5th Grade Division Mrs. Berish

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Division Unit Topics Divisibility Rules Click on the topic to go to that section Divisibility Rules Patterns in Multiplication and Division Division of Whole Numbers Division of Decimals

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. three five 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.

Divisibility Rules Look at the last digit in the Ones Place! 2 Last digit is even-0,2,4,6,8 5 Last digit is 5 OR 0 10 Last digit is 0 Check the Sum! 3 Sum of digits is divisible by 3 6 Number is divisible by 3 AND 2 9 Sum of digits is divisible by 9 Look at Last Digits 4 Last 2 digits form a number divisible by 4

x 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 x

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 x

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 x

1 Is 198 divisible by 2? Yes No

2 Is 315 divisible by 5? Yes No

3 Is 483 divisible by 3? Yes No

4 294 divisible by 6? True False

5 3,926 is divisible by 9 True False

Some numbers are divisible by more than one digit. Using the Divisibility Rules, let's practice. 18 is divisible by how many digits? Let's see if your choices are correct. Did you guess 2, 3, 6 and 9? 165 is divisible by how many digits? Did you guess 3 and 5? Click Click

28 is divisible by how many digits? Let's see if your choices are correct. Did you guess 2 and 4? 530 is divisible by how many digits? Did you guess 2, 5, and 10? Now it's your turn...... Click Click

Complete the table using the Divisibility Rules (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 156 429 446 1,006 28,550 1,218

6 What are all the digits 15 is divisible by?

7 What are all the digits 36 is divisible by?

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

9 What are all the digits 240 are divisible by?

10 What are all the digits 64 is divisible by?

Patterns in Multiplication and Division Click to return to the table of contents

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.

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 from ten to one million. 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 10 100,000 10 x 10 x 10 x 10 x 10 105 1,000,000 10 x 10 x 10 x 10 x 10 x 10 106 (greater than 1) 4

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

Use a pattern when multiplying by powers of 10. If you have memorized the basic multiplication facts, you can solve problems mentally. Use a pattern when multiplying by powers of 10. steps 1. Multiply the digits to the left of the zeros in each factor. 50 x 100 5 x 1 = 5 2. Count the number of zeros in each factor. 50 x 100 3. Write the same number of zeros in the product. 5,000 50 x 100 = 5,000 50 x 100 5,000

60 x 400 = _______ steps 1. Multiply the digits to the left of the zeros in each factor. 6 x 4 = 24 2. Count the number of zeros in each factor. 3. Write the same number of zeros in the product.

60 x 400 = _______ 3. Write the same number of zeros in the product. steps 1. Multiply the digits to the left of the zeros in each factor. 6 x 4 = 24 2. Count the number of zeros in each factor. 60 x 400 3. Write the same number of zeros in the product.

60 x 400 = _______ steps 1. Multiply the digits to the left of the zeros in each factor. 6 x 4 = 24 2. Count the number of zeros in each factor. 60 x 400 3. Write the same number of zeros in the product. 60 x 400 = 24,000

500 x 70,000 = _______ steps 1. 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. 3. Write the same number of zeros in the product.

500 x 70,000 = _______ steps 1. 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 = _______ Steps 1. 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

Your Turn.... Write a rule. Input Output 50 15,000 7 2,100 300 90,000 20 6,000 rule

Write a rule. Input Output 20 18,000 7 6,300 9,000 8,100,000 80 72,000 rule

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 0 42,000 / 100 = 420 Take off two 0s 42,000 / 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 steps Cross out the same number of 0s in the dividend as in the divisor. 2. Complete the division fact.

700 / 10 700 / 10 = 70 8,000 / 10 8,000 / 10 = 800 9,000 / 100 9,000 / 100 = 90 More Examples:

. 120 / 30 120 / 30 = 4 1,400 / 700 1,400 / 700 = 2 44,600 / 200 44,600 / 200 = 223 This pattern can be used in other problems

Rule: Divide by 50 Input Output 150 250 3,000 Your Turn.... Complete. Follow the rule. Rule: Divide by 50 Input Output 150 250 3,000

Complete. Find the rule. Find the rule. Input Output 120 40 240 8 2,700 90

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) Standard Notation Product of 0.1 Exponential 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 104 103 102 101 100 10-1 10-2 10-3 10-4 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 = ? Steps Locate the decimal point in the power of 10. Move the decimal point LEFT until you get to the number 1. 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 =

. . Let's look at how to divide a decimal by a Power of 10 (less than 1) Example: 45.6 / 1,000 Steps Locate the decimal point in the power of 10. Move the decimal point LEFT until you get to the number 1. 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 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 Click to return to the table of contents

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 20 ÷ 5 = 4

Estimating the Quotient helps to break whole numbers into groups.

x 8) 689 Estimating: One-Digit Divisor Divide 8) 68 8 8)689 Write 0 in remaining place. 80 8)689 80 is the estimate.

Let's Practice: One-Digit Estimation Estimate: 9)507 Remember to divide 50 by 9 Then write 0 in remaining place in quotient. Is your estimate 50 or 40? Yes, it is 40. Click

Then write 0 in remaining place in quotient. Estimate: 5)451 Remember to divide 45 by 5 Then write 0 in remaining place in quotient. Is your estimate 90 or 80? Yes, it is 90 Click

36 The estimation for 8)241 is 40? True False

37 Estimate 663 ÷ 7

38 Estimate 4)345

39 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?

x Estimating: Two-Digit Divisor 26)6,498 Round 26 to its greatest place. x 30)6,498 Divide 30)64 2 30) 6,498 Write 0 in remaining places. 200 30)6,498 200 is the estimate.

Let's Practice Two-Digit Estimation Estimate: 31)637 Remember to round 31 to its greatest place 30 Then divided 63 by 30 Finally, write 0's in remaining places in quotient. Is your estimate 20 or 30? Yes, it is 20 Click

Remember to round 87 to its greatest place 90 Then divide 93 by 90 Estimate: 87)9,321 Remember to round 87 to its greatest place 90 Then divide 93 by 90 Finally, write 0's in remaining places in quotient. Is your estimate 100 or 1,000? Yes, it is 100 Click

40 The estimation for 17)489 is 2? True False

41 Estimate 5,145 ÷ 25.

42 Estimate 41)2,130

43 Estimate 31)7,264

44 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 4 - 12 1 3 x 4 = 12 13 - 12 = 1 Compare 1 < 3 4 Click for step 1 3 132 2 - 12 Step 2: Bring down the 2. Can 3 go into 12, yes Click for step 2 3 x 4 = 12 12 - 12 = 0 Compare 0 < 3

Step 3: Check your answer. 44 x 3 132

45 Divide and Check 8)296

46 Divide and Check 9)315

47 Divide and check 252 ÷ 6 =

48 Divide and check 9470 ÷ 2 =

49 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?

50 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?

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

52 How many numbers between 23 and 41 have NO remainder when divided by 3? A 4 B 5 C 6 D 11

Sometimes when we break apart a whole number into groups there is an amount left over. For example: 4 7)30 -28 2 We say there are 2 left over because you can not make a group of 7 out of 2.

For example: 4 7)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: 23 15)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

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

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

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

56 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

2 4 7)30 -28 2 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. 7

More examples of the remainder written as a fraction: 5 7 6 The Remainder means that there is 5 left over that can't be put in a group containing 6 6)47 -42 5 To Check the answer, use multiplication and addition. 7 x 6 + 5 = 42 + 5 = 47

Example: 37 )264 -21 54 -49 5 Check the answer using multiplication and addition. Way 1: Way 2: 37 quotient x 7 x divisor 259 + 5 + remainder 264 dividend 5 7 7 37 x 7 + 5 = 259 + 5 = 264

57 Divide and Check 4)43 (Put answer in as a mixed number.)

58 Divide and check 61 ÷ 3 = (Put answer in as a mixed number.)

59 Divide and check 145 ÷ 7 (Put answer in as a mixed number.)

60 Divide and Check 2)811 (Put answer in as a mixed number.)

61 Divide and check 309 ÷ 2 = (Put answer in as a mixed number.)

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. STEPS Divide Multiply Subtract Compare Bring down next number

Find 4575 25 Click for step 3 Click for step 1 Click for step 2 1 8 3 Step 1: Can 25 go into 4, no so can 25 go into 45, yes 25 4575 - 25 20 25 x 1 = 25 45 - 25 = 20 Compare 20 < 25 7 - 200 75 - 75 Step 2: Bring down the 7. Can 25 go into 207, yes 25 x 8 = 200 207 - 200 = 7 Compare 7 < 25 25 x 3 = 75 75 - 75 = 0 Compare 0 < 25 Step 3: Bring down the 5. Can 25 go into 75, yes

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?

Each student will work 11 shifts at the school store. 23)253 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.

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.

62 A candy factory produces 984 pounds of chocolate in 24 hours. How many pounds of chocolate does the factory produce in 1 hour? A 38 B 40 C 41 D 45

63 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 $320 C $325

64 Solve 16)176

65 Solve 329 ÷ 47

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

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

68 Solve 46)3,588

69 Solve 3,672 ÷ 72

Write the Remainder as a Fraction. top number Difference 62 When dividing by a Two-Digit Divisor there may be a Remainder. Follow the Division Steps . Divide Multiply Subtract Compare Bring Down Repeat 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 62 bottom number Divisor 77 This means there are 62 "left over" that does not make a full group of 77. 5 62 77 Use Multiplication and Addition to check you Answer. 5 x 77 + 62 = 447 77 x 5 Problem: 77) 447 -385 62 OR 375 + 62 447

Divide, Multiply, Subtract, Compare, Bring Down Let's Practice Remember your Steps: Divide, Multiply, Subtract, Compare, Bring Down Write the Remainder as a Fraction Solve 633 ÷ 36 Check your work CHECK Divisor x Quotient + Remainder = Dividend 17 36 x 102 510 + 612 21 633 36) 633 36 - 273 252 17 21

What is the remainder when 402 is divided by 56? 70 What is the remainder when 402 is divided by 56? A 8 B 7 C 19 D 10

What is the remainder when 993 is divided by 38? 71 What is the remainder when 993 is divided by 38? A 5 B 8 C 13 D 26

72 Divide 80) 104 (Put answer in as a mixed number.)

73 Divide 556 ÷ 35 (Put answer in as a mixed number.)

74 Divide 45)1442 (Put answer in as a mixed number.)

75 Divide 4453 ÷ 55 (Put answer in as a mixed number.)

76 Divide 83)8537 (Put answer in as a mixed number.)

Interpreting the Remainder 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. 11 5) 58 -5 08 5 people will each get 11 pencils - 5 and there will be 3 left over. 3

What does the remainder below mean? Violet is packing books. She has 246 books and 24 fit in a box. How many boxes does she need? 10 24) 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.

77 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? A 47 B 48 C 49 D 50

78 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? A 6 B 7 C 8 D 9

79 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? A 10 B 11 C 12 D 13

80 For the class trip, 51 people fit on a bus and 267 people are going. How many buses will be needed? A 10 B 11 C 12 D 13

Division of Decimals Click to return to the table of contents

Divide decimals 63.93 21 31 3 To divide a decimal by a whole number: Use long division. Bring the decimal point up in the answer 63.93 21 31 3

Match the quotient to the correct problem. 4 8.12 4 81.2 4 0.812 4 0.0812 0.0203 0.203 2.03 20.3

5 64.25 81 Which answer has the decimal point in the correct location? B 1.285 C 12.85 5 64.25 D 128.5

4 224.4 82 Which answer has the decimal point in the correct location? B 56.1 224.4 C 5.61 4 D 0.561

9 0.459 83 Which answer has the decimal point in the correct location? B 5.1 C 0.51 9 0.459 D 0.051

84 Select the answer with the decimal point in the correct location. A 0.1234 B 1.234 C 12.34 3 37.02 D 123.4 E 1234

85 Select the answer with the decimal point in the correct location. A 501 B 50.1 C 5.01 5 .2505 D 0.501 E 0.0501

86 6 20.52

87 4 321.6

88 7 2.198

89 11 70.62

90 4 251.2

Be careful, sometimes a zero needs to be used as a place holder. 5.08 7 35.56 -35 0 56 - 56 7 can't go into 5, so put a 0 and bring the 6 down.

9. 3 27.21 -27 0 2 91 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

0.6 5 3.205 - 30 2 92 What is the next step in this division problem? A Put a 0 in the quotient B Put a 2 in the quotient C Bring down the 0

8. 8 64.48 -64 0 4 93 What is the next step in this division problem? A Put a 0 in the quotient B Put a 4 in the quotient C Put a 2 in the quotient

94 6 0.636

95 3 2.406

Be careful! Sometimes there is not enough to make a group so put a zero in the quotient. .076 8 0.608 -56 48 -48

6 .468 96 What is the first step in this division problem? 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

24 .1104 97 What is the first step in this division problem? A Put a 0 in the quotient in the tenths and hundredths place 0 B Put a 0 in the quotient in the ones place C Put a 4 in the quotient

98 5 .435

Instead of writing a remainder, continue to divide the remainder by the divisor (by adding zeros) to get additional decimal points. 9.4 8 75.6 -72 3 6 -32 4 Instead of leaving the 4 as a remainder, add a zero to the dividend.

9.45 8 75.60 -72 3 6 - 32 40 - 40 Add a zero to the dividend. Add a zero to the dividend. No remainder now.

99 5 3.26

100 2 87.3

101 6 0.795

102 30 0.843

103 15 0.363

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 = _____

56 5 $284 - 25 34 - 30 4 Don't leave it as remainder 4, or as 4/5 add a decimal point and zeros.

56.8 5 $284.0 - 25 34 - 30 4 0 - 4 0 Since the answer is Since the answer is in money, write the answer as $56.80

11.714 7 $82.000 - 7 12 5 0 - 4 9 10 30 -28 2 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

104 5 $63

105 $782 ÷ 9 =

106 7 $593

107 4 $352

108 $48 ÷ 22 =

To divide a number by a decimal: 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 Divisor Dividend

2.4 15.696 24 156.96 Multiply by 10, so that 2.4 becomes 24 15.696 must also be multiplied by 10 .64 6.4 64 640 Multiply by 100, so that .64 becomes 64 6.4 must also be multiplied by 100

By what power of 10 should the divisor and dividend be multiplied? .007 4.9 0.3 42.69

÷ ÷ By what power of 10 should the divisor and dividend be multiplied? 7.59 2.2 means ÷ 2.0826 0.06 means

109 0.3 42.48

110 Divide 2.592 0.08 = ÷

111 0.3 0.6876

112 20 divided by 0.25

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