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Grade 3 · 3.MD.C.7C · 3.OA.B.5

The distributive property

You know 6 \times 5 and you know 6 \times 3. Is that enough to find 6 \times 8?

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Why this exists

Ronja is planting a garden bed. It has 6 rows, with 8 lettuce plants in each row. How many plants is that? It is 6 \times 8, but Ronja does not know that fact yet. She knows her fives, her twos and her tens, and she can work out 6 \times 3 by skip-counting: 3, 6, 9, 12, 15, 18.

She could count all the plants one by one, or add 8 + 8 + 8 + 8 + 8 + 8. Both work, and both are slow. Here is a faster way. Lay a string across the bed after the fifth plant in every row. On the left of the string are 6 rows of 5 plants. On the right are 6 rows of 3. Those are two facts Ronja knows: 6 \times 5 = 30 and 6 \times 3 = 18. Together that is 30 + 18 = 48 plants.

That move, cutting a product you do not know into products you do, is the distributive property. It is how most of the multiplication table gets built from a few easy facts, and later it is the idea under every way of multiplying bigger numbers.

Definition

Multiplying a number by a sum gives the same result as multiplying that number by each part of the sum and then adding the products. For Ronja’s bed:

6 \times (5 + 3) = (6 \times 5) + (6 \times 3)

The parentheses say “work this out first, and treat it as one number”. So 6 \times (5 + 3) is 6 \times 8. On the right side, each product in parentheses is found first, and then the two are added: 30 + 18.

One factor stays whole and the other is split into parts. The factor that stays whole, here the 6, multiplies every part. The word distribute means to hand out: the 6 is handed out to the 5 and to the 3.

A reminder from What multiplication means: 6 \times 8 means 6 equal groups of 8. Set out as an array, as in Arrays, it is 6 rows with 8 in each row.

Intuition

Part 1: cutting an array in two

Draw Ronja’s bed as an array, one dot for each plant. The dashed line is the string.

Figure 1One cut turns 6 \times 8 into two facts Ronja already knows.

The cut does not add a plant or take one away. It only sorts them. Every row of 8 is now 5 on the left and 3 on the right. The left part is an array of its own, 6 rows of 5, so it holds 6 \times 5 plants. The right part is 6 rows of 3, so it holds 6 \times 3. The two parts together are the whole bed. Written as one chain:

\begin{aligned} 6 \times 8 &= 6 \times (5 + 3) \\ &= (6 \times 5) + (6 \times 3) \\ &= 30 + 18 \\ &= 48 \end{aligned}

The string could go between any two columns. After the fourth column it makes (6 \times 4) + (6 \times 4), which is 24 + 24 = 48 again. Every cut gives 48, because every cut sorts the same 48 plants.

Part 2: building a new fact from two known ones

Part 1 used the property to count an array. Its everyday use is to find a fact you do not know yet. Pick a cut so that both pieces are facts you do know.

By now the known facts are the twos, fives and tens from Facts for two, five and ten, and their turnarounds from Properties of multiplication (5 \times 6 tells you 6 \times 5). The most useful part is 5, because each of the numbers 6 to 9 is 5 and a little more:

Number	Split with a 5
6	5 + 1
7	5 + 2
8	5 + 3
9	5 + 4

So a hard fact becomes a five fact plus a smaller fact. When the smaller piece is a one or a two, both pieces are known at once. Other splits work too: 8 is 4 + 4, and 6 is 2 + 2 + 2. Any split is correct. A good split is one whose pieces you can do quickly.

Part 3: a one-digit number times a teen number

Alibek bakes cookies on a cookie sheet that holds 3 rows of 14. Now the split comes from place value: 14 is 1 ten and 4 ones, so 14 = 10 + 4.

Figure 2A teen number is a ten and some ones, so the cut goes at the ten.

\begin{aligned} 3 \times 14 &= 3 \times (10 + 4) \\ &= (3 \times 10) + (3 \times 4) \\ &= 30 + 12 \\ &= 42 \end{aligned}

The ten is the best place to cut because a tens fact is the easiest fact there is: 3 tens is 30. Every teen number from 11 to 19 is a ten and some ones, so every one-digit number times a teen number splits the same way: one tens fact plus one small fact.

Part 4: splitting either factor, into any number of parts

So far the cut has always gone between columns, which splits the second factor. A cut can also go between rows, which splits the first factor. An array of 8 rows of 9 can be cut after the fourth row:

Figure 3Cutting between rows splits the other factor.

(4 + 4) \times 9 = (4 \times 9) + (4 \times 9)

The 9 now stays whole and the 8 is split. This is the same property with the array turned on its side: 8 \times 9 and 9 \times 8 are the same array read two ways, by the commutative property, so splitting the rows of one is splitting the columns of the other.

A factor can also be split into more than two parts. Each part still gets multiplied:

\begin{aligned} 2 \times (10 + 5 + 2) &= (2 \times 10) + (2 \times 5) + (2 \times 2) \\ &= 20 + 10 + 4 \\ &= 34 \end{aligned}

and 2 \times 17 is indeed 34.

Finally, the property reads backward. Two products that share a factor can be joined into one, the way two arrays with the same number of rows can be pushed together side by side. Example 8 does this.

Results

The distributive property. Multiplying a number by a sum is the same as multiplying it by each part of the sum and adding the products. For example,

6 \times (5 + 3) = (6 \times 5) + (6 \times 3)

Either factor may be split. Splitting the first factor works the same way: (4 + 4) \times 9 = (4 \times 9) + (4 \times 9).

Any number of parts. 2 \times (10 + 5 + 2) is the same as (2 \times 10) + (2 \times 5) + (2 \times 2), three products added.

Every split gives the same product, as long as the parts add up to the factor being split.

It reads both ways. (4 \times 7) + (4 \times 3) = 4 \times (7 + 3) = 4 \times 10.

Why it is true

From the array. An array of 6 rows of 8 cut after the fifth column is two arrays side by side: 6 rows of 5 and 6 rows of 3. Every dot of the big array is in exactly one of the two small ones, and no dot is in both. So the number of dots in the big array is the sum of the dots in the small ones: 6 \times 8 = (6 \times 5) + (6 \times 3).

From equal groups. 6 \times 8 means six groups of 8:

8 + 8 + 8 + 8 + 8 + 8

Write each 8 as 5 + 3. That gives six 5s and six 3s, mixed together. Adding in any order gives the same total (the commutative property of addition), and so does grouping the numbers in any way (the associative property of addition). So put the six 5s together and the six 3s together. Six 5s is 6 \times 5 and six 3s is 6 \times 3.

Neither argument used anything special about 6, 5 and 3. The same sorting works for any whole numbers, any cut, and any number of parts, which is why the property holds every time. \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

A radish bed cut after five

Chidera’s radish bed has 3 rows of 8 radishes. She lays a string across the bed after the fifth radish in each row. Write the two smaller products the string makes, and use them to find how many radishes there are.

Solution

The string splits every row of 8 into 5 on the left and 3 on the right, so it splits 8 as 5 + 3. The rows are not cut, so both pieces still have 3 rows.

The left piece is 3 rows of 5: 3 \times 5 = 15. The right piece is 3 rows of 3: 3 \times 3 = 9, which skip-counting by threes gives (3, 6, 9).

\begin{aligned} 3 \times 8 &= 3 \times (5 + 3) \\ &= (3 \times 5) + (3 \times 3) \\ &= 15 + 9 = 24 \end{aligned}

Chidera has 24 radishes. Check by adding the rows: 8 + 8 + 8 = 24.

Example 2

Does it matter where the cut goes?

Alibek’s bed has 4 rows of 9 seedlings. He cuts it after the fourth column. Ronja cuts the same bed after the fifth column. Write each person’s two products and find each total.

Solution

Figure 4Two different cuts of the same bed. The dots do not change, so the total does not either.

Alibek’s cut (picture A) splits 9 as 4 + 5:

\begin{aligned} 4 \times (4 + 5) &= (4 \times 4) + (4 \times 5) \\ &= 16 + 20 = 36 \end{aligned}

Ronja’s cut (picture B) splits 9 as 5 + 4:

\begin{aligned} 4 \times (5 + 4) &= (4 \times 5) + (4 \times 4) \\ &= 20 + 16 = 36 \end{aligned}

Both get 36. The pieces are the same sizes, just in the other order, and adding in any order gives the same sum. Even a different cut, after the second column, gives (4 \times 2) + (4 \times 7) = 8 + 28 = 36. The bed has 36 seedlings however it is cut.

Example 3

Eight sevens from two easy facts

Find 8 \times 7 using only facts for fives and twos.

Solution

Split the 7 so that both pieces are known: 7 = 5 + 2. The 8 stays whole and multiplies each piece.

\begin{aligned} 8 \times 7 &= 8 \times (5 + 2) \\ &= (8 \times 5) + (8 \times 2) \\ &= 40 + 16 \\ &= 56 \end{aligned}

Here 8 \times 5 = 40 is the turnaround of 5 \times 8, a five fact, and 8 \times 2 = 16 is the turnaround of the two fact 2 \times 8. So 8 \times 7 = 56.

Example 4

Choosing a split

Find 9 \times 6. Compare splitting 6 as 5 + 1 with splitting it as 2 + 2 + 2.

Solution

With 6 = 5 + 1, the 9 multiplies each part:

\begin{aligned} 9 \times (5 + 1) &= (9 \times 5) + (9 \times 1) \\ &= 45 + 9 = 54 \end{aligned}

The piece 9 \times 1 is just one group of 9, so this split needs only one real fact, 9 \times 5 = 45.

With 6 = 2 + 2 + 2, there are three parts, and the 9 multiplies each one:

\begin{aligned} 9 \times (2 + 2 + 2) &= (9 \times 2) + (9 \times 2) + (9 \times 2) \\ &= 18 + 18 + 18 = 54 \end{aligned}

Both splits give 54, as they must. The first is quicker: one five fact and a single 9, against adding three 18s. So 9 \times 6 = 54.

Example 5

Alibek’s cookie sheet

Alibek’s cookie sheet holds 3 rows of 14 cookies. How many cookies fill the sheet?

Solution

The sheet is the array in Part 3. Split 14 by place value, 14 = 10 + 4, so the 3 multiplies the ten and the four:

\begin{aligned} 3 \times (10 + 4) &= (3 \times 10) + (3 \times 4) \\ &= 30 + 12 = 42 \end{aligned}

3 \times 10 = 30 is 3 tens. 3 \times 4 = 12 is the turnaround of 4 \times 3, three groups of four. The sheet holds 42 cookies. Check by adding the rows: 14 + 14 = 28 and 28 + 14 = 42.

Example 6

A long onion bed

Ronja’s long bed has 5 rows of 17 onions. How many onions are in the bed?

Solution

Drawing 85 dots is a chore, so draw a plain rectangle instead and write the numbers on its sides. The split is still 17 = 10 + 7.

Figure 5The same split without drawing every dot.

\begin{aligned} 5 \times 17 &= 5 \times (10 + 7) \\ &= (5 \times 10) + (5 \times 7) \\ &= 50 + 35 \\ &= 85 \end{aligned}

Both pieces are five facts: 5 tens is 50, and 5 \times 7 = 35. The bed has 85 onions. The rectangle stands for the array; it only saves drawing the dots.

Example 7

Splitting the rows

A seed tray has 8 rows of 9 seedlings. Example 2 found that 4 \times 9 = 36. Use that to find 8 \times 9, then check by splitting the 9 instead.

Solution

Cut between the fourth and fifth rows, as in the picture in Part 4. That splits the 8 as 4 + 4, and the 9 stays whole:

\begin{aligned} (4 + 4) \times 9 &= (4 \times 9) + (4 \times 9) \\ &= 36 + 36 = 72 \end{aligned}

Check by splitting the 9 as 5 + 4 with the 8 whole:

\begin{aligned} 8 \times (5 + 4) &= (8 \times 5) + (8 \times 4) \\ &= 40 + 32 = 72 \end{aligned}

where 8 \times 4 = 32 is itself two copies of 8 \times 2: 16 + 16 = 32. Both ways give 72, so the tray holds 72 seedlings.

Example 8

Two sheets pushed together

Alibek has one cookie sheet with 4 rows of 7 cookies and another with 4 rows of 3. He pushes them together side by side, so the rows line up. Find the total two ways.

Solution

Figure 6Read backward, the property joins two products into one.

One way: find each sheet and add. 4 \times 7 = (4 \times 5) + (4 \times 2) = 20 + 8 = 28, and 4 \times 3 = 12. Together 28 + 12 = 40.

The other way reads the property backward. Both sheets have 4 rows, so pushed together they make one array of 4 rows of 7 + 3 = 10:

\begin{aligned} (4 \times 7) + (4 \times 3) &= 4 \times (7 + 3) \\ &= 4 \times 10 = 40 \end{aligned}

Both ways give 40 cookies. The second needed only one tens fact, because the 7 and the 3 joined into a ten.

Boundaries

The parts must make the factor exactly. Splitting 8 as 5 + 3 works because 5 + 3 = 8. Pieces that add to 7 count only 7 columns: (6 \times 5) + (6 \times 2) = 42, which is 6 \times 7, not 6 \times 8.
The whole factor multiplies every part. 6 \times (5 + 3) is (6 \times 5) + (6 \times 3). Multiplying only one part, as in (6 \times 5) + 3 = 33, leaves out five of the six rows of the right-hand piece.
It spreads over adding, not over multiplying. 2 \times (3 \times 4) is 2 \times 12 = 24. Handing the 2 to both numbers gives (2 \times 3) \times (2 \times 4) = 6 \times 8 = 48, which is a different number.
It belongs to multiplying, not to adding. 3 + (4 + 5) = 12, but (3 + 4) + (3 + 5) = 15. Adding a number to a sum does not hand the number out to each part.
In this lesson the parts are added. A factor can also be written as a difference, such as 10 - 1, and the property still works; that version comes in Facts for seven, eight and nine. Here every number is a whole number; the property also holds for fractions, decimals and negative numbers, which come much later.

Where this goes next

The next two lessons use the property to fill in the rest of the table. In Facts for three, four and six a four fact is two twos, and a six fact is a five fact plus one more group. In Facts for seven, eight and nine a nine fact is a ten fact minus one group, the same property with a part taken away instead of added. Together with the commutative property, a handful of easy facts is enough to rebuild every fact up to 10 \times 10; that is the work of Multiplication facts.

The split rectangle in Example 6 comes back when area is measured: a rectangle cut in two has the area of its two pieces added, which is the area model and part of Area of rectangles and squares. Splitting a teen number at the ten is the first step of break-apart mental strategies and of Multi-digit multiplication: when both factors have tens and ones, both are split by place value and every pair of parts is multiplied, which is Partial products. And when one of the numbers is a letter standing for any number, the same property becomes the distributive property with variables, used in Equivalent expressions: the distributive property and like terms to rewrite expressions.

Exercises

Exercises 1 to 6 go with Part 1, 7 to 11 with Part 2, 12 to 17 with Part 3, and 18 to 22 with Part 4. The reasoning group draws on all four.

Part 1: cutting an array in two

An array has 2 rows of 8. It is cut after the third column. Write the two smaller products and find the total.
Fill in the boxes: 5 \times 9 = 5 \times (4 + 5), which is (5 \times 4) + (5 \times \square), which is \square + \square, which is \square.
Ronja’s pepper bed has 4 rows of 6 plants. She cuts it after the first column. Write the two products and find how many plants there are.
Which of these equal 4 \times 8? (a) (4 \times 5) + (4 \times 3) (b) (4 \times 5) + 3 (c) (4 \times 6) + (4 \times 2) (d) (4 \times 4) + (4 \times 4)
A bed of 5 rows of 6 tomato plants is cut down the middle, after the third column. Write the two products and find the total.
A cut array shows (3 \times 4) + (3 \times 5). How many rows and columns does the whole array have, and how many dots?

Part 2: building a new fact

Find 8 \times 6 by splitting 6 as 5 + 1.
You know 8 \times 3 = 24. Find 8 \times 8 by splitting 8 as 5 + 3.
You know 9 \times 4 = 36. Find 9 \times 9 by splitting one 9 as 5 + 4.
Find 6 \times 6 by splitting one 6 as 5 + 1.
Alibek knows 3 \times 5 = 15 and 3 \times 2 = 6. Which fact can he build from these two, and what is it?

Part 3: a one-digit number times a teen number

Find 2 \times 13 by splitting 13 at the ten.
Find 4 \times 12.
Chidera’s cookie sheet holds 5 rows of 13 cookies. How many cookies fill it?
Find 6 \times 14. (Hint: 6 \times 4 is (6 \times 2) + (6 \times 2).)
Find 8 \times 12.
Fill in the boxes: 3 \times 16 = (3 \times 10) + (3 \times \square) = 30 + \square = \square.

Part 4: either factor, any number of parts

An array of 9 rows of 4 is cut after the fifth row. Write the two products and find the total.
Find 4 \times 13 by splitting the 4 as 2 + 2. Then check by splitting the 13 at the ten.
Find 5 \times 12 by splitting 12 as 5 + 5 + 2.
Find (9 \times 6) + (9 \times 4) by joining it into one product.
Find (2 \times 8) + (3 \times 8) by joining it into one product.

Reasoning

Ronja finds 7 \times 8 by splitting 8 as 4 + 4. Chidera splits it as 5 + 3. Without working out either one, explain why they must get the same answer.
Explain why 10 is a good place to split a teen number such as 16. Would a split of 16 as 8 + 8 also give the right product?
Is 2 \times (3 \times 4) equal to (2 \times 3) + (2 \times 4)? Work out both sides and say why the distributive property does not apply here.

Answers

(2 \times 3) + (2 \times 5) is 6 + 10 = 16. The cut splits 8 as 3 + 5.
The boxes are 5, then 20 + 25, then 45: (5 \times 4) + (5 \times 5) is 20 + 25 = 45.
(4 \times 1) + (4 \times 5) is 4 + 20 = 24 plants. The cut splits 6 as 1 + 5.
(a), (c) and (d) all equal 32: 20 + 12, 24 + 8 and 16 + 16. (b) is 20 + 3 = 23, because the 3 was never multiplied by 4.
(5 \times 3) + (5 \times 3) is 15 + 15 = 30 plants.
3 rows and 4 + 5 = 9 columns, so 3 \times 9, which is 12 + 15 = 27 dots.
8 \times 6 = (8 \times 5) + (8 \times 1), which is 40 + 8 = 48.
8 \times 8 = (8 \times 5) + (8 \times 3), which is 40 + 24 = 64.
9 \times 9 = (9 \times 5) + (9 \times 4), which is 45 + 36 = 81.
6 \times 6 = (6 \times 5) + (6 \times 1), which is 30 + 6 = 36.
3 \times 7, because 5 + 2 = 7. Then 3 \times 7 = (3 \times 5) + (3 \times 2), which is 15 + 6 = 21.
2 \times 13 = (2 \times 10) + (2 \times 3), which is 20 + 6 = 26.
4 \times 12 = (4 \times 10) + (4 \times 2), which is 40 + 8 = 48.
5 \times 13 = (5 \times 10) + (5 \times 3), which is 50 + 15 = 65 cookies.
6 \times 4 = 12 + 12 = 24. So 6 \times 14 = (6 \times 10) + (6 \times 4), which is 60 + 24 = 84.
8 \times 12 = (8 \times 10) + (8 \times 2), which is 80 + 16 = 96.
The boxes are 6, 18 and 48: 3 \times 6 = 18 and 30 + 18 = 48.
(5 \times 4) + (4 \times 4) is 20 + 16 = 36. The cut splits the 9 rows as 5 + 4.
(2 + 2) \times 13 = (2 \times 13) + (2 \times 13), which is 26 + 26 = 52. Check: (4 \times 10) + (4 \times 3) is 40 + 12 = 52.
(5 \times 5) + (5 \times 5) + (5 \times 2) is 25 + 25 + 10 = 60.
Both products have 9 rows, so they join into 9 \times (6 + 4), and that is 9 \times 10 = 90. Check: 54 + 36 = 90.
Both have 8 columns, so the rows join into (2 + 3) \times 8, and that is 5 \times 8 = 40. Check: 16 + 24 = 40.
Both splits cut the same array of 7 rows of 8, only in different places. A cut never adds a dot or removes one, so the two pieces always hold all 56 dots, whichever split is used.
A teen number is 1 ten and some ones, so the split 16 = 10 + 6 gives a tens fact, which is the easiest kind, plus one small fact. The split 8 + 8 is also correct, because 8 + 8 = 16: (3 \times 8) + (3 \times 8) = 24 + 24 = 48 for 3 \times 16, for example. It is right, just less quick.
No. 2 \times (3 \times 4) = 2 \times 12 = 24, while (2 \times 3) + (2 \times 4) = 6 + 8 = 14. The property hands a factor out to the parts of a sum. Inside the parentheses here is a product, 3 \times 4, not a sum, so there are no parts to hand out to.