LESSON
6.3
Simplest Form
FOCUS
COHERENCE
RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction
(n × a)/(n × b) by using visual fraction models, with attention to how
the number and size of the parts differ even though the two fractions themselves are
the same size. Use this principle to recognize and generate equivalent fractions.
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.A.1 5.NF.A.1
3.NF.A.3b
5.NF.A.2
3.NF.A.3c
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 and identify equivalent fractions in
simplest form.
Language Objective
Student pairs prepare a step-by-step
explanation on how to write a fraction as an
equivalent fraction in simplest form.
Materials
MathBoard, color pencils
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 325J.
About the Math
Professional Development
Teaching for Depth
Students use fraction models of equivalent fractions to
explore how to find the simplest form of a fraction. They
find that a fraction is in simplest form if it is modeled
using as few equal-size parts of a whole as possible.
After exploring with models, students learn to list the
factors of the numerator and the denominator, identify
the common factors, and divide the numerator and
denominator by their greatest common factor. They
conclude that a fraction is in simplest form when 1 is the
only common factor of the numerator and denominator.
Professional Development Videos
339A
Chapter 6
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 6.3
_1 .
4
Two fractions are equivalent to The
fractions have the denominators 8 and 16.
4
What are the fractions? 2_8 ; __
16
Vocabulary simplest form
™Interactive Student Edition
™Multimedia eGlossary
with the Interactive Student Edition
Essential Question
How can you write a fraction as an equivalent fraction in
simplest form?
Making Connections
Invite students to tell what they know about fractions.
What does the fraction 5_8 represent? Possible answer: it means five parts
out of eight equal parts in a whole. What are equivalent fractions?
Possible answer: fractions that have the same value
Learning Activity
Connect the story to the problem. Ask the following questions.
Vocabulary Builder
• How many critters are on the log? 16
2
• What fraction of the critters on the log are spiders? __
16
Semantic Mapping can be used to:
__
• What fraction of the critters on the log are ants? 10
16
• learn the
meaning
and uses of
new words;
• What problem are you asked to solve? the fraction of beetles on the
log in simplest form
• see words
in new
contexts;
Literacy and Mathematics
simplest
form
• analyze the relationship between words.
Have students consider the new vocabulary
word and list as many related words or
phrases as possible. This will help students
connect the new word and associate its
concept to previous knowledge.
Choose one or more of the following activities.
• Have students make fraction strips divided into 16 equal parts to
model the fractional parts of the critters that are ants, spiders, and
beetles. Have them shade and label the three different parts using
a different color.
• Ask students to write story problems about a group of three
different types of animals in a zoo. Have them tell how many
animals are in the whole group, and the numbers of each type
of animal. Then, have them write fractions that represent the
different parts of the group and draw fraction strips that model
the three parts.
How can you write
a fraction as an
equivalent fraction in
simplest form?
Lesson 6.3
339B
LESSON
6.3
2 EXPLORE
4.NF.A.1 Explain why amust
fraction
a/b is through
equivalent
to ainfo"
fraction (n × a)/(n × b) by using visual fraction models, with attention
DO
NOT EDIT--Changes
be made
"File
to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle
CorrectionKey=B
to recognize and generate equivalent fractions.
Unlock the Problem
Simplest Form
To introduce the lesson, have students watch
the Real World Video, Ice Cream Production.
Brainstorm with students how fractions are
used for ice cream production. Discuss how
equivalent fractions might be used in recipes;
for example, you may use equivalent fractions
to double the batch of a recipe.
Unlock
Unlock the
the Problem
Problem
Vicki made a fruit tart and cut it into 6 equal pieces.
Vicki, Silvia, and Elena each took 2 pieces of the tart
home. Vicki says she and each of her friends took 1_3
of the tart home. Is Vicki correct?
take?
2
Vicki took home.
• How do the number of parts in the model
for ​ _13 ​ compare to the number of parts in the
model for ​ _26 ​? There are half as many.
For the bulleted question, discuss how
students know ​ _36 ​is equivalent to ​ _12 ​.
ELL Strategy:
Develop Meanings
Write on the board: simple, simpler, simplest.
6
Then write: ​ 3_9 ​= ​ 1_3 ​= ​ __
18  ​.
•Have students copy the fractions and then
label them as simple, simpler, or simplest.
B
STEP 2 Use a red pencil to shade the pieces
Silvia took home.
B
Y
STEP 3 Use a yellow pencil to shade the pieces
R
Y
Elena took home.
R
6 equal-size pieces. The 3 colors on the
The tart is divided into _
model show how to combine sixth-size pieces to make
3 equal third-size pieces.
_
1
So, Vicki is correct. Vicki, Silvia, and Elena each took ____ of
3
the tart home.
Math
Talk
© Houghton Mifflin Harcourt Publishing Company
• What if Vicki took 3 pieces of the tart home and Elena
took 3 pieces of the tart home. How could you combine
the pieces to write a fraction that represents the part
each friend took home? Explain.
MATHEMATICAL PRACTICES 4
Interpret a Result
Compare the models
for 2_6 and 1_3 . Explain how
the sizes of the parts are
related.
Possible explanation: you could combine 6 sixth-size pieces into 2 equal groups of 3.
Since each friend took 1 group (3 pieces) and 2 groups make the whole, each friend
took _12 of the tart home.
Math Talk: Possible explanation: the sizes of the parts in the
model for 1_3 are twice the sizes of the parts in the model for 2_6 .
Math
Talk
for ​ 1_3 ​ compare to the sizes of the parts in
the model for ​ 2_6 ​? They are 2 times as large.
6
• How many pieces did each girl
STEP 1 Use a blue pencil to shade the pieces
parts of the whole, and 2 parts are shaded blue.
Use Math Talk to focus on
students’ understanding of
the relationship between models and
equivalent fractions.
• How do the sizes of the parts in the model
tart cut?
Materials ■ color pencils
Work through the activity with the class. Have
students count the number of pieces of tart.
Then guide students to shade two pieces for
each of the three girls in a different color.
After students have shaded the tart, ask:
MP2 Reason abstractly and quantitatively.
• How can you write a fraction for the blue
shaded part of the tart using the total
number of pieces? Explain. ​ _26 ​; there are 6 equal
• What are the two ways to represent the
amount of the tart Vicki took home using
fractions? ​ _62 ​ and ​ _13 ​
Point out that ​ 2_6 ​and ​ 1_3 ​are equivalent.
• Into how many pieces was the
Activity
Activity
parts of the whole, and 1 part is shaded blue.
Number and Operations—
Fractions—4.NF.A.1
MATHEMATICAL PRACTICES
MP4, MP6, MP7
Essential Question How can you write a fraction as an equivalent
fraction in simplest form?
MATHEMATICAL PRACTICES
• How can you write a fraction for the blue
shaded part of the tart using the number
of different colors? Explain. ​ _13 ​; there are 3 equal
Lesson 6.3
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Lesson 6.3
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Instruction
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Lesson 6.3
Enrich
Name
Simplest Form
To Simplify or Not To Simplify?
Tell whether each fraction below is in simplest form. If a fraction is
in simplest form, write Simplest form. If a fraction is not in simplest
form, write it in simplest form.
A fraction is in simplest form when 1 is the only factor that the
numerator and denominator have in common.
7 is in simplest form.
Tell whether the fraction _
8
Look for common factors in the numerator and the denominator.
7 is 7. List all the
Step 1 The numerator of _
8
factors of 7.
13757
7 is 8. List all
Step 2 The denominator of _
8
the factors of 8.
13858
23458
1.
4
_
8
1
_
2
2.
10
__
3.
4
_
5
Simplest
form
4.
9
__
10
Simplest
form
5.
2
_
3
Simplest
form
6.
19
___
Simplest
form
7.
6
_
8
3
_
4
8.
1
_
6
Simplest
form
9.
7
__
12
Simplest
form
10.
5
__
12
Simplest
form
11.
4
_
6
2
_
3
12.
3
_
4
Simplest
form
13.
2
_
5
Simplest
form
14.
2
_
8
1
_
4
15.
5
_
8
Simplest
form
16.
3
__
Simplest
form
17.
47
___
18.
4
__
1
_
3
5
_
6
12
The factors of 7 are 1 and 7.
100
The factors of 8 are 1, 2, 4, and 8.
Step 3 Check if the numerator and
_ have any common
denominator of 7
8
factors greater than 1.
The only common factor of 7 and 8 is 1.
_ is in simplest form.
So, 7
8
Tell whether the fraction is in simplest form. Write yes or no.
1. ___
​4​​​
__
2. 2
​​
10
__
3. 3
​​
8
no
5
yes
no
10
Simplest
form
100
12
Write the fraction in simplest form.
4. ___
​4​​​
12
1
_
3
5. ___
​6​​​
10
3
_
5
6. __
​3​
6
19.
1
_
2
Describe how you determined which fractions were
already in simplest form.
Possible answer: I looked for common factors in the numerator and
the denominator of each fraction.
339 Chapter 6
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
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18/02/14 12:54 AM
Simplest Form
Explain to students that they can use models
and division to generate a special type of
equivalent fraction, called the simplest form
of a fraction.
Simplest Form A fraction is in simplest form when you can
represent it using as few equal parts of a whole as possible. You need to
describe the part you have in equal-size parts. If you can’t describe the
part you have using fewer parts, then you cannot simplify the fraction.
One Way
• For the first row, why do you divide the
numerator and denominator by 2? Possible
One Way Use models to write an equivalent
fraction in simplest form.
MODEL
WRITE EQUIVALENT
FRACTIONS
RELATE EQUIVALENT
FRACTIONS
2 = __
1
__
8 4
2 ÷ 2 = __
1
______
3
6 = __
___
3
6 ÷ 2 = __
________
1
6 = __
___
1
6 ÷ 6 = __
________
10
8÷2
10 ÷ 2
5
12
12 ÷ 6
2
answer: because the shaded part that shows 2_8 is
the same as the shaded part for 1_4 . You can divide
eighth-size pieces into 4 equal groups, if you group by 2s.
• In the third row, why do you divide by 6
instead of 2? Possible answer: because the shaded
4
6
1
_
parts show that __
12 is the same as the shaded part for 2 .
You can divide twelfth-size pieces into 2 equal groups,
if you group by 6s.
5
MP6 Attend to precision.
• In the third row, suppose you divided
by 3. Would the fraction you write be
an equivalent fraction in simplest form?
Explain. Possible answer: it would be the equivalent
2
6 , you can combine tenth-size parts into equal
To simplify __
10
groups with 2 parts each.
3
6 = ________
6 ÷ 2 = ____
.
So, ___
10 10 ÷ 2
5
fraction 2_4 , but this is not in simplest form because 4 is
not the fewest parts that you can divide twelfth-size
parts into.
6
Another Way Use common factors to write ___
10
in simplest form.
Another Way
A fraction is in simplest form when 1 is the only factor that the
numerator and denominator have in common. The parts of the
whole cannot be combined into fewer equal-size parts to show
the same fraction.
denominator. Circle common factors.
STEP 2 Divide the numerator and denominator
by a common factor greater than 1.
1
2
3
6
Factors of 6: _, _, _, _
1
2
5
10
Factors of 10: _, _, _, _
3
6 = ________
6 ÷ 2 = ___
___
10
10 ÷ 2
5
_3
5
Since 1 is the only factor that 3 and 5 have in common, _
is written in simplest form.
340
© Houghton Mifflin Harcourt Publishing Company
STEP 1 List the factors of the numerator and
• How do you generate a list of factors?
I think of all the pairs of numbers that can be multiplied
to equal the product.
• Why are 3 and 6 only on the factor list for
6? I cannot multiply any number by 3 or 6 and have
a product of 10.
• 1 is a common factor of both 6 and 10. Why
doesn’t it make sense to divide by 1 to find
a fraction in simplest form? Possible answer:
any number divided by 1 is itself, so the new fraction
would be the same as the given fraction.
Use the final statement on the page to check
that students understand how to check for a
fraction in simplest form without using models.
Advanced Learners
Logical / Mathematical
Partner
COMMON ERRORS
Materials 5 to 10 index cards per student
• Challenge students to use mental math
to write fractions in simplest form.
• Give each student 5 to 10 index cards.
Ask students to write a fraction on each
index card that is not in its simplest
form.
5
15
8
122
6
12
144
28
8
2
22
• Have partners exchange cards. Using mental math, each
student will write the simplest form of the fraction on
the card.
• Have partners review their completed cards, and
discuss any suspected errors. They should use models or
common factors to verify any questionable answers.
Error Students only divide the denominator
by the common factor when the numerator
and common factor are equal.
3 3
Example _ = _
9
3
Springboard to Learning Remind students
that they always need to divide the numerator
and denominator by the same factor when
dividing to find simplest form. Suggest that
they check to be sure the numerators of their
equivalent fractions are different. If they are
the same, then they know there is an error.
Lesson 6.3
340
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=A
Name
3 EXPLAIN
MATH
Share
Share and
and Show
Show
BOARD
8 in simplest form.
__
1. Write 10
Share and Show
8 = _______
8 ÷ 2 = ___
4
___
MATH
10 ÷ 2
10
BOARD
Use the checked exercises for Quick Check.
Write the fraction in simplest form.
6
2. ___
2
3. ___
12
Quick Check
Quick Check
If
If
Then
3
2
31
2
1
5
Rt I
Rt I
1
_
2
__
a student misses the checked
exercises
__
4. 6
10
__
5. 4
8
1
_
5
__
6
3
_
Possible explanation: the numerator and denominator
have no common factors other than 1.
9
6. ___
341 Chapter 6
___
8. 10
8
3
_
4
__
20
9. ____
12
1
_
2
__
100
5
_
6
__
1
_
5
__
Tell whether the fraction is in simplest form.
Write yes or no.
__
10. 2
8
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
For Exercises 6­–9, make sure students divide
the numerator and denominator by the same
number. For any fraction to be written in
simplest form, students must verify that 1 is
the only number that can be divided evenly
into the numerator and denominator.
__
7. 4
12
Use Math Talk to focus on students’
understanding of how to identify a
fraction in simplest form.
• What do the numerator and denominator of
a fraction in simplest form have in common?
On Your Own
MATHEMATICAL PRACTICES 6
Explain how you know
a fraction is in simplest
form.
Write the fraction in simplest form.
Math
Talk
Their only common factor is 1.
3
__
Math
Talk
On
On Your
Your Own
Own
Differentiate Instruction with
• Reteach 6.3
• Personal Math Trainer 4.NF.A.1
• RtI Tier 1 Activity (online)
2
_
4
__
no
__
14.
9
11. ___
12
no
__
__
12. 5
6
yes
__
4
13. ___
10
no
__
DEEPER
There are 18 students in Jacob’s homeroom.
Six students bring their lunch to school. The rest eat lunch
in the cafeteria. In simplest form, what fraction of students
eat lunch in the cafeteria?
2
_ of students
3
Chapter 6 • Lesson 3
4_MNLESE342248_C06L03.indd 341
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28/02/14 2:17 PM
MATHEMATICAL PRACTICES
COMMUNICA5&t1&34E7&3&tCONSTRUCT ARGUMENTS
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Use the map for 15−16.
15.
7 Identify Relationships What
fraction of the states in the southwest
region share a border with Mexico? Is this
fraction in simplest form?
_3 ;
4
16.
Regions of the United States
MATHEMATICAL
PRACTICE
Northeast
Southeast
Middle West
Southwest
West
yes
what fraction of the states in the
southeast are on the Gulf of Mexico?
WRITE
SMARTER
Pete says that to write 4_6 as 32_ , you combine
8 , you break apart pieces. Does
pieces, but to write 4_6 as __
12
this make sense? Explain.
DEEPER
Exercise 16 requires students to create a
problem that has the given answer.
Yes. Possible explanation: to write 4_6 as 2_3 , combine sixth-size
÷2
_2
pieces into equal groups of 2. Then _46 = 4____
6 ÷ 2 = 3 . To write
Math on the Spot
Video Tutor
8
4
4
×2
8
_
____
__
as __
12 , break each sixth-piece into 2 pieces. Then 6 = 6 × 2 = 12 .
Use this video to help students model and
solve this type of Think Smarter problem.
Personal Math Trainer
9 of the
In Michelle’s homeroom, __
15
4
__
students ride the bus to school, 12 get a car ride, and
2 walk to school. For numbers 18a–18c, select
__
30
True or False for each statement.
SMARTER
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
18a.
In simplest form, 3_5 of the
students ride the bus to school.
True
False
18b.
In simplest form, 1_4 of the
students get a car ride to school.
True
False
1 of the
__
In simplest form, 15
students walk to school.
True
18c.
© Houghton Mifflin Harcourt Publishing Company
18.
MATHEMATICAL PRACTICES
Math
Show Your Work
_4
6
Problem Solving • Application
MP7 Look for and make use of structure.
For Exercise 15, have students name the
states in the southwest region and the states
that share a border with Mexico to help
them find the numerator and denominator
of the fraction.
What's the Question?
1
_ of the states in this region are on the
3
Gulf of Mexico.
SMARTER
Possible question: in simplest form,
17.
4 ELABORATE
False
342
DIFFERENTIATED INSTRUCTION
D
SMARTER
Personal Math Trainer
Be sure to assign Exercise 18 to students in
the Personal Math Trainer. It features an
animation to help them model and answer
the problem. This item assesses a student’s
ability to write fractions in simplest form.
For students who do not select the correct
answer, ask them what number they used to
divide the numerator and denominator of
each fraction to write it in simplest form.
INDEPENDENT ACTIVITIES
5 EVALUATE Formative
Assessment
Differentiated Centers Kit
Activities
Ruler Challenge
Literature
Sleeping Half the
Day Away
Essential Question
Games
Fraction Action
(BNFT
Students complete
blue Activity
Card 6 by finding
equivalent fractions
using a ruler.
Students read
about what
fraction of the day
different kinds of
animals sleep.
Students practice
matching
fractions with
pictorial models.
Using the Language Objective
Reflect Have student pairs prepare a
step-by-step explanation to answer the
Essential Question.
How can you write a fraction as an
equivalent fraction in simplest form? Possible
answer: I can divide the numerator and denominator of a
fraction by a common factor greater than 1 until the only
common factor is 1.
Math Journal
WRITE
Math
Explain using words or drawings how to
write 6_9 in simplest form.
Lesson 6.3
342
Practice and Homework
Lesson 6.3
Name
Simplest Form
COMMON CORE STANDARD—4.NF.A.1
Extend understanding of fraction equivalence
and ordering.
Write the fraction in simplest form.
Practice and Homework
6
1. __
10
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.
2. _6
8
___
6÷2
3
_
10 = 10 ÷ 2 = 5
6
__
___
5. 100
100
3. _5
5
3
_
4
__
6. _2
6
1
_ or 1
1
__
8
4. __
12
2
_
1
_ or 1
1
__
3
__
4
8. __
10
7. 2_
8
1
_
3
__
2
_
1
_
5
__
4
__
Tell whether the fractions are equivalent.
Write = or ≠.
6 ≠ __
1
9. __
12
12
10. _3 ≠ _5
4
6
6 = _3
11. __
10
5
3 ≠ _1
12. __
12
3
Problem
Problem Solving
Solving
13. At Memorial Hospital, 9 of the 12 babies
© Houghton Mifflin Harcourt Publishing Company
born on Tuesday were boys. In simplest
form, what fraction of the babies born on
Tuesday were boys?
14. Cristina uses a ruler to measure the length
of her math textbook. She says that the book
4
__
is 10
meter long. Is her measurement in
simplest form? If not, what is the length of
the book in simplest form?
3
_
4
15.
_ meter
No; 2
5
Math Explain using words or drawings how to
WRITE
write 6_9 in simplest form.
Check students’ work.
Chapter 6
343 Chapter 6
343
Lesson Check (4.NF.A.1)
1. Six out of the 12 members of the school
10
__ in simplest form.
2. Write 12
choir are boys. In simplest form, what
fraction of the choir is boys?
1
_
2
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.
5
_
6
Spiral Review (4.OA.A.3, 4.OA.B.4, 4.NBT.B.5, 4.NF.A.1)
3. Each of the 23 students in Ms. Evans’ class
4. List two common factors of 36 and 48.
raised $45 for the school by selling coupon
books. How much money did the
class raise?
Possible answer: 4 and 6
3
5. Bart uses __
12 cup milk to make muffins.
3
Write a fraction that is equivalent to __
12 .
_
Possible answer: 1
4
6. Ashley bought 4 packages of juice boxes.
There are 6 juice boxes in each package.
She gave 2 juice boxes to each of 3 friends.
How many juice boxes does Ashley
have left?
18 juice boxes
© Houghton Mifflin Harcourt Publishing Company
$1,035
FOR MORE PRACTICE
GO TO THE
344
Personal Math Trainer
Lesson 6.3
344