Using Benchmarks with Fraction
Addition and Subtraction
Objective To review estimating with fractions using
benchmarks.
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Key Concepts and Skills
Reviewing Fractions and Division
• Identify benchmarks on a number line. Math Journal 1, pp. 192A and 192B
Student Reference Book, pp. 62, 63,
78C, and 78D
Students review the relationship
between, and solve problems
involving, fractions and division.
[Number and Numeration Goal 6]
• Add and subtract fractions and mixed
numbers with like and unlike denominators. [Operations and Computation Goal 4]
• Use benchmarks to estimate sums and
differences. [Operations and Computation Goal 6]
Key Activities
Students estimate solutions to number
stories involving addition and subtraction of
fractions with like and unlike denominators
that refer to the same whole. Students use
benchmarks, number lines, and number
sense of fractions to estimate mentally and
assess reasonableness of answers.
Ongoing Assessment:
Informing Instruction See page 421.
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Math Boxes 6 8
Math Journal 1, p. 193
Students practice and maintain skills
through Math Box problems.
Study Link 6 8
Math Masters, p. 174
Students practice and maintain skills
through Study Link activities.
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Comparing Fractions with
1
_
2
Math Masters, p. 175
Math Journal 2, Activity Sheets 5–7
Students sort Fraction Cards using
benchmarks and number sense.
EXTRA PRACTICE
Writing Fraction Number Stories
Math Masters, p. 176
Students create and solve fraction number
stories and estimate solutions to problems
with sums and differences of fractions.
ENRICHMENT
Playing Fraction Top-It
(Advanced Version)
Math Journal 2, Activity Sheets 5–7
Student Reference Book, p. 316
Math Masters, p. 493
Students use benchmarks to play an
advanced version of Fraction Top-It.
Use journal pages 191 and 192. [Operations and Computation Goal 6]
Materials
Math Journal 1, pp. 191 and 192
Math Journal 2, Activity Sheets 5–7
Advance Preparation
Students will need the Fraction Cards from Activity Sheets 5–7 that were first used in Lesson 5-1.
Lesson 6 8
417
Mathematical Practices
SMP1, SMP2, SMP3, SMP5, SMP6, SMP7
Content Standards
Getting Started
5.NF.1, 5.NF.2, 5.NF.3
Mental Math and Reflexes
Math Message
Pose fraction addition and subtraction problems. Have students estimate
1
whether the sum or difference is closest to 0, _
, or 1. Students should be
2
prepared to explain their thinking.
Draw and label the
benchmarks on the number line in
Problem 1, Math Journal 1, page 191. Be
prepared to explain your reasoning for the
placement.
1
4
_
+_
1
12
4
3 _
2
1
_
_
+
12
12 2
3
2
_
+_
1
4
8
3
1 1
_
-_ _
2
4
_
+_
1
3
9
9
14
_
_
- 1
5 2
4
11
7
_
_
0
12
8
3
2
_
-_
1
2
4
7
8
4
7 _
1
_
1_
7
6 2
Study Link 6 7
Follow-Up
Have partners compare
answers and resolve differences.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Journal 2, Activity Sheets 5–7)
NOTE The Fraction Cards on Activity
Sheets 5–7 were cut out in Lesson 5-1.
Ask students to share how they decided to label the benchmarks
on the number line. Distribute the Fraction Cards from Activity
Sheets 5–7. Ask each partnership to find Fraction Cards that are
1 , and 1 to use as a frame of reference. Ask: How can
equal to 0, _
2
benchmark fractions be useful when operating with fractions?
Sample answer: A benchmark is useful for comparing fractions. For
12 is almost half because the numerator (12) is almost
example, _
25
1 as the benchmark, you know
half of the whole (25); so using _
2
12
1
_
_
that 25 is almost 2 .
1 , 1, 1_
1,
Draw and label a number line using the benchmarks 0, _
2
2
and 2.
0
1
2
1
1 12
2
Ask students where they would place an X to represent the
3.
4 and _
approximate sum for the problem _
12
12
418
Unit 6
●
7 Students should recognize that this is
What is the sum? _
12
1.
about _
2
●
Should the X be closer to 0 or closer to 1? 1 Why? Sample
7 is _
7 from 0 and _
5 from 1. _
7 is greater than _
1 , so
answers: _
12
12
12
12
2
it is closer to 1.
●
7
1? _
1 Why? Sample answers: _
Should the X be closer to 0 or _
2 2
12
7
1 than to 0, because it is only _
1 away from _
1 and _
is closer to _
2
12
2
12
7 is greater than _
1 , so it is closer to _
1 than to 0.
away from 0. _
12
2
2
●
Where should the X be placed? A little more than halfway
7
_
between 0 and 1 Ask a volunteer to place the X to show where 12
would be on the number line.
Using Data; Addition and Subtraction of Fractions
Student Page
Date
Adjusting the Activity
Time
LESSON
Using Benchmarks and Estimation to Solve Problems
68
Have students label the benchmarks on the number line with equivalent
0
6
_1
_
fractions of twelfths. For example, under 0, record _
12 ; under 2 , record 12 ; and so
on. Students can see the equivalent fractions with the benchmarks.
Math Message
1.
1
1
_
Place the benchmarks 0, _
2 , 1, 1 2 , and 2 on the number line below.
0
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
1
2
1 1 12 2
V I S U A L
Use a number line, Fraction Cards, or a mental strategy to estimate the solution
to each problem. Circle the answer.
2.
3 cards from the Fraction
4 and _
Ask students to take out the _
12
1
About _
2 mile
12
Cards. Ask: How could I use shading on the two cards to show
3 is closest to 0, _
4 and _
1 , or 1? Sample
whether or not the sum of _
12
12
2
answer: You can lay the blue shaded parts of the two cards
1.
together and see that the sum is about _
2
If necessary, show students how to match the shaded parts of the
Fraction Cards.
1
1
_
Samuel ran _
5 mile in the morning and 2 mile in the afternoon. How much of a mile
did he run?
3.
1
A recipe for bread calls for 1 cup of flour. If Ryan has _
2 cup of white flour and
3
_
cup of whole-wheat flour, about how much flour would he have left after he
4
makes the bread dough?
1
Less than _
2 cup
About 1 cup
More than 1 cup
3
1
_
Simone has _
4 cup of whole milk and 8 cup of 2% milk. About how much milk
does she have?
4.
1
About _
2 cup
1
2
1
About 1 _
2 miles
About 1 mile
1
About 1 _
2 cups
About 1 cup
1
Math Journal 1, p. 191
EM3MJ1_G5_U06_164-204.indd 191
1/20/11 8:35 AM
1 is the actual sum? _
1
Ask: How much greater than _
2
12
1?
Ask: Where would you place an X to represent the sum of _54 and _
4
Ask a volunteer to place the X to show where the sum would be.
Expect that students will know that the sum of the two fractions
is about 1, but do not expect students to find the exact sum of
the two fractions. Ask students how they know that the sum is
1 and one
about 1. Sample answers: One fraction is greater than _
2
1
_
fraction is less than 2 ; when they are added together, they must
1. _
4 is only _
1 away from 1, so adding _
1 would give
be greater than _
2 5
5
4
a sum greater than 1.
4 and _
1 cards from the Fraction
Ask students to take out the _
5
4
Cards and to use the cards to show that the sum of the two cards
is close to 1.
Student Page
Date
LESSON
68
Time
Using Benchmarks and Estimation to Solve Problems cont.
3
1
1
_
_
Ms. Lyons bought _
3 lb turkey, 2 lb ham, and 4 lb salami. About how many pounds
of meat did she purchase?
5.
1
About _
2 lb
1
2
1
1
About _
2 yard
Pose additional problems as necessary. Encourage students to
use their slates to draw number lines or use Fraction Cards to
estimate solutions using benchmark fractions.
7.
1
About 1 _
2 lb
3
1
_
Kevin has 1 _
2 yards of fabric. He will use 8 of a yard to make a pillow. About how
much fabric will he have left?
6.
About 1 lb
About 1 yard
About 2 yards
Write your own fraction number story similar to the ones in Problems 2– 6.
Answers vary.
Try This
8.
5
1
_
A candle is 2 _
2 inches tall. When it burns, it goes down about 8 inch each hour.
About how much of the candle will be left after 3 hours? Explain your answer.
1
Sample answer: About _
2 inch will be left. In 3 hours,
5
5
5
15
16
_
_
_
_
+
+
=
will
be
used. Because _
8
8
8
8
8 = 2, almost 2 inches
1
1
1
_
_
will be used. 2 _
2 - 2 = 2 , so about 2 inch will be left.
9.
9
The perimeter of a triangle is 10 inches. One side is 3 _
16 inches long. Another side
5
is 4 _
8 inches long. About how long is the third side? Show how you estimated.
9
5
1
_
_
Sample answer: 3 _
16 is close to 3 2 and 4 8 is close to
1
4_
2 , so the length of these two sides is about 8. Then
the third side is about 10 - 8, or about 2 inches long.
Math Journal 1, p. 192
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2/11/11 12:47 PM
Lesson 6 8
419
Student Page
Date
▶ Using Benchmarks to
Time
LESSON
Fractions as Division
68
Estimate Sums and Differences
of Fractions
Refer to pages 62 and 63 of your Student Reference Book to review renaming
fractions as mixed numbers or whole numbers.
Write or complete the fractions or mixed numbers for each problem.
8
_
1. 3
=
2
2_
3
27
2.
3
1
_
_
21 3 6 , or 3 2
_
=
4.
5.
6
_=
3
9
34
_
=
4
6_
41
_
6
= 5_
7
6.
_= _
9 34
4
7
39
5
5
3.
Write each of the following division problems as a mixed number.
2
11 _
3
35 3 =
7.
8.
29 4 =
1
7_
4
9.
52 10 =
Have students use the Fraction Cards, number lines, or a mental
strategy to estimate sums and differences for fraction number
1 , and 1 as the benchmarks as
stories. Remind students to use 0, _
2
they solve the problems. Pose the following problems:
2
1
_
5_
10 , or 5 5
Find the missing number for each number sentence below.
10.
53 5
3
= 10 _
5
11.
17
4 = 4 _14
12.
38
7 = 5 _37
Solve.
13.
●
Juan and Liz ordered two small pizzas, one cheese and one
5 of the cheese pizza, and Liz ate _
1 of the
veggie. Juan ate _
6
8
veggie pizza. Estimate the amount of pizza they ate together.
About 1 pizza
●
3 hours of
1 hours of TV on Saturday and 1_
Jake watched 1_
3
4
TV on Sunday. About how much TV did he watch over the
weekend? About 3 hours
●
7
1 yards of fabric to make a blanket. He used 1_
Mark bought 2_
2
8
1 yard
yards. About how much fabric does he have left? About _
2
●
1 yards long. She
McKenna has a piece of yarn that is 3_
8
7
needs about _
of
a
yard
to
make
a
necklace.
About how many
8
necklaces can she make? About 3
Carlos was having a party. He had 5 large granola bars to share equally among
the 8 boys at the party. How much of a granola bar will each boy get?
5
_
8 of a granola bar
5
5÷8= _
8
Number Model:
14.
Debbie, Rachel, and Meg wanted to share 2 foot-long sandwiches equally.
How much of a sandwich will each girl get?
2
_
3 sandwich
2
2÷3= _
Number Model:
3
1
There are 36 balloons in a package. _
4 of them are blue. How many blue
balloons are in the package?
15.
9 balloons
36 ÷ 4 = 9
Number Model:
Math Journal 1, p. 192A
164-204_EMCS_G5_S_G5_SMJ_U06_57637X.indd 192A
2/11/11 12:47 PM
▶ Using Benchmarks and
to encourage the use of estimation rather
than finding common denominators and
computing exact answers.
Pose the following problems orally. Have students justify their
solutions as they share their responses and reasoning.
Student Page
●
1 +_
1 closer to 0 or to _
1 ? Sample answer: Both _
1 and
Is _
12
35
2
12
1 are very close to 0, so their sum is closer to 0 than _
1.
_
35
2
●
7 closer to 1 or to 2? Sample answer: Closer to
12 + _
Is _
13
8
7 is close to 1
12 is close to 1 and _
2 because _
13
8
●
1 +_
1 closer to 0 or to 1? Sample answer: Closer to
Is _
9
12
1 and _
1 are close to 0
0 because _
9
12
●
7 closer to _
1 −_
1 or to 0? Sample answer: Closer to
Is _
2
17
2
7
1
_
0 because 17 is almost _
2
●
18 + _
23 closer to 2 or to 3? Sample answer: Closer to
Is _
19
12
23 is close to 2 and _
18 is close to 1
3 because _
12
19
Time
LESSON
Fractions as Division
68
continued
Write the letter for the expression that makes sense for the given situation.
5
_
16. 3
17.
22
_
8 is the same as which expression?
18.
19.
20.
is the same as which expression?
5 friends share 3 pizzas. How much pizza
does each friend get?
$8 is equally shared among 22 people.
How much money does each person get?
b
a
d
a.
3÷5
b.
5÷3
c.
8 ÷ 22
d.
22 ÷ 8
c
3
3
_
Noreen said that Picture A shows _
4 . Milton says that Picture B shows 4 . With whom
do you agree? Explain. Use the value of the WHOLE in your explanation.
1
Sample answer: They are both right. Noreen sees _
4 shaded
3
three times which is the same as _
4 . The WHOLE is one
9
_
square. Milton sees 12 of the 3 squares shaded, and that
3
is the same as _
4 . The WHOLE is the group of 3 squares.
Picture A
21.
WHOLE-CLASS
ACTIVITY
Number Sense to Mentally
Estimate Solutions
NOTE Work quickly through these problems
Date
WHOLE-CLASS
ACTIVITY
Picture B
▶ Using Benchmarks and Number
Write a number story for this number sentence: 8 ÷ 3 = ? Draw a picture to show
how you can solve your number story.
Answers vary.
PARTNER
ACTIVITY
Sense to Solve Fraction Problems
(Math Journal 1, pp. 191 and 192)
Try This
Solve and show your work.
22.
Mrs. Wilson’s class decided to bake cookies for their spring
celebration. They baked 3 cookie sheets filled with small cookies.
1
There were 24 students in the class. Each student got _
24 of the
cookies to eat or take home. What fraction of the cookies on 1 cookie
sheet did each student get?
Students work with a partner. Have students use benchmarks
to estimate the sums or differences for fraction number stories
on journal pages 191 and 192. Bring the class together to share
solution strategies.
3
1
_
_
24 , or 8 of the cookies on 1 cookie sheet
Math Journal 1, p. 192B
164-204_EMCS_G5_S_G5_SMJ_U06_57637X.indd 192B
420
Unit 6
8/29/11 10:19 AM
Using Data; Addition and Subtraction of Fractions
Student Page
Date
Time
LESSON
Math Boxes
6 8
Ongoing Assessment: Informing Instruction
1. a.
What is 30% of $1.00?
2.
$0.30, or 30¢
Watch for students who do not understand how to use benchmarks. Guide them
to find equivalent fractions. Consider having them write equivalent fractions next
to the benchmarks for each problem.
b.
c.
What is 75% of $1.00?
d.
How many cents is 5% of $1.00?
Decimal
Percent
3
7
10
11
8
15
7
9
8
14
0.43
0.91
0.53
0.78
0.57
43%
91%
53%
78%
57%
$0.75, or 75¢
$0.05, or 5¢
51
89 90
Use your calculator to complete the table.
Journal Pages
191 and 192
Problems
3–6
Exponential
Notation
Product
of Factors
Standard
Notation
94
9999
6,561
124
12 12 12 12
20,736
Use journal pages 191 and 192, Problems 3–6 to assess students’ ability to
estimate solutions to addition and subtraction problems with fractions. Students
are making adequate progress if they can correctly indicate that the solution is
less than, greater than, or equal to 1. Some students may be able to explain
their reasoning using benchmarks.
Fraction
How many cents is 60% of $1.00?
$0.60, or 60¢
3.
Ongoing Assessment:
Recognizing Student Achievement
Use a calculator to complete the table.
(Round decimals to the nearest hundredth.)
8888
4,096
11 11 11 11 11 161,051
10 10 10 1,000
84
5
11
103
5 6
4.
Complete the “What’s My Rule?”
table and state the rule.
in
Rule
out in 7
[Operations and Computation Goal 6]
5.
out
Make true sentences by inserting
parentheses.
27
20
a.
5 (4 2) 10
13
6
b.
25 (8 7) 81
5
2
10
c.
36 /(6 5) 36
d.
(45 / 9) 6 11
e.
45 /(9 6) 3
3
222 223
231 232
Math Journal 1, p. 193
2 Ongoing Learning & Practice
▶ Reviewing Fractions and Division
PARTNER
ACTIVITY
(Math Journal 1, pp. 192A and 192B; Student Reference
Book, pp. 62, 63, 78C, and 78D)
Students review the relationship between fractions and division.
Refer students to pages 62, 63, 78C, and 78D of the Student
Reference Book to review improper fractions, mixed numbers,
and connections between fractions and division. Have students
complete journal pages 192A and 192B to practice solving
problems that involve division and fractions.
Study Link Master
Name
STUDY LINK
68
Date
Time
Estimating with Fractions
Circle the best estimate for each situation described below.
▶ Math Boxes 6 8
INDEPENDENT
ACTIVITY
1.
3
18
The sum of _4 and _
19 is closest to
2.
1
1
_
The sum of _
11 and 15 is closest to
3.
9
1
_
The sum of _
10 and 32 is
4.
Use the circle below to draw a spinner as follows:
0
(Math Journal 1, p. 193)
0
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 6-10. The skills in Problem 5
previews Unit 7 content.
▶ Study Link 6 8
1
2
1
less than 1
2
greater than 1
_
Shade a red sector that is more than _
8 of the circle, but less than 4 of the circle.
1
1
1
_1
Shade a blue sector that is more than _
4 of the circle, but less than 2 the circle.
Sample answer:
red
blue
INDEPENDENT
ACTIVITY
The total amount of the circle that is shaded is
(Math Masters, p. 174)
less than _43
5.
Students practice estimating sums when adding fractions.
3
equal to _4
3
greater than _4
6
_1
The number line below shows an estimate for the sum of _
13 and 8 .
1
Explain why the sum is greater than _2 .
0
1
2
1
6
1
_
_1
_
13 is less than 2 . If you added 13
6
7
1
1
_
_
_
to 13 , you would get 13 , which is greater than 2 . Because _8
6
1
1
1
_
_
_
_
and is greater than .
is greater than , the sum of
Sample answer:
13
13
8
2
Math Masters, p. 174
EM3MM_G5_U06_157-186.indd 174
1/19/11 1:36 PM
Lesson 6 8
421
Teaching Master
Name
Date
LESSON
68
Time
1
Comparing Fractions with _
2
3 Differentiation Options
Use the Fraction Cards from Math Journal 2, Activity Sheets 5–7.
Sort the cards into three piles.
Fractions less than _12
Fractions equal to _
2
1
READINESS
Fractions greater than _
2
1
Place the cards next to one another to check your work. When you are finished,
write the fractions in each pile in the correct box below.
1
Less than _
1
Equal to _
2
2
_1 _1 _0 _1 _2
3, 4, 5, 5,
_2 _1 _2 _3
6, 8, 8, 8,
0 __
2 __
4
__
10 , 10 , 10 ,
0 __
4 __
2
__
12 , 16 , 16 ,
_1 _2 _3 _4
5,
_3
9,
3
__
12 ,
6
__
16
2, 4, 6, 8,
5 __
6 __
8
__
10 , 12 , 16
1
Greater than _
2
▶ Comparing Fractions with _12
15–30 Min
(Math Masters, p. 175; Math Journal 2, Activity Sheets 5–7)
_2 _2 _3 _3 _4
2, 3, 3, 4,
_3 _4 _5 _4
5, 5, 5, 6,
_6 _7 _8 _6
8, 8, 8, 9,
8 __
10 __
8
__
10 , 10 , 12 ,
10 __
16
14 __
__
16 , 16 , 16
PARTNER
ACTIVITY
4,
_6 _5
6, 8,
6
__
10 ,
9
__
12 ,
Partners work together to sort the Fraction Cards into three
1 , less than _
1 , or greater than
groups: fractions that are equal to _
2
2
_1 . First have students locate the _1 card and place
it face up.
2
2
Remind students that the blue shading on the card is a model or
1 . Tell students they will use the _
1 card as
picture for the area of _
2
2
1,
a benchmark to determine if each Fraction Card is less than _
2
1 , or equal to _
1.
greater than _
2
2
Math Masters, p. 175
EM3MM_G5_U06_157-186.indd 175
1/19/11 1:36 PM
EXTRA PRACTICE
▶ Writing Fraction Number Stories
PARTNER
ACTIVITY
5–15 Min
(Math Masters, p. 176)
Students use Math Masters, page 176 to write a number story
using fraction addition or subtraction. Ask students to exchange
and solve each other’s problems using benchmarks to estimate the
solution. Students should draw a picture or model for how they
solved the problem.
Teaching Master
Name
LESSON
68
Date
Math Log
PARTNER
ACTIVITY
ENRICHMENT
Time
▶ Playing Fraction Top-It
5–15 Min
(Advanced Version)
(Math Journal 2, Activity Sheets 5–7; Student Reference Book, p. 316;
Math Masters, p. 493)
To further explore using benchmarks, have students play an
advanced version of Fraction Top-It. In playing this version,
students apply their knowledge of benchmarks to estimate sums
and compare them. Each player draws two Fraction Cards from the
deck and records them on Math Masters, page 493. Players then
decide which symbol, <, >, or =, should be used to compare the two
expressions.
Example:
Round
1
Math Masters, p. 176
EM3MM_G5_U06_157-186.indd 176
422
Unit 6
1/19/11 1:36 PM
Using Data; Addition and Subtraction of Fractions
Player 1
_1 + _5
2
6
<, >, or =
>
Player 2
_3 + _0
4
5
Name
STUDY LINK
68
Date
Time
Estimating with Fractions
Circle the best estimate for each situation described below.
1.
3
18
The sum of _4 and _
19 is closest to
0
2.
1
2
9
1
_
The sum of _
10 and 32 is
less than 1
4.
2
1
1
_
The sum of _
11 and 15 is closest to
0
3.
1
greater than 1
Use the circle below to draw a spinner as follows:
_
Shade a red sector that is more than _
8 of the circle, but less than 4 of the circle.
1
1
_
Shade a blue sector that is more than _
4 of the circle, but less than 2 the circle.
1
1
less than _34
5.
3
equal to _4
3
greater than _4
6
1
The number line below shows an estimate for the sum of _
and _8 .
13
1
Explain why the sum is greater than _2 .
0
174
1
2
1
Copyright © Wright Group/McGraw-Hill
The total amount of the circle that is shaded is
Name
Date
LESSON
68
Time
1
Comparing Fractions with _
2
Use the Fraction Cards from Math Journal 2, Activity Sheets 5–7.
Sort the cards into three piles.
Fractions less than _12
Fractions equal to _
2
1
Fractions greater than _
2
1
Place the cards next to one another to check your work. When you are finished,
write the fractions in each pile in the correct box below.
1
Less than _
1
Equal to _
2
1
Greater than _
2
Copyright © Wright Group/McGraw-Hill
2
175
68
Copyright © Wright Group/McGraw-Hill
176
Math Log
LESSON
Time
Date
Name
Name
Date
Time
1 2
4 3
Top-It Record Sheet
Round
Player 1
>, <, =
Player 2
Sample
1
2
3
4
5
Name
Date
Time
1 2
4 3
Top-It Record Sheet
Copyright © Wright Group/McGraw-Hill
Round
Player 1
>, <, =
Player 2
Sample
1
2
3
4
5
493