 2 \times 9 = 18. Check: 18 \div 2 = 9.
- 24 \div 3 = 8: the 24 buttons split into 3 equal rows have 8 in each row (sharing). 24 \div 8 = 3: each row is a group of 8, and there are 3 rows (grouping). Both come from 3 \times 8 = 24.
- \square = 10, because 40 split into 10 equal groups has 4 in each: 10 \times 4 = 40.
- 3 \times \square = 15, so \square = 15 \div 3 = 5 buttons on each card. Check: 3 \times 5 = 15.
- \square \times 5 = 25, so \square = 25 \div 5 = 5 strands. Check: 5 \times 5 = 25.
- Set the beads in 5 rows of 7. Each row is one of 5 equal groups, with 7 in it, which shows 35 \div 5 = 7. Each column is a group of 5, and there are 7 columns, which shows 35 \div 7 = 5. One picture shows both because both divisions come from the one fact 5 \times 7 = 35, read by rows or by columns.
- 2 friends get 6 each, 3 friends get 4 each, 4 friends get 3 each, and 6 friends get 2 each. The total stays 12, so cutting it into more equal groups leaves less in each group; in every case friends times share is 12.
- No. 10 \div 2 shares 10 things between 2 people, and each gets 5. 2 \div 10 shares 2 things among 10 people, and there are not enough to give each person one whole thing, so it has no whole-number answer. Division is not commutative.
- Each subtraction of 4 takes away exactly the buttons for one card. So the number of subtractions it takes to reach 0 is the number of cards filled. Both count the same groups of 4, and both give 5, since 5 \times 4 = 20.