8.6
Two-Step
Multiplication Problems
PROBLEM SOLVING •
?
Essential Question How can you use the strategy make a table to solve one-step
and two-step multiplication problems?
Texas Essential
Knowledge and Skills
How can you use the strategy
make a table to solve one-step
and t wo-step multiplication
problems?
Number and Operations—3.4.K
Solve one-step and two-step problems involving
multiplication and division within 100 using strategies based
on objects; pictorial models, including arrays, area models,
and equal groups; properties of operations; or recall of facts.
MATHEMATICAL PROCESSES
3.1.B Use a problem-solving model
3.1.F Analyze mathematical relationships
Are You Ready?
Access Prior Knowledge
Use the Are You Ready? 8.6 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Lesson Opener
Making Connections
Go to Multimedia eGlossary at
thinkcentral.com
Invite students to tell you what they know about reading tables.
Why do we use tables? (To organize information) What are the parts of a table called?
(Rows are horizontal and columns are vertical.) Have you ever used a table before?
Tell us about it.
Using the Digital Lesson
You may wish to show students a blank table on the board. Ask them to identify
what parts are columns and what part are rows.
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
• How many tall cactus plants does Doc see? (5)
• What desert friends are hiding in the tall cactus plants? (spiders, beetles, and ants)
• How many spiders are hiding in each tall cactus plant? (3)
• How many beetles are hiding in each tall cactus plant? (2)
• How many ants are hiding in each tall cactus plant? (4)
• What question does Doc want to answer? (how many desert friends are in all 5 tall
cactus plants in total)
Literacy and Mathematics
Resources
For the student
For the teacher
Interactive
Student Edition
provides students
with an interactive learning
environment!
Digital Management
Center organizes program
resources by TEKS!
eTeacher
Edition
Math on the Spot
Video Tutor
Online Assessment
System
• Ask students to identify parts of the story that tell them what problem needs to
be solved.
iTools Virtual
Manipulatives
• Invite students to draw the 5 tall cactus plants and place the correct number of desert
friends on each cactus.
Soar to Success Math
Online Intervention
Lesson 8.6 267A
Name
8.6
?
Unlock the Problem
After students read the problem, ask them to put it
in their own words. Then work through the steps in
the graphic organizer. Be sure students understand
that the column for the number of pages with
1 stamp has to start with 2 and continue by 2s
because the total, 18, is an even number of stamps.
What do I need to find?
how many different ways Scott could put
all 18 stamps in an album with 1 stamp on
some pages and 2 stamps on others
I used the table and a pattern to find every way that the
stamps could be arranged by multiplying each of the
numbers in the first column by 2 and adding it to the
number in the second column. Each row equals
18 stamps, the total number of stamps Scott has.
Leveled Activities
ELPS
Beginning: Activity 20
1.A.1, 3.G.2, 4.C.3
Intermediate: Activity 22
1.A.1, 3.G.1, 4.C.1
Advanced: Activity 47
2.D.2, 2.I.5, 3.D.1
Advanced High: Activity 16
4.C.1, 4.F.3, 4.F.10,
4.G.3
Go to thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
267 Module 8
Make a table to show the number
of pages with 1 stamp and with
2 stamps. Each row must equal
18 , the total number of stamps.
_
Pages with
2 Stamps
Pages with
1 Stamp
Total
Stamps
8
2
18
18
Scott has _
stamps. Some album
7
4
18
1
pages have _
stamp on them, and
6
6
18
5
8
18
2
the other pages have _
stamps.
Plan
What is my plan or strategy?
table showing all the
I will make a _
different ways of arranging the stamps
in the album.
© Houghton Mifflin Harcourt Publishing Company
• How would the table change if Scott had 20
stamps? The numbers in the first and third columns
English Language Learners
Solve
What information am I given?
• How do you know your answer is reasonable?
2 stamps and 0 pages with 1 stamp; 0 pages with 2 stamps
and 18 pages with 1 stamp
How can you use the strategy make a table to solve
one-step and two-step multiplication problems?
Read
1 stamp or 2 stamps that will equal 18.
Ask students to describe two additional ways Scott
could put the 18 stamps in his album. 9 pages with
Essential Question
Scott has a stamp album. Some pages have 1 stamp on them,
and other pages have 2 stamps on them. If Scott has 18 stamps,
show how many different ways he could put them in the album.
Use the graphic organizer below to solve the problem.
• What do the pairs of numbers in the table
represent? All the possible number of pages containing
Go Deeper
Multiplication Problems
Unlock
Unlock the
the Problem
Problem
Have volunteers explain how the numbers in the
table are used to make a total of 18. Students should
realize that they need to begin by multiplying the
number in the first column by 2 because there are
2 stamps on each of the pages, and that they can
then add to the number of stamps in the second
column.
would change and there would be one more row.
Number and
Operations—3.4.K
MATHEMATICAL PROCESSES
3.1.B, 3.1.F
PROBLEM SOLVING • Two-Step
1.
4
10
18
3
12
18
2
14
18
1
16
18
8 different ways.
So, there are _
What number patterns do you see in the table?
Possible answers: in the first column, the pattern is to count back by 1; in the second
column, the pattern is to count by twos.
ELL Language Support
Module 8 267
Visual
Small Group
ELPS 1.A.2, 1.D, 3.G.1
Strategy: Model Concepts
Materials: postage stamps, crayons
• Students can use modeling to understand unfamiliar concepts.
• Explain that some people collect postage stamps.
• Ask students to describe a photo album. Help
students see how a photo album and a stamp
album are similar.
• Have students draw different ways to display
stamps in an album. How many stamps are on
each page? Answers will vary. If there are 10 pages,
how many stamps do you have? Accept all correct
responses.
Try
Try Another
Another Problem
Problem
What if Scott bought 3 more stamps and now has
21 stamps? Some album pages have 1 stamp and some
pages have 2 stamps. Show how many different ways he
could put the odd number of stamps in the album.
Read
Try Another Problem
Have students answer the questions in the graphic
organizer and solve the problem. Invite them to
share the tables and the patterns they found. Be sure
students understand that since the total is odd, the
number of pages with 1 stamp needs to be an odd
number.
Solve
What do I need to find?
how many ways Scott could put the stamps
in the album if he bought 3 more stamps
Pages with
2 Stamps
Pages with
1 Stamp
Total
Stamps
10
1
21
9
3
21
What information am I given?
8
5
21
Scott has 21 stamps. Some pages in his
album have 1 stamp and some have
2 stamps.
7
7
21
6
9
21
5
11
21
4
13
21
What is my plan or strategy?
3
15
21
I will make a table showing all the different
ways of showing 1 or 2 stamps on a page.
2
17
21
1
19
21
Plan
• What if Scott’s album had no pages with 2 stamps?
How many pages would have 1 stamp? 21 pages
• What if Scott got more stamps and now has 24
stamps? Would the second column of the table
have odd or even numbers? Explain. Even; Possible
explanation: because the total is even, the pages with
1 stamp would need to be in pairs.
Go Deeper
Have students use counters or drawings to show
that the following statements are true or false. Ask
students to justify their answers.
10 different ways.
So, there are _
2.
• The sum of two even numbers is always even.
True; the total number of counters can be paired into
groups of 2.
What patterns do you see in this table?
• The sum of two odd numbers is always odd.
Possible answer: in the first column: count back by 1; in the second column:
False; the total number of counters can be paired into
groups of 2.
Possible explanation: on page 267, Scott had an even
number of stamps, so the pages with 1 stamp had
even numbers beginning with 2. On this page, he has
an odd number of stamps, so the pages with 1 stamp
are odd numbers beginning with 1.
Math Talk
Mathematical Processes
Explain how these
patterns are different
from the patterns in the
table on page 267.
268
© Houghton Mifflin Harcourt Publishing Company
count by twos with odd numbers.
• The sum of an odd and even number is always
even. False; the total number of counters when paired
into groups of 2 will have one left over.
COMMON ERRORS
C
E
Error
Students may have difficulty making a table
from information in a word problem because they
have difficulty understanding the relationship
between the numbers.
Enrich
Example Students may think they need to perform
an operation using the numbers given in the
problem.
Logical / Mathematical
Individual
Display this table on the board.
?
?
1
2
3
4
5
?
4
8
12
16
20
Springboard to Learning Have students underline
the question. Ask them to explain how they could
find the different ways they could use the numbers
to solve the problem. Guide them to realize that
the numbers given describe a relationship. Once
students understand the relationship, they can
make a table to solve the problem.
• Have students choose a title and labels for the table and challenge
them to write a word problem that can be solved by using the
information displayed. Possible answer: Title: Cars; Labels: Number of Cars;
Number of Tires: Problem: There are 5 cars in the parking lot. Each car has 4 tires.
How many tires are there in all?
• Have students share their problems with the class and have students
solve the problems.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Lesson 8.6
268
Name
Share
Share and
and Show
Show
Share and Show
1.
The first problem connects to the learning model. Have
students use the MathBoard to explain their thinking.
Also, Problem 1 is scaffolded to help students set up
a table and understand the relationship among the
numbers.
Aaron’s mother is making lemonade. For each pitcher,
she uses 1 cup of lemon juice, 1 cup of sugar, and 6 cups
of water. What is the total number of cups of ingredients
she will use to make 5 pitchers of lemonade?
First, make a table to show the number of cups of lemon
juice, sugar, and water that are in 1 pitcher of lemonade.
Next, multiply to find the number of cups of water
needed for each pitcher of lemonade.
Check students’ tables to be sure they have used the
correct relationships between the numbers.
Last, use the table to solve the problem.
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on the
MathBoard.
3
2
Quick Check
1
IF
Think: For every pitcher,
the number of cups of
water increases by 6.
a student misses the checked exercises
Number of Pitchers
1
2
3
4
5
Cups of Lemon Juice
1
2
3
4
5
Cups of Sugar
1
2
3
4
5
Cups of Water
6
12
18
24
30
Total Number of Cups of Ingredients
8
16
24
32
40
5 cups of
So, in 5 pitchers of lemonade, there are _
THEN
Differentiate Instruction with
RtI Tier 1 Lesson 61
5 cups of sugar, and _
30 cups of water.
lemon juice, _
40 cups of ingredients.
This makes a total of _
2.
Go Deeper
shows that 5 cups of lemon juice are needed for 5 pitchers.
You can multiply 5 × 4 to find the number of lemons needed
for 5 pitchers.
Problem Solving
of ingredients? Possible answer: the total number of cups increases by 8 cups
for each pitcher: 8 cups for 1 pitcher, 16 cups for 2, 24 for 3, 32 for 4, and 40 for 5.
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
Have students describe how they can use the table
to help them find how many lemons it would take
to make 5 pitchers of lemonade if it takes 4 lemons
to make 1 cup. 20 lemons; Possible explanation: the table
What pattern do you see in the total number of cups
3.
Multi-Step Cammi needs 36 postcards.
She buys 4 packages of 10 postcards. How many
postcards will Cammi have left over? Explain.
4 postcards; Possible explanation: Cammi buys 4 × 10 = 40 postcards. She only needs 36.
40 − 36 = 4, so she has 4 postcards left over.
Module 8 • Lesson 6 269
Problem
Problem 3 is a multi-step problem that requires
students to first find the number of postcards Cammi
bought and then subtract the number she needs
to find how many are left over.
3
RtI Tier 1 Lesson 61
2
1
Enrich 41
Name
Name
M
Math
on the Spot
V
Video Tutor
LESSON
61
269 Module 8
Using Tables
OBJECTIVE Solve one-step and two-step multiplication problems by using the
strategy make a table.
Solve. Use a table to organize the information for 1–2.
What information am I given?
1
Lucy’s mother uses
fruit juice,
1
1.
Solve
What do I need to find? I need to
find how much of each ingredient
Lucy’s mother needs to make 5
pitchers of punch.
can of
bottle of ginger
6
scoops of sherbet
ale, and
for each pitcher.
Plan
What is my plan or strategy?
© Houghton Mifflin Harcourt Publishing Company
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
3.4.K
Lucy’s mother is making punch for Lucy’s class. For each pitcher, she
uses 1 can of fruit juice, 1 bottle of ginger ale, and 6 scoops of sherbet.
How much of each ingredient will she need to make 5 pitchers of punch?
Read
Through the Math on the Spot Video Tutor,
students will be guided through an interactive
solving of this type of H.O.T. problem. Use this
video to also help students solve the H.O.T.
problem in the Interactive Student Edition. With
these videos and the H.O.T. problems, students
will build skills needed in the TEXAS assessment.
Enrich 41
1
Problem Solving • Two-Step
Multiplication Problems
I will make a table to show the
total amounts of each ingredient
Lucy’s mother needs.
First, make a table with the
information.
Number of Pitchers
1
2
3
4
5
Cans of Fruit Juice
1
2
3
4
5
Bottles of Ginger Ale
1
Scoops of Sherbet
2 3 4 5
6 12 18 24 30
Next, look for information in the table
that will help you solve the problem.
Look for a pattern. The cans of fruit
juice and the bottles of ginger ale
increase by 1. The scoops of sherbet
increase by 6. Complete the table.
2.
Miguel puts photos in an album.
The album has 5 right-hand and
5 left-hand pages. Each right-hand
page holds 4 photos. Each lefthand page holds 3 photos. How
many photos will fit in the album?
Number of
Paintings
1
2
3
4
Blue Paint
1
2
3
4
Number of
Pairs of Pages
1
2
3
Yellow Paint
2
4
6
8
Right Page
4
8
12 16 20
Red Paint
1
2
3
4
Left Page
3
6
9
3.
Renee earns 2 dimes each time
she helps her mother. How many
dimes will she earn if she helps her
mother 4 times a day for a week?
5.
Stretch Your Thinking Suppose Exercise 2 asked
So, Lucy’s mother will need 5 cans
of fruit juice, 5 bottles of ginger ale,
and 30 scoops of sherbet.
4
5
12 15
35 photos
16 bottles
4.
Henry earns $6 every time he
waters Mr. Young’s lawn. How much
will Henry earn if he waters the
lawn 4 times in July and
4 times in August?
56 dimes
$48
how many photos fit on the first 6 pages of Miguel’s
album. Explain how you could find the answer.
1. Suppose Lucy’s mother decides to make 2 more pitchers
of punch. How many scoops of sherbet would she need for
7 pitchers of punch? Explain your answer.
42 scoops; possible
explanation: the table shows that 6 scoops
are needed for 1 pitcher. Multiply 7 × 6
to find the number of scoops needed for
7 pitchers.
Number and Operations
Marcy uses 1 bottle of blue paint, 2
bottles of yellow paint, and
1 bottle of red paint for one
painting. How many bottles of
paint will she need to make 4 of the
same paintings?
121
Possible answer: there are 3 pairs of rightand left-hand pages in the first 6 pages. One
pair of pages holds 4 1 3 5 7 photos. 3 3 7
5 21, so 6 pages would hold 21 photos.
Enrich
© Houghton Mifflin Harcourt Publishing Company
E41
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Fill in the bubble for the correct answer choice.
4.
5.
6.
Daily Assessment Task
There are 5 boats in a race. Each boat has 6 rowers
and 1 coach. How many people are in the race? Use the
table to solve the problem.
Number of Boats
1
2
Number of Rowers
6
12
Number of Coaches
1
2
Total Number of People
7
3
4
5
C
12
B
30
D
35
7 cups
C
8 cups
B
3 cups
D
12 cups
YES
3
C
4
B
2
D
1
Soar to Success Math
Enrich 41
• Homework and Practice
•
Lesson 8.6
TEXAS Test Prep Coach
Multi-Step Sheri puts 16 pictures in an album. Some
pages have 2 pictures. The other pages have 4 pictures.
There are 3 pages with 4 pictures. How many pages have
2 pictures?
A
•
Warm-up 61.02
Apply For each pan of bread, you need 2 cups of white
flour and 1 cup of wheat flour. What is the total number
of cups of flour you will need to make 4 pans of bread?
A
THEN
IF
NO
7
1
Can students use the strategy make a table
to solve one-step and two-step multiplication
problems?
3
A
2
In the Test Prep exercise, if students selected:
A They added 8 and 6, and subtracted the sum
from 50.
B They added the number of years to the number of
states visited each year.
C They only found the number of states visited.
7.
Each year for 8 years, the Williams family visited
6 different states on vacation. How many more states
do they need to visit before they have seen all 50 states?
A
36
C
48
B
14
D
2
270
© Houghton Mifflin Harcourt Publishing Company
TEXAS Test Prep
?
Essential Question
Write
Math
How can you use the strategy make a table to solve
one-step and two-step multiplication problems?
Making a table helps me organize the information, makes
patterns easier to see, and ensures I have found all the
possible solutions.
Differentiated Centers Kit
Games
Games
Guess My
Numbers
Students practice
multiplication
facts between
1 and 9 by
guessing the
factors to make
a given product.
Literature
The Workshop
Students read about
how Amanda uses
multiplication to find the
number of wooden stool
legs she needs to carve.
Activities
Diamond Derby
Students complete
purple Activity Card
15 by practicing
multiplication facts
through 10 by 10.
Lesson 8.6
270
5
Number and Operations—3.4.K
MATHEMATICAL PROCESSES 3.1.B, 3.1.F
Ho mewo rk
and Practice
8.6
Name
4.
PROBLEM SOLVING • Two-Step
Multiplication Problems
Make a table to solve the problem.
1.
Janey is making party mix. For each bowl of mix, she uses
2 cups of raisins, 1 cup of peanuts, and 4 cups of granola.
What is the total number of cups of ingredients she will
use to make 6 bowls of party mix?
Number of Bowls
TEXAS Test Prep
Lesson
Lesson Check
Check
1
2
3
4
5
6
Cups of Raisins
2
4
6
8
10
12
Cups of Peanuts
1
2
3
4
5
6
Cups of Granola
4
8
12
16
20
24
Total Number of Cups
7
14
21
28
35
42
6.
12 cups of
So, in 6 bowls of party mix, there are _
24 cups of granola.
6 cups of peanuts, and _
raisins, _
There are 6 relay teams at a race.
Each team has 4 runners and
1 captain. How many people are in
the race?
5.
Sara makes 7 fruit baskets for the
senior center. She puts 3 apples and
4 pears in each basket. How many
pieces of fruit are there?
A
11
A
14
B
30
B
21
C
24
C
49
D
25
D
28
Thomas wants to collect 40 toy cars
for his collection. If he has 6 boxes of
4 toy cars each, how many more cars
does Thomas need?
7.
Vernon needs to sell 60 granola bars
for his scout troop. If he sells 4 boxes
of 8 bars each, how many more bars
does Vernon need to sell?
A
16
A
48
B
24
B
32
C
50
C
28
D
34
D
30
42 cups of ingredients.
This makes a total of _
To make an apple pie, Mimi needs
2 cups of flour and 3 cups of apples.
How many cups of ingredients does
Mimi need to make 5 pies?
© Houghton Mifflin Harcourt Publishing Company
2.
3.
To make muffins, Seth needs 3 cups
of mix and 1 cup of raisins. How
many cups of ingredients does Seth
need to make 3 batches of muffins?
25 cups
12 cups
Module 8 • Lesson 6
Homework and Practice
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
271-272
Module 8
271
272
Multi-Step Jessie has 4 boxes of
8 crayons and 3 boxes of 5 markers
and 2 single markers. How many
more crayons than markers does
Jessie have?
A
17
B
22
C
15
D
20
9.
Multi-Step To make a salad, Alan
needs 3 cups of lettuce and 1 cup
of tomatoes. Alan makes 6 bowls
of salad. How many more cups of
lettuce than tomatoes does Alan
need?
A
10 cups
B
12 cups
C
2 cups
D
15 cups
© Houghton Mifflin Harcourt Publishing Company
8.
Problem
Problem Solving
Solving
Module 8 Assessment
Formative Assessment
Vocabulary
Vocabulary
Choose the best term from the box to complete
the sentence.
1.
Use the Module Assessment to assess students’
learning and progress in the module. The
formative assessment provides the opportunity to
adjust teaching methods for individual or whole
class instruction.
Vocabulary
Associative Property
of Multiplication
Distributive Property
Associative
The ___
Property of Multiplication
states that when the grouping of factors is changed,
the product is the same. (p. 243)
Concepts
Concepts and
and Skills
Skills
Use arrays to show each property.
2.
q TEKS 3.4.F, 3.5.B
Distributive Property
3.
4.
6
×8
_
© Houghton Mifflin Harcourt Publishing Company
q TEKS 3.4.F, 3.5.B
7
×4
_
5.
48
Associative Property of Multiplication
Circle groups to show 3 × (2 × 3).
Break apart the array to show
8 × 6 = (4 × 6) + (4 × 6).
Find the product
Possible answers are given.
6.
28
5
×9
_
7.
45
6
×7
_
8.
7
×8
_
42
9.
56
8
×8
_
64
10.
0
9×0=_
11.
56
7×8=_
12.
32
4×8=_
13.
63
9×7=_
14.
81
9×9=_
15.
35
7×5=_
16.
50
5 × 10 = _
17.
21
7×3=_
Module 8
273
Data-Driven Decision Making
3
2
1
Based on the results of the Module 8 Assessment, use the following
i
resources to strengthen
individual or whole class instruction.
Item
Lesson
2–3
8.1, 8.2,
8.4, 8.5
4–17
8.1, 8.2,
8.4, 8.5
Common Error
Intervene With
RtI* Tier 1 Lessons
Soar to
Success Math
3.4.E,
3.4.K,
3.5.B
May find an incorrect product when breaking apart the
array or circling groups
36, 37, 38, 60
12.29, 12.30,
12.31, 12.50
3.4.F, 3.5.B
May find an incorrect product
36, 37, 38, 60
12.29, 12.30,
12.31, 12.50
TEKS*
*TEKS—Texas Essential Knowledge and Skills; RtI—Response to Intervention
Module 8 Assessment 273
Module 8 Assessment
Name
Module 8 Assessment
18.
Adele needs 5 pieces of ribbon, each
8 centimeters long. How many
centimeters of ribbon does she need
altogether? q TEKS 3.4.K
Record your answer and fill in the
bubbles on the grid. Be sure to use
the correct place value.
4
0
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
TEXAS Test Prep
Fill in the bubble for the correct answer choice.
20.
21.
Depth of Knowledge
DOK Level
Items
1
2–3, 4–17, 18,
19
2
20, 21, 22
22.
When Chloe finds the multiplication facts
for 2, which digit will NOT be in the ones
place of the products? q TEKS 3.4.E
A
4
C
6
B
8
D
3
A honeybee is an insect. It has 6 legs. How many more
legs do 9 honeybees have than 7 honeybees? q TEKS 3.4.K
A
54
C
12
B
42
D
16
Jody has bags of shells. Each bag has 6 shells. She gives
3 bags each to 2 friends. How many shells did Jody give
away? q TEKS 3.4.F, 3.4.K, 3.5.B
A
36
C
18
B
6
D
12
Austin was on summer vacation for 7 weeks. He spent
3 weeks at band camp and the rest of the time at home.
How many days did he spend at home? q TEKS 3.4.K, 3.5.B
A
10 days
C
21 days
B
28 days
D
4 days
© Houghton Mifflin Harcourt Publishing Company
19.
274
Data-Driven Decision Making
3
2
1
Intervene With
RtI* Tier 1 Lessons
Soar to
Success Math
May find an incorrect product
36
12.29
3.4.E
May incorrectly identify the pattern
35
12.38
8.6
3.4.K
May not subtract the products
61
61.02
21
8.2
3.4.F, 3.4.K
May multiply by only 2 numbers
37
12.50
22
8.6
3.4.K
May forget to subtract the weeks before multiplying
61
61.02
Item
Lesson
TEKS*
18
8.4
3.4.K
19
8.3
20
Common Error
*TEKS—Texas Essential Knowledge and Skills; RtI—Response to Intervention
274
Module 8 Assessment