Comparison
Number Stories
Objective To introduce comparison number stories by using
comparison diagrams.
c
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Share number story solution strategies. [Operations and Computation Goal 2]
• Describe and solve comparison number
stories. [Operations and Computation Goal 4]
• Write number models to summarize
addition and subtraction number stories. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Addition Top-It
My Reference Book, pp. 122 and 123
Math Masters, pp. 449 and 473
(optional)
per partnership: 4 each of number
cards 0–20 (from the Everything Math
Deck, if available)
Children practice adding three or
more numbers.
Math Boxes 6 2
Children are introduced to the comparison
diagram as a tool for recording known and
missing information in a comparison number
story. Children then solve comparison
number stories by using comparison
diagrams and writing number models.
Ongoing Assessment:
Informing Instruction See page 387.
Math Journal 1, p. 136
Children practice and maintain skills
through Math Box problems.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Doing a Penny Grab Activity
30 pennies or other small counters
Children compare numbers using a
concrete model.
ENRICHMENT
Comparing Number Stories
Math Masters, p. 161
Children compare two number stories by
writing number models and answering
related questions.
Home Link 6 2
Math Masters, pp. 159 and 160
Children practice and maintain skills
through Home Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 134. [Operations and Computation Goal 4]
Key Vocabulary
comparison number story difference comparison diagram
Materials
Math Journal 1, pp. 134 and 135
My Reference Book, pp. 110 and 111
Home Link 61
transparency of Math Masters, p. 436
(optional) number line number grid
(optional)
Advance Preparation
For the Solving Comparison Number Stories activity in Part 1, decide how you will display a comparison
diagram. Some possibilities:
• Make an overhead transparency of Math Masters, page 436.
• Draw and erase comparison diagrams on the board as needed.
Teacher’s Reference Manual, Grades 1–3 pp. 84–94, 225–227
384
Unit 6
Whole-Number Operations and Number Stories
384_EMCS_T_TLG1_U06_L02_550524.indd 384
3/4/11 2:19 PM
Getting Started
Mental Math and Reflexes
Math Message
Write multiple-addend problems like the following on
the board. Encourage children to look for combinations
that will make the addition easier.
Fish A is 14 inches long. Fish B is 6 inches long.
3 + 9 + 7 = ? 19
14 + 8 + 6 = ? 28
6 + 4 + 8 + 2 = ? 20
? = 21 + 5 + 9 35
34 + 6 + 7 = ? 47
57 + 3 + 10 + 5 = ? 75
? = 8 + 5 + 12 + 5 30
22 + 28 + 8 + 3 = ? 61
63 + 27 + 9 = ? 99
How many more inches long is Fish A than Fish B?
8 inches
Home Link 6 1 Follow-Up
Ask volunteers to share how they found the
answer to each problem and why they chose the
order in which they added the three numbers.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Masters, p. 436)
Algebraic Thinking Use the Math Message problem to start a
discussion of comparison number stories. Comparison stories
involve the difference between two quantities. The Math Message
number story compares two things that are measured.
You can also compare things that are counted.
Example:
Beth scored 14 points. Ivy scored 8 points. So Beth scored 6
more points than Ivy. (Alternately, Ivy scored 6 fewer points
than Beth.) The difference in points scored is 6 points.
Draw a picture on the board and show how children can solve the
Math Message problem by matching the two quantities one to one.
(See margin.) Discuss the meaning of the word quantity in the
diagram. The quantity that is left unmatched is the difference. In
the Math Message problem, the difference tells how many more
inches Fish A is than Fish B.
Display a transparency of a comparison diagram (Math Masters,
page 436) or draw one on the board. In it, write the numbers 14, 6,
and 8. (See margin.) Say that the diagram is a convenient way to
represent the fish comparison story. The longer Quantity cell
shows the larger number of inches of Fish A. The shorter Quantity
cell shows the smaller number of inches of Fish B. The Difference
cell shows how many more inches Fish A is than Fish B.
NOTE Point out that the quantity box on the
top is as long as the quantity and difference
boxes on the bottom. This often provides a
good visual for children.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
difference
Note: This ruler is not to scale.
Quantity
Fish A—14 inches
Quantity
Fish B—
6 inches
8
Difference
Lesson 6 2
385-389_EMCS_T_TLG1_U06_L02_550524.indd 385
385
3/4/11 2:20 PM
Adjusting the Activity
Whenever a comparison diagram is
provided, children should always write the
known and missing information (shown with
a question mark) on the diagram. Have
children write words or short phrases on the
diagram as a reminder of what the numbers
mean. For the Math Message problem, the
words “Fish A” and “Fish B” might be written
as reminders.
AUDITORY
KINESTHETIC
TACTILE
VISUAL
Write a number model that represents the problem using a
question mark to represent what children have to find out to solve
the problem (or what they don’t know). Tell them that the number
story might be modeled using any of four possible number models.
The choice of a number model depends on how a child thinks
about the problem. Write the remaining number models below the
diagram (in all, 14 – 6 = ?, 14 – ? = 8, ? + 6 = 14, and 6 + ? = 14).
Then, below the open sentences, write number models for the
solution to the problem with 8 substituted for the question mark.
Solving Comparison
Number Stories
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
(My Reference Book, pp. 110 and 111)
Algebraic Thinking You may wish to read about comparison
diagrams in My Reference Book, pages 110 and 111 with your
class. Display a comparison diagram. Work with the children to
solve several comparison stories.
Quantity
30 Joey
Example 1
Quantity
10 Max
?
Difference
Joey scored 30 points. Max scored 10 points. How many more
points did Joey score than Max? 20 points
Fill in the comparison diagram as shown in the margin. Write ?
for the difference, which is the number to be found. Invite
children to share mental-arithmetic strategies for finding the
difference between 30 and 10. Sample strategies:
Think, “What must I add to 10 to get 30?”
Think of the comparison diagram as a Fact Triangle. Think,
“30 - 10 is the difference I want.”
As children share their solution strategies, write a number model
below the diagram to represent one of the strategies using ? for
the unknown number. Then, below the open sentence, write a
number model for the solution to the problem with 20 substituted
for the question mark. For example, for the first strategy, write
10 + ? = 30 and 10 + 20 = 30.
Quantity
Example 2
$47 radio
A radio costs $47. A watch costs $20. How much more does the
radio cost? $27
Quantity
$20 watch
?
Difference
As children share their solution strategies, write a number model
below the diagram to represent one of the strategies using ? for the
unknown number. Then, below the open sentence, write a number
model for the solution to the problem with $27 substituted for the
question mark.
Quantity
$47 radio
Quantity
Example 3
? calculator
$12
Difference
386
This problem is similar to the previous example. With the class, fill
in the comparison diagram as shown in the margin. Invite children
to share their strategies for solving the problem.
A radio costs $47. A calculator costs $12 less than the radio. How
much does the calculator cost? $35 Fill in the diagram as shown in
the margin.
Unit 6 Whole-Number Operations and Number Stories
385-389_EMCS_T_TLG1_U06_L02_550524.indd 386
3/4/11 2:20 PM
Student Page
This example differs from the previous ones—this time, the
smaller quantity is not known, but the difference is known. Make
sure that children understand this. The strategies for solving this
problem are the same as before, except that counting up is
replaced by counting back (start at 47, count back 12).
Date
Time
LESSON
Comparison Number Stories
62
For each number story:
Write the numbers you know in the comparison diagram.
Write ? for the number you want to find.
Solve the problem.
Write a number model.
Number models vary.
1. Barb scored 27 points.
Solving Comparison
PARTNER
ACTIVITY
Number Stories
Barb scored
than Cindy.
PROBLEM
PR
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SOLVING
SO
S
OL
O
LV
VIIIN
VIN
NG
N
G
(Math Journal 1, pp. 134 and 135)
Quantity
Cindy scored 10 points.
17
Number model:
27
more points
Quantity
10
27 - 10 = 17
2. Frisky lives on the 16th floor.
?
Difference
Quantity
Fido lives on the 7th floor.
Algebraic Thinking Partners complete the problems on the
journal pages. Check that children are recording the known
information in the comparison diagram and that they are writing
a question mark to represent the unknown number. For most
problems, the difference will be the unknown number. For some
problems, however, the difference will be known, and one of the
two quantities will be unknown.
Frisky lives
than Fido.
9
16
floors higher
Quantity
7
16 - 7 = 9
Number model:
3. Ida is 36 years old. Bob is 20 years old.
16
Ida is
?
Difference
Quantity
36
years older than Bob.
Number model:
36 - 20 = 16
Quantity
20
?
Difference
Math Journal 1, p. 134
Ongoing Assessment:
Recognizing Student Achievement
Journal page 134
Problems 1
and 2
EM3MJ1_G2_U06_131_158.indd 134
1/29/11 9:43 AM
Use journal page 134, Problems 1 and 2 to assess children’s ability to solve
comparison number stories. Children are making adequate progress if they can
solve the problems using the number grid, number line, drawings, or manipulatives.
Some children may be able to write a number model that summarizes the problem.
Others may be able to do Problems 5, 6, and 7 with an unknown quantity.
[Operations and Computation Goal 4]
Ongoing Assessment: Informing Instruction
Watch for children who have difficulty writing a number model; have them think
of 17, 8, and 9 as the numbers on a Fact Triangle. Ask them what addition and
subtraction sentences they can make using the three numbers.
Student Page
Date
Time
LESSON
Comparison Number Stories
62
continued
4. A jacket costs $75.
Children may use number lines, number grids, or any other
learning tool. It is fine for children to draw pictures and doodles.
Quantity
Pants cost $20.
The pants cost $
than the jacket.
Number model:
55
75
less
Quantity
75 - 20 = 55
20
?
Difference
Try This
5. Jack scored 13 points. Jack scored 6
2 Ongoing Learning & Practice
PARTNER
ACTIVITY
(My Reference Book, pp. 122 and 123;
Math Masters, p. 449)
7
Eli scored
13
points.
Quantity
7 + 6 = 13
13 - 6 = 7
Number model:
or
Playing Addition Top-It
Quantity
more points than Eli.
?
6. Billy is 16 years old. Paul is
Quantity
6 years younger than Billy.
Paul is
10
16
years old.
Quantity
6 + 10 = 16
=
16 - 6 10
Number model:
or
?
7. Marcie is 56 inches tall.
Addition Top-It was introduced in Lesson 1-4 to practice facts and
compare sums. A domino version was played in Lesson 2-5. An
optional record sheet can be found on Math Masters, page 449.
14
70
inches shorter than Nick.
56 + 14 = 70
70 - 56 = 14
Number model:
or
6
Difference
Quantity
Nick is 70 inches tall.
Marcie is
6
Difference
Quantity
56
?
Difference
Math Journal 1, p. 135
EM3MJ1_G2_U06_131_158.indd 135
1/28/11 4:41 PM
Lesson 6 2
385-389_EMCS_T_TLG1_U06_L02_550524.indd 387
387
1/30/11 11:04 AM
Student Page
Date
Time
LESSON
6 2
䉬
Math Boxes
1. Fill in
⫹2
the
frames.
17
19
14
16
10
8
13
11
3. Write ⬍, ⬎, or ⫽.
11 ⫹ 6 ⫹ 4
⬎
⬍
32
19
show 30¢ using ‰, Í, Â.
‰Â
ÍÍÍ
ÍÍÂÂ
ÍÂÂÂÂ
ÂÂÂÂÂÂ
NOTE Some children may prefer to play Three Addends from Lesson 6-1 again.
Math Masters, page 473 is available for children to record their numbers and
number models.
98 99
4. Draw a line segment that is
Unit
⬎
⬍
7 ⫹ 5 ⫹ 30
2. Draw all the possible ways to
⫺5
Children now play another version with number cards 0 through
20. They draw 3 or 4 cards at each turn to practice addition with
3 or 4 numbers.
about 5 cm long.
Math Boxes 6 2
40
(Math Journal 1, p. 136)
26
18 ⫹ 7 ⫹ 2
13 ⫹ 9 ⫹ 1
6. Halve.
2
12
1
4
2
9
3
8
8
4
7
7
6
:
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 6-4. The skill in Problem 6
previews Unit 7 content.
9
5. Write the time.
11
10
INDEPENDENT
ACTIVITY
1
2
4
5
7
25
10
14
50
Writing/Reasoning Have children draw, write, or
verbalize their answers to the following: Explain how you
know you have written all the possible coin combinations
for 30¢ in Problem 2. Sample answer: I started with a quarter and
a nickel. I then used 3 dimes to make 30¢ and then I kept
breaking the dimes into 2 nickels until I used all nickels. This is
the smallest coin I could use.
5
25
Math Journal 1, p. 136
Home Link 6 2
INDEPENDENT
ACTIVITY
(Math Masters, pp. 159 and 160)
Home Connection Children solve comparison number
stories. They fill in a comparison diagram and write a
number model for each problem. The explanation in the
Family Note will help parents and guardians feel comfortable
when participating in these activities with their children.
Home Link Master
Home Link Master
Name
Date
Time
Name
62
䉬
Family
Note
62
䉬
Today your child learned about a device that is useful when solving number stories. We
call it a comparison diagram. Diagrams like these can help your child organize the
information in a problem. When the information is organized, it is easier to decide which
operation (⫹, ⫺, ⫻, or ⫼) to use to solve the problem.
Comparison diagrams are used to represent problems in which two quantities are given and
the question is how much more or less one quantity is than the other (the difference).
Example 1: There are 49 fourth graders
and 38 third graders. How many more fourth
graders are there than third graders?
Note that the number of fourth graders is
being compared with the number of third
graders.
• Answer: There are 11 more fourth graders
than third graders.
• Possible number models: Children who think
of the problem in terms of subtraction
will write 49 ⫺ 38 ⫽ 11. Other children
may think of the problem in terms of
addition: “Which number added to 38
will give me 49?” They will write the
number model as 38 ⫹ 11 ⫽ 49.
Example 2: There are 53 second graders.
There are 10 more second graders than
first graders. How many first graders are there?
Note that sometimes the difference is known
and that one of the two quantities is unknown.
Date
HOME LINK
Comparison Number Stories
HOME LINK
Quantity
49 fourth graders
continued
In each number story:
䉬 Write the numbers you know in the comparison diagram.
䉬 Write ? for the number you want to find.
Sample number
models are given.
䉬 Solve the problem. Then write a number model.
Quantity
1. Ross has $29. Omeida has $10.
29
Ross has $ 19 more than Omeida.
Number model: 29 ⫺ 10 ⫽ 19
Quantity
38 third
graders
Comparison Stories
Time
?
Quantity
10
?
Difference
Difference
Your child may write words in the diagram
as a reminder of what the numbers mean.
2. Omar swam 35 laps in the pool.
Quantity
Anthony swam
than Omar.
53
Number model:
Quantity
?
• Answer: There are 43 first graders.
10
15
35
Quantity
fewer laps
20
20 ⫹ 15 ⫽ 35
3. Claudia’s birthday is June 10.
p
For Problems 1 and 2, ask your child to explain the number model that he or she wrote.
Also ask your child to explain the steps needed to solve Problems 4–6.
Please return the second page of this Home Link to school tomorrow.
py g
110 111
?
Difference
Quantity
?
Tisha’s birthday is 12 days later.
Difference
g
• Possible number models:
53 ⫺ 10 ⫽ 43 or 10 ⫹ 43 ⫽ 53
Quantity
Anthony swam 20 laps.
22 .
Number model: 10 ⫹ 12 ⫽ 22
Tisha’s birthday is June
Quantity
10
Practice
12
Difference
Unit
Add. Use the partial-sums method.
4.
39
⫹ 62
90
⫹ 11
101 101
Math Masters, p. 159
388
5.
48
⫹ 7
55
40
⫹ 15
55
6.
33
⫹ 54
87
80
⫹ 7
87
Math Masters, p. 160
Unit 6 Whole-Number Operations and Number Stories
385-389_EMCS_T_TLG1_U06_L02_550524.indd 388
1/25/11 8:46 AM
3 Differentiation Options
READINESS
Doing a Penny Grab Activity
PARTNER
ACTIVITY
5–15 Min
To provide experience with comparing numbers using a concrete
model, have children do the following Penny Grab activity.
1. Place a pile of pennies or other counters on a table between
two children.
2. Each child grabs a handful of pennies, counts them, and
records the amount.
3. Partners compare their amounts by lining up the pennies
side-by-side and finding the difference.
4. Partners work together to record number models that describe
the lined-up pennies. For example, 9 - 6 = 3.
Encourage such remarks as, “I have 3 more than you. You have
3 less than I do.”
The difference between 9 and 6 is 3.
ENRICHMENT
Comparing Number Stories
INDEPENDENT
ACTIVITY
5–15 Min
(Math Masters, p. 161)
To further explore comparison number stories, have children
complete Math Masters, page 161. Children compare number
stories by writing number models and answering related
questions.
Teaching Master
Name
LESSON
62
䉬
Date
Time
Comparing Number Stories
Solve each number story. Be sure to write a number model.
Then answer the questions at the bottom of the page.
1. There were 23 children in the classroom. 17 went to the
computer lab. How many were left in the classroom?
Number Model:
23 ⫺ 17 ⫽ 6
2. There were 6 children in the classroom. 17 came back
from the computer lab. How many children are in the
classroom now?
Number Model:
6 ⫹ 17 ⫽ 23
How are the problems alike?
Sample answer: They use the same
3 numbers.
How might solving Problem 1 help you solve Problem 2?
Explain your thinking.
Sample answer: If you know that the
three numbers go together in a subtraction
problem, then they are a fact family and
will go together in an addition problem.
Math Masters, p. 161
Lesson 6 2
385-389_EMCS_T_TLG1_U06_L02_550524.indd 389
389
1/25/11 8:46 AM