A Guide for Solving
Number Stories
Objectives To introduce a simplified approach to solving number
stories;
and to provide practice solving number stories.
s
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Teaching the Lesson
Key Concepts and Skills
• Solve addition and subtraction
number stories. [Operations and Computation Goal 2]
• Explain strategies for solving addition and
subtraction number stories. [Operations and Computation Goal 2]
• Use a table of air distance data. [Data and Chance Goal 2]
• Write number models to represent addition
and subtraction number stories. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Students learn and apply a four-step
approach that can be used to solve number
stories. They focus on addition and
subtraction number stories.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing High-Number Toss
Student Reference Book, p. 252
Math Masters, p. 487
per partnership: 1 six-sided die
Students practice place-value skills
and compare numbers.
Math Boxes 3 8
Math Journal 1, p. 68
Students practice and maintain skills
through Math Box problems.
Study Link 3 8
Math Masters, p. 91
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Reviewing Situation Diagrams
Student Reference Book, p. 178
Math Masters, p. 92
Students explore situation diagrams as visual
organizers for solving number stories.
ENRICHMENT
Writing Number Stories
Math Masters, p. 93
Students solve a number story and write an
additional one.
ELL SUPPORT
Building Vocabulary
chart paper
Students generate a list of words commonly
used in number stories that may indicate
an operation.
Use a Math Log or Exit Slip (Math
Masters, page 388 or 389). [Operations and Computation Goal 2]
Materials
Math Journal 1, pp. 66 and 67
Student Reference Book, p. 175
Study Link 37
Math Masters, p. 388 or 389 (optional);
pp. 422 and 423 (optional)
transparency of Math Masters, p. 422
(optional)
Advance Preparation
For Part 1, read page 175 of the Student Reference Book before class. You can enlarge Math Masters, page 422
to use as a poster of “A Guide for Solving Number Stories.”
Teacher’s Reference Manual, Grades 4–6 pp. 19, 104–107, 298–309
198
Unit 3
Multiplication and Division; Number Sentences and Algebra
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6
Getting Started
Content Standards
4.OA.3, 4.NBT.2, 4.MD.2
Mental Math and Reflexes
Math Message
Pose number stories. Suggestions:
J.T. traveled from Colorado
to Maine. He traveled 586 miles on
Monday, 487 miles on Tuesday,
630 miles on Wednesday, and 373 miles
on Thursday. How many total miles did
he travel? 2,076 miles
Anthony has saved $7 toward a ticket to a theme park. Tickets are $12.
How much more does he need? $5
Michael, Meredith, and 3 of their friends are going to the aquarium. Tickets cost
$2 each. What is the total cost? $10
Ben and Wilson built a model airplane. Wilson spent $3.50 for supplies, and Ben
spent $6.50. How much did all the supplies cost? $10
Alyssa runs 3 miles every day. About how many miles does she run in a month?
About 90 miles
Nander’s farm has three silos. One silo holds 28 bushels of corn, another holds
34 bushels, and the third holds 36 bushels. How many bushels of corn can
Nander store on his farm? 98 bushels
Hannah has $400. She spent $175 on groceries and $50 on clothes. How much
money does she have left? $175
Study Link 3 7
Follow-Up
Partners compare answers.
Note that an incorrect real
distance may be due to an initial
measurement error rather than an
error in the conversion.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Student Reference Book, p. 175; Math Masters, p. 422)
Briefly discuss “A Guide for Solving Number Stories” on Student
Reference Book, page 175. You may want to show a transparency or
a poster of Math Masters, page 422.
Use the problem in the Math Message to illustrate the steps.
1. Understand the problem.
Student Page
Ask students to explain what J.T. is doing. Taking a trip
Ask what the question in the problem is. How far did he
travel?
Ask what details are given about J.T.’s trip. Amounts he
traveled each day: 586, 487, 630, and 373
2. Plan what to do.
Ask for suggestions on how to solve the problem. Make a
table, draw a picture, make a list, make a diagram, and so on.
Ask for an estimate of the answer.
3. Carry out the plan.
Tell students to use an approach that makes sense to them
to solve the problem. Give them a few minutes to work.
Problem Solving
A Guide for Solving Number Stories
Learning to solve problems is the main reason for studying
mathematics. One way you learn to solve problems is by
solving number stories. A number story is a story with
a problem that can be solved with arithmetic.
1. Understand the problem.
♦ Read the problem. Can you retell it in your own words?
♦ What do you want to find out?
♦ What do you know?
♦ Do you have all the information needed to solve the problem?
2. Plan what to do.
♦ Is the problem like one that you solved before?
♦ Is there a pattern you can use?
♦ Can you draw a picture or a diagram?
♦ Can you write a number model or make a table?
♦ Can you use counters, base-10 blocks, or some other tool?
♦ Can you estimate the answer and check if you are right?
3. Carry out the plan.
♦ After you decide what to do, do it. Be careful.
♦ Make a written record of what you do.
♦ Answer the question.
A Guide for
Number Stories
1. Understand the problem.
2. Plan what to do.
3. Carry out the plan.
4. Look back.
Note
Understanding the
problem is a very
important step. Good
problem solvers take
time to make sure they
really understand
the problem.
Note
Sometimes it’s easy
to know what to do.
Other times you need
to be creative.
4. Look back.
♦ Does your answer make sense?
♦ Does your answer agree with your estimate?
♦ Can you write a number model for the problem?
♦ Can you solve the problem in another way?
Use the Guide for Solving Number Stories to help you solve the following problems.
Explain your thinking at each step. Also explain your answer(s).
1. Mr. Cline walked 2 miles per day for
2 days in a row, 3 miles per day for
the next 3 days, and so on, until he
walked 6 miles per day. On which
day did he first walk 6 miles?
2. Lisa cut a rectangle with a perimeter
of 24 inches into two squares.
What were the length and width of
the original rectangle?
Check your answers on page 346.
Student Reference Book, p. 175
Lesson 3 8
199
Student Page
Date
Time
LESSON
4. Look back.
Flying to Cairo
38
175–178
Pretend that you are flying from
Washington, D.C., to Cairo, Egypt.
You have a choice of flying by way of
Amsterdam, in the Netherlands, or by
way of Rome, Italy. The air distances
are shown in the table.
Ask students if their answers agree with their estimates.
Air Distance between Cities (in miles)
Amsterdam
Rome
Washington, D.C.
3,851
4,497
Cairo
2,035
1,326
Compare different approaches, being sure to emphasize that
the answers obtained from using different approaches should
agree. J.T. traveled 2,076 miles.
Solve each problem below. Record a number
model for the problem using a letter for the
unknown. Then write a summary number
model with your answer in place of the letter.
Ask students if their answers make sense. Ask how they can
tell. Sample answer: Yes. The total number of miles traveled
is more than any one day.
Sample number models are given.
1.
What is the total distance from
Washington, D.C., to Cairo by
way of Amsterdam?
5,886 miles
3,851 + 2,035 = m
Answer: About
Ask students how to write a number model for the problem
that uses a letter for the unknown information. This will
symbolically represent the number story. Sample answer:
586 + 487 + 630 + 373 = m
(number model with unknown)
3,851 + 2,035 = 5,886
(number model with answer)
2.
About how many more miles is it from
Washington, D.C., to Rome than from
Washington, D.C., to Amsterdam?
646 miles
4,497 – 3,851 = m
Answer: About
Ask students if their answers make the number model true.
(number model with unknown)
4,497 – 3,851 = 646
Write a summary number model on the board with the answer,
2,076, in place of m: 586 + 487 + 630 + 373 = 2,076.
(number model with answer)
Math Journal 1, p. 66
EM3MJ1_G4_U03_53-77.indd 66
1/6/11 10:43 AM
Pose another problem to the class: Amelia’s goal is to run 30 miles
every week. So far this week, she has gone for a 6-mile run, a 4-mile
run, a 7-mile run, and a 5-mile run. How much more does she need
to run to meet her goal?
Have students solve the problem in small groups. Ask:
●
What is the answer? 8 miles Have students share their solution
strategies.
●
Does the answer make sense? yes How can you tell? Sample
answer: I can estimate that she has run more than 20 miles, so
the answer should be less than 10.
Have volunteers suggest number models for the number story that
use a letter for the unknown.
Sample answer: 6 + 4 + 7 + 5 + m = 30
Student Page
Date
Flying to Cairo
38
3.
NOTE Some students may suggest 30 – 6 + 4 + 7 + 5 = m. Although students
Time
LESSON
will not work with parentheses until Lesson 3-10, acknowledge this number model
with the unknown and write it with parentheses: 30 - (6 + 4 + 7 + 5) = m.
Explain that the numbers inside the parentheses are added first, then the sum is
subtracted from 30 to find the missing part. Other students may suggest two
separate number models: 6 + 4 + 7 + 5 = 22, then 22 + m = 30. This is
acceptable.
continued
What is the total distance from Washington, D.C.,
to Cairo by way of Rome?
5,823 miles
4,497 + 1,326 = m
Answer: About
(number model with unknown)
4,497 + 1,326 = 5,823
(number model with answer)
4.
Ask students if their answers make the number model true.
About how many fewer miles will you fly
if you go by way of Rome rather than
Amsterdam?
Write a summary number model on the board with the answer, 8,
in place of m: 6 + 4 + 7 + 5 + 8 = 30.
63 miles
5,886 - 5,823 = m
Answer: About
(number model with unknown)
5,886 - 5,823 = 63
(number model with answer)
Try This
5.
If you fly first class, your ticket will cost $9,250. If
you fly economy class, you will save $6,500. You
bought 2 first class tickets and 4 economy tickets.
What is the total cost?
Answer:
$29,500
$9,250 - $6,500 = $2,750;
$9,250 º 2 + $2,750 º 4 = t
(number model(s) with unknown)
$9,250 – $6,500 = $2,750;
$9,250 º 2 + $2,750 º 4 = $29,500
(number model(s) with answer)
67
Math Journal 1, p. 67
EM3MJ1_G4_U03_53-77.indd 67
200
11/10/10 3:15 PM
Unit 3 Multiplication and Division; Number Sentences and Algebra
Student Page
Solving Number Stories
PARTNER
ACTIVITY
about Air Distances
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IN
NG
G
Date
Time
LESSON
Math Boxes
3 8
䉬
1.
Complete the name-collection box.
A number has
2.
Sample answers:
1
3
5
7
9
2
4
480
(Math Journal 1, pp. 66 and 67; Math Masters, p. 423)
60 ⫻ 8
400 ⫹ 80
⫺10 ⫹ 490
500 ⫺ 20
2,400 ⫼ 5
Encourage students to refer to “A Guide for Solving Number
Stories” as they work on journal pages 66 and 67. When most
students have finished, discuss their strategies.
the
the
the
the
the
the
the
tens place,
hundreds place,
ones place,
hundred-thousands place,
thousands place,
ten-thousands place, and
millions place.
Write the number:
4, 7 2 9, 3 1 5
149
3. a.
in
in
in
in
in
in
in
B
L
About
Ongoing Assessment:
Recognizing Student Achievement
Math Log or
Exit Slip
b.
4
Measure the line segment to the nearest inch.
5
in.
1
2
Measure the line segment to the nearest ᎏᎏ inch.
R
T
1
About
4ᎏ2ᎏ
in.
128
Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess
students’ ability to use and explain a strategy for solving an addition number
story. Have students explain how they used the table of air distances to solve
Problem 3 on journal page 67. Students are making adequate progress if their
explanations involve adding the distance between Washington, D.C., and Rome
to the distance between Rome and Cairo. Some students may demonstrate how
subtraction can be used to check their answers.
4.
Solve with a paper-and-pencil algorithm.
604
⫹ 817
a.
b.
1,421
5.
Sara estimated the length of her arm.
Circle the best estimate.
3,005
⫺ 686
A
5 centimeters
2,319
B
50 centimeters
C
100 centimeters
D
150 centimeters
130
10–15
[Operations and Computation Goal 2]
Math Journal 1, p. 68
2 Ongoing Learning & Practice
Playing High-Number Toss
Adjusting
the Activity
PARTNER
ACTIVITY
Have situation diagrams (Math Masters,
page 423) available for student use.
(Student Reference Book, p. 252; Math Masters, p. 487)
AUDITORY
Students play High-Number Toss to practice place-value skills
and comparing large numbers. See Lesson 2-7 for
additional information.
Math Boxes 3 8
Name
TACTILE
Date
STUDY LINK
VISUAL
Time
Addition and Subtraction Number Stories
38
Sample number models for Problems 1–4
1.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 3-6. The skill in Problem 5
previews Unit 4 content.
Writing/Reasoning Have students write a response to
the following: Explain why you chose the measurement
for Problem 5. Sample answer: Five centimeters is
about the length of my little finger. 100 centimeters is as long
as a meterstick. So I picked 50 centimeters because my arm is
close to half a meterstick in length.
KINESTHETIC
Study Link Master
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 68)
Study Link 3 8
Rhode Island, the smallest state in the United States, has an area of 1,545
square miles. The area of the second-smallest state, Delaware, is 2,489 square
miles. What is the combined area of these two states?
4,034
Answer:
square miles
153
1,545 + 2,489 = s
(number model with unknown)
1,545 + 2,489 = 4,034
(number model with answer)
2.
A polar bear can weigh as much as 700 kilograms. An American black bear can
weigh as much as 227 kilograms. How much more can a polar bear weigh than
an American black bear?
Answer:
473 kilograms more
700 - 227 = k
(number model with unknown)
=
700 227 473
(number model with answer)
3.
The Pacific leatherback turtle’s maximum weight is about 1,552 pounds. The
Atlantic leatherback turtle’s maximum weight is about 1,018 pounds. What is the
difference between the turtles’ maximum weights?
Answer:
534
pounds
1,552 - 1,018 = p
(number model with unknown)
1,552 - 1,018 = 534
(number model with answer)
4.
According to the National Register of Historic Places, New York City has
the largest number of historic places in the United States with 624 sites.
Philadelphia is second with 470 sites, and Washington, D.C., is third with 336
sites. How many historic sites are there in these three cities?
1,430 historic sites 624 + 470 + 336 = s
(number model with unknown)
624 + 470 + 336 = 1,430
Answer:
INDEPENDENT
ACTIVITY
(Math Masters, p. 91)
Home Connection Students solve addition and
subtraction number stories. Students write a number
model with an unknown and a summary number model.
(number model with answer)
Practice
5.
The numbers 81, 27, and 45 are multiples of
6.
List the first ten multiples of 6.
6
,
3 or 9 .
12 , 18 , 24 , 30 , 36 , 42 , 48 , 54 , 60
Math Masters, p. 91
EM3MM_G4_U03_072-105.indd 91
1/6/11 10:24 AM
Lesson 3 8
201
Teaching Master
Name
Date
LESSON
38
3 Differentiation Options
Time
Situation Diagrams
Situation diagrams can help you organize the information in a number story
and can help you decide what to do to solve the problem.
152 178
Decide which diagram to use for each problem. Complete the diagram.
Then solve the problem.
Sample answers:
1.
283 students attended the football game. 371 students attended the
soccer game. How many more students attended the soccer game?
Change
Start
Reviewing Situation
students
Quantity
End
371
283
?
(Student Reference Book, p. 178; Math Masters, p. 92)
Difference
Shawn had $145.00 in his bank account. He took out some money to buy
a new bike. Now he has $85.00 in his account. How much did his bike cost?
Change
Start
145
$60.00
To explore solving number stories using a visual organizer, have
students use situation diagrams to model number stories. Some
students will like the diagrams and use them often. Other
students will approach problems in other ways. Encourage
students to use a question mark to represent the missing
information.
Quantity
End
?
85
Quantity
Difference
3.
Aldo bought milk for 55 cents and a peanut butter and
jelly sandwich for $1.25. How much money did he spend?
$1.80
Change
Start
?
5–15 Min
Diagrams
Quantity
2.
INDEPENDENT
ACTIVITY
READINESS
88
Quantity
End
Quantity
$0.55 $1.25
INDEPENDENT
ACTIVITY
ENRICHMENT
Writing Number Stories
Difference
Math Masters, p. 92
EM3MM_G4_U03_072-105.indd 92
15–30 Min
(Math Masters, p. 93)
11/10/10 1:57 PM
ELL
Social Studies Link To apply students’ understanding of
solving number stories, have them read a short essay and
solve and write a number story based on data in the essay.
Remind students that often more information is provided than
is necessary to solve a problem.
To support English language learners, provide an opportunity
for students to share and revise their writing. For example:
Read problems aloud or have students read their own
problems aloud.
Have students read and comment on each other’s drafts.
Teaching Master
Name
Date
LESSON
Time
The Great Pyramid and the Gateway Arch
38
䉬
Great Pyramid of Giza
Building Vocabulary
Gateway Arch
Originally the Great Pyramid was about 147 meters tall. Over the years, stones were
removed from its surface, reducing its height by about 9 meters. The Great Pyramid was the
tallest structure in the world for 44 centuries, until the Eiffel Tower was built in Paris in 1889.
1.
How much taller is the Gateway Arch than the Great Pyramid?
About
54
meters
2.
Underline the information in the essay that you used to solve Problem 1.
3.
Use data from the essay above to write your own problem. Ask a partner
to solve it.
Answers vary.
Operations
+
Addition
●
●
●
●
●
●
altogether
added
got
total
in all
sum
Math Masters, p. 93
202
5–15 Min
To provide language support for number stories, have students
make and display a poster like the one shown below. With
the students, generate words and phrases that may help
determine which operation to use when solving a number story.
The Great Pyramid of Giza is perhaps the greatest structure ever built by humans. It stands
in the desert near Cairo, Egypt. Completed about 2580 B.C., it was the tomb of Pharaoh
(king) Khufu. It is estimated that 100,000 workers labored more than 20 years to build the
Great Pyramid. It contains more than 2 million stone blocks with an average weight of about
2ᎏ12ᎏ tons each.
Today the tallest monument in the United States is the Gateway Arch, also known as the
Jefferson National Expansion Memorial, in St. Louis, Missouri. The Gateway Arch was
completed in 1965 and is about 192 meters tall.
SMALL-GROUP
ACTIVITY
ELL SUPPORT
Unit 3 Multiplication and Division; Number Sentences and Algebra
Subtraction
●
●
●
●
●
left
more or less
difference
lost
shorter or
longer
∗
Multiplication
●
●
●
●
some number
of equal
groups
each has
times
by (with
dimensions)
÷
Division
●
●
●
●
share
How many in
each group?
How many
groups?
divide
Name
Date
STUDY LINK
Addition and Subtraction Number Stories
38
1.
Time
Rhode Island, the smallest state in the United States, has an area of 1,545
square miles. The area of the second-smallest state, Delaware, is 2,489 square
miles. What is the combined area of these two states?
Answer:
153
square miles
(number model with unknown)
(number model with answer)
2.
A polar bear can weigh as much as 700 kilograms. An American black bear can
weigh as much as 227 kilograms. How much more can a polar bear weigh than
an American black bear?
Answer:
kilograms more
(number model with unknown)
(number model with answer)
3.
The Pacific leatherback turtle’s maximum weight is about 1,552 pounds. The
Atlantic leatherback turtle’s maximum weight is about 1,018 pounds. What is the
difference between the turtles’ maximum weights?
Answer:
pounds
(number model with unknown)
(number model with answer)
Copyright © Wright Group/McGraw-Hill
4.
According to the National Register of Historic Places, New York City has
the largest number of historic places in the United States with 624 sites.
Philadelphia is second with 470 sites, and Washington, D.C., is third with 336
sites. How many historic sites are there in these three cities?
Answer:
historic sites
(number model with unknown)
(number model with answer)
Practice
5.
The numbers 81, 27, and 45 are multiples of
6.
List the first ten multiples of 6.
,
,
,
,
,
.
,
,
,
,
91
Name
LESSON
38
Date
Time
Situation Diagrams
Situation diagrams can help you organize the information in a number story
and can help you decide what to do to solve the problem.
152 178
Decide which diagram to use for each problem. Complete the diagram.
Then solve the problem.
1.
283 students attended the football game. 371 students attended the
soccer game. How many more students attended the soccer game?
Change
Start
students
Quantity
End
Quantity
Difference
2.
Shawn had $145.00 in his bank account. He took out some money to buy
a new bike. Now he has $85.00 in his account. How much did his bike cost?
Change
Start
Quantity
End
Quantity
3.
Aldo bought milk for 55 cents and a peanut butter and
jelly sandwich for $1.25. How much money did he spend?
Change
Start
Quantity
End
Quantity
Difference
92
Copyright © Wright Group/McGraw-Hill
Difference