Multiplication of Whole
Numbers and Decimals
Objectives To review the partial-products method for whole
numbers and decimals.
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Key Concepts and Skills
Solving Number Stories
• Solve whole-number and decimal problems
using the partial-products algorithm. Math Journal 1, p. 52
Students solve addition and
subtraction number stories by writing
and solving open number sentences.
[Operations and Computation Goal 3]
• Make magnitude estimates. [Operations and Computation Goal 6]
• Use magnitude estimates to place the
decimal point in products. [Operations and Computation Goal 6]
Curriculum
Focal Points
Math Boxes 2 8
Math Journal 1, p. 53
Students practice and maintain skills
through Math Box problems.
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Modeling the Partial-Products Method
Math Masters, pp. 56, 416, and 417
per group: transparencies of Math Masters,
pp. 416 and 417; base-10 blocks; dry erase
marker; eraser
Students use base-10 blocks to practice
multiplication using a concrete model.
ENRICHMENT
Key Activities
Study Link 2 8
Multiplying Numbers That End in 9
Students review the partial-products method
for whole numbers. They use magnitude
estimates to solve multiplication problems
involving whole numbers and decimals.
Math Masters, p. 55
Students practice and maintain skills
through Study Link activities.
Math Masters, p. 57
Students explore a calculation strategy for
multiplying by 9.
Ongoing Assessment:
Informing Instruction See page 122.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 50. [Operations and Computation Goal 6]
Key Vocabulary
partial-products method magnitude estimate ballpark estimate
Materials
Math Journal 1, pp. 50 and 51
Student Reference Book, pp. 19, 38, and 39
Study Link 27
Math Masters, p. 415 (optional)
Class Data Pad
Advance Preparation
Make copies of the computation grid (Math Masters, page 415) available for students’ use throughout
Part 1. For the optional Readiness activity in Part 3, make transparencies of Math Masters, pages 416
and 417, cut them out, and tape them together with clear tape.
Teacher’s Reference Manual, Grades 4–6 pp. 126–132
120
Unit 2
Estimation and Computation
Mathematical Practices
SMP2, SMP3, SMP5, SMP6, SMP8
Getting Started
Content Standards
5.NBT.2, 5.NBT.3a, 5.NBT.4, 5.NBT.7
Math Message
Mental Math and Reflexes
Write problems on the board or the Class Data Pad, so students
can visually recognize the patterns. Ask students to share the patterns
of decimal point placement and the value of the digits in the products of
each set. Sample answer: In the first set of problems, the decimal point
1
moves to the left and the value of each digit is _
10 the value it was in the
previous problem.
3 ∗ 8 24
6 ∗ 8 48
7 ∗ 9 63
∗
∗
3 0.8 2.4
6 0.8 4.8
0.7 ∗ 0.9 0.63
3 ∗ 0.08 0.24
0.6 ∗ 0.8 0.48
0.7 ∗ 0.09 0.063
Estimate the solution to this problem.
Write a number sentence showing how you
found your estimate. Be prepared to discuss
how you used rounding in your estimate.
3.7 ∗ 6.2 4 ∗ 6 = 24
Study Link 2 7 Follow-Up
Have partners compare answers and
correct any errors. Ask students to share
solutions to Problem 6.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
ELL
Ask volunteers to share their estimates and number models. Ask:
How does estimating products of decimals differ from estimating
products of whole numbers? Emphasize that when the decimal is
less than 1, comparisons to other decimals can help students make
an appropriate estimate. For example, they can compare the
decimal to 0.5 or 0.33 or compare it to 1.
Emphasize that when estimating with decimals, rounding can also
help make an appropriate estimate. For example, in this problem,
rounding each number to the nearest whole number is helpful;
3.7 is rounded to 4, and 6.2 is rounded to 6.
Algorithm Project The focus of this
lesson is the partial-products method for
multiplying whole numbers and decimals.
To teach U.S. traditional multiplication with
whole numbers and with decimals, see
Algorithm Project 5 on page A22 and
Algorithm Project 6 on page A27.
NOTE The partial-products
method is a direct application of
the Distributive Property of
Multiplication over Addition. For
more practice with the Distributive Property,
see www.everydaymathonline.com.
NOTE Working from left to right is consistent
▶ Reviewing the Partial-Products
WHOLE-CLASS
DISCUSSION
Method with Whole Numbers
(Student Reference Book, p. 19)
with the process of estimating products. Point
out for students that when they have found
the partial product for the leftmost digits, they
have a ballpark estimate for the product. To
support English language learners, discuss
the meaning of ballpark estimate.
The partial-products method for multiplication has been
stressed since Third Grade Everyday Mathematics. It is an
algorithm that all students are expected to know because it helps
students develop a good understanding of place-value and
multiplication concepts. It has the added benefit of facilitating
student use of mental arithmetic as they solve problems.
Refer students to page 19 of the Student Reference Book. With the
partial-products method, each part of one factor is multiplied by
each part of the other factor. Each partial product is written on a
separate line. These partial products are then added. This process
is usually fairly simple and has the additional benefit of providing
practice with column addition.
Lesson 2 8
121
Student Page
Date
Multiplication of Whole Numbers
28
For each problem, make a magnitude estimate. Circle the appropriate box.
Do not solve the problems.
6 ∗ 543
1.
3 ∗ 284
2.
10s
100s
1,000s 10,000s
10s
10 * 500 = 5,000
100s
How I estimated
100s
1,000s 10,000s
10s
100s
How I estimated
100s
1,000s 10,000s
10s
30 ∗ 40 = 1,200
100s
Multiply each part of 26
1,000s 10,000s
60 ∗ 400 = 24,000
How I estimated
by each part of 43:
How I estimated
Solve each problem above for which your estimate is at least 1,000. Use the
partial-products method for at least one problem. Show your work on the grid.
7.
Sample answers:
5
3
4 3
× 6
3
7
× 2
5
0
0
6 0
0
4
0
1 4
0
1
8
1 5
0
5
8
3
5
9 2
5
0
2
3
2
4 0
× 5
2 0
2
6
0
0
0
4
5
0
4
2 2
9
9
10s 1s
2
6
4
3
40 ∗ 20 → 8
0
0
40 ∗ 6 → 2
4
0
3 ∗ 20 →
6
0
3∗ 6 →
1
8
1
8
∗
Think of 43 as 40 + 3:
How I estimated
56 ∗ 409
6.
100s
Think of 26 as 20 + 6:
1,000s 10,000s
4 ∗ 200 = 800
25 ∗ 37
10s
Example: 43 ∗ 26 = ?
How I estimated
50 ∗ 100 = 5,000
5.
1,000s 10,000s
4 ∗ 204
4.
10s
3 * 300 = 900
46 ∗ 97
3.
Go through the following example with the class. Then distribute
computation grids (Math Masters, page 415) and have students
solve the Check Your Understanding problems using this method.
Time
LESSON
Add four partial products:
11
0
0
5
4
NOTE Make sure the digits students write are properly aligned in columns. It will
0
4
also help if they write place-value reminders (such as 100s, 10s, and 1s) above
the columns.
Math Journal 1, p. 50
EM3cuG5MJ1_U02_029-059.indd 50
1/11/11 11:32 AM
Ongoing Assessment: Informing Instruction
Watch for students who do not recognize the value of the digits in a number.
Have them write the factors in expanded notation.
NOTE To provide additional practice
using the Distributive Property, have students
play the game Multiplication Wrestling.
See Student Reference Book, page 324
for directions.
▶ Reviewing Multiplication
WHOLE-CLASS
ACTIVITY
of Decimals
(Student Reference Book, pp. 38 and 39)
Ask students to solve the following problem: 1.3 ∗ 5. After a couple
of minutes, have them share their solution strategies and explain
their reasoning. Encourage students to use drawings or strategies
based on their place-value knowledge. Expect that they may have
difficulty because one of the factors is a decimal.
Student Page
Date
Time
LESSON
Multiplication of Decimals
28
For each problem, make a magnitude estimate. Circle the appropriate box.
Do not solve the problems.
1.
2.4 ∗ 63
2.
0.1s
1s
10s
100s
0.1s
2 ∗ 60 = 120
1s
4.
1s
10s
100s
3.58 ∗ 2.1
0.1s
10 ∗ 0.3 = 3
1s
7.84 ∗ 6.05
0.1s
1s
100s
8 ∗ 6 = 48
2.8 ∗ 93.6
0.1s
1s
10s
2. Multiply the numbers as though they were whole numbers.
100s
3 ∗ 90 = 270
How I estimated
7.
1. First make a magnitude estimate of the product.
100s
How I estimated
6.
10s
10s
4∗2=8
How I estimated
5.
100s
How I estimated
13.4 ∗ 0.3
0.1s
10s
7∗1=7
How I estimated
3.
Explain that one way to solve multiplication problems containing
decimal factors is to multiply as though both factors were whole
numbers and then adjust the product. Specifically:
7.2 ∗ 0.6
3. Then use the magnitude estimate as a guide to inserting the
decimal point at the correct location in the answer.
How I estimated
Solve each problem above for which your estimate is at least 10. Show your work
on the grid below.
Sample answers:
6
1
×
2 .4
2
0
2
1
5
6 .0
5
×
7 .8
4
0
0
0
3
0
6
0
4
0
1
2
4
2
4
3
5
0
0
8
0
0
0
4
0
0
4
0
0
2
0
2
0
2
1 .2
4
0
7 .4
3
9
×
1
8
7
2
6
Example: 1.3 ∗ 5 = ?
3 .6
2 .8
0
0
0
6
0
0
1
2
0
2
0
0
2
4
0
4
8
2 .0
8
1. Make a magnitude estimate: 1.3 rounds to 1; because 1 ∗ 5 = 5,
the product will be in the ones. Ask students to justify how they
would round 1.3. Answers vary.
2. Ignore the decimal point and multiply 13 ∗ 5 as though both
factors were whole numbers: 13 ∗ 5 = 65.
Math Journal 1, p. 51
EM3cuG5MJ1_U02_029-059.indd 51
122
Unit 2
1/11/11 11:32 AM
Estimation and Computation
Student Page
3. Since the magnitude estimate is in the ones, the product must
be in the ones. The answer must be 6.5. So, 1.3 ∗ 5 = 6.5.
Date
Time
LESSON
Solving Number Stories
28
For each problem, fill in the blanks and solve the problem.
Ask partners to solve several multiplication problems in which
one of the factors is a decimal. Suggestions: 25 ∗ 0.6 15;
400 ∗ 1.7 680.
Next use examples like those on pages 38 and 39 of the Student
Reference Book to demonstrate how to find the product of two
decimals.
1.
2.
Example: 3.4 ∗ 4.6 = ?
1. Round 3.4 to 3 and 4.6 to 5. Since 3 ∗ 5 = 15, the product will
be in the tens.
3.
2. Ignore the decimal points and multiply 34 ∗ 46 as though both
factors were whole numbers: 34 ∗ 46 = 1,564.
Refer students to Student Reference Book, page 39. Introduce the
strategy of placing the decimal point in a product by counting
the number of decimal places to the right of the decimal point in
each factor. Have students try this strategy to solve the Check
Your Understanding problems on page 39 of the Student Reference
Book. This strategy is especially useful with numbers that have
many decimal places.
Ask partners to solve several multiplication problems in which both
factors are decimals. Suggestions: 6.3 ∗ 1.8 11.34; 0.71 ∗ 3.2 2.272.
12.40 and 15.88
The total amount of money
a.
List the numbers needed to solve the problem.
b.
Describe what you want to find.
c.
Open sentence:
d.
Solution:
m = 28.28
12.40 + 15.88 = m
$28.28
Answer:
e.
If the video game cost $22.65, how much money did they have left?
22.65 and 28.28
The amount of money left
a.
List the numbers needed to solve the problem.
b.
Describe what you want to find.
c.
Open sentence:
d.
Solution:
m = 5.63
28.28 – 22.65 = m
$5.63
Answer:
e.
Linell and Ben borrowed money so they could also buy a CD for $13.79. How much
did they have to borrow so they would have enough money to buy the CD?
13.79 and 5.63
The amount of money
a.
List the numbers needed to solve the problem.
b.
Describe what you want to find.
c.
Open sentence:
d.
Solution:
borrowed
4.
3. Since the magnitude estimate is in the tens, the product must
be in the tens. The answer must be 15.64. Thus,
3.4 ∗ 4.6 = 15.64.
Linell and Ben pooled their money to buy a video game. Linell had $12.40 and Ben
had $15.88. How much money did they have in all?
m = 8.16
5.63 + m = 13.79
$8.16
Answer:
e.
How much more did the video game cost than the CD?
22.65 and 13.79
How much more the video
a.
List the numbers needed to solve the problem.
b.
Describe what you want to find out.
c.
Open sentence:
d.
Solution:
game costs
m = 8.86
22.65 – 13.79 = m
$8.86
Answer:
e.
Math Journal 1, p. 52
EM3cuG5MJ1_U02_029-059.indd 52
1/11/11 11:32 AM
NOTE Some students will use friendly
numbers to make an estimate rather than
rounding. Making the appropriate magnitude
estimate is the important concept, not whether
the student uses rounding to make an
estimate.
NOTE There are borderline cases where a magnitude estimate is not accurate
enough to guide the correct placement of the decimal point in a product. For
example, 3.4 ∗ 3.4 → 3 ∗ 3 = 9. Place the decimal point to make the product as
close to 9 as possible: 34 ∗ 34 = 1,156; 3.4 ∗ 3.4 = 11.56. Remind students that
the placement of the decimal point should result in a product that is reasonable.
Student Page
▶ Practicing Multiplication
PARTNER
ACTIVITY
of Whole Numbers and Decimals
Date
Time
LESSON
1.
(Math Journal 1, pp. 50 and 51)
Use the map on page 386 of your Student
Reference Book to answer the following
questions.
2. a.
Choose the best answer.
a.
Students estimate the answers to Problems 1–6 on journal page
50 and Problems 1–6 on journal page 51. They will find the exact
answer only for some, not all, of these problems. Students may use
whatever method they prefer to make a ballpark estimate. They
should write the number sentence they used to make their estimate
on the line and then circle the magnitude of their estimate.
b.
About how many miles is it from
Juneau, Alaska, to the Arctic Circle?
maximum: 18
mode: 7
range: 13
median: 12
200 mi
Sample answer: 5, 6, 7, 7, 7,
7, 12, 14, 15, 16, 16, 18, 18
187.5 mi
525 mi
b.
1,000 mi
750 mi
900 mi
350 mi
Find the missing numbers and landmarks
for the set of numbers below:
18, 20, 20, 24, 27, 27,
range: 22
b.
mode: 27
c.
minimum:
d.
maximum:
4
3
2
1
0
5 6 7 8 9 10 11 12 13 14 15 16 17 18
27 , 30, 33, 34,
40
a.
Make a bar graph of the data.
Sample answer:
About how far is it from the center of
the California-Oregon border to the
center of the California-Mexico border?
36, 36,
Make up a data set of at least
12 numbers that have the following
landmarks.
150 mi
211
3.
Have students complete both pages. Then have partners check each
other’s solutions. When they have finished, write Problem 6 from
Math Journal 1, page 51 on the board. Ask students to explain
the strategies and reasoning they used to estimate and solve the
problem. Explanations should include how they placed the decimal
point and should reflect their understanding of decimal place values.
Math Boxes
28
119 122
4.
Acute angles measure greater than
0 degrees and less than 90 degrees.
Circle all the acute angles below.
18
40
139
119
Math Journal 1, p. 53
EM3cuG5MJ1_U02_029-059.indd 53
1/11/11 11:32 AM
Lesson 2 8
123
Study Link Master
Name
Date
STUDY LINK
28
䉬
䉬
For each problem, make a magnitude estimate.
䉬
Circle the appropriate box. Do not solve the problem.
䉬
Then choose 3 problems to solve. Show your work on the grid.
10s
100s
Journal
Page 50
Problems 1–6
247
Use journal page 50, Problems 1–6 to assess students’ understanding of how
to make magnitude estimates. Students are making adequate progress if they
make reasonable magnitude estimates based on their number sentences.
152
8 º 19
1.
Ongoing Assessment:
Recognizing Student Achievement
Time
Estimating and Multiplying
1,000s 10,000s
8 º 20 ⫽ 160
[Operations and Computation Goal 6]
How I estimated
930
155 º 6
2.
10s
100s
1,000s 10,000s
150 º 6 ⫽ 900
How I estimated
37 º 58
3.
2 Ongoing Learning & Practice
2,146
100s
10s
1,000s 10,000s
40 º 60 ⫽ 2,400
How I estimated
10s
▶ Solving Number Stories
21
5 º 4.2
4.
100s
1,000s 10,000s
5 º 4 ⫽ 20
(Math Journal 1, p. 52)
PROBLEM
PRO
PR
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
IN
ING
How I estimated
26.04
9.3 º 2.8
5.
10s
INDEPENDENT
ACTIVITY
100s
Algebraic Thinking Students solve addition and subtraction number
stories. They write an open number sentence for each problem and
solve the open sentence to find the answer to the problem.
1,000s 10,000s
9 º 3 ⫽ 27
How I estimated
Math Masters, p. 55
▶ Math Boxes 2 8
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 53)
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 2-10. The skill in Problem 4
previews Unit 3 content.
▶ Study Link 2 8
INDEPENDENT
ACTIVITY
(Math Masters, p. 55)
Teaching Master
Name
Date
LESSON
Time
Model the Partial-Products Method
28
䉬
Materials
Home Connection Students make magnitude estimates
for multiplication problems in which the factors are whole
numbers and/or decimals. They then select 3 problems to
solve for exact answers.
䊐 array grid (Math Masters, pp. 416 and 417)
䊐 base-10 blocks
Directions
䉬 Draw a line around rows and columns on the grid to model each problem.
䉬 Cover the array you made using as few base-10 blocks as possible.
䉬 Solve using the partial-products method.
䉬 Then match each part of the array with a partial product.
䉬 Record the solution, filling in the sentences to match the blocks you used.
1.
138
6 º 23 ⫽
In each of
6 rows
there are…
2
3
There are
2.
26 º 18 ⫽
In each
of 20 rows
there are…
In each
of 6 rows
there are…
longs, so there are
cubes, so there are
138
120
18
Write the problem showing
cubes. the partial products.
cubes.
120 ⫹ 18 ⫽ 138
cubes in all.
468
1
8
1
8
There are
longs, so there are
cubes, so there are
longs, so there are
cubes, so there are
468
200
160
60
48
cubes.
cubes.
Unit 2
READINESS
▶ Modeling the
Write the problem showing
the partial products.
Partial-Products Method
200 ⫹ 160 ⫹ 60
⫹ 48 ⫽ 468
(Math Masters, pp. 56, 416, and 417)
SMALL-GROUP
ACTIVITY
5–15 Min
cubes.
cubes.
cubes in all.
Math Masters, p. 56
124
3 Differentiation Options
Estimation and Computation
To provide experience with multiplication using a concrete model,
have students solve multidigit multiplication problems with
base-10 blocks. Use transparencies of Math Masters, pages 416
and 417. (See Advance Preparation.) Place the assembled grids on
a table. Do not use an overhead projector. Gather a small group of
students around the table. Use an overhead marker to show a
4-by-28 array. (See Figure 1.)
Have students cover the array using as few base-10 blocks as
possible.
Guide students’ use of the partial-products method to solve 4 ∗ 28.
Match each part of the 4-by-28 array with a partial product.
(See Figure 2.)
Figure 1: Array model of 4 ∗ 28
1. There are 2 longs in each of 4 rows, so there are 80 cubes.
2. There are 8 cubes in each of 4 rows, so there are 32 cubes.
3. There are 80 + 32, or 112 cubes in all.
Clear the masters. Now use an overhead marker to mark off a
17-by-32 array.
Ask students to cover the array using as few base-10 blocks (flats,
rods, and cubes) as possible.
Guide students’ use of the partial-products method to solve 17 ∗ 32.
Now match each part of the 17-by-32 array with a partial product.
1. There are 10 rows with 30 cubes in each row (3 flats).
2. There are 7 rows with 30 cubes in each row (21 longs).
Figure 2: Base-10 block model of 4 ∗ 28
3. There are 10 rows with 2 cubes in each row (2 longs).
4. There are 7 rows with 2 cubes in each row (14 cubes).
5. There are 300 + 210 + 20 + 14, or 544 cubes in all.
Ask students to work with partners using base-10 blocks to solve
the multiplication problems on Math Masters, p. 56.
ENRICHMENT
▶ Multiplying Numbers That End in 9
INDEPENDENT
ACTIVITY
Teaching Master
Name
LESSON
(Math Masters, p. 57)
28
䉬
Date
Time
A Mental Calculation Strategy
When you multiply a number that ends in 9, you can simplify the calculation by
changing it into an easier problem. Then adjust the result.
To further explore multiplication strategies, have students solve
problems using a mental multiplication strategy. Students read
Math Masters, page 57 and use the mental math strategy given to
answer the questions on the page. If necessary, read and discuss
Example 1 as a class.
Example 1:
2 º 99 ⫽ ?
䉬
Change 2 º 99 into 2 º 100.
䉬
Find the answer: 2 º 100 ⫽ 200
䉬
Ask: How is the answer to 2 º 100 different from the answer to 2 º 99?
100 is 1 more than 99, and you multiplied by 2.
So 200 is 2 more than the answer to 2 º 99.
䉬
Adjust the answer to 2 º 100 to find the answer to 2 º 99:
200 ⫺ 2 ⫽ 198. So 2 º 99 ⫽ 198.
Example 2: 3 º 149 ⫽ ?
Change 3 º 149 into 3 º 150.
䉬
䉬
Find the answer: 3 º 150 ⫽ (3 º 100) ⫹ (3 º 50) ⫽ 450.
䉬
Ask: How is the answer to 3 º 150 different from the answer to 3 º 149?
150 is 1 more than 149, and you multiplied by 3.
So 450 is 3 more than the answer to 3 º 149.
䉬
Adjust: 450 ⫺ 3 ⫽ 447. So 3 º 149 ⫽ 447.
Use this strategy to calculate these products mentally.
1.
5 º 49
3.
8 º 99
5.
2 º 119
245
792
238
2.
5 º 99
4.
4 º 199
6.
3 º 98
495
796
294
Math Masters, p. 57
Lesson 2 8
125