Develop Skills and Strategies
Lesson 6
Part 1: Introduction
CCSS
Multiplication and Division in Word Problems
4.OA.A.2
In Lesson 5, you thought about equations that compare numbers using
multiplication. In this lesson, you will solve those types of problems. Take a
look at this problem.
Hannah scored 3 goals last season. She scored 4 times as many goals this
season. How many goals did Hannah score this season?
Last season
This season
Explore It
Use the math you already know to solve the problem.
How many goals did Hannah score last season? Count to find the number of goals she scored this season. How can you use skip-counting to find the number of goals Hannah scored this
season? Besides addition, what operation can you use to solve the problem?
What is 4 times as many as 3? 44
L6: Multiplication and Division in Word Problems
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Part 1: Introduction
Lesson 6
Find Out More
You will often need to solve for some unknown amount, like when you found the
number of goals Hannah scored. You used skip counting to find 4 times 3. You can
also use a bar model.
Last season
3
This season
3
3
3
3
?
The bar model can help you write an equation to solve the problem.
4 3 goals last season 5 goals this season
Goals last season is known (3). Goals this season is unknown. You can use a symbol,
such as an empty box or a question mark, to stand for the unknown number in the
equation.
4 3 3 5
Reflect
1 What can you think of that has two times as many objects as something else?
L6: Multiplication and Division in Word Problems
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45
Part 2: Modeled Instruction
Lesson 6
Read the problem below. Then explore different ways to understand it.
Janelle’s Market sells bags of 8 oranges. Simone needs 5 times that amount.
Write and solve an equation to find the number of oranges Simone needs.
Model It
You can use models to help understand the problem.
Number in one bag
8
Number Simone needs
8
8
8
8
8
?
Skip-count to find the total Simone needs: 8, 16, 24, 32, 40.
Model It
You can use the bar model to make an equation to help understand the
problem.
5 3 oranges in one bag 5 total oranges needed
The number of oranges in one bag is known (8). The total oranges needed is not
known.
5 3 8 5
46
L6: Multiplication and Division in Word Problems
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Part 2: Guided Instruction
Lesson 6
Connect It
Now you will solve the problem from the previous page using an equation.
2 You don’t know how many oranges Simone needs. What part of the bar model
shows how many she needs? 3 How does the bar model show how many oranges are in one bag?
4 How does the bar model show how many bags Simone needs?
5 How can you find “5 times as many” as 8? 6 Write an equation using numbers to show how many oranges Simone needs.
Simone needs oranges.
7 Explain how you can write a multiplication equation from a bar model.
Try It
Use what you just learned to solve these problems.
8 Neil and Vincent are collecting cans. Neil has collected 10 cans and Vincent has
collected 3 times as many cans as Neil. Write and solve an equation to find the
number of cans Vincent has collected.
9 Mr. Cherry ate 6 times as many raisins as Mary. Mary ate 11 raisins. Write and solve
an equation to find the number of raisins Mr. Cherry ate. L6: Multiplication and Division in Word Problems
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47
Part 3: Modeled Instruction
Lesson 6
Read the problem below. Then explore different ways to understand it.
Juan found 3 times as many seashells at the beach as Jeremy found. Juan found
24 shells. Write and solve an equation to find the number of shells Jeremy found.
Model It
You can use a model to help understand the problem.
Jeremy found one group of seashells. Juan found 3 times as many shells.
Jeremy’s shells
?
Juan’s shells
?
?
?
24
Divide 24 by 3 to find the number of seashells in each group: 8.
Solve It
You can use the model to make an equation to help understand the problem.
3 3 Jeremy’s shells 5 Juan’s shells
The number of shells Juan found is known (24). The number Jeremy found is not
known.
3 3
48
5 24
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Part 3: Guided Instruction
Lesson 6
Connect It
Now you will solve the problem from the previous page using an equation.
10 You don’t know the number of shells Jeremy found. In the bar model, what part
shows the number of shells Jeremy found?
11 How does the bar model show how many shells Juan found?
12 How does the bar model show that 24 is 3 times another number?
13 How can you find what number times 3 is 24?
14 Write a division equation using numbers to show how many shells Jeremy found.
Jeremy found shells.
15 Explain how you can write a division equation from a model.
Try It
Use what you just learned to solve these problems.
16 Monique and Wint are both reading the same book. Monique read 63 pages last
weekend. She read 7 times as many pages as Wint. Write and solve an equation to
find the number of pages Wint read. 17 The winning baseball team scored 4 times as many runs as their opponent. The
winning team scored 8 runs. Write and solve an equation to find the number of
runs their opponent scored. L6: Multiplication and Division in Word Problems
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49
Part 4: Guided Practice
Lesson 6
Study the model below. Then solve problems 18–20.
Student Model
There are twice as many
boxes in Karina’s model
as in her cousin’s model.
Karina is 6 feet tall. Her cousin is 3 feet tall. How many times
as tall as her cousin is Karina?
Look at how you could show your work using a bar model.
Cousin’s height
3
Karina’s height
3
3
6
Pair/Share
How else could you
solve this problem?
What does it mean when
the problem says 3
times as many?
33
5 6;
52
Solution: Karina is 2 times as tall as her cousin.
18 A small shrimp taco has 5 shrimp. There are 3 times as many
shrimp in a large taco. How many shrimp are in a large taco? Write
and solve an equation to find the answer.
Show your work.
Pair/Share
Did you and your
partner write the same,
or different, equations?
50
Solution: L6: Multiplication and Division in Word Problems
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Part 4: Guided Practice
19 Christina read 7 pages in a magazine. She read 5 times as many
pages in a book. How many pages did Christina read altogether?
Show your work.
Lesson 6
I remember that
multiplication and
division are opposite
operations!
Pair/Share
Solution: 20 Aida swam 7 laps in a pool. Kaya swam 28 laps. How many times
the number of laps Aida swam did Kaya swim? Circle the letter of
the correct answer.
How can you check
your answer?
Does Jae Ho’s answer
make sense?
A4
B21
C35
D196
Jae Ho chose D as the correct answer. How did he get that
answer?
Solution: L6: Multiplication and Division in Word Problems
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Pair/Share
How did you and your
partner know what
operation to use?
51
Part 5: Common Core Practice
Lesson 6
Solve the problems.
1 Kyle sold 28 boxes of fruit for a fundraiser. Omar sold 2 times as many boxes of fruit as
Kyle sold. What is the total number of boxes that Kyle and Omar sold?
A84
B56
C42
D14
2 Raoul biked 11 miles last week. Jackson biked 22 miles last week. Jackson biked
how many times as many miles as Raoul? Which equation can help you answer the
question?
A22 2 11 5 h
B22 4 11 5 h
C11 3 22 5 h
D11 1 22 5 h
3 Which problems can be solved using the equation 3 3 9 5 A? Circle the letter of all
that apply.
52
A
Pam is 9 years old. She is 3 times as old as Kate. How old is Kate?
B
Marco is making 9 apple tortes. He needs 3 apples for each torte. How many
apples does he need?
C
Three groups of actors are performing plays at a festival. There are 9 actors in
each group. How many actors are performing?
D
An art class meets 3 times a week for 9 weeks. How many times does the class
meet?
E
Judy found 3 acorns. Aaron found 3 times as many acorns as Judy. How many
acorns did Aaron find?
L6: Multiplication and Division in Word Problems
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Part 5: Common Core Practice
Lesson 6
4 Maria has 32 postcards. Henry has h postcards. Maria has 4 times as many postcards as
Henry.
Choose Yes or No to indicate whether each statement is true.
a. The number of Henry’s postcards can be
represented by the expression 32 4 4.
Yes
No
b. Henry has 6 postcards.
Yes
No
Yes
No
c.
The number of Henry’s postcards can be
found by solving the equation 32 5 4 3 h.
5 Viet learned 25 new spelling words last week. He learned 5 times as many words as
Max. How many words did Max learn? Draw a bar model to find the number of words
Max learned.
Show your work.
Answer Max learned new spelling words last week.
6 Mr. Naik traveled 18 hours on vacation last summer. Miss Cooper traveled 3 hours
on vacation last summer. How many times as many hours did Mr. Naik travel as
Miss Cooper? Write an equation to find the answer.
Show your work.
Answer Mr. Naik traveled times as many hours on vacation as Miss Cooper.
Self Check Go back and see what you can check off on the Self Check on page 37.
L6: Multiplication and Division in Word Problems
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Develop Skills and Strategies
Lesson 6
(Student Book pages 44–53)
Multiplication and Division in Word Problems
Lesson Objectives
The Learning Progression
•Use drawings and symbols to represent a
multiplicative comparison problem.
In this lesson, students will apply what they know
about multiplicative comparisons to solve
multiplication and division problems. Students will
model problems involving “times as many,” write a
number sentence using symbols, and solve for
the unknown.
•Use an equation to solve for the unknown in a
multiplicative comparison problem.
PrerequisiTe Skills
•Recall basic multiplication and division facts.
•Interpret products of whole numbers, including
multiplicative comparisons.
•Use arrays, drawings, number lines, and equations
to solve multiplication and division problems.
•Determine the unknown whole number in a
multiplication or division equation.
•Understand the relationship between multiplication
and division.
Vocabulary
unknown: a missing number in an equation
symbol: an object used to stand for an unknown
number in an equation
For example, students may solve a multiplication
problem in which they know the number of rings that
Sonja has (4) and are told that Mia has 3 times as many.
Or, students may solve a division problem to find the
number of rings that Sonja has. They create models and
write number sentences using symbols to solve for
the unknown.
Developing student understanding and application of
multiplicative comparisons in Grade 4 develops a
foundation for understanding multiplication as scaling
when multiplying by a fraction; for example, students
will later understand one fourth times a number as one
fourth as many.
Teacher Toolbox
Review the following key terms.
multiplication: an operation used to find the total
number of items in equal-sized groups
Ready Lessons
product: the result of multiplying numbers together
Tools for Instruction
factors: the numbers you multiply
Interactive Tutorials
Teacher-Toolbox.com
Prerequisite
Skills
4.OA.A.2
4.OA.2
✓✓
✓
✓
✓
✓✓
✓
division: an operation used to separate a number of
items into equal-sized groups
equation: a mathematical sentence that uses an
equal sign (5) to show that two expressions have the
same value
CCSS Focus
4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g. by using drawings and equations with a symbol
for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 3, 4, 5, 7 (See page A9 for full text.)
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L6: Multiplication and Division in Word Problems
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Part 1: Introduction
Lesson 6
At a Glance
Students review thinking about multiplication as a
comparison (times as many). They use models and skip
counting to solve a multiplication problem.
Develop skills and strategies
Lesson 6
Part 1: introduction
ccss
4.oa.a.2
Multiplication and Division in Word Problems
in Lesson 5, you thought about equations that compare numbers using
multiplication. in this lesson, you will solve those types of problems. take a
look at this problem.
Step By Step
•Tell students that this page models how to find the
product for a problem that compares two numbers
using the words “times as many.”
Hannah scored 3 goals last season. She scored 4 times as many goals this
season. How many goals did Hannah score this season?
Last season
This season
•Have students read the problem at the top of the page.
•Work through Explore It as a class. Ask students to
work with a partner to answer the first two questions
and have them share their responses.
explore it
•Draw 3 soccer balls on the board. Use the drawing
to model 3 goals last season by circling the group of
3 soccer balls. Add 3 more balls, circle them, and say,
“2 times as many.” Continue until students see that
4 3 3 is this season’s score (12) and that it is 4 times
as large as last season’s score.
•Ask student pairs to explain their answers for
the last three questions on the page. Use the
Mathematical Discourse questions to help clarify and
extend students’ thinking about the problem.
use the math you already know to solve the problem.
How many goals did Hannah score last season?
3
Count to find the number of goals she scored this season.
12
How can you use skip-counting to find the number of goals Hannah scored this
season? 3, 6, 9, 12
Besides addition, what operation can you use to solve the problem?
multiplication
What is 4 times as many as 3?
44
12
L6: Multiplication and Division in Word Problems
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•You may wish to assess students, understanding by
asking, What does the product tell you in this problem?
Concept Extension
Compare different meanings of a product in a
multiplication problem.
•Point out to students that when they find 4 times
3, they are finding the product of 4 and 3. Review
what the product 12 tells us in the soccer ball
problem. [It tells us what number is 4 times as
many as 3. 12 is the number of goals Hannah
scored this year, compared to last year (3).]
•Give students this problem and ask what the
product tells you: Chloe scored 3 goals in each
game. She played 4 games. How many points did
she score in all? The product tells the total
number of goals for all 4 games.
Mathematical Discourse
•What are you comparing in this problem?
You are comparing last year’s goals (3) to this
year’s goals.
•Which would be more: 4 times as many or 8 times as
many? How do you know?
Look for answers that show students
understand that 8 times as many is more
because it means 8 copies of some amount
instead of only 4 copies. 8 copies is more than
4 copies because 8 is more than 4.
•Ask: How is the meaning of the products in the
Hannah and Chloe problems different? [In the
Hannah problem, the product tells what is 4 times
as many as 3. In the Chloe problem, the product
tells the total number of goals.]
L6: Multiplication and Division in Word Problems
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51
Part 1: Introduction
Lesson 6
At a Glance
Students see a bar model that represents the
multiplication comparison and then see how to write an
equation from that model.
Step By Step
•Read Find Out More as a class.
•Help students connect the 4 bars of length 3 to the
4 sets of 3 soccer balls.
•Point out that a number sentence uses numbers and
symbols to describe a problem and helps make it
clear what you know and what you need to find.
•Use the problems on the page to point out what is
“known,” what the “unknown” is, and how “boxes”
or a question mark can stand for the unknown.
Part 1: introduction
Lesson 6
Find out More
You will often need to solve for some unknown amount, like when you found the
number of goals Hannah scored. You used skip counting to find 4 times 3. You can
also use a bar model.
Last season
3
This season
3
3
3
3
?
The bar model can help you write an equation to solve the problem.
4 3 goals last season 5 goals this season
Goals last season is known (3). Goals this season is unknown. You can use a symbol,
such as an empty box or a question mark, to stand for the unknown number in the
equation.
4335
reflect
1 What can you think of that has two times as many objects as something else?
Possible answer: a car has 2 times as many wheels as a bicycle.
•Have students read and reply to the Reflect directive.
L6: Multiplication and Division in Word Problems
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Visual Model
From Models to Symbols to Products
Materials: White boards or paper and markers
•Draw 2 small circles on the board. Underneath,
write “4 times as many.”
•Ask students to work in pairs to make a drawing
of 4 times as many circles as you drew. Have them
write a number sentence underneath using a
“box” to show the unknown (product). Ask pairs
to hold up their white board or paper and say the
number sentence aloud. [4 3 2 5 8 or 2 3 4 5 8]
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45
Real-World Connection
Encourage students to think of times when they
could have a certain number of “times as many”
more of something than someone else. Ask a few
students to share situations. Students may describe
situations comparing with friends or siblings
amounts of money, treats, time, distance, and so
forth. If students need help getting started, remind
them that “twice as many” is a multiplication
comparison.
•Draw 3 circles with the label “two times as many”
and have pairs create drawings and write number
sentences. [2 3 3 5 6 or 3 3 2 5 6]
•Draw 2 circles with the label “6 times as many”
and have students do the same. [6 3 2 5 12 or
2 3 6 5 12]
52
L6: Multiplication and Division in Word Problems
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Part 2: Modeled Instruction
Lesson 6
At a Glance
Part 2: Modeled instruction
Students use models and an equation to help them
understand a problem.
read the problem below. then explore different ways to understand it.
Janelle’s Market sells bags of 8 oranges. Simone needs 5 times that amount.
Step By Step
Write and solve an equation to find the number of oranges Simone needs.
•Read the problem at the top of the page as a class.
Model it
you can use models to help understand the problem.
•Read the first Model It. Make a simple drawing to
show one bag of 8 oranges on the board and point
out that Simone needs “5 times as many.”
•Read the second Model It together. Underneath
the drawing on the board, write the equation
8 3 5 5 . Explain that both the picture and
the number sentence describe the problem. There
are 8 oranges and Simone wants “5 times as many.”
•Note: Students may find it easier to write the
equation starting with the number “8” because that
is the quantity in one bag, and then write “3 5.”
Point out another way to say the same thing: “5 times
as many as 8.” Write the sentence as 5 3 8. Remind
students that, either way, the product is 40.
L6: Multiplication and Division in Word Problems
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Number in one bag
8
Number Simone needs
8
8
8
8
8
?
•Quickly draw 4 more bags labeled “8” and ask a
volunteer to count to find the total number of
oranges Simone needs. Help students make the
connection to using skip counting (counting by 8s)
as a way to find the product. Ask students, When
you are skip counting to find a product, how do you
know when to stop counting? [When I’ve skipped a
certain number of times. The problem tells me how
many times.]
•Ask students to compare the pictorial representation
on the board to the bar model in their books. Be sure
they see that each segment of the bar model
represents one bag of oranges. Bar models are
sometimes easier to draw than pictures, but they
serve the same purpose.
Lesson 6
Skip-count to find the total Simone needs: 8, 16, 24, 32, 40.
Model it
you can use the bar model to make an equation to help understand the
problem.
5 3 oranges in one bag 5 total oranges needed
The number of oranges in one bag is known (8). The total oranges needed is not
known.
5385
46
L6: Multiplication and Division in Word Problems
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Concept Extension
Multiplying Strategies: Dots and Circles
Materials: paper or white board and markers
•Explain to students that there are other quick
strategies they can use to find multiplication
products if they don’t remember a multiplication
fact or if skip counting becomes difficult (like
counting by 7s or 8s).
•Write the problem 5 3 8 on the board. Draw
5 circles and quickly draw 8 dots in each. Explain
that if you don’t know a product, you can make a
quick “circle and dots” drawing to model the
problem and then quickly count to find the
product. Show them that they could also draw
5 dots and circle it, then add another 5 dots and
circle it, until they have 8 groups of 5 dots.
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Part 2: Guided Instruction
Lesson 6
At a Glance
Students revisit the problem on page 46 to solve the
problem using an equation.
Step By Step
•Read the Connect It questions. Ask students to
answer the questions on their own or with a partner.
Ask students to share their answers for problems
3 and 4 with the class.
•For problem 6, ask a volunteer to model what they
thought or said to themselves to create an equation
from the bar model. Emphasize that whatever the
drawing shows, they can show the same thing using
numbers and symbols in a number sentence.
Part 2: guided instruction
Lesson 6
connect it
now you will solve the problem from the previous page using an equation.
2 You don’t know how many oranges Simone needs. What part of the bar model
shows how many she needs? the question mark.
3 How does the bar model show how many oranges are in one bag?
each little box has an 8 in it.
4 How does the bar model show how many bags Simone needs?
there are 5 little boxes.
5 How can you find “5 times as many” as 8? Multiply 5 3 8.
6 Write an equation using numbers to show how many oranges Simone needs.
5 3 8 5 40
Simone needs
40
oranges.
7 Explain how you can write a multiplication equation from a bar model.
Possible explanation: Look at how many little boxes there are and how
many are in each box. Multiply how many boxes by how many are in each
box to get the total.
SMP Tip: Asking students to think carefully and
explain how they created a number sentence from a
drawing and problem helps students think deeply
about the meaning of the quantities and their
relationships. (SMP 2)
try it
use what you just learned to solve these problems.
8 Neil and Vincent are collecting cans. Neil has collected 10 cans and Vincent has
collected 3 times as many cans as Neil. Write and solve an equation to find the
number of cans Vincent has collected.
10 3 3 5 v; v 5 30; vincent has collected 30 cans
9 Mr. Cherry ate 6 times as many raisins as Mary. Mary ate 11 raisins. Write and solve
an equation to find the number of raisins Mr. Cherry ate.
11 3 6 5 c; c 5 66; Mr. cherry ate 66 raisins
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TRY IT solutions
8Solution: 30 cans; Students may see the problem
as 3 “times as many” cans as 10 and write
3 3 10 5 30, or they may write 10 3 3 5 30 (that’s
10 three times), knowing they can change the order
of factors.
Concept Extension
Promote metacognition.
When students explain how they wrote an equation
from a model, use it as an opportunity to list some
questions students can ask themselves when they are
trying to make sense out of a problem, such as What
does each number in the problem tell me? Do I know the
total amount? Am I trying to find the total amount? Do
I see groups in the drawing? Do I see how many are in
each group? List the questions in the room and
encourage students to ask them as they make sense
of problems.
54
9Solution: 66 raisins; Students may see the problem
as 6 “times as many” as 11 raisins and write
6 3 11 5 66, or they may write 11 3 6 5 66 (that’s
11 six times), knowing they can change the order
of factors.
ERROR ALERT: Students who wrote an incorrect
product may not be using a workable strategy, such
as using a quick drawing to find the large product or
using repeated addition when they do not know the
multiplication fact 6 3 11 5 66.
L6: Multiplication and Division in Word Problems
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Part 3: Modeled Instruction
Lesson 6
At a Glance
Part 3: Modeled instruction
Students explore “times as many” situations in which
they find a missing factor or use division to solve
the problem.
Lesson 6
read the problem below. then explore different ways to understand it.
Juan found 3 times as many seashells at the beach as Jeremy found. Juan found
24 shells. Write and solve an equation to find the number of shells Jeremy found.
Step By Step
Model it
•In this lesson, this is the first time that students
encounter a “times as many” problem in which they
must find a missing factor or use division to solve
the problem.
you can use a model to help understand the problem.
Jeremy found one group of seashells. Juan found 3 times as many shells.
Jeremy’s shells
?
Juan’s shells
?
?
?
24
•Read the problem. Explain to students that they need
to think about what they know and don’t know in the
problem. Guide their thinking by asking questions
such as, Do we know how many shells Juan found?
[24 shells. That’s 3 times as many as Jeremy.] Do we
know how many Jeremy found? [no] Do either of the
boys have more than 24 shells? [no] How do you know?
Divide 24 by 3 to find the number of seashells in each group: 8.
solve it
you can use the model to make an equation to help understand the problem.
3 3 Jeremy’s shells 5 Juan’s shells
The number of shells Juan found is known (24). The number Jeremy found is not
known.
33
5 24
•Draw the model in Model It on the board. Help
students understand why you know that there are
3 equal groups. Ask students how to figure how
many should be in each group.
•Underneath the picture, write the equation from
Solve It and make a clear connection that each
number and symbol describes what the picture and
problem describes.
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L6: Multiplication and Division in Word Problems
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SMP Tip: Use the following Concept Extension
activity to give students an opportunity to practice
constructing arguments and reasoning
quantitatively to show their understanding. (SMP 3)
Concept Extension
Do You Agree?
Have students work in pairs to discuss whether or
not they agree with the following statement, and
why. If possible, ask them to give an example to
prove their point.
Statement: When you see the words “times as many”
in a problem, you always must find the product.
Students should be able to explain that in a “times as
many” problem they could be finding the product
(like the “bags of oranges” problem) or dividing to
find a missing factor (like the seashell problem).
L6: Multiplication and Division in Word Problems
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Mathematical Discourse
•How could the number 24 in this problem be both a
product and a dividend?
If you think of the problem as having a missing
factor, then 24 is the product. If you think of it
as dividing, then 24 is the dividend. In both
cases, you want to know the number of shells
Jeremy found.
•How is this problem different from the “times as
many” problems you solved earlier in this lesson?
In this problem, division is used to find the
answer. The answer wasn’t the product but was
a missing factor.
55
Part 3: Guided Instruction
Lesson 6
At a Glance
Students solve a division and multiplication problem
involving “times as many” using an equation.
Step By Step
•Read the Connect It questions. Ask students to
answer the questions on their own or with a partner.
Have students to share their answers for problems
10 through 12 with the class.
•Be sure students see that each section of the bar
model represents the same amount: the number of
shells Jeremy found.
•Point students’ attention to problem 13. After inviting
volunteers to share their thinking, explain that you
can think about the problem in two different ways:
one is to find the missing factor (What number times
3 is equal to 24?) or to find the size of a group
(What is 24 shared equally in 3 groups?) and get 8.
•Invite students to share and discuss their answers for
problem 15. Help students acknowledge the different
ways to think about a problem and still arrive at the
same answer mathematically.
Part 3: guided instruction
connect it
now you will solve the problem from the previous page using an equation.
10 You don’t know the number of shells Jeremy found. In the bar model, what part
shows the number of shells Jeremy found?
the little box with the question mark.
11 How does the bar model show how many shells Juan found?
the bar with his name on it is labeled 24.
12 How does the bar model show that 24 is 3 times another number?
the 3 little boxes altogether are the same length as the 24 bar.
13 How can you find what number times 3 is 24?
Find 3 3
To help students make connections between what
to think, say, and write for an equation that has a
missing factor, write the following sentence frame
on the board or wall: “What number times 5 ?”
Visual Model
Represent the problem with a picture.
To help students visualize the problem in another
way, draw 24 shells or dots on the board to stand for
24 shells. Tell students that “3 times as many”
means that there must be 3 times 1 group. Draw
3 empty (large) circles. Explain that they know Juan
found 24 shells. They are comparing what Jeremy
found (in one circle) to what Juan found (24). Write
the equation 3 3 5 24. Help them to think
of a multiplication fact to find the missing factor.
Relate the missing factor to the number in one group
and to the number of shells Jeremy collected.
56
5 24 or find 24 4 3 =
.
14 Write a division equation using numbers to show how many shells Jeremy found.
24 4 3 5 8
Jeremy found
8
shells.
15 Explain how you can write a division equation from a model.
Possible explanation: i look at how many little boxes there are and how
much they equal altogether. i can divide the total by the number of little
boxes to see how much each box is.
try it
use what you just learned to solve these problems.
16 Monique and Wint are both reading the same book. Monique read 63 pages last
weekend. She read 7 times as many pages as Wint. Write and solve an equation to
find the number of pages Wint read. 63 5 7 3 w or 63 4 7 5 w; w 5 9; 9 pages
17 The winning baseball team scored 4 times as many runs as their opponent. The
winning team scored 8 runs. Write and solve an equation to find the number of
runs their opponent scored. 4 3 r 5 8 or 8 4 2 5 r; r 5 2; 2 runs
L6: Multiplication and Division in Word Problems
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ELL Support
Lesson 6
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49
TRY IT solutions
16 Solution: 9 pages; Students may write a
multiplication equation to find the missing factor,
7 3 w 5 63, or a division equation, 63 4 7 5 w, to
show how many pages times 7 is 63 pages.
17 Solution: 2 runs; Students may use a model to show
8 runs as 4 times the opponent’s score and write an
equation 8 4 4 5 r or 4 3 r 5 8.
ERROR ALERT: Students may see “times as many”
and multiply 4 3 8. Help students see that the score
of 8 is the winning team’s score, which is 4 times
their opponent’s score. The unknown is the
opponent’s score.
L6: Multiplication and Division in Word Problems
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Part 4: Guided Practice
Lesson 6
Part 4: guided Practice
Lesson 6
study the model below. then solve problems 18–20.
Student Model
There are twice as many
boxes in Karina’s model
as in her cousin’s model.
Karina is 6 feet tall. Her cousin is 3 feet tall. How many times
Part 4: guided Practice
Lesson 6
19 Christina read 7 pages in a magazine. She read 5 times as many
pages in a book. How many pages did Christina read altogether?
Show your work.
as tall as her cousin is Karina?
Magazine pages
7
Book pages
7
I remember that
multiplication and
division are opposite
operations!
Look at how you could show your work using a bar model.
Cousin’s height
3
Karina’s height
3
7
7
7
7
35
3
7 1 35 5 42
Pair/share
6
Pair/share
How else could you
solve this problem?
What does it mean when
the problem says 3
times as many?
33
5 6;
Solution: 42 pages
52
Solution: karina is 2 times as tall as her cousin.
20 Aida swam 7 laps in a pool. Kaya swam 28 laps. How many times
18 A small shrimp taco has 5 shrimp. There are 3 times as many
shrimp in a large taco. How many shrimp are in a large taco? Write
and solve an equation to find the answer.
Show your work.
50
a 4
b
21
c
35
Jae Ho chose D as the correct answer. How did he get that
answer?
Solution: jae ho multiplied the number of laps each girl
swam instead of dividing them.
Solution: 15 shrimp
L6: Multiplication and Division in Word Problems
L6: Multiplication and Division in Word Problems
©Curriculum Associates, LLC
Does Jae Ho’s answer
make sense?
D 196
5 shrimp in a small taco
33
15 in a large taco
Pair/share
Did you and your
partner write the same,
or different, equations?
the number of laps Aida swam did Kaya swim? Circle the letter of
the correct answer.
How can you check
your answer?
Copying is not permitted.
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Copying is not permitted.
Pair/share
How did you and your
partner know what
operation to use?
51
At a Glance
solutions
Students use equations to solve multiplication and
division problems involving “times as many.” They
show understanding of multiplication and division as
opposite operations to check their answers.
Ex Solution: Karina is 2 times as tall as her cousin; the
student model shows using a bar model to find how
many 3s make 6. Students may also see 6 4 3 5 2
from the model.
Step By Step
18 Solution: 5 3 3 5 15 shrimp; Students may also
write 3 3 5 5 15 for “3 times as many of the
5 shrimp.” (DOK 1)
•Ask students to solve the problems individually.
•When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner and be ready to share with the class.
•Be sure students know what is meant by “operation”
and “opposite operations” and how they can use the
opposite operation to check their answers. For
example, students find the quotient for 15 4 3 (5).
They can multiply 3 3 5 and should get the
product 15.
L6: Multiplication and Division in Word Problems
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19 Solution: 42 pages; See sample student work above.
(DOK 1)
20 Solution: A; Students may have divided 28 by 7 or
thought, “how many times as many as 7 is 28?”
Explain to students why the other two answer
choices are not correct:
B is not correct because it is the difference between
7 and 28.
C is not correct because it is 7 greater than 28.
(DOK 3)
57
Part 5: Common Core Practice
Part 5: common core Practice
Lesson 6
Solve the problems.
1
Lesson 6
Part 5: common core Practice
4
84
B
56
C
42
D
14
5
2
3
Raoul biked 11 miles last week. Jackson biked 22 miles last week. Jackson biked
how many times as many miles as Raoul? Which equation can help you answer the
question?
A
22 2 11 5 h
B
22 4 11 5 h
C
11 3 22 5 h
D
11 1 22 5 h
Maria has 32 postcards. Henry has h postcards. Maria has 4 times as many postcards as
Henry.
Choose Yes or No to indicate whether each statement is true.
Kyle sold 28 boxes of fruit for a fundraiser. Omar sold 2 times as many boxes of fruit as
Kyle sold. What is the total number of boxes that Kyle and Omar sold?
A
Lesson 6
a.
The number of Henry’s postcards can be
represented by the expression 32 4 4.
b.
Henry has 6 postcards.
c.
The number of Henry’s postcards can be
found by solving the equation 32 5 4 3 h.
3 Yes
No
Yes
3 No
3 Yes
No
Viet learned 25 new spelling words last week. He learned 5 times as many words as
Max. How many words did Max learn? Draw a bar model to find the number of words
Max learned.
Show your work.
Max
5
Viet
5
5
5
5
5
25
Which problems can be solved using the equation 3 3 9 5 A? Circle the letter of all
that apply.
A
Pam is 9 years old. She is 3 times as old as Kate. How old is Kate?
B
Marco is making 9 apple tortes. He needs 3 apples for each torte. How many
apples does he need?
C
Three groups of actors are performing plays at a festival. There are 9 actors in
each group. How many actors are performing?
D
An art class meets 3 times a week for 9 weeks. How many times does the class
meet?
E
Judy found 3 acorns. Aaron found 3 times as many acorns as Judy. How many
acorns did Aaron find?
Answer Max learned
6
5
new spelling words last week.
Mr. Naik traveled 18 hours on vacation last summer. Miss Cooper traveled 3 hours
on vacation last summer. How many times as many hours did Mr. Naik travel as
Miss Cooper? Write an equation to find the answer.
Show your work.
Possible equation: 18 4 3 5
Answer Mr. Naik traveled
6
or 3 3
5 18
times as many hours on vacation as Miss Cooper.
self check Go back and see what you can check off on the Self Check on page 37.
52
L6: Multiplication and Division in Word Problems
L6: Multiplication and Division in Word Problems
©Curriculum Associates, LLC
Copying is not permitted.
©Curriculum Associates, LLC
Copying is not permitted.
53
At a Glance
4Solution: a. Yes; b. No; c. Yes (DOK 2)
Students multiply and divide to solve word problems
involving multiplicative comparison that might appear
on a mathematics test.
5Solution: 5; See sample student work above. 25 is
5 times the number of words Max learned, so Max
learned 5 words. (DOK 1)
solutions
6Solution: 6; 18 4 3 5 6 or 3 3 6 5 18 or 6 3 3 5 18
(DOK 1)
1Solution: A; Students may see the problem as
28 3 2 5 56 and 56 1 28 5 84. (DOK 2)
2Solution: B; Students may think 22 divided by a
number equals 11. (DOK 1)
3Solution: B; If there are 3 apples for each of 9 tortes,
then 3 3 9 is equal to the total apples, A.
C; If there are 9 actors in each of 3 groups, then
3 3 9 is equal to the total actors, A.
D; If there are 3 art classes each week for 9 weeks,
then 3 3 9 is equal to the total art classes, A.
(DOK 2)
58
L6: Multiplication and Division in Word Problems
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Differentiated Instruction
Lesson 6
Assessment and Remediation
•Ask students to write an equation to solve this problem: Jake has 12 bean plants in his garden. He has
3 times as many as Simone has in her garden. How many plants does Simone have? [4]
•For students who are struggling, use the chart below to guide remediation.
•After providing remediation, check students’ understanding. Ask students to solve this problem: Sophia
counted 10 cars going by. That is 5 times as many cars as Carla counted. How many cars did Carla count? [2]
•If a student is still having difficulty, use Ready Instruction, Level 4, Lesson 5.
If the error is . . .
Students may . . .
To remediate . . .
12 3 3 5 36
think that “times as
many” only indicates
multiplication.
Have students underline each number in the problem. Ask what
each number tells them. Help students verbalize what is not
known in the problem. Ask what it means when the problem
says “Jake has 3 times as many as Simone.” [12 is 3 times as
many.] Then use drawings and an equation to find the quotient.
12 2 3 5 9
think a comparison
requires subtraction.
Remind students that some comparisons involve multiplication,
like when one number is a certain “times as many” as another
number. Use number lines and drawings to show what “times as
many” means. 4 “times as many” as 3 means that they will see
“3” objects and then multiply that quantity by 4 because another
number is “4 times as many.”
12 1 3 5 15
Add to find a larger
number. Instead, they
should multiply to
compare and
understand that one
number is a certain
“times as many” as
another number.
Use number lines and drawings to show what “times as many”
means. 4 “times as many” as 3 means that they will see “3”
objects and then multiply that quantity by 4 because another
number is “4 times as many.”
Hands-On Activity
Challenge Activity
Use counters to solve “times as many” problems.
Write a problem for a given product or dividend.
Materials: Counters
Write these products on the board: 36, 24, and 45.
Challenge students to choose one of these numbers
to be a product or dividend in a “times as many”
problem. Direct them to write the problem for a
partner to solve. The partner will write an equation
and solve the problem. The problem author should
ask the partner, “How do you know that the equation
describes the problem? How do you know the
solution is correct?”
Distribute counters to each student. Write the
following sentence frame on the board: What if you
have counters and I have times as many
counters as you have? Then I have counters.
Direct each student to think of numbers to correctly
complete the sentence. Have them use counters to
model the problem and say the sentence aloud to a
partner. Ask a few students to share and have a
volunteer write the number sentence on the board
that matches the student’s sentence frame.
L6: Multiplication and Division in Word Problems
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59