Grade 3 · 3.OA.C.7

Division facts

Fifty-four books go onto 9 shelves, the same number on each. Is there a way to know how many go on a shelf without dealing them out one at a time?

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Multiplication factsGrade 3 What division meansGrade 3

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Doubling, halving and quick productsGrade 4 Equal group problemsGrade 3 Multiplication and division undo each otherGrade 3

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Division facts

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When you reach for it

What division means gave division its two meanings. Sharing asks how many are in each group; grouping asks how many groups there are. Either way the answer could be found by dealing things out, or by taking the same number away again and again. That always works, and it is slow.

A division fact is a division you can know rather than work out, the same way 9 \times 6 = 54 is a fact you know rather than count. The division facts are the quotients that match the Multiplication facts: a total up to 100, divided by a number from 1 to 10, with a whole number answer from 0 to 10. Examples are 54 \div 9 = 6, 48 \div 8 = 6 and 7 \div 7 = 1.

Reach for them whenever a total has to be split evenly. Books shelved the same number to a shelf, chairs set out in equal rows for a music recital, cards dealt to players, a class split into teams. Reach for them also when checking a multiplication, and later inside every longer division, where each step is one of these facts.

The key idea, in one sentence: every division fact is a multiplication fact with one factor missing. To find 54 \div 9, ask “nine times what makes fifty-four?” The child already knows the multiplication facts, so the division facts come with them.

This lesson takes four days, one part each. Part 1 turns a division into a missing factor. Part 2 gives ways to find the missing factor when it does not come to mind at once. Part 3 covers the special divisions with 1 and 0. Part 4 puts the box in other places and uses the facts to test which equal splits are possible.

The procedure

The reasoning behind every part is the statement from What division means: for whole numbers a, b and c, with b not zero,

a \div b = c \quad \text{exactly when} \quad b \times c = a

Part 1: a division is a missing factor

At the class library, Zenzi has 54 books to put on 9 shelves, the same number on each. This is sharing: 9 groups, and the size of each group is what we want. Written as a division it is 54 \div 9. Written as multiplication it is

9 \times \square = 54

because 9 shelves of \square books each must make the 54 books. Now the question is one the child has met before. Which number times 9 makes 54? Running through the nines, 9, 18, 27, 36, 45, 54, the sixth one is 54, and a child who knows 9 \times 6 = 54 does not even need to run through them. So \square = 6 and

54 \div 9 = 6

Figure 1Fifty-four books on 9 shelves. The picture is 9 \times 6 = 54, so 54 \div 9 = 6.

The picture explains why this is allowed. Nine shelves of 6 books is an array of 9 rows of 6 (see Arrays). The same array is both facts at once: read as a multiplication, it says 9 \times 6 = 54; read as a sharing, it says 54 books on 9 shelves puts 6 on each. Division did not need a new picture. It needed the old picture with a different number hidden.

A grouping story turns into a missing factor in the same way. If the books go 9 to a shelf and we ask how many shelves they fill, the question is \square \times 9 = 54: some number of shelves, 9 books each, 54 in all. The box sits in the first place instead of the second, but by the Properties of multiplication \square \times 9 and 9 \times \square are the same product, so the same fact answers it: 6 shelves.

That gives the habit that runs through the whole lesson. To divide, say the division as a multiplication with a box: “divisor times what is the dividend?” Then the answer comes with its own check, because multiplying the quotient by the divisor must give back the dividend. If 9 \times 6 were not 54, the 6 would be wrong.

Part 2: finding the missing factor

Some facts come to mind at once. Others need a moment’s work, and there are honest ways to do that work without dealing out objects.

Count by the divisor. For 48 \div 8, ask 8 \times \square = 48 and count by eights, keeping track of how many eights have been said: 8, 16, 24, 32, 40, 48. That is six eights, so 48 \div 8 = 6. On a number line each count is one jump of 8, and the number of jumps is the quotient.

Figure 2Counting by eights from 0 reaches 48 after six jumps, so 48 \div 8 = 6.

Counting by the divisor is grouping in disguise: each jump is one group of 8, and we count groups until the total is reached.

Start from a fact you know. The fives and tens facts are the easiest to know, so they make good starting points. For 48 \div 8 again: 8 \times 5 = 40, which is 8 short of 48. Eight more is one more eight, so add one group: 5 + 1 = 6 groups. The move is allowed because adding a whole group of 8 to 5 groups of 8 makes one more group of 8 (the The distributive property says the same in symbols: 8 \times 6 = 8 \times 5 + 8 \times 1). Tens work the same way from above: 8 \times 10 = 80 is 32 too many, which is four eights, so 10 - 4 = 6.

Read the times table backward. A multiplication table holds every product. To find 54 \div 9, go along the row for 9 until 54 appears, then read the number at the top of that column.

9 \times	1	2	3	4	5	6
product	9	18	27	36	45	54

The 54 sits under the 6. Reading the table forward is multiplying; reading it backward is dividing. This is the plainest view of why there is nothing new to memorize: the division facts are the multiplication table read the other way.

Which way to use is a choice, and a child with a strong fact like 8 \times 6 = 48 skips all of them. The aim over time is the same as for multiplication: to know the fact.

Part 3: dividing with one and with zero

Three kinds of division come out the same way every time, and each one is explained by its missing factor.

A number divided by itself is 1. Vibeke has 7 chairs to set out in 7 equal rows for the recital. Each row gets 1 chair. As a missing factor, 7 \times \square = 7, and only 1 works, because 7 \times 1 = 7. The same is true for any number of chairs, as long as there is at least one: n things in n equal rows is 1 in each row.

A number divided by 1 is the number itself. If Vibeke puts all 7 chairs in 1 row, that row holds all 7. As a missing factor, 1 \times \square = 7, and 1 \times 7 = 7, so 7 \div 1 = 7. One group holds the whole total.

Figure 3The same 7 chairs two ways. In 7 rows there is 1 chair in each row, so 7 \div 7 = 1. In 1 row all 7 chairs sit together, so 7 \div 1 = 7.

Zero divided by a number is 0. If there are no books to share among 5 shelves, every shelf gets none. As a missing factor, 5 \times \square = 0, and the only number that works is 0, because 5 groups of anything more than 0 would be more than 0. So 0 \div 5 = 0.

Figure 4No books shared among 5 shelves leaves every shelf empty, so 0 \div 5 = 0.

In symbols, for any whole number n:

\begin{aligned} n \div 1 &= n \\ 0 \div n &= 0 \quad (n \neq 0) \\ n \div n &= 1 \quad (n \neq 0) \end{aligned}

The order matters in the last two. 0 \div 5 shares nothing among five shelves and is 0. The other way round, dividing a number by zero, has no answer at all, and why is the business of Multiplication and division undo each other.

Part 4: the box can go anywhere, and some splits do not work

So far the unknown was the quotient. A division fact can hide any one of its three numbers. In \square \div 4 = 9, the dividend is missing: some total split into 4 equal groups gives 9 in each, so the total is 4 groups of 9, \square = 4 \times 9 = 36. In 32 \div \square = 4, the divisor is missing: 32 split into \square equal groups gives 4 in each, so \square \times 4 = 32 and \square = 8. Each one turns into a multiplication fact with a box, just as in Part 1.

The facts also say when a split does not work. Twenty-four chairs cannot be set in 5 equal rows, because no whole number times 5 makes 24: 5 \times 4 = 20 is too few and 5 \times 5 = 25 is too many. When a total falls between two products like that, the division does not come out even. That case belongs to Remainders and what they mean; here it tells us the split is impossible.

Worked examples

Examples 1 and 2 go with Part 1, 3 and 4 with Part 2, 5 and 6 with Part 3, and 7 and 8 with Part 4.

Example 1

Fifty-four books on nine shelves

Zenzi shelves 54 library books on 9 shelves, the same number on each shelf. How many books go on each shelf?

Solution

This is sharing: the number of groups (9 shelves) is known, and the size of each group is not. So the question is 54 \div 9.

Say it as a multiplication with a box. Nine shelves of \square books each make 54 books:

9 \times \square = 54

The nines go 9, 18, 27, 36, 45, 54, and 54 is the sixth, so 9 \times 6 = 54 and \square = 6. Each shelf holds 6 books, and 54 \div 9 = 6.

Check: 9 shelves of 6 books is 9 \times 6 = 54 books, the number Zenzi started with.

Example 2

Saying three divisions as multiplications

Write each division as a multiplication with a box, then find it: 42 \div 7, 27 \div 9 and 70 \div 10.

Solution

Each division asks “the divisor times what is the dividend?”

42 \div 7 asks 7 \times \square = 42. Since 7 \times 6 = 42, \square = 6, so 42 \div 7 = 6.
27 \div 9 asks 9 \times \square = 27. Since 9 \times 3 = 27, \square = 3, so 27 \div 9 = 3.
70 \div 10 asks 10 \times \square = 70. Ten times a number is that number with a zero on the end, and 70 is 7 with a zero, so \square = 7 and 70 \div 10 = 7.

In each case the multiplication fact is the check: multiplying the quotient by the divisor gives back the dividend.

Example 3

Forty-eight chairs in rows of eight

Vibeke sets out 48 chairs for the recital in rows of 8. How many rows does she make? Find it by counting by eights, then again from the fact 8 \times 5 = 40.

Solution

This is grouping: the size of each group (8 chairs) is known, and the number of groups is not. So the question is 48 \div 8, which asks \square \times 8 = 48, the same product as 8 \times \square.

Counting by eights and keeping a tally: 8, 16, 24, 32, 40, 48. That is 6 eights, so 48 \div 8 = 6. The number line in Part 2 shows these six jumps.

From a known fact: 5 rows of 8 would be 5 \times 8 = 40 chairs. That is 48 - 40 = 8 chairs short, and 8 chairs is 1 more row of 8. Adding that row gives 5 + 1 = 6 rows.

Both ways give 6 rows. Check: 6 rows of 8 chairs is 6 \times 8 = 48.

Example 4

Sixty-three books from a tens fact

Torben shares 63 books equally among 7 shelves. How many books go on each shelf? Use the fact 7 \times 10 = 70.

Solution

The question is 63 \div 7, which asks 7 \times \square = 63.

If each shelf held 10 books, the 7 shelves would hold 7 \times 10 = 70. That is 70 - 63 = 7 books too many. Seven books is exactly one book for each of the 7 shelves, so take one book off every shelf: each shelf holds 10 - 1 = 9.

So 63 \div 7 = 9. Check: 7 \times 9 = 63.

The tens fact worked well here because 63 is just one group of 7 short of 70.

Example 5

Seven chairs, two ways

Vibeke has 7 chairs. How many chairs are in each row if she sets them in 7 equal rows? In 1 row?

Solution

In 7 equal rows, the question is 7 \div 7, which asks 7 \times \square = 7. Seven groups of 1 make 7, and seven groups of 2 would already make 14, so \square = 1. There is 1 chair in each row: 7 \div 7 = 1.

In 1 row, the question is 7 \div 1, which asks 1 \times \square = 7. One group of \square is just \square, so \square = 7. All 7 chairs are in the one row: 7 \div 1 = 7.

The figure in Part 3 shows both arrangements of the same 7 chairs. The two answers are not coincidences: any number of chairs (at least one) in that many rows is 1 per row, and any number in 1 row is all of them.

Example 6

Nothing to share

No books came back to the library this morning, and the returns are shared equally among 5 shelves. How many returned books go on each shelf? Compare this with 5 returned books on the same 5 shelves.

Solution

With no books, the question is 0 \div 5, which asks 5 \times \square = 0. Five groups of 1 would already be 5, so no number above 0 works. Five groups of 0 is 0, so \square = 0. Each shelf gets no books: 0 \div 5 = 0.

With 5 books, the question is 5 \div 5, which asks 5 \times \square = 5, and 5 \times 1 = 5. Each shelf gets 1 book: 5 \div 5 = 1.

The two divisions look alike but say different things. Sharing nothing leaves nothing on each shelf; sharing a number among that many shelves leaves one on each.

Example 7

A box in the other places

Find the number in each box, and say what each one means in a recital story. (a) \square \div 6 = 7. (b) 45 \div \square = 5.

Solution

(a) The dividend is missing. A total split into 6 equal rows gives 7 chairs in each row. Six rows of 7 is 6 \times 7 = 42, so \square = 42. In a story: 42 chairs in 6 equal rows have 7 in each row. Check: 42 \div 6 = 7 because 6 \times 7 = 42.

(b) The divisor is missing. Forty-five chairs split into \square equal rows give 5 in each row, so \square \times 5 = 45. Counting by fives, 5, 10, 15, 20, 25, 30, 35, 40, 45 is nine fives, so \square = 9. In a story: 45 chairs set out 5 to a row fill 9 rows. Check: 9 \times 5 = 45.

Example 8

Which rows work for twenty-four chairs?

Vibeke wants 24 chairs in equal rows, with at least 2 rows and at most 9 rows, and no more than 9 chairs in any row. Which numbers of rows can she use?

Solution

For each number of rows from 2 to 9, ask whether that number times some whole number makes 24, and whether that whole number is 9 or less.

Rows	Missing factor	Works?
2	2 \times 12 = 24	no, 12 chairs in a row is more than 9
3	3 \times 8 = 24	yes, 8 in each row
4	4 \times 6 = 24	yes, 6 in each row
5	5 \times 4 = 20, 5 \times 5 = 25	no, 24 falls between
6	6 \times 4 = 24	yes, 4 in each row
7	7 \times 3 = 21, 7 \times 4 = 28	no, 24 falls between
8	8 \times 3 = 24	yes, 3 in each row
9	9 \times 2 = 18, 9 \times 3 = 27	no, 24 falls between

Vibeke can use 3, 4, 6 or 8 rows: 24 \div 3 = 8, 24 \div 4 = 6, 24 \div 6 = 4 and 24 \div 8 = 3.

Notice the pairs. Three rows of 8 and eight rows of 3 are the same array turned on its side, and so are 4 rows of 6 and 6 rows of 4.

Boundaries

The divisor is never zero. Every statement here says “b not zero”. A division by zero has no answer; Multiplication and division undo each other shows why.
Dividing a number by itself gives 1 only when the number is not zero. 0 \div 0 is a division by zero and has no answer, so “n \div n = 1” needs n \neq 0.
A division fact comes out even. The dividend must be a product of the divisor and a whole number. 24 \div 5 is not a fact: 24 falls between 5 \times 4 and 5 \times 5. Divisions like that leave something over, which is the subject of Remainders and what they mean.
The facts stop at the table. A division fact has a divisor from 1 to 10 and a quotient from 0 to 10, so its dividend is at most 10 \times 10 = 100. A division such as 96 \div 4 is not a single fact; it is built from facts in Dividing by a single digit.
Order matters. 0 \div 5 = 0, but 5 \div 0 has no answer. 7 \div 1 = 7, but 1 \div 7 is not a whole number. Swapping the dividend and divisor changes the question.

Where this goes next

The next lesson, Multiplication and division undo each other, makes the link this lesson leans on into the main subject: multiplying by a number and dividing by it undo each other, and the multiplication and division facts that share three numbers belong together. After that, Equal group problems uses the facts to solve stories with the unknown in any place, which Part 4 started. Further on, Doubling, halving and quick products reshapes a division into an easier one, and Dividing by a single digit divides totals far past 100 by using one division fact for each place.

Exercises

Part 1: a division is a missing factor

Write 20 \div 4 as a multiplication with a box, then find it.
Write 45 \div 9 as a multiplication with a box, then find it.
Write 60 \div 10 as a multiplication with a box, then find it.
Zenzi puts 18 books equally on 3 shelves. Write a multiplication with a box, and find how many books are on each shelf.
Vibeke sets out 21 chairs in rows of 7. Write a multiplication with a box, and find how many rows she makes.
Is 40 \div 5 = 9? Use a multiplication fact to decide, and give the right quotient if it is not.

Part 2: finding the missing factor

Find 24 \div 4 by counting by fours. Say how many fours you counted.
Find 35 \div 7 by counting by sevens.
Find 54 \div 6 starting from the fact 6 \times 10 = 60.
Find 72 \div 9 starting from the fact 9 \times 10 = 90.
Vibeke sets out 64 chairs in rows of 8. How many rows?
Torben shelves 36 books, 6 on each shelf. How many shelves does he fill?

Part 3: dividing with one and with zero

Find 9 \div 9 and say the multiplication that proves it.
Find 6 \div 1 and say the multiplication that proves it.
Find 0 \div 7 and say the multiplication that proves it.
Find 10 \div 10 and 1 \div 1.
Vibeke puts 8 chairs in 1 row. How many chairs are in the row? Write the division.
(a) No new books arrive, and the new books are shared among 4 shelves. How many go on each shelf? (b) Four new books are shared among the same 4 shelves. How many go on each shelf? Write a division for each.

Part 4: the box can go anywhere

Find the number in the box: \square \div 7 = 8.
Find the number in the box: 40 \div \square = 8.
Find the number in the box: \square \div 9 = 0.
Zenzi wants 18 chairs in equal rows, with at least 2 rows and at most 9 rows, and no more than 9 chairs in any row. Which numbers of rows can she use?

Explaining and reasoning

Explain, using chairs in rows, why any number of chairs other than zero divided by itself is 1.
Are 0 \div 6 and 6 \div 6 equal? Explain with a sharing story for each.
Explain why a child who knows every multiplication fact up to 10 \times 10 also knows every division fact, without learning anything new.

Answers

4 \times \square = 20. Since 4 \times 5 = 20, 20 \div 4 = 5.
9 \times \square = 45. Since 9 \times 5 = 45, 45 \div 9 = 5.
10 \times \square = 60. Since 10 \times 6 = 60, 60 \div 10 = 6.
3 \times \square = 18: 3 shelves of \square books make 18. Since 3 \times 6 = 18, there are 6 books on each shelf, and 18 \div 3 = 6.
\square \times 7 = 21: \square rows of 7 chairs make 21. Since 3 \times 7 = 21, she makes 3 rows, and 21 \div 7 = 3.
No. 5 \times 9 = 45, not 40. Since 5 \times 8 = 40, the right quotient is 40 \div 5 = 8.
4, 8, 12, 16, 20, 24 is 6 fours, so 24 \div 4 = 6.
7, 14, 21, 28, 35 is 5 sevens, so 35 \div 7 = 5.
6 \times 10 = 60 is 60 - 54 = 6 too many, which is one group of 6. So take one group off: 10 - 1 = 9, and 54 \div 6 = 9. Check: 6 \times 9 = 54.
9 \times 10 = 90 is 90 - 72 = 18 too many, which is two groups of 9. So 10 - 2 = 8, and 72 \div 9 = 8. Check: 9 \times 8 = 72.
8 rows. The question is 8 \times \square = 64, and 8 \times 8 = 64, so 64 \div 8 = 8.
6 shelves. The question is \square \times 6 = 36, and 6 \times 6 = 36, so 36 \div 6 = 6.
9 \div 9 = 1, because 9 \times 1 = 9.
6 \div 1 = 6, because 1 \times 6 = 6.
0 \div 7 = 0, because 7 \times 0 = 0.
10 \div 10 = 1, because 10 \times 1 = 10; and 1 \div 1 = 1, because 1 \times 1 = 1.
All 8 chairs are in the row: 8 \div 1 = 8, because 1 \times 8 = 8.
(a) 0 books on each shelf: 0 \div 4 = 0, because 4 \times 0 = 0. (b) 1 book on each shelf: 4 \div 4 = 1, because 4 \times 1 = 4.
\square = 56. A total split into 7 equal groups with 8 in each is 7 \times 8 = 56. Check: 56 \div 7 = 8.
\square = 5. The question is \square \times 8 = 40, and 5 \times 8 = 40. Check: 40 \div 5 = 8.
\square = 0. A total split into 9 equal groups with 0 in each is 9 \times 0 = 0. Check: 0 \div 9 = 0.
2, 3, 6 or 9 rows. 2 \times 9 = 18, 3 \times 6 = 18, 6 \times 3 = 18 and 9 \times 2 = 18 all work with 9 or fewer chairs in a row. The others do not come out even: 18 falls between 4 \times 4 = 16 and 4 \times 5 = 20, between 5 \times 3 = 15 and 5 \times 4 = 20, between 7 \times 2 = 14 and 7 \times 3 = 21, and between 8 \times 2 = 16 and 8 \times 3 = 24.
Set out the chairs in as many rows as there are chairs. Put 1 chair in each row: that uses exactly all the chairs, one row each. Two in each row would need twice as many chairs as there are. So each row holds 1 chair. In multiplication, n \times 1 = n, so n \div n = 1 for any n that is not zero.
No. 0 \div 6 = 0: sharing no stickers among 6 friends gives each friend none. 6 \div 6 = 1: sharing 6 stickers among 6 friends gives each friend 1. The divisor is the same, but the totals being shared are different.
Every division fact a \div b asks “b times what is a?” With a divisor from 1 to 10 and a quotient from 0 to 10, that is a multiplication fact up to 10 \times 10 with one factor hidden. Knowing the multiplication fact means knowing the hidden factor, which is the quotient. The times table read backward holds every division fact.