Unit 4 Number and Operations in Base Ten:
Multiplying and Dividing Decimals
Introduction
In this unit, students will learn how to multiply and divide decimals, and
learn the algorithm for dividing whole numbers and decimals by a twodigit 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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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.
Number and Operations in Base Ten
O-1
NBT5-54 Multiplying Decimals by Whole Numbers
Pages 54–56
STANDARDS
5.NBT.B.7, 5.NBT.B.5
Vocabulary
Goals
Students will multiply decimals up to the hundredths place by
a whole number.
PRIOR KNOWLEDGE REQUIRED
associative property
decimals
hundredths
hundredths block
ones block
regrouping
tenths
tenths block
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
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 students to model the decimal 2.13 on their desks with base ten
materials. (see diagram below)
(MP.4)
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)
O-2
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ASK: How many hundredths are in 1? (100) How many tenths are in 1? (10)
How many hundredths are in 1 tenth? (10)
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
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.4)
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
Number and Operations in Base Ten 5-54
O-3
(MP.4)
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.
(MP.4)
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
Replace
10 pennies with
1 dime.
$1
$1
D
P
P
P
P
P
P
D
$1
$1
D
P
P
P
P
P
P
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)
O-4
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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)
(MP.4)
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
D
D
D
D
D
D
P
P
P
Replace 10 dimes
with 1 dollar.
$1
D
D
D
D
D
D
P
P
P
$1
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
(MP.7)
Multiplying a decimal number involving regrouping using place values.
Write on the board:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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:
2.63 = 2 ones + 6 tenths + 2 hundredths
×2
×2
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
Number and Operations in Base Ten 5-54
O-5
Write on the board:
2.48 = 2 ones + 4 tenths + 8 hundredths
×2
×2
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 Bonus
e)2.76 × 3
f) 3.48 × 5
c) 1.28 × 3
d) 5.29 × 2
Answers: a) 7.83, b) 4.56, c) 3.84, d) 10.58, Bonus: e) 8.28, f) 17.40
(MP.7)
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
×
2 3 7
2
4 7 4
1
×
2 3 7
2
4 7 4
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 .
O-6
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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.
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
Extensions
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(MP.1)
(MP.1)
1. 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
Number and Operations in Base Ten 5-54
O-7
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
(MP.1)
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?
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer: $133.35
O-8
Teacher’s Guide for AP Book 5.2
NBT5-55 Multiplying Decimals by Decimals
Pages 57–58
(Introduction)
STANDARDS
5.NBT.B.7, 5.NBT.A.3
Goals
Students will multiply decimal fractions and the corresponding decimals.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can convert decimal fractions into decimals
decimals
denominator
hundredths
hundredths block
ones block
tenths
tenths block
MATERIALS
base ten blocks
overhead or digital projector
BLM Ones, Tenths, Hundredths (p. O-57)
(MP.4)
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.
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a)
b)
(2/10, 0.2)
(MP.7)
Figure 1
Figure 2
(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)
Number and Operations in Base Ten 5-55
O-9
Write on the board:
3
4
×
10
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)
2
10
5
=
×
10
100 10
c)
3
2
6
×
=
100 10
10
7
35
5
×
=
10
100
10
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
7
9
3
9
23
4
4
×
= b)
×
= c)
×
= d)
×
=
10
10
100
100
10
10
10
10
(MP.2)
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)
(MP.7)
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:
3
10
Number of zeroes
in denominator
×
+
4
10
=
12
100
=
Ask a student to fill in the bottom row of the chart.
O-10
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 12/100, b) 63/100, c) 27/1,000, d) 92/1,000
Write on the board:
0.3
Number of
decimal digits
×
0.4
+
=
0.12
=
Ask a student to fill in the bottom row of the chart.
(MP.2)
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)
Exercises: Multiply.
a)0.3 × 0.5
b) 0.7 × 0.8
Bonus
d)0.002 × 0.03
e) 0.003 × 0.004
c) 0.05 × 0.3
Answers: a) 0.15, b) 0.56, c) 0.015, Bonus: d) 0.00006, e) 0.000012
Extensions
(MP.1)
1.Find as many pairs of decimal fractions as you can that have
the product.
9
100
a)
b)
8
36
16
c)d)
1, 000
1, 000
100
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Sample answers
a)
9
3
3
18
1
5
×
,
×
, an uncommon answer:
×
10 10
10
10
10
100
b)
2
8
4
4
×
,
×
10
10
10 10
c)
2
8
4
2
1
8
1
4
,
,
×
,
×
×
×
10
10
100
10
100
100
100
10
d)
9
6
4
9
6
36
4
1 36
1
×
,
×
,
×
,
,
×
×
10 10
100
100
100 100
10 10
10 100
Number and Operations in Base Ten 5-55
O-11
NBT5-56 Multiplying Decimals by Decimals
Pages 59–60
STANDARDS
5.NBT.B.7, 5.NBT.B.5
Vocabulary
Goals
Students will multiply decimals where the product has up to
3 decimal digits.
PRIOR KNOWLEDGE REQUIRED
decimal digits
denominator
hundredths
tenths
Can multiply multi-digit numbers by 2-digit numbers
MATERIALS
calculators
Finding patterns in the number of decimal digits when multiplying
decimals. Write on the board:
3
10
×
7
100
=
Number of zeroes
in denominator
+
=
Fraction as
decimal
×
=
+
=
21
1, 000
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 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)
(MP.2)
O-12
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)
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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)
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
× 0.2
34
×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.
Exercises: Multiply.
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
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Ask students to find the product using their calculators. (7.99)
×
2 3 5
3 4
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)
Number and Operations in Base Ten 5-56
O-13
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
Extensions
(MP.1)
1.
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
(MP.1)
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
(MP.1)
Put the same number in each box.
3.
 ×  ×  ×  ×  = 0.00032
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer: 0.2
O-14
Teacher’s Guide for AP Book 5.2
NBT5-57 Decimal Word Problems—Multiplication
Page 61
STANDARDS
5.NBT.B.7, 5.NBT.B.5
Vocabulary
Goals
Students will solve word problems involving multiplying decimals by
whole numbers and by decimals
PRIOR KNOWLEDGE REQUIRED
assists
calories
goals
quire
ream
sales tax
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
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: 3
1
¢ is between 3¢ and 4¢
2
ASK: What decimal is exactly in between 0.030 and 0.040? (0.035)
Write on the board:
So, 3
1
¢ = $0.035
2
Exercises: Write the amount in dollars and cents notation.
a) 7
1
¢
2
b) 9
1
¢
2
c) 16
1
¢
2
Bonus: 9
7
¢
10
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) $0.075, b) $0.095, c) $0.165, Bonus: $0.097
Extensions
(MP.1)
1.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
Number and Operations in Base Ten 5-57
O-15
(MP.4)
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
(MP.4)
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.
(MP.4)
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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
nswer: $86 for 8 h, $96.75 for 9 h; yes, the answer for Question 5
A
($91.375) is between the figures for 8 h and 9 h.
O-16
Teacher’s Guide for AP Book 5.2
NBT5-58 Dividing Decimals by Whole Numbers
Pages 62–63
(Introduction)
STANDARDS
5.NBT.B.7, 5.NBT.B.6
Vocabulary
Goals
Students will divide decimals by whole numbers using base ten
materials and place values where no regrouping is required.
PRIOR KNOWLEDGE REQUIRED
decimal
hundredths
ones
tenths
whole number
Knows how to multiply whole numbers using base ten materials
and place values
MATERIALS
base ten materials
plastic money
Review base ten materials. Remind students that when using base ten
materials to represent decimals:
=1
= 0.1 or
1
10
= 0.01 or
1
100
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))
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a)1.43
(MP.4)
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)
Number and Operations in Base Ten 5-58
O-17
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
(MP.8)
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
(MP.4)
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:
O-18
$1
$1
$1
$1
$1
$1
D
D
D
P
P
P
P
P
P
P
P
P
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 2.14, b) 3.12, c) 1.21
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)
(MP.4)
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
(MP.8)
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
−
3 2 4
2 6 4 8
− 6
0 4
4
−
0 8
8
0
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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Write on the board:
8.24 ÷ 2
4 1 2
2 8 2 4
− 8
0 2
2
−
0 4
4
0
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)
Number and Operations in Base Ten 5-58
O-19
Exercises
(MP.4)
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
(MP.8)
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
Extensions
(MP.1)
1.
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
(MP.1)
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?
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)
O-20
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
b)How much farther would the Benitez family travel on 8 gallons if
their car drove as far as the website said?
NBT5-59 Dividing Decimals by Whole Numbers
Pages 64–65
STANDARDS
5.NBT.B.7, 5.NBT.B.6
Vocabulary
Goals
Students will divide decimals by whole numbers using base ten
materials, money, and the division algorithm.
PRIOR KNOWLEDGE REQUIRED
dividend
divisor
quotient
Knows how to multiply whole numbers using base ten materials
and place values
MATERIALS
base ten materials
plastic money
(MP.4)
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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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.
Number and Operations in Base Ten 5-59
O-21
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)
3
2 7 34
−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 34
−6
1 3
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
2 7
−6
1
−1
O-22
6
34
3
2
1
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Step 3: SAY: Divide the tenths into two equal groups.
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 6
2 7 34
−6
1 3
−1 2
14
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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 67
2 7 34
−6
1 3
−1 2
14
− 14
0
Number and Operations in Base Ten 5-59
O-23
ASK: What decimal is represented in each group of base ten materials? (3.67)
(MP.4)
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
D
D
$1
D
$1
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
$1
$1
$1
$1
D
D
P
P
D
D
P
P
D
P
D
P
D
P
D
P
D
P
P
D
P
P
D
P
D
P
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
b) 126.14 ÷ 2 c) 11,246.48 ÷ 4
Bonus: 11,111.04 ÷ 9
2. a) 8.2143 ÷ 3
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
O-24
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ASK: How much money is in each group? ($3.67)
Extensions
(MP.1)
1.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.
1
2
1
1
d)
5
8
1
1
1
1
Answers: a)
= 0.500, b)
= 0.250, c)
= 0.200, d)
= 0.125
2
4
5
8
a)
(MP.1)
b) 1
4
c)
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
Derek
3
8
Melky
1
4
Jose
2
5
Average
Answer: Derek 0.375, Melky 0.250, Jose 0.400. Jose has the highest
average.
(MP.3)
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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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.
Number and Operations in Base Ten 5-59
O-25
NBT5-60 Dividing Decimals by Decimals
Pages 66–67
STANDARDS
5.NBT.B.7, 5.NBT.A.2
Vocabulary
dividend
divisor
quotient
Goals
Students will divide decimals by decimals by first multiplying the divisor
and dividend by the power of 10 to eliminate the decimal in the divisor.
PRIOR KNOWLEDGE REQUIRED
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
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:
3×?
6
=
8
4 ×?
SAY: Remember that you can write the fraction 3/4 as 3 ÷ 4.
Write on the board:
3÷4=
3×2
3
=
4×2
4
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)
O-26
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ASK: What can we multiply the numerator and denominator by to see that
3/4 = 6/8? (multiply both by 2)
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)
(MP.8)
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
4 36
0.9
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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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)
Number and Operations in Base Ten 5-60
O-27
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)
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
(MP.8)
O-28
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.
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Continue writing on the board and highlight the 1,000:
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)
1.5
7 1 0.5
− 7
3 5
− 3 5
0
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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
Number and Operations in Base Ten 5-60
O-29
Extensions
(MP.1)
1. S
ometimes 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
(MP.1, MP.4)
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
(MP1, MP.8)
avin has a jar of pennies. The pennies weigh 180.3 g. One penny
3. G
weighs 3.1 g. Gavin estimates there are 80 pennies. Is his estimate
reasonable? Explain.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer
No, because 80 × 3 = 240 g, but the pennies only weigh 180.3 g.
O-30
Teacher’s Guide for AP Book 5.2
NBT5-61 Decimal Word Problems—Division by
Page 68
a Single Digit
Goals
STANDARDS
5.NBT.B.7
Students will solve word problems involving division of decimals by
a single-digit decimal.
Vocabulary
dividend
divisor
quotient
PRIOR KNOWLEDGE REQUIRED
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
Word problems. Lesson NBT5-61 provides practice with decimal division
using word problems.
Extensions
(MP.1, MP.4)
1. M
ile 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.
family traveling north on I-65 passes mile marker 83.7 at 12:00 noon.
A
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
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.1, MP.4)
2. The price of gasoline in the United States is sometimes written using
6
both decimals and fractions. If the price per gallon of gas is $3.87
10
6
at a gas station, it means that a gallon costs 3 dollars and 87
cents.
10
6
a) Convert 87
into a decimal.
10
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
Number and Operations in Base Ten 5-61
O-31
(MP.1, MP.4)
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?
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 1,921¢, b) 384.2¢, c) $3.842, d) $3.84 2/10
O-32
Teacher’s Guide for AP Book 5.2
NBT5-62 Division Review
Pages 69–70
STANDARDS
5.NBT.B.7, 5.NBT.B.6
Vocabulary
dividend
divisor
fact family
factor
groups or sets
items in each set
quotient
Goals
Students 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.
PRIOR KNOWLEDGE REQUIRED
Knows how to determine the number of sets and the number of items
in each set
(MP.4)
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:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
= 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 20
Ask a student to come to the board to label the divisor, dividend,
and quotient. (see answer below)
divisor
Number and Operations in Base Ten 5-62
5
4 20
quotient
dividend
O-33
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)
divisor
(MP.4, MP.7)
80
4 320
quotient
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
(MP.4)
Find the members of a fact family given a division question. Write on
the board:
5
6 30
ASK: How many sets are there? (6) How many items are in each set? (5)
How many items are there altogether? (30)
O-34
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Draw the diagram shown in the margin on large grid paper, and tape to
the board.
5
ASK: Why can we use this diagram to represent 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)
6
What division equation can it represent? (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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
3
14
32
a) 8 2 4 b)
c)
12 1 6 8
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 24
SAY: If we don’t know the answer to the division equation, think of
a multiplication equation in the fact family:
8 × ? = 24
3
SAY: Because 8 × 3 = 24, we know that 8 2 4 .
Number and Operations in Base Ten 5-62
O-35
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
Extensions
(MP.1)
1.
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
(MP.1)
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)
18b)
36c)
24d)
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
(MP.3)
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?
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
nswer: For 2. a), c), and d), the factors can be grouped in pairs (e.g.,
A
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.
O-36
Teacher’s Guide for AP Book 5.2
NBT5-63 2-Digit Division (Introduction)
Pages 71–72
STANDARDS
5.NBT.B.7, 5.NBT.B.6
Vocabulary
dividend
divisor
estimate
groups
multiples
quotient
round
skip count
Goals
Students will estimate the quotient by rounding the divisor to the nearest
ten and finding the number of tens in the dividend.
PRIOR KNOWLEDGE REQUIRED
Knows how to round a number to the nearest ten
MATERIALS
base ten materials
(MP.4)
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
(MP.4)
24 7
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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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:
14
12 1 6 8
Ask a student to write the corresponding division equation for the division
question on the board. (168 ÷ 12 = 14)
Number and Operations in Base Ten 5-63
O-37
Ask two students to come to the board and label the divisor, dividend, and
quotient for each division question. (see answers below)
dividendquotient
14
quotient
168 ÷ 12 = 14
divisor
12 1 6 8
dividend 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.
14
14
13
a) 13 1 8 2 b)
17 2 3 8 c)
18 2 3 4
Answers: a) 13, 18; b) 17, 23; c) 18, 23
(MP.4)
Review rounding numbers to the nearest ten. Draw on the board:
4050 60 7080
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
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.
O-38
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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:
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.
Continue writing on the board:
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
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 2, b) 3, c) 2
Extensions
(MP.1)
1.
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
(MP.1)
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
Number and Operations in Base Ten 5-63
O-39
NBT5-64 2-Digit Division
Pages 73–74
STANDARDS
5.NBT.B.7, 5.NBT.B.6
Vocabulary
dividend
divisor
estimate
quotient
round
Goals
Students 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.
PRIOR KNOWLEDGE REQUIRED
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
(MP.4)
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
$10
$1
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
O-40
$1
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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: 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)
$10
(MP.8)
$1
Use the division algorithm to divide by a two-digit divisor. Write on
the board:
21 7 1 4
−
, , , COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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)
3
×
21 7 1 4
−63
8
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)
Number and Operations in Base Ten 5-64
O-41
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)
3
×
21 7 1 4
−63
84
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.
34
×
21 7 1 4
−63
84
− 84
0
, , , , 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)
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.
O-42
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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)
Extensions
(MP.1, MP.4)
1.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
(MP.1, MP.4)
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
(MP.1, MP.3)
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.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer: 143 = 13 × 11
Number and Operations in Base Ten 5-64
O-43
NBT5-65 2-Digit Division—Guess and Check
Pages 75–76
STANDARDS
5.NBT.B.7, 5.NBT.B.6
Goals
Students 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.
Vocabulary
dividend
divisor
greater than (>)
less than (<)
quotient
PRIOR KNOWLEDGE REQUIRED
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)
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)
O-44
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.8)
4
23 8 5 1
−92
1
23
× 4
92
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)
4
×
23 8 5 1
−92
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.
3
×
23 8 5 1
−69
16
23
× 3
69
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
−
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
, , , 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)
2
27 8 6 4
−54
Number and Operations in Base Ten 5-65
1
27
× 2
54
O-45
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)
2
×
27 8 6 4
−54
32
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.
3
×
27 8 6 4
−81
5
2
27
× 3
81
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)
(MP.8)
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.
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
O-46
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a) 18 2,2 1 4 b)
18 1,1 7 0 c)
18 4 1 4
(MP.8)
Dividing by a two-digit divisor using the division algorithm. SAY: Twentythree 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.
(MP.6)
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)
7
23 1,4 2 6
− 16 1
2
23
7
×
161
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:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
6
23 1, 4 2 6
−1 38
4
1
×
23
6
138
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.
6
23 1, 4 2 6
−1 38
46
Number and Operations in Base Ten 5-65
O-47
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.
62
23 1, 4 2 6
−1 38
46
46
−
0
23
× 2
46
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
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:
38
22 8 3 6
−66
176
−176
0
O-48
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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)
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
Extensions
(MP.1, MP.4)
1.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
(MP.1)
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:
5 1
4  5
 4
a)  1, 8 8 7 b)5  1, 4  6 c)
1   , 0 
4 8
1 1 8
 
3 0
2 3 6
3 7
2 4
 
3 7
0
6 1
0

COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
1
425
51
24
Answers: a) 37 1,8 8 7 , b) 59 1,4 1 6 , c) 12 5,1 0 1
48
1 85
1 18
30
37
2 36
24
37
2 36
61
0
0
60
1
Number and Operations in Base Ten 5-65
O-49
NBT5-66 Word Problems—Division by
Page 77
2-Digit Numbers
STANDARDS
5.NBT.B.7, 5.NBT.B.6
Vocabulary
dividend
divisor
quotient
Goals
Students will solve word problems involving dividing decimals by
two-digit decimals.
PRIOR KNOWLEDGE REQUIRED
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
Word problems. Lesson NBT5-66 provides practice with division of
two-digit decimal numbers using word problems.
Extensions
(MP.1, MP.4)
1. 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
(MP.1, MP.4)
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
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
O-50
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
2 tablespoons of sugar
NBT5-67 Mental Math—Decimals
Pages 78–80
Goals
STANDARDS
5.NBT.B.7
Students will perform mental math involving multiplication and division
of decimal numbers.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
dividend
divisor
greater than (>)
less than (<)
quotient
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
(MP.8)
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)
Continue writing on the board:
=
3
5
×
10
10
=
15
100
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
= 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.
Number and Operations in Base Ten 5-67
O-51
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.
ASK: How much should each person get? ($0.25) What division question
0.2 5
can you use to get the answer? ( 3 0.7 5 ) SAY: I have 75 pennies to share
among three people. ASK: How many pennies should each person get? (25)
25
What division question can you use to get the answer? ( 3 7 5 )
Write on the board:
0.2 5
3 0.7 5
− 6
15
− 15
0
O-52
25
3 75
−6
15
15
0
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Review dividing decimals by a whole number. SAY: I have $0.75 to share
among three people.
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.)
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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
Number and Operations in Base Ten 5-67
O-53
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.
3
4
1
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.
Write on the board:
8 ×
3
8 × 1
4
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:
3
8 ×
<8 × 1
4
8 ×
O-54
3
4
<
8
Teacher’s Guide for AP Book 5.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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.)
(MP.2)
SAY: So multiplying a number by a fraction smaller than 1 gives you a
smaller number than you started with, for example:
8×
1
=4
2
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×
(MP.2)
3
= 12
2
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
(MP.2)
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:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
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
(MP.5)
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
Number and Operations in Base Ten 5-67
O-55
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
(MP.5)
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
Extensions
(MP.1, MP.5)
1.
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)
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 920.7, b) 1
O-56
Teacher’s Guide for AP Book 5.2