Objectives
To introduce the expression of remainders as
fractions or decimals; and to provide practice interpreting
remainders in division problems.
1
materials
Teaching the Lesson
Key Activities
Students express remainders in division problems as fractions that become part
of mixed-number answers or as decimals. They solve other division problems
in which the remainder is either rounded up or ignored.
Math Journal 1, pp. 148 and 149
Study Link 6 3
slate
13 sticks of gum (optional)
Key Concepts and Skills
• Use multiples to solve division problems. [Number and Numeration Goal 3]
• Solve division number stories and interpret remainders.
[Operations and Computation Goal 4]
• Use arrays to model division. [Operations and Computation Goal 7]
• Write number models to represent division number stories.
[Patterns, Functions, and Algebra Goal 2]
• Write number models containing grouping symbols.
[Patterns, Functions, and Algebra Goal 3]
Key Vocabulary mixed number
2
materials
Ongoing Learning & Practice
Students play Division Dash to practice dividing 2- or 3-digit dividends by
1-digit divisors.
Students practice and maintain skills through Math Boxes and Study Link activities.
Ongoing Assessment: Recognizing Student Achievement Use journal page 150.
[Operations and Computation Goal 2]
3
Students read A
Remainder of One,
create arrays and record
number models based
on the story.
ENRICHMENT
Students solve a division
number story by finding
multiples.
Game Master
(Math Masters,
p. 471)
per partnership:
4 each of number
cards 1–9
materials
Differentiation Options
READINESS
Math Journal 1,
p. 150
Student Reference
Book, p. 241
Study Link Master
(Math Masters,
p. 182)
EXTRA PRACTICE
Students practice solving
division problems.
Additional Information
Advance Preparation For the optional Readiness activity in Part 3, obtain the book
A Remainder of One by Elinor J. Pinczes (Houghton Mifflin, 2002).
Teaching Masters
(Math Masters,
pp. 183 and 184)
5-Minute Math,
pp. 20 and 25
A Remainder of
One
centimeter cubes;
counters (optional)
See Advance Preparation
Technology
Assessment Management System
Math Boxes, Problem 3
See the iTLG.
Lesson 6 4
419
Getting Started
Mental Math and Reflexes
Math Message
Pose division facts. Suggestions:
Three students share 13 sticks of gum. How many
sticks of gum does each student get if they receive equal shares?
12 6 = 2
16 4 = 4
30 6 = 5
70 7 = 10
21 3 = 7
32 4 = 8
27 3 = 9
24 6 = 4
72 9 = 8
42 7 = 6
36 4 = 9
63 9 = 7
students
sticks of gum per student
sticks of gum in all
3
?
13
Study Link 6 3 Follow-Up
Have students compare answers. Ask volunteers to
share different ways to solve the problems.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
Point out that 4 R1 is a correct answer to the Math Message
problem, but that this answer will not satisfy the three students
who want to know who gets the last piece of gum.
13 sticks of gum shared
equally by 3 students
13 / 3 = x
Ask students to draw a simple picture to organize the information
in the problem. Have several volunteers draw their pictures on
the board.
Ask: What do the quotient 4 and remainder 1 represent? Each
student can have 4 sticks of gum, and 1 stick will be left.
Ask: Should the 1 stick be ignored? No. That would be a waste.
The context of the problem indicates that the remainder should
be made part of the answer.
One way to do this is to divide the remainder among the
3 students. Draw a rectangle to represent one stick of gum
1
and divide it into thirds. Label each section 3.
1
3
13 sticks of gum.
3 students.
3 x = 13
1
3
1
3
ELL
Adjusting the Activity
Act out the problem using actual sticks of gum. Open the thirteenth stick
and break it into 3 equal pieces.
A U D I T O R Y
420
Unit 6 Division; Map Reference Frames; Measures of Angles
K I N E S T H E T I C
T A C T I L E
V I S U A L
1
Therefore, each student will receive 43 sticks of gum. Another
1
1
way to say this is 43 sticks of gum per student. Explain that 43 is
called a mixed number.
Students can check their answers by multiplying 3 4 and adding
the remainder of 1. (3 4) 1 13
Expressing Remainders
Links to the Future
In Unit 7 of Fourth Grade Everyday
Mathematics, students will continue their
exploration of mixed numbers in their work
with number lines, regions, and collections.
WHOLE-CLASS
ACTIVITY
as Fractions or Decimals
Tell students that in this lesson they will solve division number
stories in which something must be done with the remainder in
order to provide a useful answer. In the following examples, the
remainder is expressed as a fraction or decimal.
Ask students to draw a simple picture to organize the information
for each problem below.
Example 1: Four brothers are given 35 fruit bars. They agree to
share the bars equally. How many fruit bars will each boy get?
This is a division problem: 35 / 4 → 8 R3. If they each get 8 fruit
bars, 3 fruit bars would still need to be divided.
////\
///
////\
///
////\
///
////\
///
Sketch the 3 fruit bars on the board. Divide each into fourths to
represent the 4 boys sharing each bar.
3
If each boy takes one piece of each bar, he will have 8 4 fruit bars.
Some students will discover the shortcut for writing a remainder
as a fraction:
1. Make the remainder the numerator of the fraction.
2. Make the divisor the denominator of the fraction.
Then write the answer as a mixed number in which the remainder
is now expressed as a fraction.
In some problems, especially those involving money, it may be
preferable to change the fraction to an equivalent decimal.
Example 2: Four people split the cost of a $15 present equally.
What is each person’s share? 15 4 → 3 R3.
Using the remainder as the numerator of the fraction and the
3
divisor as the denominator leads to the answer $34, or $3.75.
Sketch a dollar bill on the board. Divide it into fourths to show
3
that 4 of $1.00 = $0.75.
///
35 fruit bars shared
equally by 4 brothers
35 / 4 = x
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
3
Each of the 4 brothers gets 4
of the remaining 3 bars.
25› 25› 25› 25›
$1.00
Lesson 6 4
421
Student Page
Date
Interpreting Remainders
Time
LESSON
Interpreting Remainders
6 4
in Problem Contexts
For each number story:
179
Draw a picture.
Write a number model.
Use a division algorithm to solve the problem.
Decide what to do about the remainder.
1. Jackson is buying balloons for a party.
Sample pictures:
2. Rosa is buying boxes to hold all 128 of her
Balloons cost $6 per bunch. How many
bunches can he buy with $75?
CDs. Each box holds 5 CDs. How many
boxes are needed to store all of her CDs?
Picture:
Picture:
5
5
5
5
5
3
$6 $6 $6 $6 $6
$6 $6 $6 $6 $6
$6 $6
Number model:
Answer:
12
75 6 ∑ 12 R3
5
5
5
5
5
Number model:
bunches
Answer:
26
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
128 5 ∑ 25 R3
What did you do about the remainder?
Circle the answer.
A. Ignored it
A. Ignored it
B. Reported it as a fraction or decimal
B. Reported it as a fraction or decimal
Why?
Remainder that is ignored
● Three children wish to divide a set of 16 toy cars equally. What
is each child’s share?
C. Rounded the answer up
Sample answer: The
Why?
The remainder in a division number story should not always
be converted to a fraction or decimal and retained as part of the
answer. Depending on the situation, the remainder might be
ignored because it is a leftover amount that cannot be split up
further. Or, it might indicate that the answer should be rounded
up. Discuss examples that illustrate these other situations. For
each problem, ask students to draw a simple picture to organize
the information.
boxes
What did you do about the remainder?
Circle the answer.
C. Rounded the answer up
WHOLE-CLASS
ACTIVITY
Sample answer: An
three dollars left isn’t enough
additional box was needed
to buy another bunch.
to store the remaining 3 CDs.
148
Math Journal 1, p. 148
Each child can have 5 cars, and there is 1 car left over. Unlike the
1 stick of gum in the Math Message problem that could be cut into
equal parts, the 1 remaining toy car cannot be divided up.
Therefore, the remainder is considered a “leftover” amount.
Remainder that is ignored
● Ann has $18 to buy notebooks that cost $4 each. How many
notebooks can she buy?
The division is 18 4 → 4 R2. Ann can buy 4 notebooks and will
have $2 left. The remainder can be ignored. Note that Ann will
have $2 left, but the answer to the question is 4 (notebooks).
Student Page
Date
Time
LESSON
Interpreting Remainders
6 4
3. Lateefah won 188 candy bars in a raffle.
She decided to share them equally with 7 of
her classmates and herself. How many
candy bars did each person receive?
Picture:
Sample picture:
8 equal groups
188
candy
bars
continued
Try This
88 R3
88 130
15 R12
15 1126
5. 10冄8
苶8
苶3
苶
6. 16冄2
苶5
苶2
苶
$2
$4
19.5
188 8 ∑ 23 R4
8. 183 12 15 R3
15.25
or 2312
9. 2,067 5 413 R2
413.4
candy bars
A. Ignored it
5
5
The division is 29 6 → 4 R5, or 46. Esteban needs 46 pages to
include all 29 photos. Four pages hold only 24 photos and are not
enough. Esteban must use a fifth page to hold the last 5 photos.
5
He must use 5 pages in all, which is 46 rounded up to the next
whole number.
B. Reported it as a fraction or decimal
C. Rounded the answer up
Sample answer: The
remaining 4 candy bars can
be cut into halves and shared
evenly among the 8 people.
149
Math Journal 1, p. 149
422
$4
Remainder that indicates the answer should be rounded up
● Esteban has 29 photographs. He can fit 6 photos on each page
of his photo album. How many pages must he use to hold
all 29 photos?
What did you do about the remainder?
Circle the answer.
Why?
$4
Write each answer as a decimal.
7. 39 2 19 R1
Number model:
2348
Answer:
$4
Write each answer as a mixed number by
rewriting the remainder as a fraction.
13 R1
13 12
4. 2冄2
苶7
苶
Unit 6 Division; Map Reference Frames; Measures of Angles
Student Page
Solving Division Problems
PARTNER
ACTIVITY
and Interpreting Remainders
Date
Time
LESSON
Math Boxes
6 4
1. Joe ordered 72 plants for his patio garden.
2. Solve each open sentence.
Each pot holds 4 plants. How many pots
are needed to hold all of the plants?
(Math Journal 1, pp. 148 and 149)
Encourage students to think about each remainder and the
different ways in which it should be interpreted as they complete
journal pages 148 and 149.
pots
plants per
pot
plants
in all
?
4
72
a. (6 9) (3 A) 30
A
b. 24 8 21 B
B
c. 72 (2 C) 9
C
d. 6.2 0.79 D
D
e. 8.91 E 2.72
E
5
7
4
6.99
6.19
72 / 4 18
18 pots
Number model:
Answer:
35–37
148
3. Use a paper-and-pencil algorithm to add or subtract.
a.
0.37
0.26
2.9
5.01
b.
0.63
c.
7.91
6.79
6.55
d.
0.24
7.80
3.65
4.15
2 Ongoing Learning & Practice
34–37
1
5. Circle the fractions equivalent to .
2
4. How many centimeters are in 12 meters?
Circle the best answer.
Playing Division Dash
PARTNER
ACTIVITY
A. 0.12
B. 1.2
(Student Reference Book, p. 241; Math Masters, p. 471)
8
16
5
6
6
12
2
3
12
24
8
15
C. 120
D. 1,200
Students play Division Dash to practice dividing 2- or 3-digit
dividends by 1-digit divisors.
129
51
150
Math Journal 1, p. 150
Adjusting the Activity
Use this game variation as appropriate:
Have students place only the 2 and 5 cards in the divisor pile.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
Math Boxes 6 4
V I S U A L
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 150)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 6-2. The skill in Problem 5
previews Unit 7 content.
Study Link Master
Name
Date
STUDY LINK
Interpreting Remainders
64
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 3
1.
Mrs. Patel brought a box of 124
strawberries to the party. She wants to
divide the strawberries evenly among
8 people. How many strawberries will
each person get?
Picture:
Use Math Boxes, Problem 3 to assess students’ ability to solve decimal
addition and subtraction problems. Students are making adequate progress if
they compute the correct sums and differences. Some students may be able to
explain how to use ballpark estimates or the relationship between addition and
subtraction to check their answers.
2.
Sample picture:
124 8 ∑ 15 R4
4
Answer:
Number model: 348 16 ∑ 21 R12
strawberries
Answer:
What did you do about the remainder?
Circle the answer.
Study Link 6 4
INDEPENDENT
ACTIVITY
21
groups
What did you do about the remainder?
Circle the answer.
A.
Ignored it
A.
Ignored it
B.
Reported it as a fraction or decimal
B.
Reported it as a fraction or decimal
C.
Rounded the answer up
C.
Rounded the answer up
Why?
Sample answer: You can
Why?
cut the remaining strawberries
(Math Masters, p. 182)
Sample picture:
1
158 or 152
179
16 16 16 16 16
16 16 16 16 16
16 16 16 16 16
16 16 16 16 16 16
15 15 15 15
Number model:
Mr. Chew has a box of 348
pens. He asks Maurice to
divide the pens into groups
of 16. How many groups
can Maurice make?
Picture:
15 15 15 15
[Operations and Computation Goal 2]
Time
Sample answer: There
aren’t enough remaining pens
to form another group of 16.
into halves.
Practice
Home Connection Students practice using a division
algorithm and interpreting the remainder.
3.
68 7 468
5. 9
9 R5
52
4.
6.
18 R2 74 4
31 R2
3冄苶9
苶5
苶
Math Masters, p. 182
Lesson 6 4
423
Teaching Master
Name
Date
LESSON
64
Time
A Remainder of One
Use 25 centimeter cubes to represent the 25 ants in the story A Remainder of One.
1.
Divide the cubes into 2 equal rows.
Draw what you did.
2.
3 Differentiation Options
Divide the cubes into 3 equal rows.
Draw what you did.
READINESS
Exploring Remainders
How many cubes are in each row?
12
How many cubes are in each row?
8
cubes
How many cubes are left over?
1
3.
How many cubes are left over?
1
cube(s)
Divide the cubes into 4 equal rows.
Draw what you did.
4.
How many cubes are in each row?
6
Divide the cubes into 5 equal rows.
Draw what you did.
5
cubes
How many cubes are left over?
0
cube(s)
15–30 Min
in Literature
(Math Masters, p. 183)
cube(s)
How many cubes are in each row?
cubes
How many cubes are left over?
1
cubes
SMALL-GROUP
ACTIVITY
Literature Link To explore the concept of remainders, have
students read and discuss the book A Remainder of One
by Elinor J. Pinczes (Houghton Mifflin, 2002). For each situation
in the story, ask students to create arrays using centimeter cubes
and record their work on Math Masters, page 183. Have students
describe the arrays and tell how they determined the number of
rows. Encourage the use of vocabulary from this unit.
cube(s)
ENRICHMENT
Math Masters, p. 183
Solving a Multiples
INDEPENDENT
ACTIVITY
5–15 Min
Number Story
(Math Masters, p. 184)
To apply students’ understanding of multiples, factors,
and division and remainders, have them solve a
marble-sharing number story. Encourage students to
model the problem with counters if necessary.
EXTRA PRACTICE
Name
LESSON
64
Date
Time
Multiples Number Story
Paolo has fewer than 40 marbles in a bag. If he splits up the marbles among his 4 friends,
he’ll have 3 left over.
5-Minute Math
23
How many marbles are in the bag?
2.
Show or explain how you got your answer.
marbles
MARBLES
Sample answer: The number of marbles has to be 2 more
than a multiple of 7 (9, 16, 23, 30, 37) because when Paolo
divides the marbles evenly among his 7 friends there are
2 left over. Only 23 gives a remainder of 3 when it is
divided by 4. 23 4 ∑ 5 R3; 23 7 ∑ 3 R2.
Math Masters, page 184
424
5–15 Min
To offer students more experience with division, see 5-Minute
Math, pages 20 and 25.
If he divides them among his 7 friends, he’ll have 2 left over.
1.
SMALL-GROUP
ACTIVITY
Unit 6 Division; Map Reference Frames; Measures of Angles