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```LESSON
7.6
Rename Fractions and Mixed
Numbers
FOCUS
COHERENCE
RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
4.NF.B.3b Understand a fraction a/b with a > 1 as a sum of fractions
1/b. Decompose a fraction into a sum of fractions with the same
denominator in more than one way, recording each decomposition
by an equation. Justify decompositions, e.g., by using a visual fraction model.
Also 4.MD.A.2
MATHEMATICAL PRACTICES
MP1 Make sense of problems and persevere in solving them. MP4 Model with mathematics.
F C R Coherence:
Before
3.NF.A.1 4.NF.B.3b 5.NF.A.1
5.NF.A.2
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
Write fractions greater than 1 as mixed numbers
and write mixed numbers as fractions greater
than 1.
Language Objective
Students demonstrate and tell in their own
words how to rename mixed numbers as
fractions greater than 1 and rename fractions
greater than 1 as mixed numbers.
Materials
MathBoard
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 383J.
Professional Development
Why Teach This
In real-world situations, it is easier to understand mixed
numbers than fractions greater than 1. For example, if a
student served 2 whole pizzas and 1_3 of another pizza at a
party, you would say the student served 2 1_3 pizzas, not
7
_ pizzas. That’s because 2 1_ is easier to understand than 7_ .
3
3
3
Both topics covered in this lesson are prerequisite skills
for fraction computation. Students will rename fractions
greater than 1 when they add fractions and mixed
numbers. They will rename mixed numbers as fractions
greater than 1 when they subtract mixed numbers with
renaming. Fractions greater than 1 that were once called
improper fractions are grouped with all other fractions,
including fractions that are equal to one.
Professional Development Videos
417A
Chapter 7
Interactive Student Edition
Personal Math Trainer
Math on the Spot
Animated Math Models
iTools: Fractions
HMH Mega Math
1 ENGAGE
Daily Routines
Common Core
Problem of the Day 7.6
Write a number that rounds to 57,800.
Essential Question
How can you rename mixed numbers as fractions greater
than 1 and rename fractions greater than 1 as mixed numbers?
Making Connections
Vocabulary mixed number
Interactive Student Edition
Multimedia eGlossary
Fluency Builder
with the Interactive Student Edition
Common Core Fluency
Standard 4.OA.A.3
Mental Math When students rename a
mixed number as a fraction greater than 1,
they will learn to multiply ﬁrst and then
add. Have students use mental math to
solve problems that require multiplication
Invite students to tell you what they know about fractions.
What are the top and bottom numbers of a fraction called?
numerator and denominator What are equivalent fractions? fractions
that name the same amount How can fraction strips and number lines
the number of parts in the fraction.
Learning Activity
Connect the story to the problem. Direct students’ attention to the
mixed number 1 2_3 .
• What does the number 1 2_3 represent in the problem? the number of
buckets of sand scooped up so far
• What part of the mixed number 1 2_3 is the whole number? 1
• What part of the mixed number 1 2_3 is the fraction? 2_3
Literacy and Mathematics
1. (4 × 3) + 5 17
2. (3 × 9) + 2 29
3. (2 × 7) + 1 15
4. (1 × 6) + 7 13
Choose one or more of the following activities.
5. (6 × 4) + 3 27
6. (8 × 1) + 6 14
7. (5 × 8) + 9 49
8. (2 × 5) + 8 18
• Invite students to use mathematical language to write what the
• Have students draw two equal-sized circles or rectangles to
represent the two buckets of sand in the problem. Have them
shade the circles or rectangles to show 1 2_3 .
How can you rename
mixed numbers as fractions
greater than 1 and rename
fractions greater than 1 as
mixed numbers?
Lesson 7.6
417B
LESSON
7.6
4.NF.B.3b Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Decompose a fraction into a sum of fractions
with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions,
e.g., by using a visual fraction model.
2 EXPLORE
Lesson 7.6
Name
Unlock the Problem
Rename Fractions and Mixed Numbers
• What information do you need to ﬁnd? if
Unlock
Unlock the
the Problem
Problem
Mr. Fox has 26_3 loaves of corn bread. Each loaf
was cut into 6_1-size pieces. If he has 14 people
over for dinner, is there enough bread for each
person to have 1 piece?
Mr. Fox has at least 14 pieces of bread
Example
MP4 Model with mathematics.
• In Step 1, what does one 1 strip represent?
• In Step 2, what fraction represents one
• In Step 3, how did you ﬁnd 6_6 + 6_6 + 3_6 ?
6
• How much bread does Mr. Fox need for
14 people?
__
at least 14 sixth-size pieces, or 14
6
Example Write a mixed number as a fraction.
MODEL AND RECORD
THINK
STEP 1 Model 2 3_6 .
• How did you know that Mr. Fox has
enough bread? Because he has 15 sixth-size pieces
DEEPER
MP7 Look for and make use of structure.
• In order to ﬁnd the numerator when
writing the mixed number 2 3_6 as a fraction,
you could take the following steps:
6 × 2 + 3 = 15. Relate this to division.
6
6
STEP 3 Find the total number
of 1_6 -size pieces in 2 3_6 .
__ + 6
__ + 3
__.
Think: Find 6
6
Have students make cards showing a mixed
number on one card and the equivalent
fraction greater than 1 on another card.
• Have students sort the cards into two piles:
fractions greater than 1 and mixed numbers.
6
Possible answer: you can write 1 1_4 as 1 + 1_4
or 4_4 + 1_4 . The sum is 5_4 .
Math
Talk
15 sixth-size pieces in 26_3.
There are _
So, there is enough bread for 14 people to each have 1 piece.
MATHEMATICAL PRACTICES 7
Look for Structure Give
an example of how to
write a mixed number as
a fraction without using a
model.
Chapter 7 417
3
Reteach 7.6
Differentiated
Instruction
Enrich 7.6
2
1
Lesson 7.6
Reteach
Name
Illustrate Understanding
6
15
__ = ____
23
6
6
Possible answer: if I divide 15 by 6, the quotient would
be 2 with a remainder of 3. 2 3_6 represents the quotient
with the remainder written in fractional form.
Strategy:
3
6
3
6
6
__ = ____ + ____ + ____
23
6
6
6
6
© Houghton Mifflin Harcourt Publishing Company
_5
4
6
6
STEP 2 Find how many 1_6 -size
pieces are in each
whole. Model 2 3_6 using
only 1_6 -size pieces.
Use Math Talk to focus students’
understanding on how to write a
mixed number as a fraction.
rewritten 4_4 + _14
3
6
3
__ = _
1 +_
1 + ____
23
6
6
Math
Talk
• How can you write 1 1_4 as an addition
expression with fractions? 1 + 1_4 , which can be
1
1
and he only needs 14, so he has enough.
• Then have students match each mixed
number with its equivalent fraction and
draw a picture to show that the fractions
are equivalent.
relative to the whole?
_1 of the whole
To find how many 6_1-size pieces are in 26_3, write
26_3 as a fraction.
denominator the same.
ELL
• What is the size of 1 piece of bread
A mixed number is a number represented by
a whole number and a fraction. You can write a
mixed number as a fraction.
• What is the sum expressed as a fraction?
Numbers and Operations—
Fractions—4.NF.B.3b Also 4.MD.A.2
MATHEMATICAL PRACTICES
MP1, MP4
Essential Question How can you rename mixed numbers as fractions
greater than 1 and rename fractions greater than 1 as mixed numbers?
MATHEMATICAL PRACTICES
Lesson 7.6
Enrich
Name
Rename Fractions and
Mixed Numbers
The Rename Game
Find the missing number.
A mixed number is made up of a whole number and a fraction.
You can use multiplication and addition to rename a mixed number
as a fraction greater than 1.
Rename 2 _65 as a fraction.
First, multiply the denominator, or the number
of parts in the whole, by the whole number.
2
6 3 2 5 12
1.
256
51 ___ 5 ____
5
5
total number
17 of parts
5
=
6 number of
6
parts in the whole
3.
4.
5.
17.
5 __
1
__ 5 506
____
4
8
105
12 1 5 5 17
So,
218
72 ___ 5 ____
3
3
2
422
1 5 ____
__
2
4
2 _56
2.
1
6
63
5 5 ___
102 ___
12 12
6.
__ 5 224
____
371
3
1,229
6
You can use division to write a fraction greater than 1 as a mixed number.
7.
16
Rename __
3 as a mixed number.
__ as a mixed number, divide the numerator by
To rename 16
3
the denominator.
Use the quotient and remainder to write a mixed number.
5
3q 16
⫺ 15
1
Possible answer: I can use division to
rename fractions greater than 1 as mixed
numbers. I can use multiplication and
addition to rename mixed numbers as
fractions greater than 1.
16 5 51
_.
So, __
3
3
Write the mixed number as a fraction.
1.
25
3 __
3
11
__
3
2.
35
4__
5
23
__
5
3.
35
4__
8
35
__
8
4.
15
2__
6
13
__
6
8.
Write the fraction as a mixed number.
___
5. 32
5
5
_
62
5
___
6. 19
3
5
_
61
3
___
7. 15
4
5
_
33
4
___
8. 51
10
5
Tell how you rename fractions greater than 1 as
mixed numbers and mixed numbers as fractions greater than 1.
1
5 __
10
Stretch Your Thinking Is it possible for two fractions greater
than 1 that have different numerators and denominators to be renamed
as the same mixed number? Give an example.
112 can both
___ and ___
6
3
be renamed as 37 1_3 .
417 Chapter 7
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
7-15
Reteach
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
7-16
Enrich
Example
In this example, students use a number line
and fraction strips to write a fraction greater
than 1 as a mixed number.
• How is this problem similar to the ﬁrst
problem? I have to rename a quantity in order to
Example Write a fraction greater than 1 as a mixed number.
To weave a bracelet, Charlene needs 7 pieces of brown
solve both problems.
• How are the problems different? In the ﬁrst
Write 3_7 as a mixed number.
THINK
MODEL AND RECORD
STEP 1 Model 7_3 .
0
3
1
3
3
3
2
3
4
3
5
3
7
3
6
3
•
1
1
1
1
1
1
1
7 = ____
__
+ ____ + ____ + ____ + ____+ ____ + ____
3
3
STEP 2 Find how many wholes are in
many thirds are left over.
_7 ,
3
and how
3
0
3
1
3
3
3
3
3
2
3
STEP 3 Write _73 as a mixed number.
3
3
1
3
so write 8_5 as 5_5 + 3_5 . Replace 5_5 with 1. 5_5 + 3_5 = 1 + _35
= 1 3_5 .
1
1 + ____
1 +_
_
3
1
7 = 2 ____
__
3
3 EXPLAIN
3
MATH
BOARD
Write the unknown numbers. Write mixed numbers above
the number line and fractions greater than one below the number line.
1.
1
1 61
12
6
1 63
14
6
1 65
2
2 61
22
6
2 63
24
6
2 65
3
6
6
7
6
8
6
9
6
10
6
11
6
12
6
13
6
14
6
15
6
16
6
17
6
18
6
© Houghton Mifflin Harcourt Publishing Company
_
21
Share
Share and
and Show
Sh
3
needs 7 pieces of thread that are 1_3 yard long, so she
needs 7_3 yards of thread altogether.
• What fraction represents 1 yard in
Step 2? 3_3
MP2 Reason abstractly and
quantitatively.
• How could you write 8_5 as a mixed number
without using a model? Possible answer: 5_5 = 1,
7
3
6
3
5
3
3
3 51
3
3 51
7=
__
3
4
3
3
problem, I had to rename a mixed number as a fraction
greater than 1. In this problem, I am renaming a
fraction greater than 1 as a mixed number.
Why do you model _7 in Step 1? Charlene
Share and Show
MATH
M
A
TH
ATH
TH
BOARD
B
The ﬁrst problem connects to the learning
model. Have students use the MathBoard to
explain their thinking.
COMMON ERRORS
418
Error Students may miscount when writing a
Visual / Kinesthetic
Individual
Materials inch ruler
1
• Some rulers show smaller units of 1_4 , 1_8 , or __
16 . Have
students determine the units on their ruler by
counting how many spaces are between the whole
numbers.
• Students can use a ruler to rename fractions and
mixed numbers.
• Find 3 7_8 inches. Starting from 3, count seven
1
_ -inch marks to the right of 3.
8
• To convert 3 7_8 to a fraction greater than 1, count
the number of 1_8 -inch marks to the right of 0.
__ to a mixed number
• How would you convert 21
4
using a ruler? See right.
fraction greater than 1 as a mixed number.
_ = _1 + _1 + _1 + _1 + _1 + _1 + _1
Example 8
5
5
5
= _55 + 2_5
= 1 2_5
5
5
5
5
5
Springboard to Learning Have students
check their answers by rewriting the mixed
number as a fraction greater than 1. They
should get the original fraction.
Possible answer: count twenty-one 1_4 -inch marks to the
right of 0. Then identify the whole number to the left of
that point (5). Then count the number of 1_4 -inch marks
that are to the right of 5, and write that as a fraction. This
will give me the whole number and the fraction part: 5 1_4 .
Lesson 7.6
418
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=A
Name
Use the checked exercises for QuickCheck.
Students should show their answers for Quick
Check on the MathBoard.
Write the mixed number as a fraction.
9
_
8
Math
Talk
Use Math Talk to focus on students’
understanding of why it is easier to
compare numbers written in the same form.
• In what two ways could you rename 1​ _35 ​ and ​ _75 ​ so that both numbers are in the same
form? Rename 1​ _35 ​as a fraction; rename ​ _75 ​as a mixed
_
23
On
On Your
Own
Then
1
Differentiate Instruction with
• Reteach 7.6
• Personal Math Trainer 4.NF.B.3b
• RtI Tier 1 Activity (online)
9. 3_2
3
27
__
10
MATHEMATICAL
PRACTICE
8
If students complete the checked exercises
correctly, they may continue with the
remaining exercises.
Exercises 8–10? Rename each fraction as a mixed
number. I should get the original mixed number.
MP8 Look for and express regularity in
repeated reasoning. Exercises 11 –13 involve
higher order thinking skills because students
need to find the unknown number to make
the equation true.
Math
Talk
MATHEMATICAL PRACTICES 6
Describe how you can
compare 1 3_5 and 7_5 .
11
__
3
12.
22
__
5
DEEPER
Pen has 1_2 -cup and 1_8 -cup
measuring cups. What are two ways he could
measure out 1 3_4 cups of flour?
Possible answers: three 1_2 -cups and
two _18 -cups; fourteen _18 -cups
■
57 =
__
13. __
11 ■ 11
__
■ 5_6 = 23
6
6
14.
10. 4_2
5
Use Repeated Reasoning Algebra Find the unknown numbers.
13 = 1■
__
11. __
7
7
On Your Own 419 Chapter 7
7
8. 2__
10
Rt I
Rt I
a student misses the checked
exercises
10
Write the mixed number as a fraction.
© Houghton Mifflin Harcourt Publishing Company
If
2
3
1__
5
Possible description: I can rename the mixed number as a
fraction or rename the fraction as a mixed number. Then I
can compare them as two fractions or two mixed numbers.
© Houghton Mifflin Harcourt Publishing Company
If
31
13
7. __
10
_
11
4
1 ​ _35 ​∙ ​ _58 ​; to compare ​ _85 ​and ​ _75 ​, compare the numerators
because the denominators are the same.
• If you rename ​ 7_5 ​ as a mixed number, how
can you compare the numbers? Possible
answer: ​ _75 ​∙ 1​ _25 ​; to compare 1​ 5_3 ​and 1​ _25 ​, compare the
numerators of the fractions because the whole numbers
and the denominators of the fractions are the same.
2
5
_
3
8
_
5
6. _6
5
11
5. __
4
• If you rename 1 ​ _35 ​ as a fraction, how could
you compare the numbers? Possible answer:
3
4. 1_2
3
Write the fraction as a mixed number.
number.
Quick Check
Quick Check
3. 1_3
5
2. 1_1
8
3
15.
5; 2
DEEPER
needs 3 1_2 cups of flour. Juanita only has a
1_ -cup measuring cup. How many 1_ cups of
4
4
flour will Juanita use to prepare the bread?
fourteen _14 cups of flour
Chapter 7 • Lesson 6
4_MNLESE342255_C07L06.indd 419
419
28/02/14 7:14 PM
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=A
MATHEMATICAL PRACTICES
ANALYZE • LOOK FOR STRUCTURE • PRECISION
Problem
Problem Solving
Solving •• Applications
Applications
4 ELABORATE
Use the recipe to solve 16–18.
16.
2 Reason Quantitatively Cal is
making energy squares. How many 2_1 cups of
peanut butter are used in the recipe?
MATHEMATICAL
PRACTICE
Problem Solving • Applications
three 1_2 cups
17.
MATHEMATICAL PRACTICES
SMARTER
Suppose Cal wants to make
2 times as many energy squares as the recipe
makes. How many cups of bran cereal should he
as a fraction greater than 1 in simplest form.
WRITE
MP2 Reason abstractly and
quantitatively. To find how many ​ _12 ​cups of
peanut butter are in 1​ _12 ​cups, students need
to rename 1​ 2_1 ​as a fraction. The numerator
tells the number of ​ _12 ​cups.
_ cups; 13
__ cups
61
2
2
18. Cal added 28_3 cups of raisins. Write this mixed
SMARTER
number as a fraction greater than 1 in simplest
form.
Exercise 17 requires students to double a
mixed number. Students can double the
whole number, double the fraction, and
simplify the fraction.
19
__
8
19.
DEEPER
Jenn is preparing brown rice.
She needs 12_1 cups of brown rice and 2 cups
of water. Jenn has only a 8_1-cup measuring cup.
How many 8_1 cups each of rice and water will
Jenn use to prepare the rice?
Math on the Spot
Video Tutor
twelve 1_8 -cups rice; sixteen 1_8 -cups water
Use this video to help students model and
solve this type of Think Smarter problem.
Draw a line to show the mixed number and
fraction that have the same value.
SMARTER
12_
23_
41_
12_
•
•
•
•
•
•
•
5
•
30
__
3
8
13
__
3
3
4
_
3
3
© Houghton Mifflin Harcourt Publishing Company
20.
8
_
5
420
4_MNLESE342255_C07L06.indd 420
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
SMARTER
In Exercise 20, students must find the pair
of fractions that have an equivalent value.
If students choose a fraction and mixed
number that have different numbers in the
denominators, they may not understand the
basic underlying concept of fractions.
28/02/14 7:18 PM
DIFFERENTIATED INSTRUCTION
INDEPENDENT ACTIVITIES
5 EVALUATE Formative
Assessment
Essential Question
Using the Language Objective
Differentiated Centers Kit
Activities
Fantastic Fractions
Activities
Pencil Me In
Games
Fraction Action
Games
Students complete
orange Activity
Card 8 by drawing
pictures of fractions
greater than 1.
Students complete
blue Activity
Card 8 by using
write, compare,
and order mixed
numbers.
Students practice
matching
fractions with
pictorial models.
Reflect Have students demonstrate and
explain in their own words how to answer
the Essential Question.
How can you rename mixed numbers
as fractions greater than 1 and rename
fractions greater than 1 as mixed numbers?
Mixed Numbers: I can model the mixed number using
fraction strips, and count how many parts I have.
Fractions: I can model the fraction using a number line,
and count the number of whole units and parts left over.
Math Journal
WRITE
Math
Draw and explain how you can use a
number line to rename a fraction greater
than 1 as a mixed number.
Lesson 7.6 420
Practice and Homework
Lesson 7.6
Name
Rename Fractions and Mixed Numbers
COMMON CORE STANDARD—4.NF.B.3b
Build fractions from unit fractions by applying
and extending previous understandings of
operations on whole numbers.
Write the mixed number as a fraction.
Practice and Homework
1. 2_3_
2. 4_1_
5
5+_
5+_
3.
Think: Find _
5 5 5
13
___
5
__
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.
3. 1_2_
3
5. 4_1_
5
7
_
5
__
13
__
3
__
7
6. 1___
8
7. 5_1_
10
33
__
8
__
17
__
10
__
Write the fraction as a mixed number.
31
20
9. ___
10. ___
6
10
_
51
6
__
2
11
__
2
__
15
11. ___
8
1 7_8
__
2
__
4. 3_2_
3
11
__
3
__
8. 2_3_
8
19
__
8
__
13
12. ___
6
_
21
6
__
Problem
Problem Solving
Solving
13. A recipe calls for 2 2_4 cups of raisins, but Julie
© Houghton Mifflin Harcourt Publishing Company
_1
4
15.
14. If Julie needs 3 1_4 cups of oatmeal, how many
only has a cup measuring cup. How many
1
_ cups does Julie need to measure out
4
2 2_4 cups of raisins?
_1
4
_ cups
ten 1
4
________
_ cups
thirteen 1
4
________
cups of oatmeal will she use?
Math Draw and explain how you can use a number
WRITE
line to rename a fraction greater than 1 as a mixed number.
Check students’ drawings.
Chapter 7
PROFESSIONAL
DEVELOPMENT
421
CCSS.Math.Practice.MP5 Use appropriate tools
strategically.
Using tools such as fraction strips can help students build their
understanding of how mixed numbers and fractions greater than 1
are related. Fraction strips provide students with a concrete way to
visualize equivalent numbers using the same model. Similarly, using
fraction strips with number lines shows number equivalence. The
models also provide students with a deeper understanding of the
meaning of the numerator and the denominator of fractions.
Discuss the models students used in this lesson in order to write mixed
numbers as fractions greater than 1 and vice versa.
• How does using fraction strips help you write a mixed number
as a fraction greater than 1? I can use fraction strips to model
the wholes and parts in a mixed number. I can use the appropriate
sized strips to model the wholes. Then I count all of the parts I have
to determine the numerator of the fraction.
• How does using fraction strips help you to write a fraction greater than 1 as a mixed number? I can use fraction strips to model
the fraction greater than 1. First I can determine how many wholes
I have. Then the number of left over parts makes up the fraction of
the mixed number.
421 Chapter 7
TEST
PREP
Lesson Check (4.NF.B.3c)
__ .
1. Write a mixed number that is equivalent to 16
3
2. Stacey filled her 1_2 cup measuring cup
seven times to have enough flour for a
cake recipe. How much flour does the cake
recipe call for?
1
3
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.
1 cups
3_
2
Spiral Review (4.NBT.B.5, 4.NBT.B.6, 4.NF.A.1, 4.NF.B.3d)
collection book. She put 14 stamps
on each page. If she completely filled
16 pages, how many stamps did she
put in the book?
224 stamps
5. During a bike challenge, riders have to
collect various colored ribbons. Each 1_2 mile
they collect a red ribbon, each 1_8 mile they
collect a green ribbon, and each 1_4 mile
they collect a blue ribbon. Which colors
of ribbons will be collected at the 3_4 mile
marker?
green and blue
4. Brian is driving 324 miles to visit some
friends. He wants to get there in 6 hours.
How many miles does he need to drive
each hour?
54 miles
6. Stephanie had 7_8 pound of bird seed. She
used 3_8 pound to fill a bird feeder. How
much bird seed does Stephanie have left?
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3. Becki put some stamps into her stamp
4 pound
_
8
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Personal Math Trainer
Lesson 7.6
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