Grade 3 · 2.G.A.2 · 2.OA.C.4

Arrays

Things set out in neat rows and columns, like panes in a window or trees in an orchard, can be counted without counting one by one. What does the shape tell you?

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What multiplication meansGrade 2

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Properties of multiplicationGrade 3 Perfect squares and perfect cubesMath 6 The area modelGrade 3

Where you are 16 in Multiplication and division

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Multiplication and division16

Arrays: rows and columns of objects What multiplication means

Arrays

Multiplication facts4 The distributive property ▾ Show all 16 (11 more)

Why this exists

Taavi looks at the big window in the hall and wants to know how many panes of glass it has. He could touch each pane and count, 1, 2, 3, \dots That works, but it is slow, and it is easy to skip a pane or count one twice.

The window has a shape that helps. Its panes sit in straight rows, one row above another, and every row has the same number of panes. So Taavi can count one row, see that it has 7 panes, count the rows, see that there are 3, and know the whole window is 3 groups of 7. That is a multiplication, 3 \times 7, and the window is a picture of it.

Objects set out like this are called an array. Arrays are everywhere: eggs in a carton, tiles on a floor, trees planted in an orchard, stamps on a page. Each one turns “how many in all?” into “how many rows, and how many in each row?” Two small counts replace one big one.

The array matters for more than counting. It is the picture that the rest of multiplication is built on: why the order of the factors does not matter, how a hard product splits into easy ones, and why the area of a rectangle is found by multiplying all come from looking at an array.

Definition

An array is a set of objects arranged in rows and columns so that every row has the same number of objects and every column has the same number of objects. A row runs across, from left to right. A column runs up and down.

An array with 3 rows and 7 objects in each row is called a 3 by 7 array. It holds

3 \times 7

objects. In this lesson the first number always counts the rows and the second counts the objects in each row. That order is a habit for reading, not a rule of mathematics; either order gives the same total, as Part 2 notes.

The numbers being multiplied are the factors, and the answer is the product. In 3 \times 7 = 21, the factors are 3 and 7 and the product is 21.

A reminder from What multiplication means: 3 \times 7 means 3 equal groups of 7, which is 7 + 7 + 7. In an array, each row is one of the equal groups.

Intuition

The lesson runs over four days. Part 1 is what an array is and how it counts. Part 2 reads an array as an equation and back again. Part 3 finds a missing number. Part 4 builds every array for a number and uses tens.

Part 1: equal rows count by multiplying

Here is Taavi’s window.

Figure 1Three rows, seven panes in each row, so 3 \times 7 panes.

Each row is a group of 7 panes. There are 3 rows, so there are 3 groups of 7. Counting row by row is adding 7 again for each row:

7 + 7 + 7 = 21

Skip-counting by sevens, 7, 14, 21, says the same thing, one number for each row (see Skip counting). And because the window is 3 equal groups of 7, it is 3 \times 7. So 3 \times 7 = 21, and the window has 21 panes.

What makes this work is that every row is the same. Change one row and the counting by rows still finds the total, but the total is no longer “so many rows of so many”, and no single multiplication describes it. That is why the definition insists on equal rows. Rows of 5, 5 and 4 add to 14, but 14 is not 3 rows of anything equal.

Part 2: from array to equation, and back

An array can be read two ways, and both are true at once.

Figure 2Rows run across. Columns run up and down.

Jomo’s orchard has 6 rows of 4 apple trees. Read by rows, it is 6 groups of 4:

6 \times 4 = 24

Read by columns, the same trees make 4 columns of 6, so 4 \times 6 is also 24; why the order never matters is the subject of the commutative property.

Going the other way, from an equation to an array, means choosing rows. To show 2 \times 8, set out 2 rows with 8 objects in each row. Any objects will do, dots, squares or trees, because the array is about the arrangement, not the things in it. Adding one more row to an array adds one more row’s worth of objects: 2 rows of 8 is 16, and 3 rows of 8 is 16 + 8 = 24.

Part 3: when part of the array is missing

Sometimes an array is not all in view, or one of its numbers is not given. An array is fixed by just two numbers, the rows and the row length, so a full view is not needed.

Figure 3The top row gives the row length; the left column gives the number of rows.

Bryndis has laid a card over most of a page in her stamp album. The top row still shows, and it has 8 stamps, so every row has 8. The left column still shows, and it has 5 stamps, so there are 5 rows. That is enough: the page is 5 rows of 8, which is 5 \times 8 = 40 stamps.

The other kind of missing number is a factor. A window has 16 panes in rows of 4. How many rows? The array gives the equation

\square \times 4 = 16

and we build rows of 4 until we reach 16: 4, 8, 12, 16. That is 4 rows, so \square = 4. Asking for a missing factor like this is division, which is the lesson What division means; multiplication and division are two readings of one array, as Multiplication and division undo each other explains.

Part 4: every array for a number, and arrays of tens

A number of objects can often be set out in more than one array. Taavi has 12 young apple trees to plant in an array. He can plant 1 row of 12, 2 rows of 6 or 3 rows of 4, and each of these can be turned on its side to give 12 rows of 1, 6 rows of 2 or 4 rows of 3.

Figure 4Twelve trees can stand in 1 row of 12, 2 rows of 6 or 3 rows of 4, and in each of these turned on its side.

He cannot plant 5 equal rows: 2 rows of 5 is 10 and 3 rows of 5 is 15, and no number of fives makes 12. Finding all the arrays of a number is the same as finding all the ways to write it as a product, which is Factors and multiples.

An array can also be built from tens. A sheet of stamps has 4 rows of 20. Each row of 20 is 2 tens, so the sheet is 4 rows of 2 tens.

Figure 5Four rows of two tens is eight tens.

That is 4 \times 2 = 8 tens, and 8 tens is 80. So 4 \times 20 = 80. The array lets a product with a multiple of ten be found from a basic fact, an idea Multiplying by tens and hundreds takes further.

Results

Counting an array. An array of r rows with c objects in each row holds

r \times c

objects.

One more row. An array of r rows of c, with one more row of c added, holds r \times c + c objects, which is (r + 1) \times c.

Two numbers fix it. Knowing the number of rows and the number in each row is enough to know the whole array. If the total and one of those two numbers are known, the other is the missing factor in r \times c = \text{total}.

Why it is true

Each row of an array has c objects, and every row has the same number, so each row is an equal group of c. There are r rows, so the array is r equal groups of c. By the meaning of multiplication from What multiplication means, r equal groups of c is r \times c. That proves the first result.

For the second, the new row is one more equal group of c. Then there are r + 1 groups of c, which is (r + 1) \times c, and counting the old array and the new row separately gives r \times c + c. Both count the same objects. \blacksquare

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.

Example 1

Counting the window

Taavi’s window has 3 rows of 7 panes. Write the count as an addition and as a multiplication, and find how many panes there are.

Solution

Each row is a group of 7 panes, and there are 3 rows. Adding one 7 for each row gives

7 + 7 + 7 = 21

The window is 3 equal groups of 7, so the same count is the multiplication 3 \times 7. The two must agree, so 3 \times 7 = 21. The window has 21 panes.

Check by skip-counting one row at a time: 7, 14, 21. Three rows, ending at 21.

Example 2

Is it an array?

Taavi sets out 14 stamps in rows: 5 in the first row, 5 in the second and 4 in the third. Is this an array? If not, what is the fewest stamps he can add to make one with 3 rows?

Figure 6Rows of 5, 5 and 4 are not an array. One more stamp makes them one.

Solution

An array needs every row to have the same number. Here the rows have 5, 5 and 4, so this is not an array. Counting still works, 5 + 5 + 4 = 14, but no “rows of” sentence describes it.

To keep 3 rows and make them equal, the short row needs one more stamp. Then every row has 5, and the stamps form 3 rows of 5:

3 \times 5 = 15

One stamp, the one in the dashed square, makes an array of 15. Check: 14 + 1 = 15. (Taking a stamp away from each of the first two rows would also make the rows equal, with 3 rows of 4, but that takes stamps away rather than adding them.)

Example 3

From an equation to an orchard

Jomo wants to plant 5 \times 9 apple trees in an array. Describe the array, say what each factor counts, and find how many trees he plants.

Solution

The first factor counts the rows and the second counts the trees in each row, so Jomo plants 5 rows with 9 trees in each row. Each row is one group of 9, and there are 5 groups. Skip-count by nines, one number per row: 9, 18, 27, 36, 45. So

5 \times 9 = 45

Jomo plants 45 trees. Check by adding the rows: 9 + 9 + 9 + 9 + 9 = 45.

Example 4

One more row

Jomo plants 7 rows of 5 apple trees. How many trees is that? Next spring he plants one more row of 5. How many trees does the orchard have then?

Solution

Seven rows of 5 is 7 \times 5. Skip-count by fives: 5, 10, 15, 20, 25, 30, 35. So 7 \times 5 = 35 trees.

The new row is one more group of 5. The orchard grows from 35 to

35 + 5 = 40

trees. Check by counting the new array as a whole: it has 8 rows of 5, and 8 \times 5 = 40. Adding a row adds one row’s worth, 5, never the number of rows.

Example 5

The page under the card

Bryndis covers most of a stamp album page with a card. The top row shows 8 stamps and the left column shows 5 stamps. The page is a full array. How many stamps are on it?

Figure 7The top row gives the row length; the left column gives the number of rows.

Solution

The page is an array, so every row has as many stamps as the top row: 8. Every row has one stamp at the left edge, so the left column has one stamp per row, and 5 stamps there means 5 rows. The page is 5 rows of 8:

5 \times 8 = 40

There are 40 stamps on the page. Check by skip-counting eights for the 5 rows: 8, 16, 24, 32, 40. Notice that the corner stamp belongs to both the top row and the left column. It is counted once in the array, which is why the answer is 5 \times 8 and not 8 + 5.

Example 6

How many rows?

A window has 18 panes, with 6 panes in each row. How many rows does it have?

Solution

The window is an array with rows of 6 and 18 panes in all, so the number of rows is the missing factor in

\square \times 6 = 18

Build the window one row at a time and keep a running count: 6 after one row, 12 after two, 18 after three. The count reaches 18 exactly at 3 rows, so \square = 3.

The window has 3 rows. Check: 3 \times 6 = 18.

Example 7

Every array of twelve

Taavi has 12 apple trees to plant in an array. List every array he can make, giving rows and trees in each row.

Solution

Try each possible number of trees in a row and ask whether the rows come out even.

Rows of 1: 12 rows, since 12 \times 1 = 12.
Rows of 2: 6 rows, since 6 \times 2 = 12.
Rows of 3: 4 rows, since 4 \times 3 = 12.
Rows of 4: 3 rows, since 3 \times 4 = 12.
Rows of 5: no. 2 \times 5 = 10 and 3 \times 5 = 15, so 12 is skipped.
Rows of 6: 2 rows, since 2 \times 6 = 12.
Rows of 7, 8, 9, 10 or 11: no. One such row is less than 12 and two are more.
Rows of 12: 1 row, since 1 \times 12 = 12.

So there are six arrays: 12 by 1, 6 by 2, 4 by 3, 3 by 4, 2 by 6 and 1 by 12. They come in pairs, each one the other turned on its side, so only three shapes are really different, the three in the figure in Part 4.

Example 8

A sheet in tens

A full sheet of stamps has 4 rows of 20 stamps. How many stamps are on the sheet?

Figure 8Four rows of two tens is eight tens.

Solution

Each row of 20 is 2 tens. So the sheet is 4 rows of 2 tens, and counting tens instead of stamps,

4 \times 2 = 8

The sheet holds 8 tens, which is 80 stamps. So 4 \times 20 = 80. Check by adding the rows: 20 + 20 + 20 + 20 = 80.

Boundaries

The rows must be equal. Objects in rows of different lengths can be counted by adding, but they are not an array, and no single product of rows and row length gives their total.
Zero rows, or rows of zero, hold nothing. An array of 0 rows of 7, or of 7 rows of 0, has no objects: 0 \times 7 = 0 and 7 \times 0 = 0. In real objects such an array is empty.
Some numbers have only one shape. 7 objects can be set out as 1 row of 7 or 7 rows of 1, and in no other array, because no number of equal rows of 2 to 6 makes 7. A number like this is called prime, which prime and composite numbers takes up.
An array counts objects, not lengths. Rows and columns of objects give a count. When the objects become unit squares with no gaps, the count becomes the area of a rectangle, which is The area model.

Where this goes next

The array is the picture behind almost every later fact about multiplication. Turned on its side, it shows the commutative property. Cut into two smaller arrays, it shows the distributive property, the idea that breaks a hard product like 6 \times 8 into easier ones. Those two properties are how the Multiplication facts are learned with understanding instead of by rote, and stories of rows and groups come back in Equal group problems.

When the objects are squares packed edge to edge, the array becomes a rectangle and the count becomes its area, the area model, which later carries multiplication of large numbers and even of fractions. An array with as many rows as columns is a square, and the numbers it holds, 1, 4, 9, 16, \dots, are the perfect squares.

Exercises

Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 18 with Part 3, and 19 to 22 with Part 4. The reasoning group draws on all four.

Part 1: equal rows count by multiplying

A window has 4 rows of 5 panes. Write an addition and a multiplication for the number of panes, and find it.
An orchard has 2 rows of 9 apple trees. Write the count as an addition and as a multiplication, and find it.
Bryndis puts stamps in rows of 6, 6, 6 and 5. Is this an array? How many stamps are there, and how many must she add to make 4 equal rows?
Describe an array that matches 8 + 8 + 8 + 8, write its multiplication, and find the total.
A small window has 3 rows of 3 panes. How many panes does it have?
Skip-count to find the number of stamps on a page of 5 rows of 6. Write the numbers you say, one for each row.

Part 2: from array to equation, and back

Describe an array for 2 \times 7 and find the product.
Jomo plants 6 rows of 7 apple trees. How many trees is that? Write the multiplication and say what each factor counts.
A window has 2 rows of 10 panes. Write a multiplication and an addition for it.
An orchard has 6 rows of 9 trees. One more row of 9 is planted. How many trees are there now?
A stamp page has 7 rows of 4 stamps. Bryndis takes out one whole row. How many stamps are left?
Which of these give the number of objects in 4 rows of 6? (a) 4 + 6 (b) 4 \times 6 (c) 6 + 6 + 6 + 6. Find the number.

Part 3: when part of the array is missing

A card covers most of a stamp page. The top row shows 9 stamps and the left column shows 3. How many stamps are on the page?
A window has 30 panes in rows of 5. How many rows does it have?
An orchard of 28 apple trees is planted in 4 equal rows. How many trees are in each row?
An array has rows of 7 and 56 objects in all. Write an equation with a \square and find the number of rows.
A stamp page holds 36 stamps in 4 equal rows. Bryndis has filled 2 rows. How many more stamps does she need to fill the page?
One window has 3 rows of 8 panes. Another has 4 rows of 6 panes. Which has more panes?

Part 4: every array for a number, and arrays of tens

List every array of 18 apple trees.
Can 11 stamps be set out in an array with more than one row and more than one stamp in each row? Explain.
A sheet of stamps has 3 rows of 30. How many stamps are on the sheet?
A large orchard has 5 rows of 60 trees. How many trees is that?

Reasoning

Explain why a multiplication can describe rows of 7, 7 and 7 but not rows of 7, 7 and 6.
A square array has the same number of rows as objects in each row. Which of 16, 20 and 25 objects can make a square array? Explain.
An array has 6 rows of 7. Explain why adding one more row adds 7 objects, not 6, and find the new total.

Answers

20 panes. 5 + 5 + 5 + 5 = 20, and 4 \times 5 = 20.
18 trees. 9 + 9 = 18, and 2 \times 9 = 18.
No, the last row is short. There are 6 + 6 + 6 + 5 = 23 stamps. Adding 1 stamp to the last row makes 4 rows of 6, and 4 \times 6 = 24.
4 rows of 8 objects: 4 \times 8 = 32.
9 panes: 3 \times 3 = 9.
6, 12, 18, 24, 30. Five rows, so 5 \times 6 = 30 stamps.
2 rows with 7 objects in each row: 2 \times 7 = 14.
42 trees: 6 \times 7 = 42. The 6 counts the rows and the 7 counts the trees in each row.
2 \times 10 = 20, and 10 + 10 = 20, one 10 for each row.
63 trees. 6 \times 9 = 54, and one more row makes 54 + 9 = 63, which is 7 \times 9.
24 stamps. 7 \times 4 = 28, and taking a row of 4 leaves 28 - 4 = 24, which is 6 \times 4.
(b) and (c). Both are 4 groups of 6, which is 24. (a) is 10, which adds a row length to a number of rows and counts nothing.
27 stamps: 3 rows of 9, and 3 \times 9 = 27.
6 rows: \square \times 5 = 30, and 5, 10, 15, 20, 25, 30 is six rows.
7 trees in each row: 4 \times \square = 28, and 4 \times 7 = 28.
\square \times 7 = 56, and 8 \times 7 = 56, so 8 rows.
18 stamps. Each row holds 9, since 4 \times 9 = 36. Two rows are left: 2 \times 9 = 18.
Neither. 3 \times 8 = 24 and 4 \times 6 = 24, so both have 24 panes.
Six arrays: 1 by 18, 2 by 9, 3 by 6, 6 by 3, 9 by 2 and 18 by 1. Rows of 4 fail (4 \times 4 = 16, 5 \times 4 = 20), and so do rows of 5, 7 and 8, and rows of 10 to 17 (one row is too few, two too many).
No. Rows of 2 skip 11 (5 \times 2 = 10, 6 \times 2 = 12), and so do rows of 3 (3 \times 3 = 9, 4 \times 3 = 12), rows of 4 (2 \times 4 = 8, 3 \times 4 = 12) and rows of 5 (2 \times 5 = 10, 3 \times 5 = 15). Two rows of 6 or more are already 12 or more. Only 1 row of 11 and 11 rows of 1 work.
90 stamps. Each row is 3 tens, so the sheet is 3 \times 3 = 9 tens, which is 90.
300 trees. Each row is 6 tens, so 5 \times 6 = 30 tens, which is 300.
Rows of 7, 7 and 7 are 3 equal groups of 7, so 3 \times 7 = 21 counts them. Rows of 7, 7 and 6 are not equal groups. They add to 20, but 20 is not 3 rows of any one number, so no “rows times row length” fits.
16 and 25. 16 = 4 \times 4 and 25 = 5 \times 5. For 20, 4 \times 4 = 16 is too few and 5 \times 5 = 25 is too many, so no square array holds 20.
A new row is one more group, and every row has 7 objects, so the new row brings 7. The 6 counts rows, not objects in a row. The total goes from 6 \times 7 = 42 to 42 + 7 = 49, which is 7 \times 7.