Broken-Calculator
Division
Objective To guide children as they explore computational
strategies for division and interpret remainders.
s
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Interpret calculator displays for
remainders in equal-sharing and
equal-grouping problems. [Operations and Computation Goal 6]
• Use equal sharing to solve division
number stories. [Operations and Computation Goal 6]
Key Activities
Children divide numbers using the division
keys on a calculator and interpret the
calculator display. They devise ways of
dividing numbers using a calculator without
using the division key and solve division
number stories with remainders.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Roll to 100
Math Masters, p. 456
Student Reference Book, pp. 307
and 308
per group: 2 six-sided dice
Children practice mental computation.
Math Boxes 9 8
Math Journal 2, p. 225
Children practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problem 2. [Patterns, Functions, and Algebra Goal 3]
Home Link 9 8
Ongoing Assessment:
Informing Instruction See pages 756
and 757.
Math Masters, p. 290
Children practice and maintain skills
through Home Link activities.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Picturing Division
Math Masters, p. 291
Children use a visual model to explore
equal-sharing and equal-grouping problems.
ENRICHMENT
Solving Division Number Stories
Math Masters, p. 292
Children solve division number stories
and express the remainders as fractions.
ELL SUPPORT
Using Calculators to Solve
Division Problems
calculator
Children discuss calculator keys they
press to solve division problems.
Materials
Math Journal 2, pp. 222 and 224
Home Link 97
calculator slate half-sheet of paper play money (optional) counters (optional)
Advance Preparation
Teacher’s Reference Manual, Grades 1– 3 pp. 23–29
754
Unit 9
Multiplication and Division
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Getting Started
Mental Math and Reflexes
Math Message
Children record answers to fraction number stories on
their slates. Encourage children to draw or use
counters as needed. Suggestions:
Solve Problems 3 through 6 on journal page 222
using your calculator. On a half-sheet of paper, write the
answers the calculator displays. Compare with your
answers on the journal page.
of the books were
Troy read 8 books over the summer. _
8
7
mysteries. How many mysteries did Troy read? 7 mysteries
Home Link 9 7 Follow-Up
of them were dimes. _
of them
Perry saved 20 coins. _
5
4
2
1
of the coins were nickels. The rest were
were quarters. _
10
3
pennies. How many of each coin did Perry save? 8 dimes,
Briefly go over the answers. Have children share
strategies for solving Problem 1.
5 quarters, 6 nickels and 1 penny
The guests at Tory and Marissa’s birthday party ate
10 slices of pie. If each pie had six slices, what fraction
, or 1_
, or 1_
pies
of pies was eaten? _
6
6
3
10
4
2
1 Teaching the Lesson
NOTE Some calculators have a key that
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 222)
In discussing calculator answers to Problems 3 through 6 on
journal page 222, point out that the calculator display must be
interpreted to fit the situation. For example, the calculator display
for 75 ÷ 6 is 12.5 (a number with one decimal place), but the
answer to $75 ÷ 6 is $12.50 (a number with two decimal places).
gives the answer to a division problem as a
whole number and a remainder. If this key
is used, the answer to 75 ÷ 6 is 12 with a
remainder of 3. When the calculator display is
interpreted to fit the situation, the remainder
of 3 represents $3. $75 ÷ 6 is $12 with $3 left
over. $3 can be evenly divided by 6, resulting
in $0.50. Another answer to $75 ÷ 6 is $12.50.
Student Page
Adjusting the Activity
ELL
Date
Time
LESSON
When using calculators, it is important to discuss the meaning of each
calculator entry. Ask questions such as: What does [number] stand for? What
does [number] represent? Why are you dividing? What does the answer
represent? What does the answer mean? What are the units?
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
97
1.
V I S U A L
If $54 is shared equally by 3 people, how much does each person get?
Number model:
b.
How many $10 bills does each person get?
c.
How many dollars are left to share? $
e.
●
1
24.00
8
How many $1 bills does each person get?
Answer: Each person gets $ 18.00 .
$10 bill(s)
$1 bill(s)
2. If $71 is shared equally by 5 people, how much does each person get?
A farmer wants to pack 246 eggs into egg cartons that hold a
dozen eggs each. How many full cartons will she have?
Number model:
b.
How many $10 bills does each person get?
c.
How many dollars are left to share? $
e.
3.
5.
$71 ÷ 5 = ?
a.
d.
Remind children that they must divide 246 by 12 to find how
many 12s there are in 246. When this is done on a calculator, the
display shows 20.5. Ask: In this problem, what does 20.5 stand
for? Cartons of eggs Is 20.5 cartons the answer to the problem? No.
20.5 is between 20 and 21, so there are 20 full cartons with some
eggs left over. Some children assume that the 5 after the decimal
point is the remainder or the number of eggs left over. Clarify that
this is not the case.
$54 ÷ 3 = ?
a.
d.
Ask children to solve the following problem with their calculators:
Sharing Money
Work with a partner. Put your play money in a bank for both of you to use.
1
21.00
4
How many $1 bills does each person get?
1
How many $1 bills are left over?
$1 bill(s)
f.
If the leftover $1 bill(s) are shared equally,
how many cents does each person get? $
g.
Answer: Each person gets $
$84 ÷ 3 = $
28.00
$181 ÷ 4 = $
45.25
14.20
$10 bill(s)
$1 bill(s)
0.20
.
4. $75 ÷ 6 = $
12.50
6. $617 ÷ 5 = $
123.40
Math Journal 2, p. 222
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Adjusting the Activity
Pose the following question: How many eggs are left over? 6 Possible
strategies: 0.5 is another name for _12 , and a half-full carton of eggs contains 6
eggs. 20 full cartons of eggs contain 20 × 12 = 240 eggs. That leaves 6
unpacked eggs (246 - 240 = 6).
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
V I S U A L
As time permits, pose and discuss additional problems in which
children must interpret the calculator display. Suggestions:
●
There are 263 pencils. A box holds 50 pencils. How many full
boxes of pencils are there? 5 boxes
●
A bus holds 36 people. 155 people are going on a field trip. How
many buses are needed? 5 buses
Ongoing Assessment: Informing Instruction
For the bus problem, the calculator display will show 4.3055555. Watch for
children who think this is a big number. Direct their attention to the decimal point
and the whole number to the left of the decimal point.
Exploring Strategies for
SMALL-GROUP
ACTIVITY
Finding Quotients
Ask children to pretend that the division key on each calculator is
broken. How would they use their broken calculators to solve the
following problem?
●
A farmer packs 576 eggs into cartons that hold a dozen eggs
each. How many full cartons does she pack?
Ask each group to write a brief report describing the strategies
they used to solve the problem. Bring the class together to share
strategies. Possible strategies:
Clear the calculator and enter 576. Subtract 12 over and over
until the display shows a number less than 12 (in this case, 0).
Keep a tally of the number of times 12 is subtracted. This tally
gives the number of full cartons. 48
Clear the calculator and enter 576. Subtract 120 (the number
of eggs in 10 full cartons) over and over until the display shows
a number less than 120 (in this case, 96). Keep a tally of the
number of times 120 is subtracted. This tally gives the number
of tens of cartons. 4 tens = 40 Subtract 12 from the number in
the display until the display shows a number less than 12. This
gives the number of additional full cartons. 8. Add 40 + 8 to get
48 cartons.
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Student Page
Use repeated estimates for the number multiplied by 12 to get
576. The repeated-estimates strategy is often referred to as
guess-and-check. For example:
Date
9 8
䉬
25 × 12 = 300—too small
1.
Ruth is buying soda for a party. There are
6 cans in a pack. She needs 44 cans.
How many 6-packs will she buy?
6-packs
40 × 12 = 480—too small
2.
50 × 12 = 600—too large, but close
Paul is buying tickets to the circus.
Each ticket costs $7. He has $47.
How many tickets can he buy?
3.
Héctor is standing in line for the roller coaster.
There are 33 people in line.
Each roller coaster car holds 4 people.
How many cars are needed to hold 33 people?
8
Pose additional broken-calculator problems as necessary for the
groups to solve. Suggestions:
4.
A baker packs 315 hamburger buns into packages of 8. How
many full packages does he have? 39
How many leftover buns? 3
The cafeteria manager plans to serve 78 cartons of yogurt for
lunch. The cartons come in packages of 6. How many 6-carton
packages must be purchased? 13
6
tickets
9
cars
Pretend that the division key on your calculator is broken.
Solve the following problems:
Regina is building a fence around her dollhouse.
She is making each fence post 5 inches tall.
The wood she bought is 36 inches long.
How many fence posts does each piece of wood make?
7
posts
Explain how you found your answer.
Sample answer: I subtracted 10 (2 posts) as many times
as I could. Then I subtracted 5 as many times as I could.
5.
●
Division with Remainders
Solve the problems below. Remember that you will have to decide
what the remainder means in order to answer the questions.
You may use your calculator, counters, play money, or pictures.
48 × 12 = 576—right on target!
●
Time
LESSON
Missy, Ann, and Herman found a $10 bill.
They want to share the money equally.
How much money will each person get?
$3.33
Sample answer:
I subtracted 3 from 10 three times. That left 1. or $1.00.
I subtracted .30 from 1. three times. That left .1 or $0.10.
I subtracted .03 from .1 three times. That left .01 or $.01.
Explain how you found your answer.
Math Journal 2, p. 224
Solving Division Number
PARTNER
ACTIVITY
Stories with Remainders
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SOLVING
SO
S
OL
O
LV
VIN
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G
(Math Journal 2, p. 224)
Partners use their calculators to solve the division problems on
journal page 224. Explain to children that they can use the
division key on Problems 1 through 3, but they will pretend it
is broken in Problems 4 and 5. When children finish, have
volunteers explain how they interpreted the calculator display
for each problem and how they solved Problems 4 and 5 without
using the division key.
Adjusting the Activity
ELL
When you introduce the idea of the broken calculator to English
language learners, it should be clear that you are pretending it is broken.
Explain that children will be asked to solve some problems without using
certain calculator keys.
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
V I S U A L
A picture of Problem 1 on journal
page 224
Ongoing Assessment: Informing Instruction
Watch for children who need support in implementing a strategy for solving the
division problems. Encourage them to draw pictures to illustrate the problems.
(See margin.)
7 [6s] = 42
Links to the Future
Ruth needs 2 more cans to make 44.
She must buy eight 6-packs.
The activities in this lesson are an early exposure to interpreting remainders.
Some children will need more practice before they develop a full understanding
of division concepts. Expressing the remainder as a whole number or fraction
appropriate to the context of the problem is a Grade 5 Goal.
Lesson 9 8
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Student Page
Date
Time
LESSON
Kevin and Naomi read to each
other for 35 minutes each day.
About how many hours do they
read to each other in one week?
2.
4 hours
Answer: About
(unit)
Fill in the circle for the best answer.
5 × (6 – 5) =
A
4
B
5
Playing Roll to 100
C 16
D 25
3.
3
Write 5 names for _
4.
4.
SMALL-GROUP
ACTIVITY
(Math Masters, p. 456; Student Reference Book,
pp. 307 and 308)
16
Use bills and coins.
Share $78 equally among 3 people.
3
_
4
Sample answers:
three-fourths
9
_
‰‰
12 _
6
‰‰
8
Children play Roll to 100 to practice mental addition.
For directions, see Lesson 2-1 or Student Reference Book,
pages 307 and 308.
$78 ÷ 3 = ?
or 3 × ? = $78
Each person gets $ 26 .
Number model:
27–30
5.
2 Ongoing Learning & Practice
Math Boxes
98
1.
73
Math Boxes 9 8
3
You and a friend are playing a game 6. Draw a line segment 1_
4 inches long.
with the spinner. You win if the
spinner lands on purple. Your friend
1
wins if the spinner lands on black.
Draw a line segment _
2 inch longer
Do you think this game is fair?
than the one you just drew.
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 225)
yes
black
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 9-6. The skill in Problem 6
previews Unit 10 content.
no
purple
143 144
Math Journal 2, p. 225
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Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 2
Use Math Boxes, Problem 2 to assess children’s progress toward understanding
that parentheses affect the order of operations. Children are making adequate
progress if they successfully complete Problem 2. Some children may be able to
write and solve their own number sentences with parentheses.
[Patterns, Functions, and Algebra Goal 3]
Home Link 9 8
INDEPENDENT
ACTIVITY
(Math Masters, p. 290)
Home Link Master
Name
Date
Home Connection Children solve number stories about
dividing quantities into equal parts and interpreting
remainders.
Time
Equal Shares and Equal Parts
HOME LINK
98
Family
Note
As the class continues to investigate division, we are looking at remainders and what they
mean. The focus of this assignment is on figuring out what to do with the remainder, NOT
on using a division algorithm. Encourage your child to draw pictures, use a calculator,
or use counters to solve the problems.
73 74
Please return this Home Link to school tomorrow.
Solve the problems below. Remember that you will have to decide what
the remainder means in order to answer the questions. You may use
your calculator, counters, or pictures to help you solve the problems.
3 Differentiation Options
1. There are 31 children in Dante’s class.
Each table in the classroom seats
4 children. How many tables are needed
to seat all of the children?
8 tables
READINESS
2. Emily and Linnea help out on their uncle’s
Picturing Division
chicken farm. One day the hens laid a total
of 85 eggs. How many cartons of a dozen
eggs could they fill?
7 cartons
project. She needs 93 markers. If markers
come in packs of 10, how many packs must
she buy?
To explore equal-sharing and equal-grouping problems
using a visual model, have children illustrate the solution
to division problems. Children record their work on Math
Masters, page 291.
10 packs
Practice
Solve each problem using the partial-products algorithm.
Use the back of this Home Link.
116
5. 85 × 5 =
425
5–15 Min
(Math Masters, p. 291)
3. Ms. Jerome is buying markers for a scout
4. 29 × 4 =
INDEPENDENT
ACTIVITY
6. 96 × 8 =
768
Math Masters, p. 290
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Unit 9 Multiplication and Division
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Teaching Master
PARTNER
ACTIVITY
ENRICHMENT
Solving Division
Number Stories
(Math Masters, p. 292)
5–15 Min
Name
Date
LESSON
98
Time
Picturing Division
For each problem—
Draw a picture.
Answer the question.
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SOLVING
SO
S
OL
O
LV
L
VIN
V
IIN
NG
Explain what you did with what was left over.
1. There are 18 children in art class. If 4 children can sit at each table,
how many tables do they need?
Picture:
To apply children’s understanding of remainders, have them solve
division problems and express the remainders as fractions on
Math Masters, page 292. Have children share solution strategies.
For Problem 2, children might draw 4 pizzas and divide each into
8 equal slices. They would shade 3 of the pizzas (for 24 slices)
and one of the slices in the fourth pizza (for a total of 25 shaded
slices). Point out that to serve everyone, the class will need to
order 4 pizzas.
Answer: They need
5
tables.
Sample answer: There are 4 children at each of
the 4 tables. 2 children are left over, so they need 5 tables.
Explanation:
2. Hot dogs come in packages of 8. If José is having a birthday party and
needs 20 hot dogs, how many packages must he buy?
Picture:
Answer: He must buy
3
packages.
Explanation: Sample answer: 2 packages have 16 hot dogs—
8 × 2 = 16. 3 packages have 24 hot dogs—8 × 3 = 24.
José must buy 3 packages of hot dogs.
Math Masters, p. 291
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4 pizzas are needed to serve 25 people.
ELL SUPPORT
Using Calculators to Solve
SMALL-GROUP
ACTIVITY
5–15 Min
Division Problems
Teaching Master
Name
LESSON
98
To provide support for solving division problems with a calculator,
discuss the keys that are pressed and the resulting displays. For
example, pose the following problem: A bus holds 36 people.
162 people are going on a field trip. How many buses are needed?
After the children have solved the problem, ask the following
questions:
●
●
●
What keys did you press on the calculator to solve the problem?
162
36
Date
Time
Pizza with Remainders
The third-grade class is having a pizza party. The class
expects 22 children, 1 teacher, and 2 parents. Each
pizza will be divided into 8 equal slices.
1. In all, how many people are coming to the party?
25 people
2. Suppose that each person who comes to the party
will eat 1 slice of pizza.
3 pizzas
1 slice
_1
a. How many whole pizzas will the people eat?
b. How many additional slices will be needed?
8
c. What fractional part of a whole pizza is that?
Less than
d. Is that more or less than half of a whole pizza?
e. How many whole pizzas should the teacher order?
_1
2
4 whole pizzas
3. Suppose instead that each person will eat 2 slices of pizza.
What number did you see in the display? 4.5
What does the 4 represent? The number of buses carrying
36 people
●
How many people will the 4 buses hold? 36 × 4 = 144
●
How many more people will need a seat? 162 - 144 = 18
●
Where will these 18 sit? On the fifth bus
a. How many slices of pizza will the people eat?
b. How many whole pizzas will the people eat?
c. How many additional slices will be needed?
50 slices
6 whole pizzas
2 extra slices
_2
8
d. What fractional part of a whole pizza is that?
e. How many whole pizzas should the teacher order?
7 pizzas
4. Lakeisha brought 2 granola bars to the party.
She decided to share them equally with her
3 best friends. What fractional part of a granola
bar did she and her friends get?
_1
2
Math Masters, p. 292
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