Using Mental Math to Multiply

Using Mental Math
to Multiply
Objective To guide children as they use mental math to multiply
1-digit numbers by multidigit numbers.
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eToolkit
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Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Use place-value concepts to
calculate products. [Number and Numeration Goal 1]
• Use strategies to solve 1-digit by
multidigit multiplication problems. [Operations and Computation Goal 4]
• Make reasonable estimates for
problems involving multiplication and
repeated addition. [Operations and Computation Goal 5]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Making Up and Solving
Poster Stories
Math Journal 2, pp. 204, 205, and 209
Children make up and solve
multiplication and division
number stories.
Math Boxes 9 2
Math Journal 2, p. 210
Children practice and maintain skills
through Math Box problems.
Home Link 9 2
• Use a multiplication/division diagram
to show multiples of equal groups. [Operations and Computation Goal 6]
Math Masters, p. 269
Children practice and maintain skills
through Home Link activities.
• Describe and apply the Associative
Property of Multiplication; apply the
Distributive Property of Multiplication over
Addition. [Patterns, Functions, and Algebra Goal 4]
Key Activities
Children devise and practice strategies
for mentally multiplying 1-digit numbers
by multidigit numbers.
Ongoing Assessment:
Informing Instruction See page 720.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 208. [Operations and Computation Goal 4]
Materials
Math Journal 2, pp. 204, 205, and 208
Home Link 9 1
slate
Advance Preparation
You may want to draw a multiplication/division diagram on the board.
Teacher’s Reference Manual, Grades 1– 3 pp. 227, 228
718
Unit 9
Multiplication and Division
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Using Multiplication/Division Diagrams
Math Masters, p. 270
Children use multiplication/division diagrams
to model equal-grouping problems.
ENRICHMENT
Solving an Allowance Problem
Math Masters, p. 271
Children calculate weekly allowances
according to three different plans.
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8
Content Standards
Getting Started
3.OA.1, 3.OA.3, 3.OA.5, 3.OA.8, 3.NBT.3, 3.NF.2, 3.NF.2a, 3.NF.2b
Mental Math and Reflexes
Math Message
Pose problems like the following.
Children answer on slates.
4 [70s] 280
50 [4s] 200
40 [70s]
50 [40s]
2,800
2,000
400 [70s]
500 [40s]
28,000
20,000
Could 6 adult harp seals weigh less than 1 ton?
Could they weigh more than 1 ton? (1 ton = 2,000 lb.) Use the
information on pages 204 and 205 in your journal. Record your
answers on your slate.
80 [90s] 7,200
800 [900s]
720,000
8,000 [900s]
7,200,000
Home Link 9 1 Follow-Up
Briefly go over the answers to the riddles.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, pp. 204 and 205)
The answer to both questions is yes. Have children share solution
strategies. Have them write number models for their strategies on
the board.
Because a harp seal could weigh between 200 and 396 lb,
6 seals could weigh as little as 1,200 lb (6 × 200 lb = 1,200 lb,
which is less than 2,000 lb).
Because the maximum weight (396 lb) is about twice the
minimum weight (200 lb), 6 seals could weigh about 2,400 lb
(2 × 1,200 lb = 2,400 lb, which is greater than 2,000 lb).
Because 396 is close to 400, 6 × 396 is close to 6 × 400,
or 2,400, which is greater than 2,000 lb.
Some children may have used repeated addition to solve the
problem.
Pose similar problems from the Adult Weights of North American
Animals poster and write them on the board. Children put their
thumbs up to show an answer of yes and down to show an answer
of no. They share solution strategies after each problem.
Suggestions:
●
Could 9 mountain goats weigh more than 1 ton? Yes; thumbs
up Less than 1 ton? Yes; thumbs up
●
Could 4 polar bears weigh more than 1 ton? Yes; thumbs up
Less than 1 ton? No; thumbs down
●
Could 7 white-tailed deer weigh more than 2 tons? No; thumbs
down
●
Could 8 beluga whales weigh more than 10 tons? Yes; thumbs
up Less than 10 tons? Yes; thumbs up
●
Could 25 snowshoe hares weigh more than 100 lb? Yes;
thumbs up Less than 100 lb? Yes; thumbs up
Lesson 9 2
719
Student Page
Date
Explain that children will use mental math to solve problems
involving animal weights.
Adult Weights of North American Animals
Ongoing Assessment: Informing Instruction
Arctic fox
7 lb to 20 lb
Walrus
2,000 lb to 3,500 lb
Raccoon
15 lb to 45 lb
91
Time
Harp seal
200 lb to 396 lb
LESSON
Mountain goat
170 lb to 240 lb
Black bear
250 lb to 600 lb
Multiplying 1-Digit Numbers
(Math Journal 2, pp. 204 and 205)
Start with a problem based on the Adult Weights of North
American Animals poster on journal pages 204 and 205.
Example:
Math Journal 2, p. 204
204-239_EMCS_S_MJ2_G3_U09_576418.indd 204
2/18/11 1:53 PM
Time
pounds per
animal
pounds in
all
6
15
?
Common dolphin
200 lb to 300 lb
Unit 9 Multiplication and Division
Possible strategies:
●
Think of 6 × 15 as 6 × (10 + 5). Use the Distributive Property
of Multiplication over Addition: 6 × (10 + 5) = (6 × 10) +
(6 × 5) = 60 + 30 = 90.
●
Think of 6 × 15 as (3 × 2) × 15. Use the Associative Property
of Multiplication: (3 × 2) × 15 = 3 × (2 × 15) = 3 × 30 = 90.
Another way to represent this thinking process: Double 15 to
get 30 and 3 [30s] are 90.
Once a child has shared a strategy, encourage other children to try
it. This validates the child who suggested the strategy and offers
the rest of the children the opportunity to expand their repertoire
of strategies.
Math Journal 2, p. 205
204-239_EMCS_S_MJ2_G3_U09_576418.indd 205
720
number of
animals
Pose additional problems. Draw and fill in a diagram for each
problem and ask for the solution. Be sure children share strategies
before going on to the next problem. If necessary give children
hints to get them started, but not until they have had ample
opportunity to devise strategies on their own.
Right whale
70,000 lb to 140,000 lb
Bottle-nosed dolphin
350 lb to 430 lb
Gray fox
9 lb to 16 lb
Sea otter
48 lb to 99 lb
Gila monster
1
2 2 lb to 4 lb
Puma
150 lb to 230 lb
White-tailed deer
50 lb to 480 lb
American alligator
200 lb to 500 lb
Pilot whale
3,200 lb to 6,400 lb
American porcupine
20 lb to 40 lb
Gray whale
45,000 lb to 72,000 lb
cont.
West Indian manatee
500 lb to 1,100 lb
Adult Weights of North American Animals
Atlantic green turtle
250 lb to 450 lb
9 1
How much do six 15-pound raccoons weigh? 90 lb
Have children find the answer—preferably mentally, but they may
use paper or a slate to keep track of their thinking. They share
solution strategies. Encourage them to use the language of multiples
(6 [15s] rather than 6 times 15) as they explain their strategies.
Student Page
LESSON
●
Draw a multiplication/division diagram on the board. Ask children
to tell you where to write the given information in the diagram,
and write ? for the information to be found.
NOTE The Distributive Property of
Multiplication over Addition relates
multiplication to a sum of numbers by
distributing a factor over the terms in the sum.
For any numbers a, b, and c, a(b + c) =
(a × b) + (a × c). The Associative Property
of Multiplication states that three numbers
can be multiplied in any order without
changing the product. For any three numbers
a, b, and c, (a × b) × c = a × (b × c).
Date
WHOLE-CLASS
ACTIVITY
By Multidigit Numbers Mentally
Northern fur seal
300 lb to 620 lb
Beluga whale
2,000 lb to 3,500 lb
Snowshoe hare
2
3 3 lb to 5 lb
Polar bear
650 lb to 1,750 lb
Beaver
20 lb to 56 lb
Watch for children who are having difficulty converting tons to pounds. If this
step is preventing children from focusing on the problems, give the target weight
in pounds.
2/18/11 1:53 PM
Student Page
Suggestions:
Date
Time
LESSON
92
●
How much do six 45-pound raccoons weigh? 270 lb
Possible strategies: Some children might reason that since 45 is
3 times as much as 15, 6 [45s] are 3 times as much as 90, or 270.
Others may apply the Distributive Property of Multiplication over
Addition and reason that 6 × 45 = 6 × (40 + 5). Then, 6 [40s] are 240
and 6 [5s] are 30, so 6 [45s] are 240 + 30, or 270.
1.
Yes
Yes
Explain the strategy you used.
Sample answer: I multiplied 5 × 20 to get 100 pounds, which is the
maximum weight. I multiplied 5 × 7 to get 35 pounds, which is the
minimum weight.
Could 12 harp seals weigh more than 1 ton?
Yes
Explain the strategy that you used. Sample
answer:
property or associative property. They should, however, begin to develop an
understanding that numbers in computations can be factored, grouped, and/or
renamed in different ways to make computations easier to solve.
Less than 1 ton?
No
I found the minimum weight by multiplying 12 × 200 to get 2,400
pounds, which is more than 1 ton.
3.
How much do eight 53-pound white-tailed deer weigh?
424 pounds
Sample answer:
I know that 53 is 50 + 3. I multiplied 8 × 50 and 8 × 3.
Explain the strategy that you used.
I added the products and got 424.
522 pounds
4. How much do six 87-pound sea otters weigh?
NOTE At this time, children are not expected to use the terms distributive
●
Could 5 arctic foxes weigh 100 pounds?
Less than 100 pounds?
2.
Still others may apply the Associative Property of Multiplication
and think of 6 as 3 × 2. Next, double 45 (or 2 [45s]) to obtain 90
and then triple 90 (or 3 [90s]) to obtain 270 pounds. One way to
represent this thinking process is 6 × 45 = (3 × 2) × 45 =
3 × (2 × 45) = 3 × 90 = 270.
Mental Multiplication
Solve these problems in your head. Use a slate and chalk, or pencil and
paper, to help you keep track of your thinking. For some of the problems,
you will need to use the information on journal pages 204 and 205.
5. How much do seven 260-pound Atlantic
1,820 pounds
green turtles weigh?
6.
6 × 54 =
8. 2 × 460 =
324
920
7.
9.
1,000
960
= 4 × 250
= 3 × 320
How much do five 180-pound pumas weigh? 900 lb
Math Journal 2, p. 208
Possible strategies: 5 [100s] plus 5 [80s] = 500 + 400 = 900;
or 5 [200s] minus 5 [20s] = 1,000 - 100 = 900.
204-239_EMCS_S_MJ2_G3_U09_576418.indd 208
2/18/11 1:53 PM
Pose problems without a context and ask for strategies.
Links to the Future
Problems
Possible strategies
Because children’s mental math
procedures often anticipate more formal
algorithms, it is important to provide
them with opportunities to devise
their own strategies before presenting
formal algorithms. Formal multiplication
algorithms will be presented in Lessons
9-4, 9-9, 9-11, and 9-12.
6 × 13 = ? 78
6 [10s] plus 6 [3s] = 60 + 18 = 78;
or 6 [15s] minus 6 [2s] = 90 - 12 = 78
9 × 49 = ? 441
9 [50s] minus 9 [1s] = 450 - 9 = 441;
or 9 [40s] plus 9 [9s] = 360 + 81 = 441;
or 10 [49s] minus 1 [49] = 490 - 49 = 441
7 × 32 = ? 224
7 [30s] plus 7 [2s] = 210 + 14 = 224
4 × 26 = ? 104
4 [25s] plus 4 [1s] = 100 + 4 = 104;
or 4 [20s] plus 4 [6s] = 80 + 24 = 104
2 × 950 = ? 1,900
2 [1,000s] minus 2 [50s] = 2,000 - 100 = 1,900;
or 2 [900s] plus 2 [50s] = 1,800 + 100 = 1,900
Practicing Mental
INDEPENDENT
ACTIVITY
Math Strategies
(Math Journal 2, p. 208)
Children complete journal page 208. Circulate and ask children to
share their strategies.
Ongoing Assessment:
Recognizing Student Achievement
Journal
page 208
Problem 1
Use journal page 208, Problem 1 to assess children’s progress toward using
strategies to solve problems involving multiplication of 2-digit numbers by a
1-digit number. Children are making adequate progress if they successfully
complete Problem 1 and explain the strategy they used. Some children may
complete other problems on the page successfully.
[Operations and Computation Goal 4]
Lesson 9 2
721
Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Number Stories
9 2
䉬
Use the Adult Weights of North American Animals poster on Math
Journal 2, pages 204 and 205. Make up multiplication and division
number stories. Ask a partner to solve your number stories.
Making Up and Solving
Answers vary.
1.
PARTNER
ACTIVITY
Poster Stories
(Math Journal 2, pp. 204, 205, and 209)
Answer:
2.
Partners make up stories for each other to solve on journal page
209. Encourage them to make up stories that use multiplication
and division. You might want to use some of these stories at
various times as Mental Math and Reflexes problems.
Answer:
3.
Math Boxes 9 2
Answer:
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 210)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 9-4. The skill in Problem 6
previews Unit 10 content.
Math Journal 2, p. 209
Writing/Reasoning Have children write an answer to the
following: Explain how you figured out which numbers to
write on the number line in Problem 5. Sample answer:
I knew that halfway between 0 and _23 on the number line was _13 .
I counted by thirds to find the missing numbers: 0, _13 , _32 , _33 , _43 .
Home Link 9 2
INDEPENDENT
ACTIVITY
(Math Masters, p. 269)
Home Connection Children practice multiplication facts
and fact extensions to reinforce work in this lesson, as
well as to prepare for the partial-products algorithm in
Lesson 9-4.
Student Page
Date
Time
LESSON
Math Boxes
9 2
䉬
2. Write the numbers.
1. Nicky has $806 in the bank.
5 tens 9 ones
Andrew has $589. How much
more money does Nicky have
than Andrew?
217
$
9
50 ⫹
Total: 59
3 tens 8 ones
30
8
⫹
Total:
38
258
1
ᎏᎏ
2
3.
1
1ᎏ2ᎏ
hour ⫽ 30 minutes
4.
hours ⫽ 90 minutes
2 hours ⫽
1
4
1ᎏᎏ hours ⫽
3
120
75
Draw a shape with a perimeter
of 14 units.
Sample answer:
minutes
minutes
hours ⫽ 180 minutes
What is the area of the shape?
247
5. Fill in the missing numbers.
6
6. Circle the most appropriate unit.
Use fractions.
length of a calculator:
inches
0
1
—
3
150 151
154 155
square units
2
3
1,
3
or —
3
1
—
3
4
or —
3
1 ,
feet
miles
weight of an adult:
ounces
pounds
tons
amount of gas in a car:
10 26
cups
pints
gallons
Math Journal 2, p. 210
722
Unit 9 Multiplication and Division
Teaching Master
Name
3 Differentiation Options
LESSON
92
Date
Time
Using Multiplication/Division Diagrams
For each number story, complete the multiplication/division diagram,
write a number model, and answer the question.
1. Tiffany keeps her button collection in a case with 10 shelves.
SMALL-GROUP
ACTIVITY
READINESS
Using Multiplication/
On each shelf there are 16 buttons. How many buttons are in
Tiffany’s collection?
10 × 16 = ?
Number model:
5–15 Min
Division Diagrams
160 buttons
Answer:
(unit)
number of
shelves
buttons
per shelf
buttons
in all
10
16
?
2. Rashida walks her neighbor’s dog every day. She gets paid $20.00
every week. If Rashida saves her money for 30 weeks, how much
money would she have?
(Math Masters, p. 270)
30 × $20 = ?
Number model:
To explore using multiplication/division diagrams to model equal
groups problems with multiples of 10, have children fill in the
diagrams and solve the problems on Math Masters, page 270.
number of
dollars
weeks
per week
$600
Answer:
(unit)
bulbs fit into each garden. How many tulip bulbs were planted?
4 × 50 = ?
Number model:
Solving an Allowance
bulbs
in all
number of bulbs per
garden
gardens
200 bulbs
(unit)
INDEPENDENT
ACTIVITY
?
$20
3. The third grade class helped plant 4 tulip gardens at school. 50 tulip
Answer:
ENRICHMENT
30
dollars
in all
4
?
50
Try This
4. There were 2,000 books collected in the book drive. Each class
received 200 books. How many classes received books?
5–15 Min
Problem
Answer:
PROBLEM
PR
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
ING
(Math Masters, p. 271)
2,000 ÷ 200 = ?
or ? × 200 = 2,000
10 classes
Number model:
books
in all
books
number of
per class
classes
?
200
2,000
(unit)
Math Masters, p. 270
267-318_EMCS_B_MM_G3_U09_576957.indd 270
3/10/11 2:42 PM
To apply their understanding of multidigit multiplication,
have children calculate weekly allowances according to
three different plans.
Home Link Master
Name
Date
HOME LINK
Teaching Master
Time
Name
LESSON
92
Multiplication Facts and Extensions
Extensions
Family
Note
Help your child practice multiplication facts and their extensions. Observe as your child
creates fact extensions, demonstrating further understanding of multiplication.
92
Date
Time
Allowance Plans
Sara is discussing a raise in allowance with her parents.
They ask her to choose one of three plans.
Please return this Home Link to school tomorrow.
Solve each problem.
56 , or 8 × 7 = 56
560 , or 8 × 70 = 560
How many 8s in 56? 7
d. How many 8s in 560? 70
8
How many 7s in 56? 8
f. How many 70s in 560?
9 [7s] = 63 , or 9 × 7 = 63
9 [70s] = 630 , or 9 × 70 = 630
How many 9s in 63? 7
d. How many 9s in 630? 70
9
How many 7s in 63? 9
f. How many 70s in 630?
8 [5s] = 40 , or 8 × 5 = 40
8 [50s] = 400 , or 8 × 50 = 400
How many 8s in 400? 50
d. How many 80s in 4,000? 50
How many 50s in 400? 8
f. How many 50s in 4,000? 80
1. a. 8 [7s] =
b. 8 [70s] =
c.
e.
2. a.
b.
c.
e.
3. a.
b.
c.
e.
4. Write a multiplication fact you are trying to learn.
Then use your fact to write some fact extensions like those above.
Sample
answer:
9 × 5 = 45
9 [5s] = 45
9 [50s] = 450 9 × 50 = 450
How many 9s in 45? 5 How many 5s in 45? 9
How many 9s in 450? 50 How many 50s in 450? 9
Math Masters, p. 269
267-318_EMCS_B_MM_G3_U09_576957.indd 269
Plan A Each week, Sara would get 1¢ on Monday, double Monday’s
amount on Tuesday, double Tuesday’s amount on Wednesday, and
so on. Her allowance would keep on doubling each day through
Sunday. Then she would start with 1¢ again on Monday.
Plan B Sara would get 32¢ on Sunday, Monday, Wednesday, and
Friday of each week. She would get nothing on Tuesday, Thursday,
and Saturday.
Plan C Sara would get 16¢ on each day of each week.
Which plan should Sara choose to get the most money?
Plan B
Show your work on the back of this page. Explain how you found your
Sample answer:
answer. Use number sentences in your explanation.
For Plan A, Sara would get $0.01 + $0.02 + $0.04 + $0.08
+ $0.16 + $0.32 + $0.64 = $1.27. For Plan B, she
would get $0.32 × 4 = $1.28. For Plan C, she would get
$0.16 × 7 = $1.12. Sara gets the most money
from Plan B.
Try This
Sample answer:
For the plan you chose, how much money would Sara earn in a year?
Since Sara makes $1.28 each week and there are 52 weeks
in one year, I would multiply $1.28 by 52 to get $66.56.
Math Masters, p. 271
2/18/11 7:36 PM
267-318_EMCS_B_MM_G3_U09_576957.indd 271
2/18/11 7:36 PM
Lesson 9 2
723
Name
LESSON
92
Date
Time
Using Multiplication/Division Diagrams
For each number story, complete the multiplication/division diagram,
write a number model, and answer the question.
1. Tiffany keeps her button collection in a case with 10 shelves.
On each shelf there are 16 buttons. How many buttons are in
Tiffany’s collection?
number of
shelves
Number model:
Answer:
buttons
per shelf
buttons
in all
(unit)
2. Rashida walks her neighbor’s dog every day. She gets paid $20.00
every week. If Rashida saves her money for 30 weeks, how much
money would she have?
number of
dollars
weeks
per week
Number model:
Answer:
dollars
in all
(unit)
3. The third grade class helped plant 4 tulip gardens at school. 50 tulip
bulbs fit into each garden. How many tulip bulbs were planted?
number of bulbs per
garden
gardens
Number model:
bulbs
in all
(unit)
Try This
4. There were 2,000 books collected in the book drive. Each class
received 200 books. How many classes received books?
books
number of
per class
classes
Number model:
Answer:
270
(unit)
books
in all
Copyright © Wright Group/McGraw-Hill
Answer: