5th grade division

5th Grade

Division

Mrs. Berish

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Division Unit Topics

• Patterns in Multiplication and Division

• Division of Whole Numbers

• Division of Decimals

Click on the topic to go to that section

• Divisibility Rules

Divisibility Rules

Click to return to the table of contents

Divisible

When one number can be divided by another and the result is an exact whole number.

Example: 15 is divisible by 3 because 15 ÷ 3 = 5 exactly

BUT 9 is not divisible by 2 because 9 ÷ 2 is 4 with one left over.

Divisibility

A number is divisible by another number when the remainder is 0.

There are rules to tell if a number is divisible by certain other numbers.

Look at the last digit in the Ones Place!

2 Last digit is even-0,2,4,6,85 Last digit is 5 OR 010 Last digit is 0

Check the Sum! 3 Sum of digits is divisible by 36 Number is divisible by 3 AND 2 9 Sum of digits is divisible by 9

Look at Last Digits4 Last 2 digits form a number divisible by 4

Divisibility Rules

Let's Practice!

Is 34 divisible by 2? Yes, because the digit in the ones place is an even number. Therefore, 34 / 2 = 17

Is 1,075 divisible by 5? Yes because the digit in the ones place is a 5.

Therefore, 1,075 / 5 = 215

Is 740 divisible by 10? Yes, because the digit in the ones place is a 0. Therefore, 740 / 10 = 74

Is 258 divisible by 3? Yes, because the sum of its digits is divisible by 3. 2 + 5 + 8 = 15 Look 15 / 3 = 5 Therefore, 258 / 3 = 86

Is 193 divisible by 6? Yes, because the sum of its digits is divisible by 3 AND 2.1 + 9 + 2 = 12 Look 12 /3 = 4 Therefore, 192 / 6 = 32

Is 6,237 divisible by 9? Yes, because the sum of its digits is divisible by 9.6 + 2 + 3 + 7 = 18 Look 18 / 9 = 2 Therefore, 6,237 /9=693

Is 520 divisible by 4? Yes, because the number made by the last two digits is divisible by 4.20 / 4 = 5 Therefore, 520 / 4 = 130

1 Is 198 divisible by 2?

4 294 divisible by 6?

5 3,926 is divisible by 9

18 is divisible by how many digits? Let's see if your choices are correct.

Did you guess 2, 3, 6 and 9?

Did you guess 3 and 5?

Some numbers are divisible by more than one digit.Using the Divisibility Rules, let's practice.

Did you guess 2 and 4?

Did you guess 2, 5, and 10?

Now it's your turn......

(Click on the cell to reveal the answer)

Divisible by2 by 3 by 4 by 5 by 6 by 9 by 10

39 no yes no no no no no156 yes yes yes no yes no no429 no yes no no no no no446 yes no no no no no no

yes yes no no yes no no

1,006 yes no no no no no no28,550 yes no no yes no no yes

Complete the table using the Divisibility Rules

6 What are all the digits 15 is divisible by?

8 What are all the digits 1,422 are divisible by?

9 What are all the digits 240 are divisible by?

Patterns in Multiplication and Division

Powers of 10

Numbers like 10, 100 and 1,000 are called powers of 10.

They are numbers that can be written as products of tens.

100 can be written as 10 x 10 or 102.

1,000 can be written as 10 x 10 x 10 or 103.

The raised digit is called the exponent. The exponent tells how many tens are multiplied.

A number written with an exponent, like 103, is in exponential notation.

A number written in a more familiar way, like 1,000 is in standard notation.

Powers of 10

Standard Product Exponential Notation of 10s Notation

10 10 101

100 10 x 10 102

1,000 10 x 10 x 10 103

10,000 10 x 10 x 10 x 10 10100,000 10 x 10 x 10 x 10 x 10 10

1,000,000 10 x 10 x 10 x 10 x 10 x 10 106

(greater than 1)

Powers of 10 from ten to one million.

It is easy to MULTIPLY a whole number by a power of 10.

Add on as many 0s as appear in the power of 10.

Examples: 28 x 10 = 280 Add on one 0

28 x 100 = 2,800 Add on two 0s

28 x 1,000 = 28,000 Add on three 0s

If you have memorized the basic multiplication facts, you can solve problems mentally.Use a pattern when multiplying by powers of 10.

steps1. Multiply the digits to the left of the zeros in each factor.

50 x 100 5 x 1 = 52. Count the number of zeros in each factor.

50 x 100

3. Write the same number of zeros in the product.

50 x 100 = 5,000

50 x 100 5,000

6 x 4 = 24

2. Count the number of zeros in each factor.

60 x 400 = _______

6 x 4 = 242. Count the number of zeros in each factor.

60 x 400

60 x 400 = 24,000

60 x 400 = _______

6 x 4 = 242. Count the number of zeros in each factor.

60 x 400

5 x 7 = 35

2. Count the number of zeros in each factor.

500 x 70,000 = _______

5 x 7 = 35

2. Count the number of zeros in each factor. 500 x 70,000

Steps1. Multiply the digits to the left of the zeros in each factor.

5 x 7 = 35

2. Count the number of zeros in each factor.500 x 70,000

3. Write the same number of zeros in the product.500 x 70,000 = 35,000,000

500 x 70,000 = _______

Your Turn....

Write a rule.

Input Output

50 15,000

7 2,100

300 90,000

20 6,000

Write a rule.

Input Output

20 18,000

7 6,300

9,000 8,100,000

80 72,000

11 30 x 10 =

12 800 x 1,000 =

13 900 x 10,000 =

14 700 x 5,100 =

15 70 x 8,000 =

16 40 x 500 =

17 1,200 x 3,000 =

18 35 x 1,000 =

It is easy to DIVIDE a whole number by a power of 10.

Take off as many 0s as appear in the power of 10.

Example: 42,000 / 10 = 4,200 Take off one 042,000 / 100 = 420 Take off two 0s42,000 / 1,000 = 42 Take off three 0s

If you have memorized the basic division facts, you can solve problems mentally.

Use a pattern when dividing by powers of 10.

60 / 10 =60 / 10 = 6

steps1. Cross out the same number of 0s in the dividend as in the divisor.2. Complete the division fact.

700 / 10700 / 10 = 70

8,000 / 10 8,000 / 10 = 800

9,000 / 1009,000 / 100 = 90

More Examples:

.120 / 30120 / 30 = 4

1,400 / 7001,400 / 700 = 2

44,600 / 20044,600 / 200 = 223

This pattern can be used in other problems

Your Turn....

Complete. Follow the rule.

Rule: Divide by 50

Input Output

Find the rule.

Input Output

120 40

2,700 90

Complete. Find the rule.

19 800 / 10 =

20 16,000 / 100 =

21 1,640 / 10 =

22 210 / 30 =

23 80 / 40 =

24 640 / 80 =

25 4,500 / 50 =

Remember Powers of 10 (greater than 1)

Let's look at Powers of 10 (less than 1)

Powers of 10 (less than 1)

StandardNotation

Product of 0.1

ExponentialNotation

0.1 0.1 10-1

0.01 0.1 x 0.1 10-2

0.001 0.1 x 0.1 x 0.1 10-3

0.0001 0.1 x 0.1 x 0.1 x 0.1 10-4

0.00001 0.1 x 0.1 x 0.1 x 0.1 x 0.1 10-5

0.000001 0.1 x 0.1 x 0.1 x 0.1 x 0.1 x 0.1 10-6

The number 1 is also called a Power of 10

because 1 = 100

10,000s 1,000s 100s 10s 1s 0.1s 0.01s 0.001s 0.0001s

Each exponent is 1 less than the exponent in the place to its left. This is why mathematicians defined 100 to be equal to 1.

Let's look at how to multiply a decimal by a Power of 10 (greater than 1)

Example: 1,000 x 45.6 = ?Steps1. Locate the decimal point in the power of

2. Move the decimal point LEFT until you get to the number 1.

3. Move the decimal point in the other factor the same number of places, but to the RIGHT. Insert 0s as needed.

That's your answer.

So, 1,000 x 45.6 = 45,000

1,000 = 1,000.

1 0 0 0 . (3 places)

4 5 . 6 0 0

Let's try some together.

10,000 x 0.28 =

$4.50 x 1,000 =

1.04 x 10 =

26 100 x 3.67 =

27 0.28 x 10,000 =

28 1,000 x $8.98 =

29 7.08 x 10 =

Steps 1. Locate the decimal point in the power of 10.

2. Move the decimal point LEFT until you get to the number 1.

3. Move the decimal point in the other number the same number of places to the LEFT. Insert 0s as needed.

So, 45.6 / 1,000 = 0.00456

Let's look at how to divide a decimal by a Power of 10 (less than 1)

Example: 45.6 / 1,000

1,000 = 1,000.

1 0 0 0

(3 places)

0 0 4 5 . 6

Let's try some together.

56.7 / 10 =

0.47 / 100 =

$290 / 1,000 =

30 73.8 / 10 =

31 0.35 / 100 =

32 $456 / 1,000 =

33 60 / 10,000 =

34 $89 / 10 =

35 321.9 / 100 =

Division of Whole Numbers

Some division terms to remember....

• The number to be divided into is known as the dividend

• The number which divides the other number is known as the divisor

• The answer to a division problem is called the quotient

divisor 5 20 dividend

4 quotient

20 ÷ 5 = 4

Estimating the Quotient helps to break whole numbers into groups.

Estimating: One-Digit Divisor

Divide 8) 68

8)6898

8)68980

Write 0 in remaining place.

80 is the estimate.

Let's Practice: One-Digit Estimation

Estimate:

Remember to divide 50 by 9Then write 0 in remaining place in quotient.

Is your estimate 50 or 40?

Yes, it is 40.Click

Estimate:

Remember to divide 45 by 5Then write 0 in remaining place in quotient.

Yes, it is 90 Click

The estimation for 8)241 is 40?

Estimate 663 ÷ 737

Estimate 4)345

Solve using Estimation Marta baby-sat for four hours and earned $19. ABOUT how much money did Marta earn each hour that she baby-sat?

26)6,498Round 26 to its greatest place.

30)6,498

Divide 30)64

30) 6,4982

30)6,498200

Write 0 in remaining places.

200 is the estimate.

Estimating: Two-Digit Divisor

Let's Practice Two-Digit Estimation

Estimate:

31)637

Remember to round 31 to its greatest place 30Then divided 63 by 30 Finally, write 0's in remaining places in quotient.

Yes, it is 20 Click

Estimate:

87)9,321

Remember to round 87 to its greatest place 90Then divide 93 by 90Finally, write 0's in remaining places in quotient.

Is your estimate 100 or 1,000?

Yes, it is 100 Click

The estimation for 17)489 is 2?

41 Estimate 5,145 ÷ 25.

Estimate 41)2,13042

Estimate 31)7,26443

Solve using Estimation Brandon bought cookies to pack in his lunch. He bought a box with 28 cookies. If he packs five cookies in his lunch each day, ABOUT how many days will the cookies last?

When we are dividing, we are breaking apart into equal groups

Find 132 3

Step 1: Can 3 go into 1, no so can 3 go into 13, yes

- 12 1

3 x 4 = 1213 - 12 = 1Compare 1 < 3

3 x 4 = 1212 - 12 = 0Compare 0 < 3

- 12 0

Step 2: Bring down the 2. Can 3 go into 12, yes

Click for step 1

Click for step 2

Step 3: Check your answer.

44 x 3

Divide and Check 8)29645

47 Divide and check 252 ÷ 6 =

Adam has a wire that is 434 inches long. He cuts the wire into 7-inch lengths. How many pieces of wire will he have?

Bill and 8 friends each sold the same number of tickets. They sold 117 tickets in all. How many tickets were sold by each person?

There are 6 outs in an inning. How many innings would have to be played to get 348 outs?

How many numbers between 23 and 41 have NO remainder when divided by 3?

B 5C 6D 11

Sometimes when we break apart a whole number into groups there is an amount left over.

For example: 47)30 -28 2 We say there are 2 left over

because you can not make a group of 7 out of 2.

For example:

47)30 30 ÷ 7 = 4 R 2 -28 2

This is the way you may have previously written it,

with the R meaning the remainder.

Another example:

2315)358 -30 58 -45 13 We say there are 13 left over (R)

because you can not make a group of 15 out of 13.

358 ÷ 15 = 23 R 13

A group of six friends have 83 pretzels. If they want to evenly share them, how many will be left over?

Four teachers want to evenly share 245 pencils. How many will be left over?

Twenty students want to evenly share 48 slices of pizza. How many slices will be left over?

Suppose there are 890 packages being delivered by 6 planes. Each plane is to take the same number of packages and as many as possible. How many packages will each plane take? How many will be left over? Fill in the blanks. Each plane will take _______ packages. There will be _______ packages left over.

A 149 packages, 2 left over

B 148 packages, 2 left over

47)30 -28 2

27 Instead of writing an R for

remainder, we will write it as a fraction of the 30 that will not fit into a group of 7. So 2/7 is the remainder.

More examples of the remainder written as a fraction:

6)47 -42 5

7The Remainder means that there is 5 left over that can't be put in a group containing 6

To Check the answer, use multiplication and addition.

7 x 6 + 5 = 42 + 5 = 47

37 x 7 + 5 = 259 + 5 = 264

Example:

37)264 -21 54 -49 5

Check the answer using multiplication and addition.

Way 1:

Way 2: 37 quotientx 7 x divisor

259+ 5 + remainder

264 dividend

(Put answer in as a mixed number.)

59 Divide and check 145 ÷ 7

Divide by a 2 Digit Divisor

You can divide by two-digit divisors to find out how many groups there are or how many are in each group.

When dividing by a two-digit divisor, follow the steps you used to divide by a one-digit divisor. Repeat until you have divided all the digits of the dividend by the divisor.

STEPSDivideMultiplySubtractCompareBring down next number

Find 4575 25

Step 1: Can 25 go into 4, no so can 25 go into 45, yes

- 25 20

25 x 1 = 2545 - 25 = 20Compare 20 < 25

25 4575

25 x 8 = 200207 - 200 = 7Compare 7 < 25

7 - 200 75 - 75 0

25 x 3 = 7575 - 75 = 0Compare 0 < 25

3Click for step 1

Click for step 2

Click for step 3

Step 3: Check your answer.

183 x 25

EXAMPLE

Mr. Taylor's students take turns working shifts at the school store. If there are 23 students in his class and they work 253 shifts during the year, how many shifts will each student in the class work?

Step 1 Compare the divisor to the dividend to decide where to place the first digit in the quotient. Divide the tens.Think: What number multiplies by 23 is less than or equal to 25.

Step 2 Multiply the number of tens in the quotient times the divisor. Subtract the product from the dividend.Bring down the next number in the dividend.

Step 3 Divide the result by 23.Write the number in the ones place of the quotient.Think: What number multiplied by 23 is less than or equal to 23? Step 4 Multiply the number in the ones place of the quotient by the divisor.Subtract the product from 23.If the difference is zero, there is no remainder.

Each student will work 11 shifts at the school store.

23)253

Division Steps can be remembered using a "Silly" Sentence.

David Makes Snake Cookies By Dinner.

Divide Multiply Subtract Compare Bring Down

What is your "Silly" Sentence to remember the Division Steps?

Let's try some problems together, using our "Silly" Sentence Steps.

A candy factory produces 984 pounds of chocolate in 24 hours. How many pounds of chocolate does the factory produce in 1 hour?

B 40C 41D 45

Teresa got a loan of $7,680 for a used car. She has to make 24 equal payments. How much will each payment be?

A $230

B $320C $325

Solve 16)17664

Solve 329 ÷ 4765

If 280 chairs are arranged into 35 rows, how many chairs are in each row?

There are 52 snakes. There are 13 cages. If each cage contains the same number of snakes, how many snakes are in each cage?

Solve 46)3,58868

Solve 3,672 ÷ 7269

When dividing by a Two-Digit Divisor there may be a Remainder. Follow the Division Steps .

DivideMultiplySubtractCompareBring DownRepeat

If the Difference in the Last Step of Division is not a Zero, this is the Remainder.The definition of a Remainder is an amount "left over" that does not make a full group (Divisor).Write the Remainder as a Fraction.

top number Difference 62bottom number Divisor 77

This means there are 62 "left over" that does not make a full group of 77.

5 6277

Problem: 77) 447 -385 62

Use Multiplication and Addition to check you Answer.

5 x 77 + 62 = 447 77 x 5

375+ 62447

Let's Practice

Solve 633 ÷ 36

Remember your Steps:

Write the Remainder as a Fraction

Check your work

36) 63336 -273252-

17 2136

CHECK1736x

102510+612

+ 21633

Divisor x Quotient + Remainder = Dividend

Divide, Multiply, Subtract, Compare, Bring Down

What is the remainder when 402 is divided by 56?

B 7C 19D 10

What is the remainder when 993 is divided by 38?

B 8C 13D 26

Divide 80) 10472

Divide 556 ÷ 3573

Divide 45)144274

Divide 4453 ÷ 55

Divide 83)853776

In word problems, we need to interpret the what the remainder means.

For example: Celina has 58 pencils and wants to share them with 5 people. 115) 58 -5 08 5 people will each get 11 pencils - 5 and there will be 3 left over. 3

Interpreting the Remainder

Violet is packing books. She has 246 books and 24 fit in a box. How many boxes does she need? 1024) 246 -24 06 The remainder means she

would have 6 books that would not fit in the 10 boxes. She would need 11 boxes to fit all

the books.

What does the remainder below mean?

If you have 341 oranges to transport from Florida to New Jersey and 7 oranges are in each bag, how many bags will you need to ship all of the oranges?

B 48C 49

At the bakery, donuts are only sold in boxes of 12. If 80 donuts are needed for the teacher's meeting, how many boxes should be bought?

B 7C 8D 9

The school is ordering carry cases for the calculators. If there are 203 calculators and 16 fit in a case, how many cases need to be ordered?

B 11C 12D 13

For the class trip, 51 people fit on a bus and 267 people are going. How many buses will be needed?

A 10B 11C 12

Division of Decimals

Divide decimals

To divide a decimal by a whole number:Use long division.Bring the decimal point up in the answer

0.8124

0.08124

20.30.2030.0203

Match the quotient to the correct problem.

64.255

Which answer has the decimal point in the correct location?

A 1285B 1.285C 12.85

D 128.5

224.44

A 561B 56.1C 5.61

D 0.561

0.4599

A 51B 5.1C 0.51

D 0.051

37.023

Select the answer with the decimal point in the correct location.

A 0.1234B 1.234C 12.34

D 123.4

E 1234

.25055

Select the answer with the decimal point in the correct location.

A 501B 50.1C 5.01

D 0.501

E 0.0501

20.52686

321.6487

2.198788

70.621189

251.2490

Be careful, sometimes a zero needs to be used as a place holder.

35.56 -35 0 56 - 56 0

7 can't go into 5, so put a 0 and bring the 6 down.

27.21 -27 0 2

9.What is the next step in this division problem?

A Put a 2 in the quotient

B Put a 0 in the quotient

C Put a 1 in the quotient

3.205 - 30 2

0.6What is the next step in this division problem?

C Bring down the 0

64.48 -64 0 4

8.What is the next step in this division problem?93

0.636694

2.406395

Be careful! Sometimes there is not enough to make a group so put a zero in the quotient.

0.608 -56 48 -48 0

What is the first step in this division problem?96

A Put a 0 in the quotient in the ones place

B Put a 0 in the quotient in the tenths place C Put a 7 in the quotient

.110424

What is the first step in this division problem?97

Put a 0 in the quotient in the tenths and hundredths place 0

B Put a 0 in the quotient in the ones place

.435598

Instead of leaving the 4 as a remainder, add a zero to the dividend.

Instead of writing a remainder, continue to divide the remainder by the divisor (by adding zeros) to get additional decimal points.

75.6-72 3 6 -32 4

75.60 -72 3 6 - 32 40 - 40 0

Add a zero to the dividend.

No remainder now.

3.26599

87.32100

0.7956101

0.84330102

0.36315103

With a whole number dividend, you can add a decimal point and zeros when you have a remainder.

Example:You want to save $284 over the next 5 months. How much money do you need to save each month?

$284 ÷ 5 = _____

$284- 25 34 - 30 4

Don't leave it as remainder 4, or as 4/5 add a decimal point and zeros.

$284.0- 25 34 - 30 4 0 - 4 0 0

Since the answer is in money, write the answer as $56.80

$82.000- 7 12 - 7 5 0 - 4 9 10

- 7 30

11.714

Since the answer is in money, add a decimal point and 3 zeros. Round the answer to the nearest cent (hundredths place).

$82 ÷ 7 = $11.71

$635104

$782 ÷ 9 =105

$5937106

$3524107

$48 ÷ 22 =108

• Change the divisor to a whole number by multiplying by a power of 10

• Multiply the dividend by the same power of 10

• Divide

• Bring the decimal point up in the answer

DividendDivisor

To divide a number by a decimal:

2.4 15.696

Multiply by 10, so that 2.4 becomes 2415.696 must also be multiplied by 10

24 156.96

.64 6.4

Multiply by 100, so that .64 becomes 646.4 must also be multiplied by 100

64 640

By what power of 10 should the divisor and dividend be multiplied?

7.59 2.2 means

2.0826 means0.06

42.480.3109

Divide

2.592 0.08 =

0.68760.3111

20 divided by 0.25112

Yogurts each cost $.50 each and you have $7.25. How many can you buy?

5th grade division

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