LESSON
9.7
Compare Decimals
FOCUS
COHERENCE
RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
4.NF.C.7 Compare two decimals to hundredths by reasoning about their
size. Recognize that comparisons are valid only when the two decimals
refer to the same whole. Record the results of comparisons with the symbols >, =, or
< and justify the conclusions, e.g., by using a visual model.
Mathematical Practices
MP2 Reason abstractly and quantitatively. MP4 Model with mathematics.
MP6 Attend to precision.
F C R Coherence:
Standards Across the Grades
Before
Grade 4 After
3.NF.A.3a 4.NF.C.7 5.NBT.A.3b
3.NF.A.3d
F C R Rigor:
Level 1: Understand Concepts....................Share and Show ( Checked Items)
Level 2: Procedural Skills and Fluency.......On Your Own
Level 3: Applications..................................Think Smarter and Go Deeper
Learning Objective
Compare decimals to hundredths by reasoning
about their size.
Language Objective
Students use a 2-column graphic organizer to
compare decimals.
Materials
MathBoard
FC R
For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 493J.
About the Math
Professional Development
If Students Ask When will we compare decimals? Explain that comparing
decimals is a real-world skill and not just a math skill.
About the Math
•Many sport competitions are timed in fractions of a
second, and the
results are compared
to determine how
Professional
Development
the results are ordered.
•Scientists record and compare information about
different species of plants and animals using
decimal values.
•In many real-world situations, we compare money
amounts to find the better value. This may involve
comparing packages by weight or volume.
Being able to compare decimals is important for students
to be functionally literate in our society.
Professional Development Videos
533A Chapter 9
Interactive Student Edition
Personal Math Trainer
Math on the Spot
iTools: Fractions
iTools: Number Lines
1 ENGAGE
Daily Routines
Common Core
Problem of the Day 9.7
with the Interactive Student Edition
Essential Question
How can you compare decimals?
6
Brandon ran __
mile to warm up
10
25
mile to cool down
before practice and ___
100
Making Connections
Vocabulary
What is the name of a type of tree that you know of? Possible
answers: maple, oak, pine, elm How can you describe what the tree’s
leaves look like? Accept reasonable answers.
after practice. How far did he run?
85
___
mile
100
™Interactive Student Edition
™Multimedia eGlossary
Invite students to tell you what they know about trees and leaves.
Learning Activity
What is the problem the students are trying to solve? Connect the
story to the problem.
Fluency Builder
Common Core Fluency
Standard 4.NBT.B.5
Materials number cubes
Multiply Whole Numbers Have students
work with a partner. One student tosses
three number cubes and uses the results
to generate a 3-digit factor. The other
student tosses one number cube to
generate a 1-digit factor.
First, students estimate the product. Then,
each student solves the problem. Students
compare results and the estimate. If any
discrepancy occurs, students step through
the solution to find the error.
• What is the length across the first leaf? 7.3 centimeters
• What is the length across the second leaf? 7.09 centimeters
• How can you write these numbers in a place-value chart? Answers
will vary.
• How can you represent these numbers on a number line? Answers
will vary.
Literacy and Mathematics
• Have students write a problem situation that involves finding
two coins: one that’s worth 50 cents and one that’s worth 5 cents.
Have them write the numbers in decimal form and use a model or
a number line to compare their values. Have students share their
problems and solutions with one another.
Students repeat the activity until they have
solved five multiplication problems.
Literature Connection
From the Grab-and-Go™
Differentiated Centers Kit
How can you compare
decimals?
Students read about how
to use and order decimal
numbers to find batting
averages and team
standings.
Decimals on a Diamond
Lesson 9.7
533B
LESSON
9.7
2 EXPLORE
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only
when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify
the conclusions, e.g., by using the number line or another visual model.
Lesson 9.7
Name
Unlock the Problem
Compare Decimals
Number and Operations—
Fractions—4.NF.C.7
MATHEMATICAL PRACTICES
MP2, MP4, MP6
Essential Question How can you compare decimals?
MATHEMATICAL PRACTICES
After students have read the problem, discuss
different solution strategies. Remind them that
decimals and fractions are related, so they can
use strategies to compare decimals that are
similar to those used to compare fractions.
Unlock
Unlock the
the Problem
Problem
The city park covers 0.64 square mile.
About 0.18 of the park is covered by water,
and about 0.2 of the park is covered by
paved walkways. Is more of the park covered by
water or paved walkways?
One Way
• Cross out unnecessary information.
• Circle numbers you will use.
• What do you need to find?
if more of the park is covered by
One Way Use a model.
MP4 Model with mathematics.
• How can you tell which decimal is less
using the models? The model with fewer shaded
Shade 0.18.
water or paved walkways
Shade 0.2.
columns shows the lesser number.
Other Ways
and 0.78 is close to 1.0, so 0.4 is less than 0.78.
ELL
Restate
Provide a visual reminder that restates key
vocabulary and symbols in more familiar words.
• Write on the board: > means greater than,
< means less than, and = means equal to,
same as.
• Point out that > also means that the
number to the left is more than the number
to the right.
• Have students model or draw and then say
the comparison before they write it using
the appropriate symbol.
533 Chapter 9
0.2
Other Ways
A Use a number line.
Locate 0.18 and 0.2 on a number line.
Think: 2 tenths is equivalent to 20 hundredths.
0.0
0.10
0.30
0.20
●
0.40
0.50
Math
Talk
0.18 is closer to 0, so 0.18 < 0.2.
_
B Compare equal-size parts.
18 hundredths.
• 0.18 is _
20 hundredths.
• 0.2 is 2 tenths, which is equivalent to _
●
●
18 hundredths < 20 hundredths, so 0.18 < 0.2.
paved walkways
So, more of the park is covered by ___
_.
Before starting Part B, review the size of a
tenth and a hundredth using a decimal model.
Discuss with students that when comparing
decimal amounts, the comparison is only valid
when the decimals represent parts of the
same-size wholes.
Strategy:
<
●
0.18
© Houghton Mifflin Harcourt Publishing Company
Explain that a number line can also be used to
compare decimals. Just as with whole numbers
and fractions, a decimal closer to 0 is less than
another decimal.
After working through Part A with students,
remind them that they have used benchmark
fractions to compare fractions. Draw a number
line from 0 to 1 on the board, and divide it
into ten equal parts. Work with students to
write the benchmark fractions as decimals
(0.0, 0.5, and 1.0), and ask students to use the
benchmarks to compare 0.4 and 0.78.
• How do benchmarks help you compare the
decimals? Possible answer: I know 0.4 is less than 0.5
MATHEMATICAL PRACTICES 6
Compare How does the number
of tenths in 0.18 compare to the
number of tenths in 0.2? Explain.
Possible explanation: 0.18 has
1 tenth and 0.2 has 2 tenths.
The number of tenths in 0.18
is less than the number of
tenths in 0.2.
Chapter 9 533
3
Reteach 9.7
1
Lesson 9.7
Reteach
Name
Lesson 9.7
Enrich
Name
Compare Decimals
Comparing Decimals
Solve each problem.
Alfie found 0.2 of a dollar and Gemma found 0.23 of a dollar.
Which friend found more money?
1.
To compare decimals, you can use a number line.
Step 1 Locate each decimal on a number line.
0.0
Differentiated
Instruction
Enrich 9.7
2
0.10
0.20
Abby ran the 50-yard dash in
7.05 seconds. Barb’s time was
7.5 seconds. Chris’s time was
6.94 seconds. List the runners in
order from fastest to slowest.
2.
2.26 kg, 4.4 kg,
5.4 kg
Chris, Abby, Barb
0.30
Nick’s bag weighs 5.4 kilograms.
Amelia’s bag weighs 2.26 kilograms.
Henrik’s bag weighs 4.4 kilograms.
List the weights of the bags from
lightest to heaviest.
Step 2 The number farther to the right is greater.
0.23 . 0.2, so Gemma found more money.
3.
To compare decimals, you can compare equal-size parts.
Step 1 Write 0.2 as a decimal in hundredths.
0.2 is 2 tenths, which is equivalent to 20 hundredths.
0.2 5 0.20
Jeremy has three lengths of string.
One is 8.3 centimeters long. The
second string is 8.32 centimeters
long and the third string is 8.27
centimeters long. Order the lengths
of Jeremy’s strings from longest to
shortest.
4.
Step 2 Compare.
So, Gemma found more money.
5.
Compare. Write ,, ., or 5.
1.
0.17
. 0.13
2.
0.8
. 0.08
3.
0.36
,
0.63
4.
0.4
5.
0.75
. 0.69
6.
0.3
,
7.
0.45
. 0.37
8.
0.96
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
0.7
9-17
Group A; Group C
8.32 cm, 8.3 cm,
8.27 cm
23 hundredths is greater than 20 hundredths,
so 0.23 . 0.2.
5 0.40
. 0.78
Reteach
A science class is testing model
planes. Group A’s plane flew
9.35 meters. Group B’s plane flew
9.6 meters. Group C’s plane flew
10.04 meters. Group D’s plane flew
9.57 meters. Which group’s plane
flew the shortest distance? the
longest distance?
How do you compare decimals when the digits to
the left of the decimal point are not 0?
Possible answer: If the digits are the same,
then you compare the digits to the right of
the decimal point. If the digits to the left of
the decimal point are different, then you
compare those digits first.
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
9-18
Enrich
Place Value
Make sure students know the place-value
positions without using a place-value chart.
Students should be able to recognize decimals
to hundredths.
Place Value You can compare numbers written as
decimals by using place value. Comparing decimals
is like comparing whole numbers. Always compare
the digits in the greatest place-value position first.
Example
Example Use place value.
• How do the models show which decimal is
greater? The model with the greater amount shaded
Tim has 0.5 dollar, and Sienna has 0.05 dollar.
Who has more money?
MODEL
Tim
shows the greater decimal.
RECORD
MP7 Look for and make use of structure.
• How does the place-value chart help you
compare the money amounts? Possible answer:
Sienna
Ones
.
Tenths
Hundredths
0
0
.
5
0
5
.
Tim
Sienna
I can start by comparing the digits in the place-value
position farthest to the left, the ones. Since the ones
digits are the same, I can compare the digits in the
tenths place. 5 > 0, so 0.5 is greater than 0.05.
Think: The digits in the ones place are the same.
Compare the digits in the tenths place.
Tim has more money.
So, _
●
●
5 tenths > 0 tenths, so 0.5 > 0.05.
• Compare the size of 1 tenth to the size of 1 hundredth. How could this help
you compare 0.5 and 0.05? Explain.
Try This!
Have students shade the tenths models to
represent each fraction. Students should use
their models to compare the decimals.
• Why do you need to use more than one
tenths model to represent 1.3? Possible
Possible explanation: 1 tenth of a whole is larger than 1 hundredth of a whole.
Therefore, 5 tenths of a whole are larger than 5 hundredths of a whole,
so 0.5 > 0.05.
Try This! Compare 1.3 and 0.6. Write <, >, or =.
answer: I need to shade 1 whole tenths model to
represent 1, and I need the second tenths model to
represent 0.3, or 3 tenths.
●
1.3 > 0.6
Shade to model 0.6.
Possible answer: the greatest place value is the
ones place. 1.3 has 1 one and 0.6 has 0 ones.
Since 1 . 0 and 6 tenths is not enough to make a
one, 1.3 . 0.6.
Math
Talk
MATHEMATICAL PRACTICES 7
Look for Structure How
could you use place value
to compare 1.3 and 0.6?
534
© Houghton Mifflin Harcourt Publishing Company
Shade to model 1.3.
Math
Talk
Use Math Talk to focus on
students’ understanding of using
place value to compare decimals.
• In what place will you first compare the
digits? Explain. the ones place; it is the place-value
position farthest to the left.
• How do the digits in the ones place
compare? 1 > 0
• So, how do the decimals compare? 1.3 > 0.6
Advanced Learners
Logical / Mathematical
Partners
Materials index cards, Digit Cards (0–9) (see eTeacher Resources)
• Make a game card for each student using an index card, like
the one shown.
COMMON ERRORS
Error Students do not compare equal-sized
parts when comparing decimals.
'%
• Player 1 draws a digit card and writes the digit on his or her
game card in either open space. Players take turns and repeat
the process until both players show hundredths decimals.
• Players then compare decimals. The player with the greater
decimal earns 1 point. Play continues until one player earns
5 points.
Example Students compare 5 tenths
and 18 hundredths and conclude
that 0.5 < 0.18
Springboard to Learning Give students
decimal models and have them shade to
show each decimal. Elicit from students that
0.5 = 0.50 and 50 hundredths is greater
than 18 hundredths, not less.
Lesson 9.7
534
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=A
Name
3 EXPLAIN
MATH
Share
Share and
and Show
Show
BOARD
1. Compare 0.39 and 0.42. Write <, >, or =.
Shade the model to help.
Share and Show
●
MATH
0.39 < 0.42
BOARD
0.39
Use the checked exercises for Quick Check.
Students should show their answers for Quick
Check on the MathBoard.
Quick Check
Quick Check
If
If
Then
3
2
31
2
1
Compare. Write <, >, or =.
●
Ones
.
Tenths
Hundredths
Ones
.
Tenths
Hundredths
0
0
.
2
2
6
3
0
0
.
7
5
4
.
●
Differentiate Instruction with
• Reteach 9.7
• Personal Math Trainer 4.NF.C.7
• RtI Tier 1 Activity (online)
.
Tenths
Hundredths
Ones
.
Tenths
Hundredths
1
1
.
1
3
5
4
2
.
5
8
9
.
Compare. Write <, >, or =.
●
MATHEMATICAL
PRACTICE
2
●
because 0.39 has less tenths than 0.42, the number
of hundredths in either decimal will not change the
comparison.
On Your Own If students complete the checked exercises
correctly, they may continue with the
remaining exercises.
MP2 Reason abstractly and
quantitatively. Exercises 10–13 require
students to use higher order thinking skills to
rename a fraction as a decimal or a decimal
as a fraction in order to compare amounts
written in different forms.
535 Chapter 9
© Houghton Mifflin Harcourt Publishing Company
●
8. 0.25 < 0.3
●
9. 2.61 < 3.29
●
4 < 0.2
11. ___
100
●
1
12. 0.15 > __
10
●
13. 1_ < 0.8
8
Math Talk: Yes. Possible explanation: neither number has any ones. 0.39 has 3 tenths
and 0.42 has 4 tenths. Since 9 hundredths is not enough to make another tenth,
0.39 is less than 4 tenths. So, 0.39 < 0.42.
14.
DEEPER
Robert had $14.53 in his pocket. Ivan had $14.25 in
his pocket. Matt had $14.40 in his pocket. Who had more money,
Robert or Matt? Did Ivan have more money than either Robert
or Matt?
Robert; No, Ivan did not have more money than either Robert or Matt.
are different. 3 < 4, so 0.39 < 0.42
• Do you need to compare the digits in the
hundredths place? Explain. Possible answer: no;
●
7. 1.06 > 0.6
MATHEMATICAL PRACTICES 2
Reason Abstractly Can you compare
0.39 and 0.42 by comparing only the
tenths? Explain.
Reason Quantitatively Compare. Write <, >, or =.
3
10. 0.30 = __
10
• Can you use the digits in the tenths place to
compare the decimals? Explain. Yes; the digits
.
Math
Talk
On
On Your
Your Own
Own
© Houghton Mifflin Harcourt Publishing Company
are the same.
5. 4.5 > 2.89
Ones
6. 0.9 > 0.81
Use Math Talk to focus on
students’ understanding of using
place value to compare decimals.
• Can you use the digits in the ones place to
compare the decimals? Explain. No; the digits
.
●
4. 1.15 < 1.3
a student misses the checked
exercises
Math
Talk
●
3. 0.7 > 0.54
2. 0.26 > 0.23
Rt I
Rt I
0.42
Chapter 9 • Lesson 7
4_MNLESE342279_C09L07.indd 535
535
01/03/14 5:44 PM
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
Unlock
Unlock the
the Problem
Problem
15.
4 ELABORATE
SMARTER
Ricardo and Brandon ran a 1500-meter race.
Ricardo finished in 4.89 minutes. Brandon finished in 4.83
minutes. What was the time of the runner who finished first?
Unlock the Problem
MATHEMATICAL PRACTICES
a. What are you asked to find?– the time of the runner who finished first
b. What do you need to do to find the answer?– Possible answer: use a place-value chart to
SMARTER
compare the times to find the time that is less.
c. Solve the problem.
I wrote 4.89 and 4.83 in a place-value
chart. Then I compared the digits,
beginning with the greatest place
value.
Have students read Exercise 15. Ask them to
describe how they will solve the problem.
Students should recognize that the length of
the race is unnecessary information for the
solution of the problem.
d. What was the time of the runner who
finished first?
4.83 minutes
e. Look back. Does your answer make sense?
Explain.
Ones . Tenths Hundredths
Math on the Spot
Video Tutor
Yes. The time of the runner who finished
.
8
9
4
.
8
3
8=8
9>3
4=4
first is the
lesser time of the two. Since
WRITE
Math t Show Your Work
time of the runner who finished first.
Since 3 is less than 9, 4.83 is less
than 4.89.
16.
Use this video to help students model and
solve this type of Think Smarter problem.
4.83 , 4.89, then 4.83 minutes is the
DEEPER
The Venus flytrap closes in
0.3 second and the waterwheel plant closes
in 0.2 second. What decimal is halfway
between 0.2 and 0.3? Explain.
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
Personal Math Trainer
17.
SMARTER
SMARTER
For numbers
17a–17c, select True or False for the
inequality.
Personal Math Trainer
0.25; possible explanation: 0.2 is
17a.
0.5 > 0.53
True
False
equivalent to 20 hundredths and 0.3 is
17b.
0.35 < 0.37
True
False
equivalent to 30 hundredths. Halfway
17c.
$1.35 > $0.35
True
False
between is 25 hundredths, or 0.25.
© Houghton Mifflin Harcourt Publishing Company
4
Be sure to assign Exercise 17 to students
in the Personal Math Trainer. It features
a video to help them model and answer
the problem. This item assesses a student’s
ability to compare two decimals, including
money values. Students who answered 17a
incorrectly may not have rewritten 0.5 as
0.50 to compare the decimal amounts.
536
5 EVALUATE Formative
Assessment
DIFFERENTIATED INSTRUCTION
D
INDEPENDENT ACTIVITIES
Essential Question
Using the Language Objective
Differentiated Centers Kit
Activities
Where Is the Decimal?
Literature
Games
Decimals on a Diamond Order, Please!
(BNFT
Students complete
blue Activity Card
10 by using a
number line to
order decimals.
Students read
about how to use
and order decimal
numbers to find
batting averages
and team
standings.
Students practice
placing decimals
in order from
least to greatest.
Reflect Have students use a 2-column
graphic organizer to answer the Essential
Question.
How can you compare decimals? Possible
answer: I can use a decimal model to compare decimals
by shading grids to show the two decimals and then
determining how the decimals compare.
Math Journal
WRITE
Math
Show or describe two different ways to
complete the comparison using >, <, or =:
0.4.
0.26
Lesson 9.7
536
Practice and Homework
Lesson 9.7
Name
Compare Decimals
COMMON CORE STANDARDS—4.NF.C.7
Understand decimal notation for fractions,
and compare decimal fractions.
Compare. Write <, >, or =.
Practice and Homework
1. 0.35 < 0.53
3. 0.24 < 0.31
2. 0.6 = 0.60
Think: 3 tenths is less
than 5 tenths.
So, 0.35 < 0.53
Use the Practice and Homework pages to
provide students with more practice of the
concepts and skills presented in this lesson.
Students master their understanding as they
complete practice items and then challenge
their critical thinking skills with Problem
Solving. Use the Write Math section to
determine student’s understanding of content
for this lesson. Encourage students to use their
Math Journals to record their answers.
6. 0.45 > 0.28
5. 0.3 < 0.32
4. 0.94 > 0.9
7. 0.39 < 0.93
Use the number line to compare. Write true or false.
0
8. 0.8 > 0.78
0.1
0.2
0.3
0.4
0.5
9. 0.4 > 0.84
true
0.6
0.7
0.8
10. 0.7 < 0.70
false
0.9
1.0
11. 0.4 > 0.04
false
true
Compare. Write true or false.
12. 0.09 > 0.1
13. 0.24 = 0.42
© Houghton Mifflin Harcourt Publishing Company
false
14. 0.17 < 0.32
15. 0.85 > 0.82
true
true
false
Problem
Problem Solving
Solving
16. Kelly walks 0.7 mile to school. Mary walks
17.
0.49 mile to school. Write an inequality using
<, >, or = to compare the distances they
walk to school.
Math Show or describe two
WRITE
different ways to complete the comparison
using <, >, or =: 0.26
0.4.
Check students’ work.
Possible answer: 0.7 > 0.49
Chapter 9
PROFESSIONAL
DEVELOPMENT
Mathematical Practices in Your Classroom
CCSS.Math.Practice.MP8 Look for and express
regularity in repeated reasoning.
Mathematically proficient students look for general methods and
shortcuts that make their reasoning faster, while maintaining accuracy.
When comparing decimals, students can apply different forms of
repeated reasoning. For example, students know that the number
closer to 0 on a positive number line will be the lesser number.
537 Chapter 9
537
Ask questions such as the following to help students look for methods
of reasoning when comparing decimals:
• Jason says 0.6 is less than 0.06. How can you use reasoning to
show this is false? Possible answer: 0.6 is equal to 60 hundredths,
and 60 hundredths . 6 hundredths, so that comparison is false.
• Breanna uses a number line to compare two decimals. She plots
a point at 0.65. The other decimal is plotted farther from 0 on
the number line. Explain how you know if 0.65 is the greater or
lesser decimal. 0.65 is the lesser decimal because it is closer to 0.
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=B
Lesson Check (4.NF.C.7)
1. Bob, Cal, and Pete each made a stack of
2. Three classmates spent money at the
baseball cards. Bob’s stack was 0.2 meter
high. Cal’s stack was 0.24 meter high. Pete’s
stack was 0.18 meter high. Write a number
sentence that compares Cal’s stack of cards
to Pete’s stack of cards.
school supplies store. Mark spent 0.5 dollar,
Andre spent 0.45 dollar, and Raquel spent
0.52 dollar. Write a number sentence that
compares the money Andre spent to the
money that Mark spent.
0.24 > 0.18
0.45 < 0.5
Continue concepts and skills practice with
Lesson Check. Use Spiral Review to engage
students in previously taught concepts and
to promote content retention. Common Core
standards are correlated to each section.
Spiral Review (4.NF.B.3c, 4.NF.B.4c, 4.NF.C.5, 4.NF.C.6)
$0.40 in her pocket. How much money
do Pedro and Alice have altogether?
$0.75
5. Joel has 24 sports trophies. Of the trophies,
_1
8
are soccer trophies. How many soccer
trophies does Joel have?
3 soccer trophies
4. The measure 62 centimeters is equivalent
62
to ___
100 meter. What is this measure written as
a decimal?
0.62 meter
6. Molly’s jump rope is 6 31_ feet long. Gail’s
jump rope is 4 2_3 feet long. How much longer
is Molly’s jump rope?
2 feet
1__
3
© Houghton Mifflin Harcourt Publishing Company
3. Pedro has $0.35 in his pocket. Alice has
FOR MORE PRACTICE
GO TO THE
538
4_MNLESE342279_C09P07.indd 538
Personal Math Trainer
13/10/14 6:59 PM
Lesson 9.7 538