8.4
Represent and
Solve Multi-Step Problems
PROBLEM SOLVING •
?
Essential Question How can you represent and solve multi-step problems
using equations?
Texas Essential
Knowledge and Skills
Algebraic Reasoning—5.4.B
Represent and solve multi-step problems involving the four
operations with whole numbers using equations with a letter
standing for the unknown quantity
MATHEMATICAL PROCESSES
5.1.B Use a problem solving model
5.1.D Communicate mathematical ideas and reasoning
How can you represent
and solve multistep problems using
equations?
Are You Ready?
Access Prior Knowledge
Use the Are You Ready? 8.4 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Lesson Opener
Go to Multimedia eGlossary at
thinkcentral.com
Making Connections
Invite students to tell you what they know about equations.
What is used to represent the unknown information in a word problem? (a letter)
What are ways to help translate information from a word problem to an equation?
(underline important information, draw a picture, make a strip diagram, etc.) After the
equation is solved, what must be done to finish the problem? (Explain the meaning of
the answer in terms of the word problem.)
Using the Digital Lesson
You may choose to have two buckets filled with 18 acorns each. Remove 7 acorns to
model the problem. Discuss the fact that removing the acorns is subtraction.
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
Ask the following questions.
• How many buckets are there? (2)
• How many acorns does Sasha have in each bucket? (18)
• How many acorns did Sasha give to Ramona? (7)
• What are you trying to find? (the number of acorns remaining in Sasha’s bucket)
• How are we going to solve the problem? (Write and solve an equation)
Literacy and Mathematics
Choose one or more of the following activities.
• Have students research to find out what animals eat acorns.
• Have students make a list of synonyms for the word represent and think of examples
of when one thing represents another in other subject areas or in their everyday lives.
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
iTools Virtual
Manipulatives
Soar to Success Math
Online Intervention
Lesson 8.4 341A
Name
Unlock the Problem
8.4
Hands
On
?
Read and discuss the problem. Be sure students
understand that the solution to the problem can be
found by breaking the question into steps.
Essential Question
MATHEMATICAL PROCESSES
5.1.B, 5.1.D
How can you represent and solve multi-step problems
using equations?
Unlock
Unlock the
the Problem
Problem
Shaniqua buys 140 small beads and 30 large beads to
make bracelets. She makes 5 bracelets. She uses
13 beads on each bracelet. How many beads does
Shaniqua have left?
In order to find each step, have students underline
important information in the problem.
• Underline the important information.
Example 1 Use multiple single-step equations.
Example 1
total number of small beads
• How does the model represent the equation given
in Step 1? Each addend is represented by a section
140
shown in the strip. There are two addends; therefore,
there are two sections in the strip. The sum or total, a, is
represented by the bracket.
13
ELPS
Beginning: Activity 20
1.A.1, 3.G.2, 4.C.3
Intermediate: Activity 21
3.D.2, 3.G.1, 4.C.1
Advanced: Activity 14
4.C.4, 4.F.9, 4.G.2
Advanced High: Activity 18
4.C.4, 4.E, 4.F.7
Go to thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
total number of beads Shaniqua buys
13
13
d
13
5 bracelets with 13 beads
170 – 65 = s
s
© Houghton Mifflin Harcourt Publishing Company
105 = s
_
beads used
beads left
5 × 13 = d
65 = d
_
total number of beads Shaniqua uses
STEP 3 Find the total number of beads Shaniqua has left.
• In Step 3, why do you subtract 65 from 170? 170 is
Leveled Activities
13
170 = a
_
30
STEP 2 Find the total number of beads Shaniqua uses to make 5 bracelets.
bracelets Shaniqua makes. The number inside each section
shows how many beads she uses for each bracelet.
English Language Learners
total number of large beads
a
• How does the model represent the equation
given in Step 2? Each section represents the number of
the number of beads Shaniqua has in all, which was found
in Step 1. 65 is the number of beads Shaniqua uses, which
we found in Step 2. So, we need to subtract the number
of beads she uses from the number of beads she had.
140 + 30 = a
STEP 1 Find the total number of beads Shaniqua buys.
Discuss with students the series of single-step
equations they can model and solve in order to
answer this multi-step problem.
341 Module 8
Algebraic
Reasoning—5.4.B
Represent and Solve
Multi-Step Problems
65
170
total number of beads Shaniqua buys
105 beads left.
So, Shaniqua has _
ELL Language Support
Module 8 341
Verbal/ Linguistic
Partners
ELPS 2.I.4, 3.H.3, 3.G.1
Strategy: Creative Grouping
• Partner advanced English Learners or students who are fluent in English
with beginning and intermediate students.
• Read the Math Talk box on page 342. Have pairs work together to
explain why multiplication can be used to solve a division problem.
Have pairs use one of the problems as an example in their explanation.
• Have fluent English speakers model how to use multiplication
equations to solve division problems.
• After students solve a problem using multiplication, ask them if they
prefer to use multiplication to solve division problems and to explain
why or why not.
Try This! Sometimes you can use one multi-step equation to
solve a problem.
Miguel sorts his seashell collection into boxes. He has 3 boxes with
15 periwinkle shells in each box. He has 2 boxes with 7 clamshells
in each box. He gives his little brother 10 shells. How many shells
does he have now?
15
15
15
7
7
total number
of shells
Try This!
In this method, students will write a multi-step
equation to solve a similar problem to Example 1.
3 × 15 + 2 × 7 – 10 = n
• Compare this method to the multiple singlestep equations. Both methods require several steps
45 + _
14 – _
10 = n
_
10
to solve. This method uses one equation and the order
of operations. The first method uses several one-step
equations.
59 – _
10 = n
_
n
shells given
away
49 = n
_
shells left
Example 2
Example 2
Guide students through the solution.
Meagan has two card-collection books. The first book has 8 cards on
each of 14 pages. The second book has 6 cards on each of 15 pages.
Which of the two books has more cards?
• What type of equation is related to a division
equation? multiplication equation
STEP 1 Solve the equation b ÷ 8 = 14 to find the number of cards
in the first book.
14
14
14
14
14
14
14
• Why should we use a multiplication equation to
solve the problem? because both factors are known
14
• Explain how you can check your solution by
replacing the unknown with its value. In the
equation b ÷ 8 = 14, replace b with the solution and
then simplify. If the quotient of 112 ÷ 8 is 14, then 112 is
b
Write a related multiplication equation.
8 =b
14 × _
Math Talk: Possible answer: I can use the
relationship of multiplication to division to
rewrite the equation so it is true.
112
b=_
STEP 2 Solve the equation p ÷ 6 = 15 to find the number
of cards in the second book.
15
15
15
15
15
15
the solution.
STEP 3 Compare the number of cards in
each book.
112 cards.
The first book has _
Math Talk
Write a related multiplication equation.
6 =p
15 × _
112 > _
90 , the _
first book
Since _
has more cards.
90
p=_
Math Talk
Mathematical Processes
Explain why you can use
a related multiplication to solve
a division problem.
342
Enrich
© Houghton Mifflin Harcourt Publishing Company
90 cards.
The second book has _
p
Mathematical Processes
Use Math Talk to help students understand the
relationships between the operations.
Interpersonal
Small Groups
Materials: poster paper
• Divide students into three groups. Find the three multi-step problems
on the Problem Solving page of the Student Edition.
• Assign one problem to each group. Have the groups solve the problem,
and prepare a short presentation to explain and display their solution.
• Ask each group to present.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Lesson 8.4
342
Name
Share
Share and
and Show
Show
Share and Show
Use equations to solve each multi-step problem. Equations may vary.
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
1. A caterer is making 3 trays of 24 sandwiches
2. A baseball league has a total of 156 players.
and 2 trays of 30 sandwiches. She receives an
order for 35 sandwiches. How many sandwiches,
s, does the caterer have left?
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on
the MathBoard.
The players are divided into 12 equal teams.
Each team has 3 coaches. All players and coaches
receive 2 jerseys. How many jerseys, j, will each
team receive?
j = 32; 32 jerseys
s = 97; 97 sandwiches
Write Math
Problem
Problem Solving
Solving
3
2
Quick Check
1
Use equations to solve. Equations may vary.
3. Multi-Step A florist makes 4 floral arrangements that
a student misses the checked exercises
IF
THEN
are exactly the same. He uses 4 bunches of 8 tulips and
2 bunches of 10 daisies. How many flowers, f, are in each
arrangement?
Differentiate Instruction with
RtI Tier 1 Lesson 48
f = 13; 13 flowers
4.
Problem Solving
For Exercises 3–5, students may solve these multi-step
problems using either a series of one-step equations,
using a multi-step equation and the order of
operations, or writing a related equation.
5.
36 ÷ 6 + 3 × 7 = r
Student answer: r = 28
Springboard to Learning Tell students to look at all
of the operations before solving and make a plan
to solve using the order of operations. Students
should show each step of their work.
Multi-Step Elsa has 4 trays of
15 sedimentary rocks. She has 6 trays
of 12 metamorphic rocks. She gives away
5 sedimentary rocks for 7 metamorphic
rocks. How many rocks, r, does she have?
r = 134; 134 rocks
© Houghton Mifflin Harcourt Publishing Company
Example Find the value of r.
Multi-Step Francois is rolling coins.
He has 3 rolls of 40 quarters. He has 8 rolls of 50 dimes.
He exchanges 2 rolls of dimes for 1 roll of quarters.
How many coins, c, does Francois have?
c = 460; 460 coins
COMMON ERRORS
C
E
Error
Students do not follow the order of
operations correctly.
6. Analyze Murial alternates her exercise. She runs 5 miles
each day for 3 days and walks 4 miles each day for 2 days
in one week. She rides her bike 8 miles each day for
4 days the next week. How many more miles, m, does
Murial travel during the second week than the first week?
m = 9; 9 miles
Module 8 • Lesson 4 343
3
RtI Tier 1 Lesson 48
2
1
M
Math
on the Spot
Video Tutor
V
Enrich 51
Name
Name
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 H.O.T. problems, students will
build skills needed in the TEXAS assessment.
5.4.B
OBJECTIVE Use strip diagrams to model and solve multi-step problems.
Solve the Shopping Equations
Pierre wants to do some shopping at the mall for his friend’s birthday.
Simplify each equation by evaluating first. Then solve the equation.
Draw a line in the maze to connect the answers. Which stores does
Pierre visit?
6
6
3 rows of 6
e
:fYYc\iËjJ_f\j
(-
5
5
5
5
4 rows of 5
K_\9ffbNfid 0
)+
20
the number of trees altogether
20
© Houghton Mifflin Harcourt Publishing Company
31
d
38 trees altogether
2.
x 1 16 2 4 1 (16 1 9) 5 56
3.
x 1 (5 3 11 1 6) 2 45 5 32
4.
x 1 (64 1 16 4 2) 4 9 5 16
5.
25 3 2 5 x 2 3
6.
(20 3 2) 2 18 5 x 2 4
So,
7
x 5 16
7.
x58
9 3 9 2 17 5 x 2 (4 3 4)
x 5 26
8.
x 5 80
trees were damaged by the storm.
9.
Which stores does Pierre visit?
Solve the multi-step problem. Draw strip diagrams if you need to.
10.
1. Chenglei’s social studies classroom has 3 rows of
desks with 6 desks in each row, and 2 rows of desks
with 5 desks in each row. When no students are
absent, 8 desks are empty. How many students, s,
are in Chenglei’s social studies class?
Algebraic Reasoning
x 5 19
x 5 53
7
JgfikXe[J_fg
-/
DXcc<o`k
x 1 (22 2 2) 4 2 5 15
d = 38 – 31
d=
/'
(+
A\iipËj:;j
1.
38
STEP 4 Find d, the number of trees damaged by the storm.
)(
K_\:Xe[p;`j_
(/
=i`\e[cpG\kJ_fg
*
)-
*.
x55
t = 18 + 20
t=
t
'
+'
+
STEP 3 Find t, the number of trees altogether.
18
(*
:Xe[c\:fie\i
l=
l
-
,*
/
+)
10 3 8 4 2 1 4 3 4 5 x 2 12
x 5 68
Cobbler’s Shoes and Sport
and Shop
Stretch Your Thinking Pierre decides to visit The Candy Dish
store before leaving the mall. Write an equation that represents Pierre
going to The Candy Dish store. Include at least one 2-digit number in
your equation and at least three operations.
Check students’ work. Possible answer:
(5 1 10) 3 2 2 16 5 14
s = 20; 20 students
343 Module 8
(0
STEP 2 Find l, the number of trees which have apples that ripen late.
l= 4× 5
('
K$j_`ik:\ekiXc
)
(
18
)*
*0
,
e= 3× 6
e=
K_i`]kpJg\e[`e^J_fg
DXcc<ekiXeZ\
STEP 1 Find e, the number of trees which have apples that ripen early.
6
Enrich 51
1
Represent and Solve Multi-Step Problems
LESSON
48
In a small apple orchard, the apples on 3 rows of trees ripen early in the season,
and there are 6 trees in each row. The apples on 4 rows of trees ripen late, and
there are 5 trees in each row. Thirty-one trees were not damaged by a recent
windstorm. How many trees, d, were damaged by the storm?
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
Show Your Work
95
Enrich
© Houghton Mifflin Harcourt Publishing Company
E51