10.4
Relate Subtraction and Division
ALGEBRA
?
Essential Question How is division related to subtraction?
Texas Essential
Knowledge and Skills
Number and Operations—3.4.H
Determine the number of objects in each group when a set of
objects is partitioned into equal shares or a set of objects is
shared equally
3.4.K Solve one-step and two-step problems involving
division within 100 using pictorial models, including equal
groups
How is division related to
subtraction?
Algebraic Reasoning—3.5.B
Represent and solve one- and two-step division problems
within 100 using equations
MATHEMATICAL PROCESSES
3.1.D Communicate mathematical ideas and reasoning
3.1.E Create and use representations
3.1.F Analyze mathematical relationships
Are You Ready?
Access Prior Knowledge
Lesson Opener
Making Connections
Invite students to tell you what they know about recycling.
Have you even seen a trash can that is for recycling? How can you tell? (Sample
answer: It is a blue/yellow bin that has a recycling sign on it.) What kind of things can
be recycled? (Paper, plastics) What happens to trash that is not recycled? (Sample
answer: It goes to a landfill.)
Use the Are You Ready? 10.4 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
Go to Multimedia eGlossary at
thinkcentral.com
Using the Digital Lesson
You may want to use small items, such as beans or buttons, to model the problem.
Recycling trash cans can be one color and regular trash cans can be a different color.
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
Resources
• How many trash cans are there total? (21 trash cans)
• There are two types of trash cans. What are they? (Regular and recycling)
• How many of each kind? (2 for recycling and 1 regular)
• How many trash cans in all go to each house? (3 trash cans)
Literacy and Mathematics
Choose one or more of the following activities.
• Ask students to come up with a set of questions they would ask the people from the
trash company.
• Have students infer why the trash company is delivering the new trash cans.
For the student
For the teacher
Interactive
Student Edition
provides students
with an interactive learning
environment!
Digital Management
Center organizes program
resources by TEKS!
Math on the Spot
Video Tutor
eTeacher
Edition
Online Assessment
System
iTools Virtual
Manipulatives
Soar to Success Math
Online Intervention
Lesson 10.4 325A
Name
Unlock the Problem
10.4
Hands
On
ALGEBRA
?
Do students at your school bring in items to recycle?
Read the problem to find out what Serena and
Mandy brought to school for recycling.
Number and
Operations—3.4.H, 3.4.K
Algebraic Reasoning—
Relate Subtraction
and Division
Essential Question
3.5.B
MATHEMATICAL PROCESSES
3.1.D, 3.1.E, 3.1.F
How is division related to subtraction?
Unlock
Unlock the
the Problem
Problem
Be sure students understand that they need to first
find the total number of newspapers both girls
brought in each day and then divide to find how
many days the girls brought in newspapers.
Serena and Mandy brought a total of
12 newspapers to school for the recycling
program. Each girl brought in one
newspaper each day. For how many
days did the girls bring in newspapers?
• How many newspapers were
brought in altogether?
12 newspapers
• How many newspapers did the two
girls bring in altogether each day?
2 newspapers
One Way
One Way Use repeated subtraction.
This method uses repeated subtraction to divide.
• Start with 12.
• Explain in your own words how to subtract to
divide. Start with the dividend. Keep subtracting the
• Subtract 2 until
you reach 0.
divisor until you get to zero. To find the quotient, count
the number of times you subtracted.
• Count the number
of times you
subtract 2.
• Why is it important to keep subtracting until you
get 0? so you are sure you have all the groups
• What is an advantage of using division
to solve the problem? Possible answer: it takes
less time to write 12 ÷ 2 = 6 than it does to write
12 – 2 – 2 – 2 – 2 – 2 – 2 = 0.
12
− 2
_
10
Number of
times you
subtract 2:
1
Since you subtract 2 six times,
6 groups of 2 in 12.
there are _
10
− 2
_
8
8
−2
_
6
6
−2
_
4
4
−2
_
2
2
−2
_
0
2
3
4
5
6
ERROR Alert
Be sure to keep subtracting
2 until you are unable to
subtract 2 anymore.
So, Serena and Mandy brought in
© Houghton Mifflin Harcourt Publishing Company
6 days.
newspapers for _
English Language Learners
6
Write: 12 ÷ 2 = 6 or 2)‾
12
Read: Twelve divided by two equals six.
ELL Language Support
Leveled Activities
ELPS
Beginning: Activity 2
2.I.4, 3.C.4, 3.G.2
Intermediate: Activity 3
2.D.2, 2.E.3, 3.F.2
Strategy: Draw
Advanced: Activity 47
2.D.2, 2.I.5, 3.D.1
Materials: color pencils
Advanced High: Activity 60
2.I.3, 4.F.7, 4.G.3
Go to thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
325 Module 10
Module 10 325
Auditory / Visual
Small Group
ELPS 2.I.3, 4.F.2, 4.G.4
• Students can draw to show conceptual understanding.
• Write the division situation on the board. Miguel brought in 8 cans of
food. He brought an equal number of cans each day for 2 days.
• Have students draw a picture to represent the situation.
• Ask questions about the drawings,
such as: How many cans did Miguel
bring each day? 4 cans How many
cans did he bring in all? 8 cans
Hands
On
Another Way Count back on a number line.
• Start at 12.
• Count back by 2s as
many times as you
can. Draw the rest
of the jumps on the
number line.
• Count the number
of times you
jumped back 2.
6
0
1
5
2
3
4
4
2
3
5
6
7
8
Another Way
1
9
10
11
This method uses a number line to count back by 2s
to divide.
12
After students complete the page, ask:
Check students’ drawings.
There should be 6 jumps
of 2 on the number line.
You jumped back by 2 six times.
6 jumps of 2 in 12.
There are _
6
12 ÷ 2 = _
• Why did you start at 12 and count back by 2?
because you want to find how many groups of 2 are in 12
Math Talk
Possible explanation:
I continued to draw
jumps from 8 to 6, 6
to 4, 4 to 2, and 2 to 0.
There were 6 jumps of
2 in all, so 12 ÷ 2 = 6.
• What does the number of jumps represent? the
Mathematical Processes
Explain in your own
words how you found
the answer.
answer: how many groups of 2 are in 12, which is 6
Share and Show
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
• What do your jumps of 2 represent? Each jump represents a day the girls brought in
2 newspapers.
Possible explanation: each time you count
back the same number on the number line,
it is like subtracting the number.
Share
Share and
and Show
Show
1.
Draw the rest of the jumps on the number line to
complete the division equation.
Math Talk
Mathematical Processes
2
Explain how counting
back on a number line
is like using repeated
subtraction.
1
0 1 2 3 4 5 6 7 8 9 10 11 12
Mathematical Processes
Use Math Talk to help students recognize the
similarities between counting back on a number
line and using repeated subtraction.
Math Talk
3
12 ÷ 4 = _
3
Hands
On
Use the checked exercises for Quick Check.
3
2
Write a division equation.
10
−
5
_
5
5
−
_5
0
Quick Check
1
3.
0
1
2
10 ÷ 5 = 2
3
4
5
6
7
8
8÷2=4
326
© Houghton Mifflin Harcourt Publishing Company
2.
a student misses the checked exercises
IF
THEN
Differentiate Instruction with
RtI Tier 1 Lesson 48
COMMON ERRORS
C
E
Error
Students may not continue the jumps all the
way to zero when counting back on a number line
to divide.
Enrich
Logical / Mathematical
Individual
• Write the following riddle on the board.
Subtract 3 from me 6 times
and you will reach 0.
Example
5
4
3
2
1
0 1 2 3 4 5 6 7 8 9 10 11 12
What number am I? 18
• Have students solve the problem and then write the division equation
it represents. 18 ÷ 3 = 6
• Then have students write their own division riddles.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Springboard to Learning Remind students that
when they skip counted to multiply, they started
at 0. Have students observe that when counting
back on a number line to divide, they need to
go to 0.
Lesson 10.4 326
Name
Problem
Problem Solving
Solving
Problem Solving
Use the graph for 4–6.
4.
Problem 5 requires students to interpret information
in a bar graph to solve a multi-step problem. The
problem also requires students to explain their
solution.
5 box tops
5.
Go Deeper
To extend students' thinking for Problem 6, have
them write a division equation to represent the
problem. 5 = 15 ÷ 3
20
15
10
5
0
Paige Dwayne Alma
Student
Matt
Write Math
Show Your Work
Alma brought 20 ÷ 5 = 4 box tops each day;
5 + 4 = 9 box tops altogether each day.
6.
For Problem 7, students must write a word problem
that can be solved using a given division equation.
Give students the opportunity to share their problems
and solutions with the class.
Write Math
What’s the Question?
Dwayne puts all his box tops into bins. He
puts an equal number in each bin. The
answer is 5. What’s the question?
Possible question: How many box tops did
Dwayne put in each of 3 bins?
© Houghton Mifflin Harcourt Publishing Company
7.
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
25
brought 25 ÷ 5 = 5 box tops each day;
Problem
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.
Multi-Step Paige
brought an equal number of
box tops to school each day for
5 days. Alma also brought an
equal number of box tops each
day for 5 days. How many box
tops did the two students bring
in altogether each day? Explain.
30
9 box tops; Possible explanation: Paige
Then you might restate the problem as: What If
Dwayne puts an equal number of box tops into bins,
and the answer is 3, what is the question? How many
box tops did Dwayne put in each of 5 bins? Have students
write the corresponding equation. 3 = 15 ÷ 5
M
Math
on the Spot
Video Tutor
V
Box Top Collections
Matt puts his box tops in 2 equal piles.
How many box tops are in each pile?
Number of Box Tops
Problem
Pose a Problem Write a word
problem about collecting box tops that can
be solved by using the division equation
28 ÷ 7 = 4. Then solve the problem.
Possible problem: Bella collected 7 box tops each
day. She collected 28 box tops in all. For how
many days did Bella collect box tops? 4 days
Module 10 • Lesson 4 327
3
RtI Tier 1 Lesson 48
2
1
Enrich 48
Name
Name
LESSON
48
Enrich 48
1
Algebra • Relate Subtraction
and Division
OBJECTIVE Use base-ten blocks, repeated subtraction, and a number line to relate subtraction to division.
3.4.H,
3.4.K, 3.5.B
Jed works in a pet store that sells fish. He needs to move fish
from the old tanks and put them in the new tanks. He can
move the fish in small groups only.
Find 18 ÷ 6.
Use base-ten blocks.
Fish Tank Math
Use repeated subtraction.
Step 1 Start with
the number you are
dividing, 18.
Use repeated subtraction to solve each problem. Circle groups
of fish each time you subtract. Then write how many equal
groups Jed can make and how many fish are left over.
1.
Step 2 Subtract the
number you are
dividing by, 6.
19 4 6
18
6
_
Step 3 There are
more than 6 left.
Subtract 6 again.
12
- 6
_
18
- 6
_
6
12
2.
Students
should circle
4 groups of
5 fish.
3 groups and _
1 left over
4 groups and _
3 left over
_
Step 4 There are
6 left. Subtract 6
again.
18
- 6
_
12
- 6
_
6
- 6
_
12
6
0
3.
17 4 3
_
4.
15 4 4
Students
should circle
5 groups of
3 fish.
Step 5 Count the number of times you subtract 6.
© Houghton Mifflin Harcourt Publishing Company
23 4 5
Students
should circle
3 groups of
6 fish.
12
Students
should circle
3 groups of
4 fish.
You subtract 6 three times, so there are 3 groups of 6 in 18.
_
Write: 18 ÷ 6 = 3 or 6⟌ 18
3
1.
27
- 9
_
18
18
- 9
_
9
5.
9
- 9
_
0
3 groups and _
3 left over
5 groups and _
2 left over
_
Write a division equation.
2.
16
- 4
_
12
12
- 4
_
8
8
- 4
_
4
- 4
_
4
0
_
Stretch Your Thinking Choose one of the problems.
Change the number of fish in each group so there will not be
any fish left over. Explain why you chose that tank.
Change the number of fish in each group from 4 to 5 in Exercise 4.
Possible explanation: 15 fish can be divided into 3 equal groups or
27 ÷ 9 = 3
16 ÷ 4 = 4
5 equal groups. None of the fish in the other tanks can be divided
into equal groups.
327 Module 10
Number and Operations
95
Enrich
© Houghton Mifflin Harcourt Publishing Company
E48
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Daily Assessment Task
Fill in the bubble for the correct answer choice.
You may use models or strategies to solve.
8.
9.
8
C
6
B
4
D
2
NO
•
3
6
9
12
15
A
18 ÷ 2 = 9
C
2 × 9 = 18
B
18 ÷ 9 = 2
D
18 ÷ 6 = 3
18
3
C
10
B
4
D
7
Enrich 48
• Homework and Practice
•
Lesson 10.4
TEXAS Test Prep Coach
Multi-Step Tim has 30 grapes. He keeps 9 grapes for
himself. Then he gives 7 grapes each to some of his
friends. To how many friends does Tim give grapes?
A
Soar to Success Math
Warm-Up 13.10
YES
10.
THEN
IF
Representations What division equation is shown?
0
1
Can students explain how division is related to
subtraction?
Multi-Step There are 16 dolphins in a pod. Each pod
has the same number of males and females. The female
dolphins are swimming in pairs. How many pairs of
female dolphins are there?
A
2
Test Prep Coach helps teachers to identify common
errors that students can make.
In the Test Prep exercise, if students selected:
A They subtracted too many 6s.
C They subtracted only two 6s.
TEXAS Test Prep
D They incorrectly divided by 6.
Renee has 18 photos. She wants to put the same number
of photos on each of 6 pages in her photo album. How
many photos will be on each page?
A
4
C
2
B
3
D
5
328
© Houghton Mifflin Harcourt Publishing Company
11.
?
Essential Question
Write
Math
How is division related to subtraction? You can think of
division as repeated subtraction of the divisor.
Differentiated Centers Kit
Literature
Corey’s Cookie Caper
Students read about how Corey
and Carly divide cookies equally
among friends and family.
Activities
Dividing Nickels
Students complete blue Activity
Card 9 by using nickels to divide
by five.
Lesson 10.4 328
5
Ho mewo rk
and Practice
10.4
Number and Operations—3.4.H, 3.4.K
Algebraic Reasoning—3.5.B
MATHEMATICAL PROCESSES 3.1.D, 3.1.F
Name
Fill in the bubble completely to show your answer.
7.
Relate Subtraction and Division
Wendy draws this number line.
Write a division equation.
1.
14
–7
_
7
0
2.
7
–7
_
0
TEXAS Test Prep
Lesson
Lesson Check
Check
1
2
3
4
5
6
7
8
9
10
11
12
Which division equation does she show?
0
1
2
3
14 ÷ 7 = 2
4
5
6
7
8 9
A
12 ÷ 6 = 2
C
12 ÷ 3 = 4
B
2 × 6 = 12
D
12 ÷ 4 = 3
9÷3=3
8.
Tomas draws this number line.
Problem
Problem Solving
Solving
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Use the graph for 3–6.
4 bottles
4.
Malik and Kimi combine their bottles,
then sort an equal number of each
into 3 bins. How many bottles are in
each bin?
Which division equation does he show?
18
15
12
9.
9
6
3
0
Beth
Malik
Kimi
Student
9 bottles
5.
Kimi collects another 7 bottles and sorts an equal number
into 4 bins. How many bottles are in each bin?
11.
© Houghton Mifflin Harcourt Publishing Company
4 bottles
6.
All of the students want to combine their bottles and sort
an equal number into 6 bins. Can they do this? Explain.
No; the total number of bottles is 39 and 39 cannot be divided equally by 6.
Module 10 • Lesson 4
Homework and Practice
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
329-330
Module 10
329
330
A
16 ÷ 8 = 2
C
4 × 4 = 16
B
16 ÷ 2 = 8
D
16 ÷ 4 = 4
Kev bakes 18 mini-pies. There are
6 pies on each tray. How many trays
does Kev use?
A
12
B
6
C
4
D
3
Multi-Step Tara has 32 beads. She
takes out 4 red beads, then sorts the
rest into 4 equal groups. How many
beads are in each group?
10.
12.
Keesha has 14 shoes in all. How many
pairs of shoes does Keesha have?
A
7
B
2
C
28
D
14
Multi-Step Jeff has a booklet of
30 stickers. He uses one page of
stickers. If there are 6 stickers on each
page, how many pages were left?
A
7
C
8
A
4
C
5
B
4
D
9
B
24
D
6
© Houghton Mifflin Harcourt Publishing Company
Beth sorts an equal number of her
bottles into 3 bins. How many bottles
are in each bin?
Number of Bottles
3.
Bottles Collected
for Recycling