Why multiply?

A computer can multiply thousands of numbers in less than
a second. A human is lucky to multiply two numbers in less than

a minute. So we tend to have computers do our math.

But you should still know how to do math on paper, or even

in your head. For one thing, you have to know a little math even

to use a calculator. Besides, daily life tosses plenty of math

problems your way. Do you really want to haul out Trusty Buttons

every time you go shopping?

Of course, normal multiplication can get boring. Here's the

secret: shortcuts. You might think of numbers as a dreary line

from 0 to forever. Numbers do go on forever, but you can also

think of them as cycles. Ten ones make 10. Ten tens make 100.

Ten hundreds make 1000.

If numbers were just a straight highway, there'd be no

shortcuts. But they're more like a winding road. If you know your

way around, you can cut across the grass and save lots of time.

Multiply by 10: Just add 0

The easiest number to multiply by is 10. Just “add 0.”

3 x 10 = 30 140 x 10 = 1400

Isn't that easy? This “trick” is really just using our number

system. 3 means “3 ones.” Move 3 once to the left and you get

30, which means, “3 tens.” See how our numbers cycle in tens?

Whenever you move the digits once to the left, that's the same as

multiplying by 10.

And that's the quick way to multiply by 10. Move each digit

once to the left. Fill the last place with a 0.

Easy

Exercise A:

1. Give two reasons to get good at doing math in your head.

2. Give two situations where you might need or want to do

math in your head, not with a calculator.

3. Explain the quick way to multiply by ten.

4. Solve these problems without using a calculator.

a. 4 x 10

b. 15 x 10

c. 400 x 10

d. 23 x 10

e. 117 x 10

Multiply by 9: It All Adds Up to 9

Have you ever heard of the Amazing Facts of Nine? Let's

take a look.

2 x 9 = 18 1 + 8 = 9

3 x 9 = 27 2 + 7 = 9

4 x 9 = 36 3 + 6 = 9

5 x 9 = 45 4 + 5 = 9

6 x 9 = 54 5 + 4 = 9

7 x 9 = 63 6 + 3 = 9

8 x 9 = 72 7 + 2 = 9

9 x 9 = 81 8 + 1 = 9

See the pattern? When we multiply a single-digit number

times 9:

- The tens digit is one less than our original number.

- The tens digit plus the ones digit equals nine!

This makes it easy to multiply any single digit times 9.

Suppose you want to multiply 5 times 9. First, subtract 1 from

the original number to get the tens digit.

5 - 1 = 4 tens digit of answer

Then subtract this tens digit from 9 to get the ones digit.

9 - 4 = 5 ones digit of answer

So the answer is 45. Let's double check. Do the digits add

up to 9?

4 + 5 = 9

Yes! Isn't this a great trick?

Remember, it only works for single digits. Don't try it on

13 x 9 or 6,425 x 9! Math: Quick Tricks for Multiplication

Exercise B:

1. Explain the quick way to multiply a single digit times 9.

2. Solve the following problems without a calculator.

a. 4 x 9 b. 9 x 9

c. 9 x 8 d. 5 x 9

e. 7 x 9 f. 9 x 2

g. 3 x 9 h. 6 x 9

3. Look at these problems. Circle the problems that you could

not use this lesson's trick for. Explain why not.

a. 14 x 9

b. 9 x 9

c. 9 x 7

d. 115 x 9

e. 8 x 7

Multiply by 5: It's All 5s and 0s

Is there a trick to multiply by 5? Let's look at a few facts:

2 x 5 = 10 3 x 5 = 15

4 x 5 = 20 5 x 5 = 25

6 x 5 = 30 7 x 5 = 35

See a pattern? If we multiply by an even number, the ones

digit is 0. If the number's odd, the ones digit is 5.

So what's the shortcut? Look at the tens digit. If we multiply

by an even number, the tens digit is half that number. The ones

digit is always 0.

2 x 5 = 10 2 / 2 = 1

4 x 5 = 20 4 / 2 = 2

6 x 5 = 30 6 / 2 = 3

What if we multiply by an odd number? First subtract 1 from

that number. Then take half the answer, and that's the tens digit.

The ones digit is always 5.

3 x 5 = 15 3 – 1 = 2 2 / 2 = 1

5 x 5 = 25 5 – 1 = 4 4 / 2 = 2

7 x 5 = 35 7 – 1 = 6 6 / 2 = 3

So here's the shortcut:

To multiply 5 by an even number: Get the tens digit by

dividing the number by 2. The ones digit is 0.

To multiply 5 by an odd number: Subtract 1 from the

number. Get the tens digit by dividing that answer by 2. The

ones digit is 5.

You can use this to check your work, too. What if you

multiply 5 by 3 and get 20? Since 3 is an odd number, your

answer should end in 5, not 0. You know you made a mistake.Math: Quick Tricks for Multiplication

Exercise C:

1. What's the shortcut to multiply an even number by 5?

2. What's the shortcut to multiply an odd number by 5?

3. Solve these problems without a calculator.

a. 5 x 4

b. 5 x 12

c. 5 x 19

d. 5 x 15

e. 5 x 7

f. 5 x 20

g. 5 x 13

h. 5 x 21

4. Don't solve these problems. Instead, check whether each

answer could be right or not. Explain your answer.

Example: 4 x 5 = 25

No. An even number x 5 should equal a number that

ends in 0.

a. 11 x 5 = 50

b. 3 x 5 = 18

Math: Quick Tricks for Multiplication

Multiply by 3: It All Adds Up

Remember the Amazing Facts of Nine? When you multiply

by 9, the digits of the answer eventually add up to 9.

8 x 9 = 72 7 + 2 = 9

So how about that number 9? It's 3 times 3, isn't it? Let's

see if 3 has any special properties.

4 x 3 = 12 1 + 2 = 3

5 x 3 = 15 1 + 5 = 6

6 x 3 = 18 1 + 8 = 9

7 x 3 = 21 2 + 1 = 3

Whoa! When you multiply by 3, the digits of the answer

add up to 3, 6, or 9.

You can't really use this to multiply faster. But it is a quick

way to check your work. Say you multiply 11 by 3 and get 34.

Well, 3 + 4 = 7. Oops. You must have made a mistake. The

right answer is 33. And 3 + 3 = 6.

Of course, the trick can only show whether you're wrong. It

can't prove you're right. Let's say you multiply 11 x 3 and get 36.

Well, 3 plus 6 does equal 9, but 36 is still wrong.

Still, this is a neat trick. If you multiply any number by 3, the

digits of the answer add up to 3, 6, or 9. Even big numbers.

524 x 3 = 1572

1 + 5 + 7 + 2 = 15 and 1 + 5 = 6

91,317 x 3 = 273,951

2 + 7 + 3 + 9 + 5 + 1 = 27 and 2 + 7 = 9

You can see that math has patterns. Thanks to patterns

and cycles, the digits of your answer have to add up to three.

Math works from every angle. That's what's so cool about it. Math: Quick Tricks for Multiplication

Name: _____________________________________________________

Exercise D:

1. Explain the trick about multiplying by 3.

2. Do you use this trick to solve a problem or check your work?

3. How might you get “tricked” by this trick?

4. Don't solve these problems. Instead, circle which answers

can't be correct. Explain your answer.

a. 4 x 3 = 13

b. 19 x 3 = 57

c. 27 x 3 = 85

d. 101 x 3 = 303

e. 15 x 3 = 45

f. 64 x 3 = 192

Math: Quick Tricks for Multiplication

Shortcuts and Chunks

It's easy to multiply by 10, isn't it? Which problem would you

rather do?

20 x 7 = ? 19 x 7 = ?

The first, right? Maybe you figured it out just looking at it:

140. You forget about the 0 and multiply 2 by 7. Easy. But the

second problem is a real problem. You'll have to use a pencil and

paper to get the answer: 133. Rather hard.

Well, you can use the easy problem as a shortcut to the

harder problem.

Twenty sevens is 140. Nineteen sevens is one less seven

than 140. You don't have to figure out 19 x 7. You can jump from

20 x 7 to 140. Then go back to 19 x 7 by subtracting 7 from 140.

19 x 7 = ? Ugh.

20 x 7 = 140 This is a close, easier answer.

140 – 7 = 133 Now subtract the extra seven...

19 x 7 = 133 And you're at the right place.

So you can use an easier problem as a shortcut, then

add or subtract the difference.

A similar trick is to multiply by “chunks.” First multiply the

tens, then multiply the ones, then add these sums together.

This is all you really do on paper, but you might not realize it.

32 x 8 = ? Yikes.

30 x 8 = 240 Multiply the tens.

2 x 8 = 16 Multiply the ones.

240+16 = 256 Add the two answers.

32 x 8 = 256 And we have our final answer.

If you break a problem into smaller chunks, you can often do

it more quickly than if you try to do the whole thing at once. Math: Quick Tricks for Multiplication

Exercise E:

1. How do you use an easy problem as a shortcut for a hard

one? Write an example that shows how.

2. Write an example that shows how to multiply by chunks.

3. Solve this problem by using one of the tricks you learned in

this lesson. Show your work.

49 x 4

4. Solve these problems by using one of the tricks. Do them in

your head.

a. 13 x 6

b. 71 x 7

c. 19 x 9 Math: Quick Tricks for Multiplication

Exercise F: Review

1. Solve these problems by using one of the tricks you learned

in this unit. Show your work.

a. 11 x 7

b. 39 x 3

2. Solve these problems by using one of the tricks. Do them all

in your head.

a. 9 x 8

b. 8 x 5

c. 6 x 5

e. 3 x 9

f. 51 x 4

g. 301 x 10

h. 7 x 9

i. 19 x 7

3. Don't solve these problems. Instead, circle which answers

can't be correct. Explain why.

a. 8 x 5 = 44

b. 9 x 6 = 56

c. 21 x 3 = 61

d. 18 x 5 = 90

e. 9 x 5 = 55 Math: Quick Tricks for Multiplication

Name: _____________________________________________________

12
Answer Key:

Exercise A

1. You don't need to carry around a calculator.

It's fun/interesting/strengthening.

2. Answers vary.

3. Move each digit once to the left.

Fill the last place with a 0.

4. a. 40

b. 150

c. 4000

d. 230

e. 1170

Exercise B

1. Subtract 1 from the original number to get the tens digit.

Subtract this tens digit from 9 to get the ones digit.

Double check by adding the digits together to get 9.

2. a. 36

b. 81

c. 72

d. 45

e. 63

f. 18

g. 27

h. 54

3. a 14 isn't a single digit

d. 115 isn't a single digit

e. 8 x 7 isn't multiplying by 9!

Exercise C

1. Get the tens digit by dividing the number by 2. The ones digit is 0.

2. Subtract 1 from the number. Get the tens digit by dividing that answer by 2. The

ones digit is 5.

3. a. 20

b. 60

c. 95

d. 75

e. 35

f. 100

g. 65

h. 105

4. a. No. An odd number x 5 should end in 5.

b. No. An odd number x 5 should end in 5.

Exercise D

1. If you multiply a number by 3, the digits of the answer add up to 3, 6, or 9.

2. Check your work

3. If your wrong answer is another multiple of 3, the digits will add up to 3 and you'll

be fooled.

4. a and c. The digits don't add up to 3, 6, or 9. Math: Quick Tricks for Multiplication

Exercise E

Student examples may vary.

1. 19 x 7 = ?

20 x 7 = 140

140 – 7 = 133

19 x 7 = 133

2. 32 x 8 = ?

30 x 8 = 240

2 x 8 = 16

240 + 16 = 256

32 x 8 = 256

3. a. 49 x 4

50 x 4 = 200

200 – 4 = 196

49 x 4 = 196

4. a. 78

b. 497

c. 171

Exercise F

1. a. 11 x 7

10 x 7 = 70

1 x 7 = 7

70 + 7 = 77

11 x 7 = 77

b. 39 x 3

40 x 3 = 120

120 – 3 = 117

39 x 3 = 117

2. a. 72

b. 40

c. 30

e. 27

f. 204

g. 3,010

h. 63

i. 133

3. a. The answer doesn't end in 5.

b. The digits of the answer don't add up to 9.

c. The digits of the answer don't add up to 3, 6, or 9.

e. The digits don't add up to 9.