Unit 4 Number and Operations in Base Ten: Multiplying … for CC-Edition... · Unit 4 Number and...

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O-1Number and Operations in Base Ten

Unit 4 Number and Operations in Base Ten: Multiplying and Dividing Decimals

IntroductionIn this unit, students will learn how to multiply and divide decimals, and learn the algorithm for dividing whole numbers and decimals by a two- digit divisor.

For multiplication of decimals, students will first multiply the decimals as if they were whole numbers. They will then find the sum of the number of decimal digits in each factor, and move the decimal point in the product to the left that many places.

For division of decimals, students will learn how to multiply the divisor and the dividend by the appropriate power of 10 that will eliminate the decimal point in the divisor. They will then perform the division in the same way as they did for whole numbers, but remember to place the decimal point in the quotient in the correct position.

For division by a two-digit divisor, students will learn how to estimate each digit of the quotient by rounding the divisor to the nearest ten, and counting the number of tens in the dividend. At first this method of estimation will always work, but in a later section they will have to use the guess and check method. In this method, the original estimate may be too low, too high, or just right. Students will learn how to adjust their estimate and complete the division algorithm.

Teacher’s Guide for AP Book 5.2

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NBT5-54 Multiplying Decimals by Whole Numbers Pages 54–56

Review base ten materials. Review the use of base ten materials when using decimals.

= 1 = 0.1 = 0.1

NOTE: In this context, we are now using the hundreds block as a ones block. One column or row of the ones block is now a tenths block. One unit of the tenths block is now a hundredths block.

ASK: How many hundredths are in 1? (100) How many tenths are in 1? (10) How many hundredths are in 1 tenth? (10)

Ask students to model the decimal 2.13 on their desks with base ten materials. (see diagram below)

Model multiplying a decimal by a whole number with base ten materials and without regrouping. Write on the board:

3 × 5

ASK: What addition question can you use to find the product? Ask for a volunteer to write the answer on the board. (5 + 5 + 5)

(MP.4)

STANDARDS 5.NBT.B.7, 5.NBT.B.5

VOcABUlARy associative property decimals hundredths hundredths block ones block regrouping tenths tenths block

GoalsStudents will multiply decimals up to the hundredths place by a whole number.

PRIOR KNOWlEDGE REQUIRED

Knows how to multiply a multi-digit number by a single-digit number using the standard algorithm Can use base ten materials to model decimal arithmetic and multiplication involving regrouping Can multiply a multi-digit decimal number by multiples of 10

MATERIAlS

base ten blocks plastic money grid paper

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Write on the board:

2.13 × 3

ASK: What addition question can you use to find the product? Ask for a volunteer to write the answer on the board. (2.13 + 2.13 + 2.13) Ask students to use base ten materials to add 2.13 + 2.13 + 2.13. (6.39; see diagram below)

What could we have done to the digits in 2.13 to get the answer 6.39? (multiply each digit separately by 3)

Exercises: Write the question in your notebook and find the product mentally.

a) 3.24 × 2 b) 2.31 × 3 c) 1.43 × 2 d) 1.12 × 4

e) 4.31 × 2 f ) 2.43 × 2 g) 2.21 × 4 h) 2.31 × 2

Answers: a) 6.48, b) 6.93, c) 2.86, d) 4.48, e) 8.62, f ) 4.86, g) 8.84, h) 4.62

Multiply a decimal by a whole number using place values. ASK: How can we write 2.13 using place values? (2 ones + 1 tenth + 3 hundredths) ASK: What is 3 × 2 ones? (6 ones) What is 3 × 1 tenth? (3 tenths) What is 3 × 3 hundredths? (9 hundredths) How can we write the answers in decimal notation? (6.39)

Write on the board:

2.13 = 2 ones + 1 tenths + 3 hundredths× 3 × 3

6.39 6 ones + 3 tenths + 9 hundredths

Exercise: Multiply using place values.

a) 3.12 × 3 b) 4.12 × 2 c) 1.33 × 3

Answers: a) 9.36, b) 8.24, c) 3.99

(MP.4)

Number and Operations in Base Ten 5-54


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Using base ten materials, model multiplying a decimal by a whole number with regrouping. Point out to students that none of the questions so far have involved regrouping.

Write on the board:

2.16 × 3

Ask students to use base ten materials at their desks to calculate the product using addition. (see diagram below)

Replace with a tenth block.

ASK: How many hundredths do we have? (18) What can we use to replace 10 hundredths? (1 tenth) How many hundredths remain? (8) Ask students to replace the 10 hundredths with a tenth block, and read the answer. (6.48)

Using money, model multiplying a decimal by a whole number with regrouping hundredths for tenths (pennies for dimes). Some students will benefit from a demonstration using plastic money. Consider the decimal 2.16 as $2.16. Write $2.16 on the board. ASK: How many dollars are there? (2) How many dimes are there? (1) How many pennies are there? (6) If your class has plastic money, ask students to represent $2.16 × 3 using addition. If not, draw the following on the board and tell students that D and P represent dimes and pennies.

$1 $1 D P P P P P P

$1 $1 D P P P P P P

$1 $1 D P P P P P P

D

Replace 10 pennies with

1 dime.

ASK: What can we replace 10 pennies with? (1 dime) How many pennies are left? (8) How many dimes are there now? (4) How much money is there? ($6.48)

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Multiplying a decimal by a whole number and regrouping tenths for ones (dimes for dollars). Write on the board:

1.63 × 2

Ask students to use base ten materials or money models to find the product using addition. ASK: What can we replace 10 tenths with? (a ones block) How many ones are there now? (3) How many tenths? (2) How many ones? (6) Ask a student to read the final answer. (3.26 or $3.26; see diagrams below)

Replace 10 tenths with a one.

or

$1 P P PD D D D D D

$1 P P PD D D D D D $1

Replace 10 dimes with 1 dollar.

Exercises: Use base ten materials or plastic money to find the product.

a) 2.37 × 2 b) 2.71 × 3 c) 3.17 × 3

Answers: a) 4.74, b) 8.13, c) 9.51

Multiplying a decimal number involving regrouping using place values. Write on the board:

2.63 ×2

ASK: How do we write 2.63 using place values (2 ones + 6 tenths + 3 hundredths) What is 2 ones × 2? (4) What is 6 tenths × 2? (12 tenths) What is 3 hundredths × 2? (6 hundredths)

Write on the board:


4 ones + 12 tenths + 6 hundredths

ASK: We have 12 tenths. What can we use to replace 10 tenths? (a ones block) How many tenths are left? (2 tenths) How many ones are there altogether? (5) What is the answer in decimal form? (5.26) Write on the board:

= 5 ones + 2 tenths + 6 hundredths = 5.26

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Write on the board:


4 ones + 8 tenths + 16 hundredths

ASK: As we have 16 hundredths, what can we use to replace 10 hundredths? (a tenths block) How many hundredths are left? (6 hundredths) How many tenths are there altogether? (9) What is the answer in decimal form? (4.96)

Write on the board:

= 4 ones + 9 tenths + 6 hundredths = 4.96

Exercise: Multiply using place values.

a) 2.61 × 3 b) 1.52 × 3 c) 1.28 × 3 d) 5.29 × 2

Bonus e) 2.76 × 3 f ) 3.48 × 5

Answers: a) 7.83, b) 4.56, c) 3.84, d) 10.58, Bonus: e) 8.28, f ) 17.40

Multiplying a decimal by a whole number using a grid. Ask students to multiply 237 × 2 using a grid and compare the answer to part a) above. ASK: What is the only difference in the answers? (the answer to part a) has a decimal point) Write on the board:

The decimal points line up on the grid

1 1

2 3 7 2 3 7× 2 × 2

4 7 4 4 7 4

SAY: When you multiply a decimal number by a whole number, place the decimal point in the answer underneath the decimal point in the decimal number.

Exercises: Find the products using grid paper. You may have to regroup more than once.

a) 3.64 × 2 b) 5.28 × 3 c) 6.27 × 5 d) 4.93 × 8 e) 7.04 × 9

Bonus: 9,134.57 × 8

Answers: a) 7.28, b) 15.84, c) 31.35, d) 39.44, e) 63.36, Bonus: 73,076.56

Multiplying a decimal by multiples of 10. Write on the board:

23 × 10

ASK: What is the fastest way to multiply a number by 10? (move the decimal point one place to the right) Write on the board:

23 × 10 = 2 3 0 .

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SAY: The same rule applies to multiplying a decimal by 10. Write on the board:

2.3 × 10 = 2 3 .

SAY: We can use the associative property to help us multiply decimals by multiples of 10. Write on the board:

20 × 2.3

ASK: How do we write 20 as a multiple of 10? (2 × 10)

= (2 × 10) × 2.3

SAY: The associative property lets us move the brackets.

= 2 × (10 × 2.3)

ASK: What is 10 × 2.3? (23) SAY: Calculate 2 × 23 mentally. (46)

= 2 × 23 = 46

Exercise: Calculate using this method.

a) 30 × 1.2 b) 40 × 2.1 c) 60 × 1.1

Answers: a) 36, b) 84, c) 66

Extensions1. Marty shopped at the local grocery store. This is what he bought:

Product Unit Price Quantity

Milk $4.95 3

Bread $2.93 4

Cereal $5.99 2

How much did Marty spend altogether?

Answer: $38.55

2. Maria has relatives in Laos, Moldova, and Samoa. She calls them each month and keeps track of how many minutes each call lasts. Here are the calls Maria made last month.

country called

length of call (min)

country called

length of call (min)

Laos 2 Moldova 1

Moldova 4 Laos 3

Samoa 3 Samoa 4

Moldova 3 Laos 3

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Maria’s telephone service charges for long distance calls per minute according to the chart:

country Laos Moldova Samoa

cost per minute $1.49 $1.26 $1.29

Find the total cost of Maria’s long distance calls last month.

Answer: $31.03

3. The price per gallon of gas in New York City is $3.81. Harry’s motorcycle has a gas tank that holds 5 gallons. While on vacation, Harry filled his tank 7 times. Suppose he paid the same price per gallon on his trip as he did in New York City. How much did Harry spend on gas?

Answer: $133.35

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O-9Number and Operations in Base Ten 5-55

NBT5-55 Multiplying Decimals by Decimals Pages 57–58

(Introduction)

STANDARDS 5.NBT.B.7, 5.NBT.A.3

VOcABUlARy decimals denominator hundredths hundredths block ones block tenths tenths block

GoalsStudents will multiply decimal fractions and the corresponding decimals.


Can convert decimal fractions into decimals

MATERIAlS

base ten blocks overhead or digital projector BlM Ones, Tenths, Hundredths (p. O-57)

Use base ten materials to represent decimal fractions. Use BlM Ones, Tenths, Hundredths to display the diagram of a ones block on the board (either using a digital projector or transparency).

Shade in one square and ask students what fraction is represented. (1/100) Shade in one column and ask students what fraction is represented. (1/10) Shade in one row and ask students what fraction is represented. (1/10) See sample diagrams below:

Using base ten materials, convert fractions with denominator 100 into decimals. Erase the shading from the board. Shade the areas shown below and ask students to name a fraction and a decimal for each.

a)

(2/10, 0.2)

b)

(3/10, 0.3)

c)

(18/100, 0.18)

Multiply decimal fractions by finding overlapping shading. Erase the shading on the board. Ask a student to come to the board to shade 4/10 using columns. (see Figure 1 sample answer in margin)

Without erasing the first student’s shading, ask another student to come to the board and shade 3/10 using rows. (see Figure 2 sample answer in margin)

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Figure 1 Figure 2


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Write on the board:

310

× 4

10

SAY: We can find this product by finding 3/4 of 4/10. This is the area created where the two shadings done by students overlap (see example in margin). ASK: What decimal fraction does the overlapped shading represent? (12/100) SAY: So 3/10 × 4/10 = 12/100.

Shade in the following areas, and ask students to come to the board to write a multiplication equation for each diagram.

a) b) c)

210

× 5

10 =

10100

310

× 2

10 =

6100

510

× 7

10 =

35100

Multiplying decimal fractions. Remind students that, to multiply fractions, you can multiply the numerators and then multiply the denominators. Have students multiply the following fractions in their notebooks.

Exercises: Multiply.

a) 3

10 ×

410

= b) 7

10 ×

910

= c) 3

10 ×

9100

= d) 23

100 ×

410

=

Answers: a) 12/100, b) 63/100, c) 27/1,000, d) 92/1,000

ASK: Looking at the fraction equations, how can you predict the number of zeroes in the denominator of the product? (by finding the sum of the number of zeroes in the denominators of each factor)

changing fraction equations into decimal equations. Remind students that, for fractions with powers of 10 as denominators, the number of digits after the decimal point is the same as the number of zeroes in the denominator. Example: 43/100 = 0.43. Have students convert each of the fraction equations in the previous exercises into decimal equations. Do the first conversion with the class as an example. (a) 0.3 × 0.4 = 0.12, b) 0.7 × 0.9 = 0.63, c) 0.3 × 0.09 = 0.027, d) 0.23 × 0.4 = 0.092)

Write on the board:

310

×4

10=

12100

Number of zeroes in denominator

+ =

Ask a student to fill in the bottom row of the chart.

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Write on the board:

0.3 × 0.4 = 0.12

Number of decimal digits

+ =

Ask a student to fill in the bottom row of the chart.

ASK: Looking at the two charts, how can you predict the number of decimal digits in the second chart? (find the number of zeroes in the denominators of the corresponding fractions)


a) 0.3 × 0.5 b) 0.7 × 0.8 c) 0.05 × 0.3

Bonus d) 0.002 × 0.03 e) 0.003 × 0.004

Answers: a) 0.15, b) 0.56, c) 0.015, Bonus: d) 0.00006, e) 0.000012

Extensions1. Find as many pairs of decimal fractions as you can that have

the product.

a) 9

100 b)

16100

c) 8

1000, d) 36

1000,

Sample answers

a) 9

10 ×

110

, 3

10 ×

310

, an uncommon answer: 18100

× 5

10

b) 8

10 ×

210

, 4

10 ×

410

c) 8

100 ×

110

, 8

10 ×

1100

, 4

100 ×

210

, 4

10 ×

2100

d) 4

100 ×

910

, 4

10 ×

9100

, 6

100 ×

610

, 36

100 ×

110

, 3610

× 1

100

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Finding patterns in the number of decimal digits when multiplying decimals. Write on the board:

310

×7

100=

211000,

Number of zeroes in denominator + =

Fraction as decimal × =

Numerator

Number of decimal digits

+ =

Ask a student to come to the board and write the number of zeroes in each decimal fraction in the first chart above. (1, 2, 3)

Ask another student to come to the board and write each decimal fraction as a decimal in the second chart above. (0.3, 0.07, 0.021)

Ask another student to come to the board and write the numerator for each fraction decimal. (3, 7, 21)

Ask another student to come to the board and write the number of decimal digits in each decimal. (1, 2, 3)

ASK: What was the product of the numerators? (21) ASK: What was the product of the fractions written as a decimal? (0.021) ASK: How many times did the decimal point move to the left in 21 to get 0.021? (3) ASK: What was the sum of the decimal digits? (3)

(MP.2)

NBT5-56 Multiplying Decimals by Decimals Pages 59–60


VOcABUlARy decimal digits denominator hundredths tenths

GoalsStudents will multiply decimals where the product has up to 3 decimal digits.


Can multiply multi-digit numbers by 2-digit numbers

MATERIAlS

calculators

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Rule for placing the decimal point when multiplying decimals. Write on the board:

To multiply decimals:

1. Multiply the numbers as if they were whole numbers.

2. Count the number of digits after the decimal in each factor.

3. Add the numbers from Step 2.

4. Shift the decimal point to the left that many places.

Do the following example with the class. Write on the board:

0.34 34× 0.2 × 2

68

Point out that 0.34 has 2 digits after the decimal, and 0.2 has 1 digit after the decimal. 1 + 2 = 3, so we shift the decimal point 3 places to the left.

So, 0.34 × 0.2 = 0.068. Note to students that we needed to add a zero here!

Have the class do the following exercises. Tell them that sometimes, to multiply the numbers, they may have to use a grid.


a) 0.5 × 0.7 b) 0.4 × 0.12 c) 0.32 × 0.4 d) 2.13 × 0.8

Answers: a) 0.35, b) 0.048, c) 0.128, d) 1.704

Multiplying multi-digit decimals. Write on the board:

2.35

× 3.4

Ask students to find the product using their calculators. (7.99)

ASK: How many decimal digits are there in 2.35? (2) How many decimal digits are there in 3.4? (1) How many times should we move the decimal point to the left? (3) How did you get that? (2 + 1 = 3)

ASK: For the answer your calculator gave, 7.99, how many places did the decimal point seem to move to the left? (2) Is there a problem here? (Students will likely say there is a problem, which will lead to finding the product without a calculator.) SAY: Let’s find the product without a calculator. Draw the grid provided in the margin on the board.

Ask for a student volunteer to find the product on the board. (7.990) ASK: How many times was the decimal point shifted to the left? (3) ASK: Was the calculator wrong? Why? (no, 7.990 = 7.99)

2 3 5× 3 4


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Exercises: Use your calculator to find the product, and then check manually.

a) 2.5 × 1.8 b) 1.32 × 2.5 c) 1.275 × 3.4 d) 12.8 × 13.5

Bonus: 8.125 × 3.04

Answers: a) 4.5, b) 3.3, c) 4.335, d) 172.8, Bonus: 24.7

Extensions1. The first number in each product is missing a decimal. Place the

decimal point in the correct position.

a) 3 × 5.1 = 1.53 b) 245 × 1.3 = 31.85

c) 8 × 0.7 = 0.056 d) 34 × 0.2 = 0.68

Answer: a) 0.3, b) 24.5, c) 0.08, d) 3.4

2. Find eight different pairs of numbers with the product 0.035.

Sample answers 70 × 0.0005 7 × 0.005 0.007 × 5 0.7 × 0.05 35 × 0.001 0.035 × 1 3.5 × 0.01 0.35 × 0.1

3. Put the same number in each box.

× × × × = 0.00032

Answer: 0.2

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Dollar and cent notation. Lesson NBT5-57 provides a review of multiplying decimals involving word problems. Before having students complete the word problems, we suggest you review dollar and cent notation. In some situations involving money, amounts involving fractions of cents are used.

ASK: How can we write 3 cents using dollars and cents? ($0.03) ASK: How can we write 4 cents using dollars and cents? ($0.04)

ASK: How can we write 0.03 using 3 decimal digits? (0.030) ASK: How can we write 0.04 using 3 decimal digits? (0.040)

SAY: 312

¢ is between 3¢ and 4¢

ASK: What decimal is exactly in between 0.030 and 0.040? (0.035)

Write on the board:

So, 3

12

¢ = $0.035

Exercises: Write the amount in dollars and cents notation.

a) 712

¢ b) 912

¢ c) 1612

¢ Bonus: 97

10¢

Answers: a) $0.075, b) $0.095, c) $0.165, Bonus: $0.097

Extensions1. In ice hockey, individual players get points by scoring goals or assisting

other players in scoring goals. In the 1985–86 season, Wayne Gretzky scored 0.65 goals per game and earned 2.0375 assists per game. He played 80 games that season.

a) How many goals did he score in the 80 games?

b) How many assists did he earn?

c) To find total points, add the goals and assists. How many points in total did Gretzky get during the season?

Answers: a) 52, b) 163, c) 215

(MP.1)

NBT5-57 Decimal Word Problems—Multiplication Page 61


VOcABUlARy assists calories goals quire ream sales tax

GoalsStudents will solve word problems involving multiplying decimals by whole numbers and by decimals


Knows how to multiply a multi-digit number by a two-digit number using the standard algorithm Knows how to multiply decimals by whole numbers and by decimals


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2. Barb’s bicycle shop rents out bikes for a fee of $10.75 plus $6.80 per hour. What is the total cost of renting a bike for 4.25 hours?

Answer: $39.65

3. John’s dad is on a diet. His diet recommends that he consume up to 800 calories at lunch. For lunch today, he ate: 125 grams of bread, 45 grams of cheese, 120 grams of broccoli, and a 200 gram apple.

A gram of bread contains 2.7 calories, cheese has 4.1 calories per gram, broccoli has 0.32 calories per gram, and apples have 0.52 calories per gram. Did the lunch meet the requirements of the diet?

Answer: Yes. The lunch had 664.4 calories, which is less than the recommended 800 calorie limit.

4. John earns $10.75 per hour at a fast-food restaurant. How much does he earn if he works 8 h? 9 h? Is your answer to Question 5 on AP Book 5.2 p. 60 between these two? If not, look for a mistake.

Answer: $86 for 8 h, $96.75 for 9 h; yes, the answer for Question 5 ($91.375) is between the figures for 8 h and 9 h.

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Review base ten materials. Remind students that when using base ten materials to represent decimals:

= 1 = 0.1 or 1

10 = 0.01 or

1100

Have students review base ten materials by having them represent the following numbers at their desks. (see diagram below for sample answer to part a))

a) 1.43 b) 1.72 c) 3.16

Use base ten materials to model division of decimals by whole numbers without regrouping. Write on the board:

3.69 ÷ 3

Ask students to represent 3.69 using base ten materials. (see diagram below)

(MP.4)


VOcABUlARy decimal hundredths ones tenths whole number

GoalsStudents will divide decimals by whole numbers using base ten materials and place values where no regrouping is required.


Knows how to multiply whole numbers using base ten materials and place values

MATERIAlS

base ten materials plastic money

NBT5-58 Dividing Decimals by Whole Numbers Pages 62–63

(Introduction)


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Ask students to divide the materials into three equal groups. (see diagram below)

ASK: What is the division statement? (3.69 ÷ 3 = 1.23)

Exercises: Use base ten materials to perform the division and then write a division statement:

a) 4.26 ÷ 2 b) 8.48 ÷ 4 c) 9.36 ÷ 3

Answers: a) 4.26 ÷ 2 = 2.13, b) 8.48 ÷ 4 = 2.12, c) 9.36 ÷ 3 = 3.12

Use place values to model division of decimals by whole numbers without regrouping. Write on the board:

6.82 ÷ 2

ASK: How do we write 6.82 using place values? (6 ones + 8 tenths + 2 hundredths) ASK: What is 6 ones ÷ 2? (3 ones) What is 8 tenths ÷ 2? (4 tenths) What is 2 hundredths ÷ 2? (1 hundredth)

Write on the board and say:

6.82 ÷ 2 = (6 ones + 8 tenths + 2 hundredths) ÷ 2 = 3 ones + 4 tenths + 1 hundredth

ASK: How do we write this in decimal notation? (3.41) Write the answer on the board.

Exercises: Use place values to divide. (NOTE: Students should arrive at their answers by first noting place values, as in the example above, and then finding the decimal notation.)

a) 4.28 ÷ 2 b) 9.36 ÷ 3 c) 4.84 ÷ 4

Answers: a) 2.14, b) 3.12, c) 1.21

Use money to model division of decimals by whole numbers without regrouping. Some students will benefit from a demonstration of division using money. Write on the board:

6.39 ÷ 3

ASK: How can we represent $6.39 using dollar bills, dimes, and pennies? (6 dollar bills, 3 dimes, and 9 pennies)

Use plastic money to model or draw the following on the board:

$1 $1

$1 $1 $1

$1P

P

P

P

P

P

P

P

P

D D D

(MP.8)

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ASK: If we divide the $6 among 3 friends, how many $1 bills will each friend get? (2) If we divide 3 dimes among three friends, how many dimes does each get? (1) If we divide 9 pennies among three friends, how many pennies does each get? (3) So how much money does each friend get? ($2.13)

Exercises: Divide the money.

a) $8.46 ÷ 2 b) $6.99 ÷ 3 c) $4.84 ÷ 4

Answers: a) $4.23, b) $2.33, c) $1.21

Recognize that dividing decimals by whole numbers can be done by dividing without the decimal and later placing the decimal point. Write on the board:

2 6 4 8 6.48 ÷ 2− = (6 ones + 4 tenths + 8 hundredths) ÷ 2

Ask two students to come to the board and perform the divisions: the first using the division algorithm (see answer in margin) and the second using place values. (3 ones + 2 tenths + 4 hundredths = 3.24)

ASK: What is the same about the quotients? (same digits) What is different? (when the dividend has a decimal point, the quotient has a decimal point) What do you notice about the position of the decimal points in the quotient and the dividend in the second question? (they are in the same place)

SAY: To divide a decimal by a whole number, perform the division using the algorithm as if there were no decimal point and then place the decimal point in the correct place in the quotient.

Write on the board:

8.24 ÷ 2

Ask students to perform the division in their notebooks. When they have had enough time, ask a student to perform the division on the board. (see answer in margin)

Write on the board:

If 824 ÷ 2 = 412, then 8.24 ÷ 2 = ???

Ask a student to come to the board to complete the division equation. (8.24 ÷ 2 = 4.12)

(MP.4)

(MP.8)

3 2 42 6 4 8− 6

0 4− 4

0 880

4 1 22 8 2 4− 8

0 2− 2

0 440


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Exercises

1. Divide by using the division algorithm. First ignore the decimal point, and then place the decimal point in the quotient.

a) 6.39 ÷ 3 b) 4.28 ÷ 2 c) 5.26 ÷ 2 d) 4.23 ÷ 3

Answers: a) 2.13, b) 2.14, c) 2.63, d) 1.41

2. Use the fact that 5,284 ÷ 4 = 1,321 to divide:

a) 52.84 ÷ 4 b) 528.4 ÷ 4 c) 5.284 ÷ 4 d) 0.5284 ÷ 4

Answers: a) 13.21, b) 132.1, c) 1.321, d) 0.1321

Bonus: Use the fact that 3,173,255 ÷ 5 = 634,651 to divide 31,732.55 ÷ 5

Answer: 6,346.51

Extensions1. Ava earns $78.40 working for 7 hours at a part-time job.

a) What is her pay per hour? NOTE: When writing numbers in dollar notation, two decimal digits are required.

b) When Ava works on a holiday, she gets paid extra. She is paid 1.5 times as much per hour. What is her pay per hour on a holiday?

c) How much does Ava earn for 6 hours of work on a holiday?

Answers: a) $11.20, b) $16.80, c) $100.80

2. A website about fuel economy says that a particular car model will drive 29.7 miles per gallon of gas.

a) The Benitez family drove that same model of car for 234.4 miles using 8 gallons of gas. Was the website correct?

b) How much farther would the Benitez family travel on 8 gallons if their car drove as far as the website said?

Answers: a) no, they traveled 29.3 miles per gallon, so the website was not correct; b) 3.2 miles ((8 × 29.7) − 234.4 = 3.2)

(MP.4)

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Use base ten materials to model division of decimals by whole numbers using the division algorithm where regrouping is required. Write on the board:

2 7.3 4

Ask students to model the steps of the division algorithm at their desks using base ten materials.

SAY: Use base ten materials to represent 7.34. (see diagram below)

Ask students to follow the steps at their desks using base ten materials.

Step 1: Divide the ones blocks into two equal groups.

(MP.4)

NBT5-59 Dividing Decimals by Whole Numbers Pages 64–65


VOcABUlARy dividend divisor quotient

GoalsStudents will divide decimals by whole numbers using base ten materials, money, and the division algorithm.


Knows how to multiply whole numbers using base ten materials and place values

MATERIAlS

base ten materials plastic money


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Continue writing on the board as you ask the following questions.

ASK: How many ones are in each group? (3) How many were placed in groups? (6) How many ones remain? (1)

32 7 3 4− 6

1

Step 2: SAY: Exchange the ones block for 10 tenths.

ASK: How many tenths are there now? (13) Continue writing on the board:3

2 7 3 4− 6

1 3

Step 3: SAY: Divide the tenths into two equal groups.

Continue writing on the board as you ask the following questions. ASK: How many tenths are in each group? (6) How many tenths were placed in groups? (12) How many tenths remain? (1)

3 62 7 3 4− 6

1 3− 1 2

1

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Step 4: SAY: Exchange the ten block for 10 hundreths blocks.

ASK: How many hundredths blocks are there now? (14) Continue writing on the board:

3 62 7 3 4− 6

1 3− 1 2

1 4

Step 5: Divide the 14 hundreths blocks into two equal groups.

SAY: Place the decimal point in the quotient directly above the decimal point in the dividend so we can line up tenths with tenths and hundredths with hundredths. Add the decimal point between the 3 and the 6 in the quotient.

Continue writing on the board. ASK: How many hundredths are in each group? (7) How many hundredths were placed altogether? (14) How many hundredths are remaining? (0)

3 6 72 7 3 4− 6

1 3− 1 2

1 4− 1 4

0


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ASK: What decimal is represented in each group of base ten materials? (3.67)

Use money to model division of decimals by whole numbers using the division algorithm where regrouping is required. Some students will benefit from using a money model.

AcTIVITy

Model 2 7.3 4 using plastic money.

ASK: How can we represent $7.34 using plastic money? (see diagram below)

$1 $1

$1 $1

$1 $1 $1

D D D P P P P

Ask students to follow these steps on their own to model the division.

Step 1: Divide the dollar bills into two equal groups. Step 2: Exchange a $1 for 10 dimes. Step 3: Divide the resulting dimes into two groups. Step 4: Exchange the remaining dime for 10 pennies. Step 5: Divide the pennies into two groups.

The final model should look like this:

$1 $1 $1D D D D D D

P P P P P P P

$1 $1 $1D D D D D D

P P P P P P P

ASK: How much money is in each group? ($3.67)

Exercises: Divide. NOTE: Students should notice this is exactly like dividing using whole numbers and then putting the decimal point in the correct place.

1. a) 7.17 ÷ 3 b) 4.944 ÷ 4 c) 0.117 ÷ 9

Bonus: 11,111.04 ÷ 9

2. a) 8.2143 ÷ 3 b) 126.14 ÷ 2 c) 11,246.48 ÷ 4

Answers 1. a) 2.39, b) 1.236, c) 0.013, Bonus: 1,234.56 2. a) 2.7381, b) 63.07, c) 2,811.62

(MP.4)

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Extensions1. To turn a fraction into a decimal, write the numerator using at least three

decimal digits and then divide the numerator by the denominator. For example, 1/4 = 1.000 ÷ 4. Find decimal representations for the fraction.

a) 12

b) 14

c) 15

d) 18

Answers: a) 12

= 0.500, b) 14

= 0.250, c) 15

= 0.200, d) 18

= 0.125

2. In baseball, a batter’s average is a decimal with three decimal digits. To find the decimal, divide the number of hits by the number of times at bat. Rewrite the number of hits using three decimal places (example: 3 = 3.000). Which batter has the highest average?

Batter Number of Hits

Number of Times at Bat Average

Derek 3 8

Melky 1 4

Jose 2 5

Answer: Derek 0.375, Melky 0.250, Jose 0.400. Jose has the highest average.

3. A pack of three pens costs $5.85.

a) How much does each pen cost? Estimate, and then find the exact answer.

b) Lina estimates that 20 pens will cost $42. Is her estimate reasonable? Explain.

Answers: a) under $2 each; $1.95, b) No, because the cost of each pen is less than $2, so the cost of 20 pens should be less than $40.

(MP.1)

(MP.1)

(MP.3)



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Review finding equivalent fractions. On the board, draw the diagram shown in the margin.

Ask a student to come to the board to write a fraction for the shaded region. (3/4) Ask the student to explain how they got the answer. (3 is the number of shaded regions; 4 is the total number of regions in the whole)

Divide each of the regions on the board into two parts so that the diagram looks like the example shown in the margin.

Ask a different student to come to the board to write a fraction for the shaded region other than 3/4. (6/8) Ask the student to explain how they got the answer. (6 is the number of shaded regions; 8 is the total number of regions in the whole)

SAY: We didn’t change the amount of pie or pizza we divided, so what can we say about the fractions 3/4 and 6/8? (they are equal)

Write on the board:

34

××

??

= 68

ASK: What can we multiply the numerator and denominator by to see that 3/4 = 6/8? (multiply both by 2)

SAY: Remember that you can write the fraction 3/4 as 3 ÷ 4.

Write on the board:

3 ÷ 4 = 34

= 3 × 2 4 × 2

SAY: How can we write this last fraction using the division symbol (÷)? ((3 × 2) ÷ (4 × 2))

Write on the board:

3 ÷ 4 = (3 × 2) ÷ (4 × 2)

NBT5-60 Dividing Decimals by Decimals Pages 66–67

STANDARDS 5.NBT.B.7, 5.NBT.A.2


GoalsStudents will divide decimals by decimals by first multiplying the divisor and dividend by the power of 10 to eliminate the decimal in the divisor.


Knows how to divide decimals by whole numbers using the division algorithm Knows how to find equivalent fractions Knows how to multiply decimals by powers of 10

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Ask students to find other equivalent division statements for 3 ÷ 4. (Sample answers: 3 ÷ 4 = (3 × 5) ÷ (4 × 5), 3 ÷ 4 = (3 × 8) ÷ (4 × 8))

Review multiplying decimals by powers of 10. Write on the board:

2.13 9.48 7.34 8.219

× 10 × 10 × 100 × 100

Ask students to come to the board to write the answers on the board. (23, 94.8, 734, 821.9)

ASK: When multiplying by 10, how many places does the decimal point move? (1) and in which direction? (right)

ASK: When multiplying by 100, how many places does the decimal point move? (2), and in which direction? (right)

Find equivalent division statements by multiplying by powers of 10. SAY: For division of decimals, we will be multiplying the numerator (dividend) and denominator (the divisor) by powers of 10.

Write on the board:

3.6 ÷ 4

ASK: How can we do this calculation mentally? (ignore the decimal to divide 36 ÷ 4 and then place the decimal point in the quotient later) Write on the board:

9 0.94 3 6 4 3.6

Write on the board:

3.6 ÷ 0.4

ASK: What is the difference between this question and the previous one? (in this question, the divisor has a decimal) SAY: Let’s find an equivalent division statement for 3.6 ÷ 0.4.

Write on the board and highlight the 10 in the following:

3.6 ÷ 0.4 = (3.6 × 10) ÷ (0.4 × 10)

ASK: What is 3.6 x 10? (36) What is 0.4 x 10? (4)

Continue writing on the board, again highlighting the 10:

3.6 ÷ 0.4 = (3.6 × 10) ÷ (0.4 × 10) = 36 ÷ 4

ASK: What is 36 ÷ 4? (9)

ASK: What happened to the decimal in the divisor? (it is gone) ASK: Why did that happen? (because we multiplied the divisor by 10) ASK: Why do you think we multiplied by 10 and not 100 or 1,000? (there was only one decimal digit in the divisor so multiplying by 10 was enough to change the divisor to a whole number)

(MP.8)


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Write on the board:

0.021 ÷ 0.07

ASK: What do you think we should multiply both the divisor and dividend by this time? (100) Why? (there are two decimal places in the divisor and so multiplying by 100 will change the divisor to a whole number)

Write on the board, highlighting the 100:

0.021 ÷ 0.07 = (0.021 × 100) ÷ (0.07 × 100)

ASK: What is 0.021 × 100? (2.1) ASK: What is 0.07 × 100 ? (7) SAY: Calculate 2.1 ÷ 7 mentally. (0.3)

Continue writing on the board:

0.021 ÷ 0.07 = (0.021 × 100) ÷ (0.07 × 100) = 2.1 ÷ 7 = 0.3

Having to add zeroes in the dividend. Write on the board:

0.14 ÷ 0.007

ASK: What do you think we should multiply both the divisor and dividend by this time? (1,000) Why? (because there are three decimal digits in the divisor and so multiplying it by 1000 will change it to a whole number)

ASK: What is 0.007 × 1,000? (7) ASK: What is 0.14 × 1,000 ? (140) ASK: Why did we add a zero? (there were only two digits, but we had to move the decimal point three places) SAY: Calculate 140 ÷ 7 mentally. (20)

Continue writing on the board and highlight the 1,000:

0.14 ÷ 0.007 = (0.14 × 1,000) ÷ (0.07 × 1,000) = 140 ÷ 7 = 20

Exercises: Find an equivalent division statement and then find the answer.

a) 5.6 ÷ 0.8 b) 0.045 ÷ 0.05 c) 0.24 ÷ 0.006 d) 4.2 ÷ 0.06

Bonus: e) 0.45 ÷ 0.000005 f) 0.32 ÷ 0.0000000002

Answers: a) 7, b) 0.9, c) 40, d) 70, Bonus: e) 90,000, f ) 1,600,000,000

Finding an equivalent division question by moving the decimal point in the divisor and dividend. SAY: When it is too difficult to perform a mental calculation, we can use the division algorithm.

(MP.8)

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Write on the board:

0.4 2.8

ASK: What is the new division statement if we multiply both the divisor and dividend by 10? (4 2 8 )

Ask students to come to the board to write an equivalent division question so that the divisor has no decimals.

Exercises

a) 0.3 1.2 b) 0.05 3.5 c) 0.6 2.4 6 d) 0.08 0.0 1 2 8

Answers: a) 3 1 2 , b) 5 3 5 0 , c) 6 2 4.6 , d) 8 1.2 8

Dividing decimals by the division algorithm after moving the decimal point in the divisor and dividend. Write on the board:

1.05 ÷ 0.7

0.7 1.0 5

SAY: We need to eliminate the decimal in the division. First, what power of 10 do we need to multiply by? (10) After that, what is the new division statement? (7 1 0.5 ). Ask a student to come to the board to perform the division using the division algorithm, that is, long division. (see sample answer below)

1.57 1 0.5− 7

3 5− 3 5

0

ASK: Where do we place the decimal point in the quotient? (directly above the decimal in the dividend)

Have students perform the following in their notebooks and then ask for volunteers to present on the board. Tell them that, in some cases, they may have to add zeroes to the end of the dividend.

Exercises: Divide.

a) 4.62 ÷ 0.3 b) 1.264 ÷ 0.02 c) 7.8 ÷ 0.06 d) 2.43 ÷ 0.003

Bonus e) 0.1734 ÷ 0.3 f ) 2.61848 ÷ 0.4

Answers: a) 15.4, b) 63.2, c) 130, d) 810, Bonus: e) 0.578, f ) 6.5462


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Extensions1. Sometimes we want to find the price for a unit of something, for example,

the price for a pound of peanuts. To find a unit price, divide the amount of money by the unit of weight or volume. For example, if 3 pounds of peanuts cost $12.00, the unit price = $12.00 ÷ 3 = $4.00 per lb.

Two different stores advertise their prices. Find the lower unit price:

a) Dry roasted almonds: 0.6 lb for $5.37 or 0.8 lb for $7.08

b) Rolled oats: 0.5 lb for $0.55 or 0.6 lb for $0.78

Answers The lower unit price is underlined: a) almonds: $8.95/lb or $8.85/lb b) oats: $1.10 /lb or $ 1.30/lb

2. Sales tax varies by state. In Alabama, the sales tax is calculated by multiplying the price by 0.04. On a product with the price $80, the sales tax is 0.04 × $80 = $3.20. The total price is $80 + $3.20 = $83.20.

a) Find the price of the product with the Alabama sales tax.

i) $3.60 ii) $4.80 iii) $13.60

b) Find the total price if the Alabama sales tax is $107.20.

Answers a) i) $90, ii) $120, iii) $340 b) $2,787.20

3. Gavin has a jar of pennies. The pennies weigh 180.3 g. One penny weighs 3.1 g. Gavin estimates there are 80 pennies. Is his estimate reasonable? Explain.

Answer No, because 80 × 3 = 240 g, but the pennies only weigh 180.3 g.

(MP.1)

(MP.1, MP.4)

(MP1, MP.8)

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Word problems. Lesson NBT5-61 provides practice with decimal division using word problems.

Extensions1. Mile markers are used on interstate highways to help describe locations

on the highway. For example, mile marker 1 on I-65 is one mile north of the Kentucky state line. Mile marker 83.7 is 83.7 miles north of the Kentucky state line.

A family traveling north on I-65 passes mile marker 83.7 at 12:00 noon. The family passes mile marker 132.5 exactly 0.8 hours later.

a) How far did the family travel in that time?

b) To calculate a car’s speed, divide the distance traveled by the amount of time in which it was traveled. How fast did the family travel measured in miles per hour?

Answers: a) 48.8 miles, b) 61 miles per hour

2. The price of gasoline in the United States is sometimes written using

both decimals and fractions. If the price per gallon of gas is $3.876

10

at a gas station, it means that a gallon costs 3 dollars and 876

10 cents.

a) Convert 876

10 into a decimal.

b) How many cents are there in 3 dollars?

c) Find the price per gallon of gas written in cents.

d) Find the cost of 5 gallons of gas written in cents.

e) Find the cost of 5 gallons of gas written in dollars.

Answers: a) 87.6, b) 300¢, c) 387.6¢, d) 1,938¢, e) $19.38

(MP.1, MP.4)

(MP.1, MP.4)

STANDARDS 5.NBT.B.7


GoalsStudents will solve word problems involving division of decimals by a single-digit decimal.


Knows how to divide a multi-digit number by a whole number using the division algorithm Knows how to divide a decimal by a whole number Knows how to divide a decimal by a single-digit decimal

NBT5-61 Decimal Word Problems—Division by Page 68

a Single Digit


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3. At a gas station across the street, a customer spends $19.21 for 5 gallons of gas.

a) How many cents are in $19.21?

b) What is the cost per gallon of gas in cents?

c) What is the cost per gallon of gas in dollars?

d) What is the cost per gallon of gas written in decimals and fractions?

Answers: a) 1,921¢, b) 384.2¢, c) $3.842, d) $3.84 2/10

(MP.1, MP.4)

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Writing a multiplication statement for a given diagram. On the board, draw the diagram in the margin.

ASK: How many groups are there? (3) How many items are in each group? (4) How many items are there altogether? (12)

Ask a student to come to the board to write an addition statement for the diagram. (4 + 4 + 4 = 12) Ask a different student to come to the board to write a multiplication statement for the diagram. (3 × 4 = 12) Although 4 × 3 is also correct, tell students we are going to use 3 × 4 = 12 because we have 3 groups of 4 items each rather than 4 groups of 3 items each. Ask another student to come to the board to write a division statement for the diagram. (12 ÷ 3 = 4) Again, although 12 ÷ 4 = 3 would also be correct, tell students that we are going to use the number of groups as the divisor.

Writing a division statement for a diagram that uses base ten models. Draw on the board:

= 1,000 = 100 = 10 = 1

ASK: Ask a student to come to the board and use the diagrams to represent 1,253. (see diagram below)

5

4 2 0

Ask a student to come to the board to label the divisor, dividend, and quotient. (see answer below)

5 quotientdivisor 4 2 0 dividend

(MP.4)

NBT5-62 Division Review Pages 69–70


VOcABUlARy dividend divisor fact family factor groups or sets items in each set quotient

GoalsStudents will review how to write a division statement for items that have been grouped into sets. Students will review how to write members of a fact family for a division statement.


Knows how to determine the number of sets and the number of items in each set


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SAY: For now, we will use the divisor as the number of sets, the quotient as the items in each set, and the dividend as the total number of items.

Draw on the board:

ASK: What number is represented on the left? (320) How many groups or sets are on the right? (4) How many items are in each set? (80) How many items are there altogether? (320)

Write on the board:

Ask three different students to come to the board and fill in the divisor, the dividend, and the quotient. (see answer below)

8 0 quotientdivisor 4 3 2 0 dividend

Exercises: Write a division statement for the diagram.

a)

b)

c)

Answers: a) 240 ÷ 8 = 30, b) 126 ÷ 3 = 42, c) 1,224 ÷ 4 = 306

Find the members of a fact family given a division question. Write on the board:

56 3 0

ASK: How many sets are there? (6) How many items are in each set? (5) How many items are there altogether? (30)

(MP.4, MP.7)

(MP.4)

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Draw the diagram shown in the margin on large grid paper, and tape to the board.

ASK: Why can we use this diagram to represent 5

6 3 0 ? (it has 6 rows and 5 dots in each row) What multiplication equation can this diagram represent? (6 × 5 = 30)

Rotate the grid paper that was taped to the board by 90o.

rotate 90o

SAY: If we rotate the diagram to turn it on its side, we don’t change the number of dots. ASK: In the rotated diagram, how many rows are there? (5) How many columns? (6) How many dots are there altogether (30) What multiplication equation can the rotated diagram represent? (5 × 6 = 30)

What division equation can it represent? (6

5 3 0 or 30 ÷ 5 = 6)

SAY: So the dot diagrams lead to two multiplication equations and two division equations. Write on the board:

5 × 6 = 30 6 × 5 = 30 30 ÷ 5 = 6 30 ÷ 6 = 5

SAY: We say these equations form a fact family.

Exercises: Find all four members of the fact family.

a) 3

8 2 4 b) 1 4

12 1 6 8 c) 3 2

26 8 3 2

Answers: a) 8 × 3 = 24, 3 × 8 = 24, 24 ÷ 3 = 8, 24 ÷ 8 = 3; b) 12 × 14 = 168, 14 × 12 = 168, 168 ÷ 12 = 14, 168 ÷ 14 = 12; c) 32 × 26 = 832, 26 × 32 = 832, 832 ÷ 32 = 26, 832 ÷ 26 = 32

Solve a division equation by thinking of one of the multiplication equations in the fact family. Write on the board:

?8 2 4

SAY: If we don’t know the answer to the division equation, think of a multiplication equation in the fact family:

8 × ? = 24

SAY: Because 8 × 3 = 24, we know that 3

8 2 4 .


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Exercises: Find the quotient by solving an equivalent multiplication equation.

a) 8 × = 56 so

8 5 6 b) 9 × = 72 so

9 7 2

c) 12 × = 48 so

12 4 8 d) 11 × = 88 so

11 8 8

Extensions1. A fruit basket has fewer than 20 apples. The number of apples can be

shared equally among 3, 4, or 6 people. How many apples are in the basket?

Answer: 12

2. The number 12 has the following factors: 1, 2, 3, 4, 6, and 12. Each of these numbers divides evenly into 12. Find all the factors of:

a) 18 b) 36 c) 24 d) 45

Answers a) 1, 2, 3, 6, 9, 18 b) 1, 2, 3, 4, 6, 9, 12, 18, 36 c) 1, 2, 3, 4, 6, 8, 12, 24 d) 1, 3, 5, 9, 15, 45

3. Count the total number of factors for each part in Question 2.

a) Is the total number of factors even or odd?

Answer: 2. a), c), d) are all even; b) is odd

b) Why is the total number of factors for 2. b) odd?

Answer: For 2. a), c), and d), the factors can be grouped in pairs (e.g., for 18, the factors are 1 & 18, 2 & 9, 3 & 6). When you try to group the factors for numbers like 36 (known as perfect squares), one of the pairs has the same number repeated. Because we only write this number once in the list of factors, there will be an odd number of factors: 1 & 36, 2 & 18, 3 & 12, 4 & 9, 6.

(MP.1)

(MP.1)

(MP.3)

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Find the number of tens in a number using base ten materials. Write on the board: 247. Ask students to use their base ten materials to represent this number. (see diagram below)

Ask students to exchange each hundreds block for 10 tens blocks and count the total number of tens blocks. (24)

Exercises: Use base ten materials to find the number of tens.

a) 318 b) 274 c) 729

Answers: a) 31, b) 27, c) 72

Find the number of tens in a number by crossing out the ones digit. Write on the board: 247. Ask a student to come to the board and cross out the ones digit. Ask a different student to come to the board and circle the remaining digits, as shown in the margin.

SAY: The number circled is the same as the number of tens we found when using the base ten materials.

Exercises: Find the number of tens by crossing out the ones digit and circling the remaining digits.

a) 318 b) 274 c) 729

Answers: a) 31, b) 27, c) 72

Find the divisor and the number of tens in the dividend. Write on the board:

1 412 1 6 8

Ask a student to write the corresponding division equation for the division question on the board. (168 ÷ 12 = 14)

(MP.4)

(MP.4)

2 4 7

NBT5-63 2-Digit Division (Introduction) Pages 71–72


VOcABUlARy dividend divisor estimate groups multiples quotient round skip count

GoalsStudents will estimate the quotient by rounding the divisor to the nearest ten and finding the number of tens in the dividend.


Knows how to round a number to the nearest ten

MATERIAlS

base ten materials


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Ask two students to come to the board and label the divisor, dividend, and quotient for each division question. (see answers below)

1 4 quotientdivisor 12 1 6 8 dividend

dividend quotient

168 ÷ 12 = 14

divisor

ASK: Which word represents the total number of items? (dividend) ASK: Which word represents the number of groups or sets? (divisor) ASK: Which word represents the number of items in each group? (quotient)

Exercises: Find the number of groups and number of tens in the dividend.

a) 1 4

13 1 8 2 b) 1 4

17 2 3 8 c) 1 3

18 2 3 4

Answers: a) 13, 18; b) 17, 23; c) 18, 23

Review rounding numbers to the nearest ten. Draw on the board:

40 50 60 70 80

Ask a student to place a dot on the number line to represent 72. Ask: What multiples of 10 is the dot in between? (70, 80) ASK: Is the dot closer to 70 or 80? (70) SAY: So 72 rounded to the nearest ten is 70. ASK: How could we have found this answer without using a number line? (look at the last digit in 72) When do we round up to the nearest ten? (if the last digit is 5 or greater) When do we round down to the nearest ten? (if the last digit is 4 or smaller)

Exercises: Round to the nearest ten.

a) 84 b) 68 c) 43 d) 79

Answers: a) 80, b) 70, c) 40, d) 80

Make an initial estimate of the quotient. Write on the board:

73 ÷ 24

SAY: To make an estimate of the quotient, round the divisor to the nearest ten and find the number of tens in the dividend. Write on the board:

Number of tens in 73 = 24 rounded to nearest 10 =

Ask two different students to come to the board to write the answers. If needed, provide these hints:

Cross out the ones digit in 73. Is 24 closer to 20 or closer to 30? (20)

Write on the board:

73 ÷ 24 ≈ 70 ÷ 20

SAY: To estimate the quotient, count by 20 until you pass 70.

(MP.4)

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Write on the board:

20, 40, 60, 80 too high

ASK: How many multiples of 20 did we write before we passed 70? (3) SAY: So 3 is our estimate.


73 ÷ 24 ≈ 70 ÷ 20 ≈ 3

Exercises: Estimate the quotient by finding the number of tens in the dividend, rounding the divisor to the nearest ten, and then counting multiples of the new divisor until you pass the number of tens.

a) 64 ÷ 19 b) 84 ÷ 37 c) 127 ÷ 41

Answers: a) 3, b) 2, c) 3

Estimate the number of tens in each group. Write on the board:

18 6 1 2

ASK: How many groups or sets are there? (18) How many items are there altogether? (612)

ASK: How many tens are there in the dividend? (61) What is the divisor rounded to the nearest ten? (20)

Ask a student to count by 20s until the student passes 61. Write the answer on the board. (20, 40, 60, 80) ASK: How many multiples did we write before we passed 61? (3)

SAY: So there are 3 tens in each group.

Exercises: Estimate the number of tens in each group for the division.

a) 42 8 8 2 b) 17 6 4 4 c) 39 8 1 9

Answers: a) 2, b) 3, c) 2

Extensions1. Hot dog buns often come in packages of 10, while the actual hot dogs

usually come in packages of 12. Find the number of hot dog bun packages needed for five packages of hot dogs.

Answer: 6

2. A cashier went to the store office to exchange quarters for dimes. How many dimes should the cashier get for 324 quarters?

Answer: 810

(MP.1)

(MP.1)


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Use money to introduce division by a two-digit divisor. Ask a student to come to your desk and hand over $714 in play money. (see diagram below) SAY: A group of 21 parents are sharing the cost of a $714 ping pong table for the school. Ask for some guesses about how much each parent should contribute.

$100 $1$10

ASK: Why can’t we divide the $100 bills among the 21 parents? (there are only 7 bills, but 21 parents) SAY: We need to make change for the $100 bills. We want to find the number of $10 bills in $714. ASK: How many $10 bills are in $100? (10) SAY: Let’s replace each $100 bill with ten $10 bills. Ask seven different students to come up and each replace a $100 bill with ten $10 bills. ASK: How many $10 bills are in $700? (70) How many $10 bills are in $14? (1) How many $10 bills are there altogether in $714? (71)

ASK: What is the number of parents rounded to the nearest ten? (20) SAY: Count by 20 until you pass the number of $10 bills. (20, 40, 60, 80) ASK: How many multiples did you count before you passed the number of $10 bills? (3) SAY: We need to divide the money owed among 21 parents. Ask for 21 volunteers to act as the parents and come to the front of the class and each take three $10 bills back to their desks. How many $10 bills were shared? (63) How many $10 bills and $1 bills are left? (eight $10 bills, four $1 bills, as shown below)

$10 $1

(MP.4)

NBT5-64 2-Digit Division Pages 73–74


VOcABUlARy dividend divisor estimate quotient round

GoalsStudents will divide a multi-digit number by a two-digit divisor by rounding the divisor to the nearest ten and finding the number of tens in the dividend. The questions are designed so they do not require a “guess and check” method.


Knows how to round to the nearest ten Knows how to find the number of tens in a multi-digit number Knows how to divide a multi-digit number by a single-digit divisor

MATERIAlS

play money

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ASK: Why can’t we divide the $10 bills among the 21 parents? (there are eight $10 bills, but 21 parents) SAY: We need to exchange the $10 bills for $1 bills. ASK: How many $1 bills are in a $10 bill? (10) Ask eight different students to come up and each exchange a $10 bill for ten $1 bills. ASK: How many $1 bills are there altogether? (84) SAY: Count by 20 again until you pass the number of $1 bills. (20, 40, 60, 80, 100) ASK: How many multiples did you count before you passed the number of $1 bills? (4) Ask the “parents” from earlier on to come up and each take back four $1 bills. ASK: How much money does each parent now have? ($34, three $10 bills and four $1 bills) ASK: Is there any money left over? (no) SAY: So how much money should each parent contribute to pay for the ping pong table? ($34, as shown below)

$1$10

Use the division algorithm to divide by a two-digit divisor. Write on the board:

21 7 1 4−

, , ,

ASK: How many groups or parents are there in the problem? (21) How much money is to be paid? ($714) ASK: What is the number of groups rounded to the nearest ten? (20) What is the number of tens, or $10 bills, in 714? (71) SAY: Count by the rounded divisor, 20, until you pass the number of tens or $10 bills in 714. (20, 40, 60, 80) Ask a student to skip count on the board in the blanks provided. ASK: How many multiples were counted before you passed the number of tens? (3)

SAY: Place 3 as the first digit of the quotient in the tens column. (see diagram below)

× 321 7 1 4

− 6 38

SAY: Each parent gets three $10 bills. ASK: How many $10 bills were shared? (63) How did you get the answer? (multiply 3 × 21) ASK: How many $10 bills were not shared? (8) What operation can you perform to find this number? (subtract 63 from 71)

(MP.8)


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ASK: What remains to be shared among the 21 parents or groups? (eight $10 bills and four $1 bills) Exchange each $10 bill for ten $1 bills. ASK: Now how many $1 bills are left altogether to be shared? (84) What do we do to show that there are 84 dollar bills left to be shared? (“bring” down the 4 from the dividend; see diagram below)

× 321 7 1 4

− 6 38 4

SAY: We need to find the next digit in the quotient. ASK: What is the divisor rounded to the nearest 10? (20) What is the number of $1 bills in 84? (84) SAY: Count by 20 until you pass the number of ones. (20, 40, 60, 80, 100) Ask a student to write the multiples on the board.

× 3 421 7 1 4

− 6 38 4

− 8 40

, , , ,

ASK: How many multiples did you count before you passed the total number of ones left? (4) SAY: This is the next digit in the quotient.

ASK: If each group gets four $1 bills, how many $1 bills are shared altogether? (84) What operation did you perform? (multiply 4 × 21) How many $1 bills are left to be shared? (0) What operation did you perform? (subtract 84 from 84)

ASK: What is the division statement for this question? (714 ÷ 21 = 34) How can we use multiplication to check this answer? (multiply 34 × 21; the answer should be 714)

Exercises: Divide using the division algorithm.

a) 903 ÷ 43 b) 1,463 ÷ 34 c) 961 ÷ 29 d) 982 ÷ 37

Answers: a) 21, b) 43 R 1, c) 33 R 4, d) 26 R 20

NOTE: Questions in this lesson have been selected so that counting the multiples will give the correct estimate for the quotients. If you want to create more questions, it may best be done after the next lesson on division, NBT5-65 (2-Digit Division—Guess and Check) starting on AP Book 5.2 p. 75.

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Extensions1. Many states use a bottle deposit program to encourage recycling.

Michigan offers $1.20 for a dozen bottles. Sara has 372 bottles in her basement. How much money will she get for recycling the bottles?

Answer: $37.20

2. Billy collects baseball cards and stores them in cardboard storage boxes. Each box holds 72 cards. He can sell each box for $9.75. How much will he get if he sells his 8,856 cards?

Answer: $1,199.25

3. A prime number is divisible by only two numbers: 1 and itself. For example, 7 is prime because it is only divisible by 1 and 7. Show that the number 143 is not prime.

Answer: 143 = 13 × 11

(MP.1, MP.4)

(MP.1, MP.4)

(MP.1, MP.3)


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NOTE: In the previous lesson, students determined the next digit in the quotient by rounding the divisor to the nearest ten, finding the number of tens in the dividend, and counting multiples. In all examples in that lesson plan, counting multiples always led to the correct next digit for the quotient. This lesson plan addresses how that will not always be the case!

AcTIVITy

Determining whether a guess is too high, too low, or just right. Distribute a copy of BlM Is the Price Right? to each student. Students play the game in pairs. The object of the game is to guess an opponent’s price. After each guess, the opponent indicates whether the price is too high, too low, or just right.

Estimate a digit in the quotient where the guess is too high. SAY: Let’s divide $851 among 23 people. Write on the board:

23 8 5 1−

, , , ,

ASK: What is the divisor rounded to the nearest ten? (20) What is the number of $10 bills in the dividend? (85) Ask a student to come to the board and write down multiples of 20 until they pass 85. (20, 40, 60, 80, 100) ASK: How many multiples did we write before we passed 85? (4)

SAY: So 4 is the first digit in our quotient. Or so we think! ASK: What is the next step in the division algorithm? (multiply 4 by 23, the divisor) ASK: What is 23 × 4? Ask a student to come to the board and multiply 23 × 4. (92)

(MP.8)

NBT5-65 2-Digit Division—Guess and check Pages 75–76


VOcABUlARy dividend divisor greater than (>) less than (<) quotient

GoalsStudents will divide a multi-digit number by a two-digit divisor. For each digit in the quotient, students will determine whether their estimate was too high, too low, or just right.


Knows how to round to the nearest ten Knows how to find the number of tens in a multi-digit number Knows how to divide a multi-digit number by a single-digit divisor

MATERIAlS

BlM Is the Price Right? (p. O-58)

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423 8 5 1

− 9 2

1

2 3× 4

9 2

SAY: If we share out four $10 bills to each of the 23 groups, how many $10 bills would be shared? (92) ASK: What is the problem? (92 is greater than 85; we’ve shared too many $10 bills)

× 423 8 5 1

− 9 2

SAY: 92 is greater than 85, so our estimate of 4 for the quotient was too high! Let’s try a lower digit for the quotient. On the board, erase the 4 in the quotient, and replace it with 3. Also erase the 92. Ask a student to come to the board and multiply 3 × 23.

× 323 8 5 1

− 6 91 6

2 3× 3

6 9

ASK: How many $10 bills were shared this time? (69) How many were we supposed to share? (85) How many are left? (16) Note the results on the board.

Estimate a digit in the quotient where the guess is too low. Write the following on the board and SAY: This time let’s divide $864 among 27 people.

27 8 6 4−

, , ,

ASK: What is the divisor rounded to the nearest ten? (30) What is the number of $10 bills in the dividend? (86) Ask a student to come to the board and write down multiples of 30 until they pass 86. (30, 60, 90) ASK: How many multiples did we write before we passed 86? (2)

SAY: So 2 is our estimate for the first digit in our quotient. ASK: What is the next step in the division algorithm? (multiply 2 by 27, the divisor) ASK: What is 27 × 2? Ask a student to come to the board and multiply 27 × 2. (54)

227 8 6 4

− 5 4

1

2 7× 2

5 4


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SAY: If we share out two $10 bills to each of the 27 groups, how many $10 bills would be shared? (54) How many are left to share? (32)

× 227 8 6 4

− 5 43 2

SAY: 32 is greater than 27, so our estimate of 2 for the quotient was too low! We could have shared at least 1 more $10 bill to each of the 27 groups. Let’s try a higher digit for the quotient. Erase the 2 in the previous quotient, and replace it with 3. Ask a student to come to the board and multiply 3 × 27.

× 327 8 6 4

− 8 15

2

2 7× 3

8 1

ASK: How many $10 bills were shared this time? (81) How many were we supposed to share? (86) How many are left? (5) Could we have shared one more $10 bill among the 27 groups? (no, there are only five $10 bills left but 27 groups)

circling the first part of the dividend that is at least as big as the divisor. SAY: Before we try to make an estimate for the quotient, we must find the first part of the dividend that is greater than or equal to the divisor.Write on the board:

a) 18 2,2 1 4 b) 18 1,1 7 0 c) 18 4 1 4

SAY: We circled the number of tens in other examples, but sometimes the numbers are larger. Ask a student to circle the number of hundreds in each dividend. If a hint is needed, tell them to cross out the tens digit and the ones digit.

a) 18 2,2 1 4 b) 18 1,1 7 0 c) 18 4 1 4

SAY: If the number of hundreds is greater than the divisor, you can start the division. If not, circle the number of tens. If the number of tens is bigger than the divisor, you can start the division. If not, circle the number of ones!

SAY: In a), 22 is greater than 18, so we can start the division. But in b), 11 is less than 18 and in c), 4 is less than 18, so we will have to circle the number of tens instead. Write on the board:

a) 18 2,2 1 4 b) 18 1,1 7 0 c) 18 4 1 4

Exercises: Circle the first part of the dividend that is at least as big as the divisor.

a) 12 2 5 2 b) 23 1,6 3 3 c) 37 2,9 9 7 d) 14 1 1 2

Answers: a) 25, b) 163, c) 299, d) 112

(MP.8)

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Dividing by a two-digit divisor using the division algorithm. SAY: Twenty-three people shared a raffle ticket that won $1,426 in prize money. ASK: How many groups are there? (23) How many $1 bills are being shared? (1,426) How can we find how much each person gets? (divide)

Write on the board:

23 1,4 2 6

Ask a student to come to the board to circle the first part of the dividend that is larger than the divisor. (142) ASK: What is the divisor rounded to the nearest ten? (20) SAY: Count by 20s until you pass 142. (20, 40, 60, 80, 100, 120, 140, 160) ASK: How many multiples did we count before we passed 142? (7) SAY: Let’s try 7 as our first digit in the quotient. Add the 7 to the quotient above the tens digit.

Ask a student to come to the board to multiply 23 × 7. (161) Write the answer for 23 × 7 underneath the dividend. ASK: Is this number too high, too low, or just right? (too high) How can you tell? (because 161 is greater than 142)

723 1,4 2 6

− 16 1

2

2 3× 7

1 6 1

ASK: What number can we try in the quotient instead of 7? (6) Why is 6 a better guess than 8? (if 7 is too high, then so is 8; the quotient number must be lower than 7)

Erase the 7 and fill in 6 for the quotient. Write on the board as the discussion continues:

623 1, 4 2 6

− 1 3 84

1

2 3× 6

1 3 8

Ask a student to come to the board to multiply 23 × 6. (138) Write the answer underneath the dividend. ASK: Is this number too high, too low, or just right? (just right) How can you tell? (4 is less than 23; the number of tens left over is less than the divisor) Subtract to find how many tens are left over. (4)

SAY: Let’s continue with our division by bringing down the last digit in our dividend.

623 1, 4 2 6

− 1 3 84 6

(MP.8)

(MP.6)


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O-48

ASK: Now we have to estimate how many groups of 23 are in 46. Count by 20 until you pass 46. (20, 40, 60) How many multiples did we count before we passed 46? (2) SAY: That is our estimate for the next digit in the quotient.

6 223 1, 4 2 6

− 1 3 84 6

− 4 60

2 3× 2

4 6

Subtract to find how many ones are left to share. (0) ASK: What is the division statement? (1,426 ÷ 23 = 62) ASK: How much money does each person win? ($62)

Exercises: Divide using the division algorithm.

a) 13 3 3 8 b) 28 1,5 1 2 c) 41 1,5 5 8

Answers: a) 26, b) 54, c) 38

Bonus

a) 12 1,5 7 2 b) 23 2,8 9 8 c) 51 5 7, 2 7 3

Answers: a) 131, b) 126, c) 1,123

Dividing decimals by two-digit divisors. SAY: In the metric system, 1 kg ≈ 2.2 lb. Jane weighs 83.6 lb. To find how much Jane weighs in kilograms, we need to divide by 2.2.

Write on the board:

2.2 8 3.6

ASK: How can we get rid of the decimal in the divisor? (multiply both the divisor and the dividend by 10) ASK: What is 2.2 × 10? (22) What is 83.6 × 10? (836)

Write on the board:

22 8 3 6

Ask students to try it at their desks. When they are ready, ask a volunteer to write the solution on the board:

3 822 8 3 6

− 6 61 7 6

− 1 7 60

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Exercises: Divide.

a) 15.84 ÷ 2.4 b) 2.142 ÷ 0.42 c) 87.4 ÷ 2.3 d) 24.18 ÷ 0.31

Answers: a) 6.6, b) 5.1, c) 38, d) 78

Extensions1. A person with a taxable income between $8,925 and $36,250 calculates

federal income tax by multiplying the taxable income by 0.15.

a) Sam has a taxable income of $28,750. What federal income tax does he pay?

b) Lee multiplies by 0.15 and pays $1,912.50 in federal income tax. What is Lee’s taxable income?

Answers: a) $4,312.50, b) $12,750

2. The total team salary for a professional soccer team is $34.5 million. The team has 23 players. If each player earns an equal share of the money, how much is each player paid?

Answer: $1.5 million or $1,500,000

3. Find the missing numbers:

a) 5 1

1, 8 8 7

3 73 7

0

b) 4

5 1, 4 61 1 8

2 3 6

0

c) 4 5

1 0 4 8

3 02 4

6 1

1

,

Answers: a) 5 1

37 1,8 8 71 8 5

3 73 7

0

, b) 2 4

59 1,4 1 61 1 8

2 362 36

0

, c) 4 2 5

12 5,1 0 14 8

3 02 4

6 16 0

1

(MP.1, MP.4)

(MP.1)


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O-50

Word problems. Lesson NBT5-66 provides practice with division of two-digit decimal numbers using word problems.

Extensions1. A class of 18 students buys supplies for a party. Three students spend

$5.31 each. Seven students spend $4.65 each. Eight students spend $2.31 each.

a) How much do the students spend altogether?

b) The students want to share the cost of the party equally. How much should each student pay?

Answers: a) $66.96, b) $3.72

2. A parent council is helping to make pancakes for a breakfast party at the school. The recipe they are using will make six pancakes and calls for the following ingredients:

2.5 cups of pancake mix

2 tablespoons of sugar

0.5 teaspoons of cinnamon

0.25 teaspoons of nutmeg

0.25 teaspoons of ground ginger

2 eggs

1.5 cups of milk

The parents have plenty of the other ingredients, but only 45 cups of pancake mix and 25.5 cups of milk. Without having to go buy more ingredients, how many pancakes can they make?

Answer: 102

(MP.1, MP.4)

(MP.1, MP.4)



GoalsStudents will solve word problems involving dividing decimals by two-digit decimals.


Knows how to divide a multi-digit number by a two-digit number using the standard algorithm Knows how to divide decimals by whole numbers

NBT5-66 Word Problems—Division by Page 77

2-Digit Numbers

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Review multiplying decimals. Write on the board:

0.5 × 0.3

ASK: How can we write each decimal as a fraction? (5/10, 3/10). ASK: What is the rule for multiplying fractions? (multiply the numerators, or tops, and then the denominators, or bottoms)


= 5

10 ×

310

= 15

100

= 0.15

ASK: What is 5 × 3? (15) What is 10 × 10? (100) How do we write 15/100 as a decimal? (0.15)

If we ignore the decimals at the very beginning, what is 5 × 3? (15) SAY: notice that these are the digits in the final answer. ASK: How can we determine where to put the decimal place in the final answer? (Find the number of decimal digits in each factor and add them. Move the decimal point to the left this many digits.)

SAY: To multiply decimals, multiply as if there were no decimals. Then count the number of decimal digits in each factor and add them. Move the decimal point to the left this many places.

(MP.8)

NBT5-67 Mental Math—Decimals Pages 78–80

STANDARDS 5.NBT.B.7

VOcABUlARy dividend divisor greater than (>) less than (<) quotient

GoalsStudents will perform mental math involving multiplication and division of decimal numbers.


Knows how to multiply a decimal number by a power of ten Knows how to divide a decimal number by a power of ten Can perform mental math with whole numbers involving multiplication and division Knows how to compare numbers using the > and < symbols

MATERIAlS

BlM Mental Math Decimals Multiplication Flash cards (pp.O-59–60) BlM Mental Math Decimals Division Flash cards (pp. O-61–62) calculators


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Write on the board:

0.5 × 0.03

ASK: What is 5 × 3? (15) How many decimal digits are there in 0.5? (1) How many decimal digits in 0.03? (2) How many times to the left should the decimal point be moved? (3; 1 + 2)

Write on the board:

15

SAY: We want three decimal digits here, but we only have two digits. ASK: What can we do before placing the decimal point? (add zeroes in front)

Write on the board:

0 0 1 5

SAY: Place your pencil to the right of the ones digit, where the decimal point would be. Count back three decimal places to the left. ASK: Where can we place the decimal point? (between the 2 zeroes) What is the final answer? (0.015)

AcTIVITy

Distribute BlM Mental Math Decimals Multiplication Flash cards. Students play the game in pairs. Have students cut out the flash cards and shuffle them. Each card has a multiplication fact, followed by four decimal questions and the answers written in the wrong order. The object of the game is to match the questions to the correct answers. Player A picks a card, reads Side 1, and gives answers while Player B checks the answers against Side 2. The players then switch roles.

Review dividing decimals by a whole number. SAY: I have $0.75 to share among three people.

ASK: How much should each person get? ($0.25) What division question

can you use to get the answer? ( 0.2 5

3 0.7 5 ) SAY: I have 75 pennies to share among three people. ASK: How many pennies should each person get? (25)

What division question can you use to get the answer? (2 5

3 7 5 )

Write on the board:

0.2 53 0.7 5− 6

1 5− 1 5

0

2 53 7 5− 6

1 51 5

0

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ASK: What is the only difference between these two answers? (in one answer the decimal point is placed in the dividend and quotient)

SAY: To divide a decimal by a whole number, ignore the decimal point and divide as if the dividend had only whole numbers. Then place the decimal point back in the quotient above the decimal point in the dividend.

Exercises: Use mental math to divide.

a) 35 ÷ 7 = 5 b) 128 ÷ 4 = 32 c) 482 ÷ 2 = 241 Find 0.35 ÷ 7. Find 12.8 ÷ 4. Find 4.82 ÷ 2.

Answers: a) 0.05, b) 3.2, c) 2.41

Review dividing decimals by a decimal. SAY: John has been collecting nickels for the last month. He has $3.45 in nickels. ASK: How do we write a nickel using dollars and cents? ($0.05) What division question will help us find the number of nickels? ($3.45 ÷ $0.05)

Write on the board:

0.05 3.4 5

SAY: Here we are not dividing by a whole number. One way we can get rid of the decimal points is to change the whole question into pennies. ASK: How many pennies are in a nickel? (5) How many pennies are in $3.45? (345) What division question can we write to find the number of nickels?

(5 3 4 5 )

ASK: What can we multiply 0.05 by to get 5? (100) What can we multiply 3.45 by to get 345? (100)

SAY: To get rid of the decimal in the divisor of a division question, multiply both the divisor and the dividend by the power of 10 that will change the divisor to a whole number.

ASK: How does multiplying the divisor by a power of 10 help? (For each power of 10, the decimal point moves one place to the right. We do this until the decimal point is gone.)

Exercises

1. What would you multiply both the divisor and dividend by to get rid of the decimal point in the divisor?

a) 0.2 1.4 6 b) 0.12 1.5 6 c) 1.3 2.2 1

Answers: a) 10, b) 100, c) 10

2. Write a new division question where the decimal point in the divisor has been eliminated.

a) 0.3 1.5 3 b) 0.08 5.6 c) 1.4 0.8 4

Answers: a) 3 1 5.3 , b) 8 5 6 0 , c) 14 8.4


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3. Write a new division equation where the decimal in the divisor has been eliminated. Then perform the new division.

a) 0.6 4.2 b) 0.03 0.8 4 c) 0.09 0.0 7 2

Answers: a) 6 4 2 , 7; b) 3 8 4 , 28; c) 9 7.2 , 0.8

AcTIVITy

Students play the game in pairs. Distribute BlM Mental Math Decimals Division Flash cards. Have students cut out the flash cards and shuffle them. Each card has a division fact, followed by four decimal questions and the answers written in the wrong order. The object of the game is to match the questions to the correct answers. Player A picks a card, reads Side 1, and gives answers while Player B checks the answers against Side 2. The players then switch roles.

Multiplying by a fraction larger than one. On the board, draw the pie diagrams shown in the margin; do not include the labels.

Ask a student to come to the board and write a fraction for the diagram on the left. Ask a different student to come to the board and write a whole number for the diagram on the right (see example in the margin).

ASK: Which is smaller? (3/4) What symbol can we use to show that 3/4 is smaller than 1? (<) For students who have difficulty distinguishing between the symbols > and <, remind them that, on the number line, numbers on the left are smaller and numbers on the right are larger. The meaning of the symbol > can be remembered by pretending it is an arrow. (- - - >) Because this arrow points to the right, it is the “greater than” sign. Write the symbol < on the board between the two numbers.

SAY: I want to multiply both of the pie diagrams (earlier in the lesson plan) by 8. How can I do this? (Add seven more diagrams to each side to change from a total of one of each diagram to eight of each diagram.)

Write on the board:

8 × 34

8 × 1

ASK: What is 8 × 3/4? (6) What is 8 × 1? (8) Is 6 greater than or less than 8?

(<, less than) Write on the board:

8 × 34

< 8 × 1

8 × 34

< 8

34

1

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SAY: So multiplying a number by a fraction smaller than 1 gives you a smaller number than you started with, for example:

8 × 12

= 4

SAY: In the same way, multiplying a number by a fraction greater than 1 gives you a larger number than you started with, for example:

8 × 32

= 12

SAY: Since fractions are another way of writing decimals, the same holds true for decimals:

Multiplying a number by a decimal less than 1 gives you a smaller number than you started with.

Multiplying a number by a decimal greater than 1 gives you a larger number than you started with.

Exercises: Write T for True or F for False.

a) 4 × 1.25 > 4 b) 8 × 0.78 > 8

c) 6 × 0.24 > 6 d) 12 × 1.25 > 12

Answers: a) T, b) F, c) F, d) T

SAY: Since division is really just the opposite of multiplication, the opposite of the statements above will be true:

Dividing a number by a decimal greater than 1 gives you a smaller number than you started with, for example:

8 ÷ 2.0 = 4

Dividing a number by a decimal less than 1 give you a larger number than you started with, for example:

8 ÷ 0.1 = 80

Exercises: Write “T” for True, or “F” for False. Check your answers using a calculator.

a) 4 ÷ 0.99 > 4 b) 8 ÷ 0.78 > 8

c) 6 ÷ 0.24 < 6 d) 12 ÷ 1.25 > 12

Answers: a) T, b) T, c) F, d) F

Use rounding to help do mental math with decimals. SAY: Even when people use calculators, they sometimes make errors inputting the numbers and don’t notice their answers are wrong. It is important to know whether answers “make sense.”

Write on the board:

98.7 × 23.4

(MP.2)

(MP.2)

(MP.2)

(MP.5)


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SAY: A student using a calculator got the answer 230.958. Does this make sense?

ASK: Round 98.7 to the nearest 100. (100)

Write on the board:

98.7 × 23.4

≈ 100 × 23.4

ASK: What is the quickest way to multiply a decimal by 100? (move the decimal place to the right two places) What number do we get when we move the decimal place for 23.4 to the right two places? (2,340) ASK: Did the student’s answer seem reasonable? (No; the student’s answer was in the hundreds, but the real answer should be in the thousands!)

Exercises

Use rounding to the nearest 10 or 100 to help choose the correct answer from the list.

a) 9.4 × 345 (3.243; 32.43; 324.3; 3,243)

b) 14,609.4 ÷ 93.65 (0.156; 15.6; 156; 1,560)

Answers: a) 3,243, b) 156

Extensions1. Use rounding to the nearest 1, 10, or 100 to help choose the correct

answer from the list.

a) 111.6 × 99 ÷ 12 (0.9207; 9.207; 92.07; 920.7; 9,207)

b) (11,051.7 ÷ 985) ÷ (953.7 ÷ 85) (0.1, 1, 10, 100, 1,000)

Answers: a) 920.7, b) 1

(MP.5)

(MP.1, MP.5)

Unit 4 Number and Operations in Base Ten: Multiplying … for CC-Edition... · Unit 4 Number and...

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