Grade 3 · 3.OA.A.3

Equal group problems

Six bags hold four favors each, or fifty-six slices are shared over eight tables. How do you know whether a story asks you to multiply or to divide?

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Division factsGrade 3

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Problems using several operationsGrade 3 Using a letter for the unknown numberGrade 3

Where you are 16 in Multiplication and division

Number9

Arithmetic5

Multiplication and division16

▴ 9 above Properties of multiplication Remainders and what they mean

Equal group problems

Long division4 Multiplication and division undo each other ▾ Show all 16 (2 more)

When you reach for it

An equal groups story is a story about groups that all hold the same amount. Every story of this kind has three numbers in it:

the number of groups (for example, 6 goody bags),
the size of each group (4 party favors in each bag),
the total (24 party favors in all).

The story tells two of the three and asks for the third. That is the whole skill: picture the groups, see which of the three numbers is missing, and write one equation with a box where the missing number goes. Then find the box with a fact.

The pieces are already in place. What multiplication means gave multiplication its meaning as equal groups. What division means gave division its two questions, how many in each group and how many groups. Division facts made every division a multiplication with one factor missing, and Multiplication and division undo each other showed that multiplying and dividing by the same number undo each other. This lesson puts all of that to work on stories.

Reach for it whenever a story has equal groups in it: party favors packed into bags, marshmallows threaded onto sticks, pizza slices shared over tables, tents or chairs set out in equal rows, a roll of streamer cut into equal pieces, water carried in buckets of the same size. The groups can be bags, rows of an array, or equal measured amounts such as meters or liters.

The lesson takes four days, one part each. Part 1 finds a missing total. Part 2 finds a missing group size or a missing number of groups. Part 3 reads the same story in an array. Part 4 uses measured amounts, and puts the box in all three places with the same three numbers.

The procedure

One equation sits behind every equal groups story:

\text{groups} \times \text{size of each} = \text{total}

The habit of writing the number of groups first and the size second is only a reading habit, so that 6 \times 4 reads “6 groups of 4”. By the Properties of multiplication, 4 \times 6 is the same product, so the order never changes an answer. It keeps the equation matched to the story.

Which number the box stands for depends only on which number the story leaves out:

What is missing	Equation with a box	The same question as a division
the total	6 \times 4 = \square	none needed
the size of each group	8 \times \square = 56	56 \div 8 = \square
the number of groups	\square \times 5 = 35	35 \div 5 = \square

Part 1: the total is missing

Kazuki is filling goody bags for a party. He has 6 bags and puts 4 party favors in each. How many party favors does he need?

Start with a picture. It does not need to be art: six squares for the bags and four dots in each are enough.

Figure 1Six goody bags with 4 party favors in each. The total is missing, so the equation is 6 \times 4 = \square.

The picture shows what is known and what is not. The number of groups is known (6 bags) and the size of each group is known (4 favors). The total is what the story asks for. So the box goes on the total:

6 \times 4 = \square

This is multiplication because putting together equal groups is exactly what multiplication counts. Adding 4 + 4 + 4 + 4 + 4 + 4 gives the same answer the long way. Counting by fours, 4, 8, 12, 16, 20, 24, also works; a child who knows the fact 6 \times 4 = 24 needs neither. So \square = 24, and the answer goes back into the story as a sentence: Kazuki needs 24 party favors.

That last step matters. The number 24 is a count of favors, not of bags. Saying what the number counts is how a reader checks that the equation answered the question the story asked.

A tape diagram is a second kind of drawing that works for every equal groups story. It is a bar cut into equal parts, one part for each group. For the goody bags it is a bar of 6 equal parts, each marked 4, with a question mark over the whole bar. The bar is quicker to draw than dots, and it still works when the numbers are too big to draw one dot each.

Part 2: the size of each group or the number of groups is missing

At a campfire, Mitsuki has 35 marshmallows to roast, 5 on each stick. How many sticks does she need?

Now the size of each group is known (5 marshmallows on a stick) and the total is known (35). The number of groups is missing, so the box goes in the first place:

\square \times 5 = 35

Counting by fives and keeping a tally, 5, 10, 15, 20, 25, 30, 35 is seven fives, so \square = 7. Mitsuki needs 7 sticks. As a division the same question is 35 \div 5 = \square, the grouping meaning of division: how many fives are in 35?

Figure 2Thirty-five marshmallows, 5 to a stick, fill 7 sticks. The number of groups was missing, so the equation was \square \times 5 = 35.

At a pizza party, Noriko shares 56 slices of pizza equally over 8 tables. How many slices go on each table?

This time the number of groups is known (8 tables) and the total is known (56 slices). The size of each group is missing, so the box goes in the second place:

8 \times \square = 56

Figure 3A tape diagram for 56 slices shared over 8 tables. The whole is known and each part is the missing number, so the equation is 8 \times \square = 56.

The tape diagram shows the whole bar as 56 and asks what each of its 8 equal parts holds. The fact 8 \times 7 = 56 fills the box: \square = 7, so each table gets 7 slices. As a division it is 56 \div 8 = \square, the sharing meaning: 56 shared into 8 equal parts.

Both stories are divisions, but they ask different questions, and the picture shows which. In the marshmallow story the groups are being counted; in the pizza story the groups are being filled.

What decides the operation is the situation, not any single word in the story. The word “each” appears in Kazuki’s story (“4 party favors in each”) and in Noriko’s (“how many slices go on each table”), and one is a multiplication while the other is a division. What tells them apart is which of the three numbers is missing. When the total is missing, multiply. When the total is known and one of the other two is missing, the question is a multiplication with a box in a factor’s place, which is the same as a division.

Every division answer carries its own check: multiply back. Seven sticks of 5 is 7 \times 5 = 35 marshmallows, and 8 tables of 7 is 8 \times 7 = 56 slices, which are the totals the stories began with.

Part 3: equal groups in an array

An array is equal groups set out in rows (see Arrays). Each row is one group, so the number of rows is the number of groups and the number in each row is the size of each group.

Figure 4A sheet pizza cut into 5 rows of 6 squares. Each row is a group, so the pieces number 5 \times 6 = 30.

A sheet pizza cut into 5 rows of 6 squares has 5 \times 6 = \square pieces, and \square = 30. The same rectangle is also 6 columns of 5, so 6 \times 5 = 30 describes it just as well. Choosing rows or columns as the groups is a choice of how to read the picture; the total does not change.

An array story can hide any of its three numbers too. If the total and the number of rows are known, the missing number is how many in each row, a sharing division. If the total and the number in each row are known, the missing number is how many rows, a grouping division. Example 6 does both with the same chairs.

Part 4: measured amounts, and the box in every place

The groups in a story do not have to be things counted one by one. A group can be a measured amount: a length in meters, an amount of water in liters, a mass in kilograms.

Noriko hangs 8 streamers across the party room, each 3 meters long. How many meters of streamer does she use? Each streamer is a group of 3 meters, and there are 8 groups, so 8 \times 3 = \square and \square = 24. She uses 24 meters.

Figure 5Eight streamers, each 3 meters long, laid end to end reach 8 \times 3 = 24 meters.

Laid end to end on a meter scale, the eight streamers make eight equal jumps of 3, and the last one ends at 24. That is the same picture as counting by threes.

Notice which numbers carry a unit. The size of each group (3 meters) and the total (24 meters) are both lengths. The number of groups (8) is a plain count of streamers. That pattern holds in every measured story, and it is a quick check on an equation: the total and the size of each group are measured in the same unit, while the number of groups counts how many equal amounts there are. Liters and kilograms work the same way, as in Story problems with grams, kilograms and liters.

Finally, any three numbers that make a multiplication fact give three different stories, one with the box in each place. With 4, 9 and 36, a story can ask for the total, for the size of each group, or for the number of groups. Example 8 writes all three.

Later the box is replaced by a letter, such as n, standing for the number the story asks for; that is Using a letter for the unknown number. The thinking does not change.

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

Party favors in goody bags

Kazuki fills 6 goody bags with 4 party favors in each bag. How many party favors does he use? Draw a picture and write an equation with a box.

Solution

The picture is six bags with four dots in each, as in the figure in Part 1.

The number of groups (6 bags) and the size of each group (4 favors) are both given. The story asks for the total, so the box goes on the total:

6 \times 4 = \square

Six groups of 4: counting by fours, 4, 8, 12, 16, 20, 24. So 6 \times 4 = 24 and \square = 24.

Kazuki uses 24 party favors. The answer counts favors, which is what the story asked about.

Example 2

Writing a story for an equation

Write an equal groups story about a campfire for 3 \times 9 = \square, then find the number in the box.

Solution

The equation says 3 groups of 9 with the total missing. So the story needs 3 groups, 9 in each group, and a question about the total.

One story: Mitsuki, Noriko and Kazuki each gather 9 sticks of kindling for the campfire. How many sticks do they gather in all? There are 3 campers, which is the number of groups, and 9 sticks each, which is the size of each group.

Three nines: 9, 18, 27. So 3 \times 9 = 27 and \square = 27. They gather 27 sticks.

Any story works if it has 3 equal groups of 9 and asks for the total. A story that asked “how many campers?” would be putting the box somewhere else.

Example 3

Marshmallows on sticks

At the campfire, Mitsuki has 35 marshmallows and puts 5 on each stick. How many sticks does she fill?

Solution

The size of each group is given (5 marshmallows on a stick) and so is the total (35). The number of groups is missing, so the box goes first:

\square \times 5 = 35

Which number times 5 makes 35? Counting by fives with a tally, 5, 10, 15, 20, 25, 30, 35 is 7 fives, so 7 \times 5 = 35 and \square = 7. As a division, 35 \div 5 = 7.

Mitsuki fills 7 sticks. Check: 7 sticks with 5 on each is 7 \times 5 = 35 marshmallows, all of them.

Example 4

Pizza slices shared over tables

At the pizza party, Noriko shares 56 slices equally over 8 tables. How many slices go on each table? Draw a tape diagram.

Solution

The tape diagram is a bar for the 56 slices, cut into 8 equal parts, one for each table, with a question mark in each part. That is the figure in Part 2.

The number of groups (8 tables) and the total (56 slices) are given. The size of each group is missing, so the box goes second:

8 \times \square = 56

Eight times what makes 56? The fact 8 \times 7 = 56 answers it. If that fact does not come to mind, start from 8 \times 5 = 40: that leaves 56 - 40 = 16 slices, which is 2 more for each of the 8 tables, so each table gets 5 + 2 = 7. Either way \square = 7, and as a division 56 \div 8 = 7.

Each table gets 7 slices. Check: 8 \times 7 = 56.

Compare Example 3. Both are divisions, but there the groups were counted (how many sticks) and here the groups are filled (how many on each table).

Example 5

Tents in rows at the campsite

The tents at a campsite stand in 4 rows with 8 tents in each row. How many tents are there? Write the equation two ways, once reading rows and once reading columns.

Solution

The tents form an array. Reading the rows as the groups, there are 4 groups of 8:

4 \times 8 = \square

Counting by eights, 8, 16, 24, 32, so \square = 32.

Reading the columns as the groups instead, there are 8 columns with 4 tents in each, so 8 \times 4 = \square. By the Properties of multiplication this is the same product, 32.

There are 32 tents. The two equations read the same array two ways and agree.

Example 6

Camp chairs around the campfire

Kazuki sets out 42 camp chairs facing the campfire in 6 equal rows. How many chairs are in each row? If he set the same 42 chairs in rows of 7 instead, how many rows would he make?

Solution

First question: the number of rows (6) and the total (42) are given, and the number in each row is missing. That is the size of each group:

6 \times \square = 42

The fact 6 \times 7 = 42 fills the box, so there are 7 chairs in each row, and 42 \div 6 = 7.

Second question: now the number in each row (7) and the total (42) are given, and the number of rows is missing:

\square \times 7 = 42

The same fact, 6 \times 7 = 42, fills this box with 6. He would make 6 rows, and 42 \div 7 = 6.

Both arrangements are the same 6 by 7 array. One question asked how many chairs stand in a row, the other how many rows there are.

Example 7

A roll of streamer and buckets of water

(a) Noriko cuts a 40-meter roll of streamer into 5 equal pieces. How long is each piece? (b) After the campfire, Mitsuki pours 18 liters of water on the ashes, carried in buckets that each hold 3 liters. How many full buckets does she carry?

Solution

(a) The number of groups (5 pieces) and the total (40 meters) are given. The length of each piece is the size of each group, so

5 \times \square = 40

Since 5 \times 8 = 40, \square = 8. Each piece is 8 meters long, and 40 \div 5 = 8. The answer is a length, in the same unit as the roll.

(b) The size of each group (3 liters in a bucket) and the total (18 liters) are given. The number of buckets is missing:

\square \times 3 = 18

Counting by threes, 3, 6, 9, 12, 15, 18 is 6 threes, so \square = 6. She carries 6 buckets, and 18 \div 3 = 6. This answer is a count of buckets, not a number of liters.

Check both: 5 \times 8 = 40 meters, and 6 \times 3 = 18 liters.

Example 8

One fact, three stories

At the pizza party, cups go on tables: 4 tables, 9 cups on a table, and 36 cups in all. Write three stories with these numbers, one with the box on the total, one on the size of each group, and one on the number of groups. Solve each.

Solution

All three stories rest on the one fact 4 \times 9 = 36: 4 groups of 9 make 36.

Box on the total. Kazuki puts 9 cups on each of 4 tables. How many cups does he put out? 4 \times 9 = \square, and \square = 36 cups.

Box on the size of each group. Kazuki has 36 cups to share equally over 4 tables. How many cups go on each table? 4 \times \square = 36, and since 4 \times 9 = 36, \square = 9 cups. As a division, 36 \div 4 = 9.

Box on the number of groups. Kazuki has 36 cups and puts 9 on each table. How many tables does he use? \square \times 9 = 36, and \square = 4 tables. As a division, 36 \div 9 = 4.

The numbers never changed. Only the question changed, and with it the place of the box.

Boundaries

The groups must be equal. If one bag gets 3 favors and another 5, the story is not an equal groups story, and multiplying the number of bags by one bag’s amount gives a wrong total. Such a story is solved by adding.
Every division here comes out even. The total is always the number of groups times a whole number. If 40 marshmallows go 6 to a stick, no whole number of sticks uses them all, because 6 \times 6 = 36 and 6 \times 7 = 42. A story with something left over belongs to Remainders and what they mean.
One step only. Each story here has one equal groups situation and one missing number. A story that needs a second step, such as buying the favors and then giving some away, is Problems using several operations.
Not comparison stories. A story that compares two amounts by saying one is a number of times the other has no groups being filled or counted. That is a different situation, met in Times as many.
Within the facts. The size of each group and the number of groups here are numbers up to 10, so every total is at most 100 and every missing number is one fact away. Larger totals need Dividing by a single digit.
Units match where they should. The total and the size of each group share a unit (meters with meters, liters with liters); the number of groups is a count. An equation that multiplies meters by meters is not an equal groups story.

Where this goes next

The box in this lesson becomes a letter in Using a letter for the unknown number, where the same equations are written as 6 \times 4 = n or 8 \times n = 56. Stories that join an equal groups step to an adding or subtracting step come in Problems using several operations. Further on, Times as many meets a new kind of multiplication story, comparing one amount with another, and Remainders and what they mean takes on the division stories that do not come out even.

Exercises

Part 1: the total is missing

Kazuki fills 5 goody bags with 8 stickers in each. Write an equation with a box, and find how many stickers he uses.
At the campfire, 8 campers each roast 2 hot dogs. How many hot dogs are roasted?
Noriko orders 3 pizzas, and each pizza is cut into 8 slices. Draw a tape diagram and find how many slices there are.
Write a pizza party story for 6 \times 9 = \square, and find the number in the box.
Mitsuki buys 7 packs of balloons with 10 balloons in each pack. How many balloons does she buy?

Part 2: the size of each group or the number of groups is missing

Mitsuki has 45 marshmallows and puts 9 on each stick. How many sticks does she fill?
Kazuki shares 54 party favors equally among 6 goody bags. How many favors go in each bag?
At the pizza party there are 48 slices, and each pizza was cut into 8 slices. How many pizzas were there?
Eight campers carry 64 sticks of firewood, the same number each. How many sticks does each camper carry?
Noriko ties 72 balloons in bunches of 8. Write an equation with a box, say what the box stands for, and find it.
(a) Kazuki stacks 63 napkins in stacks of 9. How many stacks does he make? (b) Kazuki puts 63 napkins equally on 7 tables. How many napkins go on each table?
Write two campfire stories for 24 \div 6 = \square, one asking how many in each group and one asking how many groups. Find the number in the box.

Part 3: equal groups in an array

Goody bags are lined up on a table in 4 rows of 7. How many goody bags are there?
Mitsuki sets 48 goody bags in 6 equal rows. How many bags are in each row?
A sheet pizza is cut into 49 squares in 7 equal rows. How many squares are in each row?
Kazuki hangs 60 lanterns around the campsite in rows of 10. How many rows does he make?
Noriko grills corn on the cob in 3 rows of 7. Draw the array, write a multiplication for the total, and write a division that finds the number in each row.

Part 4: measured amounts

Noriko hangs 6 streamers, each 5 meters long. How many meters of streamer does she use?
Kazuki cuts a 45-meter roll of streamer into 9 equal pieces. How long is each piece?
Mitsuki carries 28 liters of water in jugs that hold 4 liters each. How many full jugs does she carry?
There are 6 bundles of firewood, and each bundle has a mass of 9 kilograms. What is the mass of all the firewood?
A cooler holds 20 liters of lemonade, poured equally into 5 jugs. How many liters go in each jug?

Explaining and reasoning

The word “each” appears in both of these: “6 bags with 4 favors in each” and “56 slices shared over 8 tables; how many on each?” One is a multiplication and the other a division. Explain what tells them apart.
Explain why every division story in this lesson could also be written as a multiplication with a box, and why the answer can always be checked by multiplying.
Mitsuki carries 7 buckets of water, each holding 2 liters, so she carries 7 \times 2 = 14 liters. Explain why the answer is a number of liters and not a number of buckets, and say what the 7 counts.

Answers

5 \times 8 = \square. Five groups of 8: 8, 16, 24, 32, 40. He uses 40 stickers.
8 \times 2 = \square, so 16 hot dogs. The 8 campers are the groups and 2 is the size of each.
The bar has 3 equal parts, each marked 8, with a question mark over the whole bar. 3 \times 8 = 24, so there are 24 slices.
One story: Noriko sets out 6 tables with 9 cups on each; how many cups in all? Any story with 6 equal groups of 9 that asks for the total works. 6 \times 9 = 54, so \square = 54.
7 \times 10 = \square, so 70 balloons.
\square \times 9 = 45. Counting by nines, 9, 18, 27, 36, 45 is 5 nines, so she fills 5 sticks, and 45 \div 9 = 5.
6 \times \square = 54. Since 6 \times 9 = 54, each bag gets 9 favors, and 54 \div 6 = 9.
\square \times 8 = 48. Since 6 \times 8 = 48, there were 6 pizzas, and 48 \div 8 = 6.
8 \times \square = 64. Since 8 \times 8 = 64, each camper carries 8 sticks, and 64 \div 8 = 8.
\square \times 8 = 72, where the box stands for the number of bunches. Since 9 \times 8 = 72, there are 9 bunches, and as a division 72 \div 8 = 9.
(a) \square \times 9 = 63, so 7 stacks, since 7 \times 9 = 63. (b) 7 \times \square = 63, so 9 napkins on each table. Same three numbers, with the box in different places.
How many in each: Kazuki shares 24 marshmallows equally among 6 campers; how many does each camper get? How many groups: Kazuki has 24 marshmallows and gives 6 to each camper; how many campers get marshmallows? In both, 6 \times 4 = 24, so \square = 4 and 24 \div 6 = 4.
4 \times 7 = \square, so 28 goody bags.
6 \times \square = 48. Since 6 \times 8 = 48, there are 8 bags in each row.
7 \times \square = 49. Since 7 \times 7 = 49, there are 7 squares in each row.
\square \times 10 = 60. Since 6 \times 10 = 60, he makes 6 rows.
The array has 3 rows of 7 ears of corn. Multiplication: 3 \times 7 = 21. Division: 21 \div 3 = 7, so 7 in each row.
6 \times 5 = \square, so 30 meters.
9 \times \square = 45. Since 9 \times 5 = 45, each piece is 5 meters long.
\square \times 4 = 28. Since 7 \times 4 = 28, she carries 7 jugs.
6 \times 9 = \square, so 54 kilograms.
5 \times \square = 20. Since 5 \times 4 = 20, each jug holds 4 liters.
The word does not decide; the missing number does. In the bags story the number of groups and the size of each group are given and the total is missing, so it is 6 \times 4 = \square, a multiplication. In the pizza story the total and the number of groups are given and the size of each group is missing, so it is 8 \times \square = 56, which is the division 56 \div 8 = \square.
Every equal groups story fits groups times size of each equals total. In a division story the total is known and one factor is missing, so it is that multiplication with a box in a factor’s place. Because the answer is the missing factor, putting it back into the multiplication must give the total; if it does not, the answer is wrong.
Each bucket is one group, and the size of each group is 2 liters. Seven groups of 2 liters is 2 + 2 + 2 + 2 + 2 + 2 + 2 liters, and adding liters to liters gives liters, so the total is 14 liters, in the same unit as the size of each group. The 7 counts buckets: it is the number of groups, a plain count with no unit of water.