Investigate • Model Equivalent Fractions
LESSON 9.6
Investigate ? Model Equivalent Fractions
FOCUS COHERENCE RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Also 3.NF.A.1, 3.NF.A.2a, 3.NF.A.2b, 3.NF.A.3, 3.NF.A.3b, 3.NF.A.3c, 3.G.A.2
MATHEMATICAL PRACTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.) MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP7 Look for and make use of structure.
F C R Coherence:
Standards Across the Grades Before Grade 3 After 2.NBT.A.4 3.NF.A.3a 4.NF.A.1
Learning Objective
Model equivalent fractions by folding paper, using area models, and using number lines.
Language Objective
Student pairs illustrate and write captions showing how you can use models to find equivalent fractions.
Materials
MathBoard, sheet of paper, crayon or color pencil, Fraction Strips (see eTeacher Resources)
F C R Rigor:
Level 1: Understand Concepts....................Share and Show ( Checked Items) Level 2: Procedural Skills and Fluency.......On Your Own, Practice and Homework Level 3: Applications..................................Think Smarter and Go Deeper
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions for this chapter, see page 505J.
About the Math
Professional Development
Modeling Equivalent Fractions
Equivalent fractions are fractions that represent the same area or number. The concept of equivalent fractions can be presented by having students compare models with different denominators that have the same areas shaded. Students learn that equal areas show equivalent fractions. The number of equal parts can change as long as the total area remains the same.
Students also explore equivalent fractions on a number
line. They learn that two fractions are equivalent if the
fractions are at the same point on a number line. The
three
models
all
show
that
3 _
4
is
equivalent
to
6 _
8
.
Professional Development Videos
0 1 2345678 888888888
0
1
2
3
4
4
4
4
4
4
3__ 4
=
6__ 8
539A Chapter 9
Daily Routines
Common Core
Problem of the Day 9.6
Juanita has 56 stuffed animals. She wants to display them on 8 shelves. Write a multiplication number sentence that can help Juanita divide the animals equally.
8 ? 7 = 56 or 7 ? 8 = 56
Vocabulary equivalent, equivalent fractions
Interactive Student Edition Multimedia eGlossary
Vocabulary Builder
Materials Word Web (see eTeacher Resources)
Have students make a word web for the term equivalent. Students should generate words that relate to each other structurally by writing words with the same root or the same suffix or prefix on the arms of the graphic.
1 ENGAGE
with the Interactive Student Edition
Essential Question
How can you use models to find equivalent fractions?
Making Connections
Invite students to tell you what they know about creatures that hop.
Can you name some insects that hop or jump? grasshoppers, crickets, fleas
Can you name some animals that hop or jump? frogs, rabbits, kangaroos, flying squirrels, dolphins
Learning Activity
Ask students to restate the problem in their own words.
? What is going on in this problem? Gable is hopping to his friend's
house but stops before he gets there.
?
What
fraction
represents
how
far
Gable
has
hopped?
2__ 3
? What is the question you have to answer? What fraction is equivalent to 2_3_?
Literacy and Mathematics
Have students work together in pairs.
1. Have one student in each pair use a ruler to draw a number line scaled from 0 to 1, divided into thirds.
2. Have that student draw a point that shows 2_3.
3. Have the other student draw a circle and divide it into thirds,
shading
2 _ 3
of
it.
4.
Then,
have
the
partners
explain
why
both
models
show
2 _ 3
.
How can you use models to find
equivalent fractions?
Lesson 9.6 539B
LESSON 9.6
2 EXPLORE
Investigate HOannds
MATHEMATICAL PRACTICES
MP5 Use appropriate tools strategically.
Work together with students to complete the steps of the activity in Part A. Be sure students fold their papers in two equal parts and label each half.
? What do the two halves equal? one whole
Have students follow the steps to fold and label the paper as directed in Part B.
? Describe how one half of the paper is now labeled. One half is divided into two equal parts. Each
part is labeled one fourth.
Have students follow the steps to fold and label the paper as directed in Part C. Students can label the eighths on the back side of the folded paper if the paper is too crowded. You may also suggest that students label the eighths by writing the fractions smaller.
? Describe how one fourth of the paper is now labeled. One fourth is divided into two equal
parts. Each part is labeled one eighth.
? How many one-eighth parts are in one half? 4
Guide students to name the equivalent fractions and to complete the statement.
MP7 Look for and make use of structure.
?
Look
at
the
fractions
1 _
2
,
2_4,
and
4 _
8
.
What
do
you notice about the relationship between
the numerator and denominator. Possible
answer: For each fraction, the denominator is twice the numerator.
ELL Strategy: Scaffold Language
Students can use sentence frames to support them in describing how to solve a problem.
? As students work through problems to find equivalent fractions, have them describe the process to a partner.
? Write the sentence frame on the board for
students to use. First, I drew a
and
divided it into
pieces. Then, I shaded
pieces. That's how I know that
and
are equivalent.
539 Chapter 9
3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Name
Model Equivalent Fractions
Essential Question How can you use models to find equivalent fractions?
IInnvveessttiiggaattee
HOannds
Materials sheet of paper crayon or color pencil
Two or more fractions that name the same amount are
called equivalent fractions. You can use a
sheet
of
paper
to
model
fractions
equivalent
to
1 _ 2
.
A. First, fold a sheet of paper into two equal parts. Open the paper and count the parts.
1 _
There are _ 2 equal parts. Each part is _ 2 of the paper.
Lesson 9.6
Number and Operations--Fractions-- 3.NF.A.3a Also 3.NF.A.1, 3.NF.A.2a, 3.NF.A.2b, 3.NF.A.3, 3.NF.A.3b, 3.NF.A.3c, 3.G.A.2 MATHEMATICAL PRACTICES MP4, MP5, MP7
Shade
one
of
the
halves.
Write
1 _ 2
on
each
of
the halves.
B. Next, fold the paper in half two times. Open the paper.
Now there are _ 4 equal parts. Each part is
1 _
_ 4 of the paper.
Write
1 _ 4
on
each
of
the
fourths.
Look
at
the
shaded
parts.
1 _ 2
=
_2__ 4
C. Last, fold the paper in half three times.
Now there are _ 8 equal parts. Each part is
1 _
_8 of the paper.
Write
1 _ 8
on
each
of
the
eighths.
Find
the
fractions
equivalent
to
1 _ 2
on
your
paper.
So,
1 _ 2
,
_2__ 4
,
and
_4__ 8
are
equivalent.
Chapter 9 539
? Houghton Mifflin Harcourt Publishing Company 1 4
Reteach 9.6
Name
Model Equivalent Fractions
Equivalent fractions are two or more fractions that name the same amount.
You can use fraction circles to model equivalent fractions.
Find a fraction that is equivalent to _21.
_ 1 2
5
__ 4
Step 1 Look at the first circle. It is divided into 2 equal parts. Shade one part to show 1_2.
Step 2 Draw a line to divide the circle into 4 equal parts because 4 is the denominator in the second fraction.
Step 3 Count the number of parts shaded now. There are 2 parts out of 4 parts shaded.
1 _ 2
5
_ 2 4
So,
1 _ 2
is
equivalent
to
2_4.
1 4
1 4
3 2 1 Lesson 9.6 Reteach
11 22
1 4
Shade the model. Then divide the pieces to find the
equivalent fraction. Possible shading is shown.
1.
2.
2 1__
4
5
____ 8
3.
4 1__
2
5
____ 8
4.
4 2__
3
5
___ 6
Chapter Resources ? Houghton Mifflin Harcourt Publishing Company
6 3__
4
5
____ 8
9-15
Reteach
Enrich 9.6
Differentiated Instruction
Name
Name Equivalent Fractions
For each of the following shapes, shade some of the parts. Write the fraction that represents the parts you shaded. Then use the shape to write an equivalent fraction for the parts you shaded.
1.
2.
Lesson 9.6 Enrich
Possible answers are given.
2 _
Fraction:
4
1 _
Equivalent Fraction:
2
3.
3 _
Fraction:
4
6 _
Equivalent Fraction:
8
4.
2 _
Fraction:
8
1 _
Equivalent Fraction:
4
5.
2 _
Fraction:
2
6 _
Equivalent Fraction:
6
6.
2 _
Fraction:
6
1 _
Equivalent Fraction:
3
1 _
Fraction:
4
2 _
Equivalent Fraction:
8
7.
Stretch Your Thinking
Draw
a
model
that
shows
_3_ 3
shaded.
Then use your drawing to find two equivalent fractions.
Check students' drawings. Possible answer:
1_1,
_ 6 6
Chapter Resources ? Houghton Mifflin Harcourt Publishing Company
9-16
Enrich
N Draw Conclusions
1. oEnxpel1_4aipnahrtoownmyoaunryp1_8appaerrt.s are equivalent to
Possible
explanation:
there
are
two
1 _ 8
parts
in
one
1 _ 4
part,
= so
2 _ 8
1 _ 4
.
Two or more numbers that have the same value or name the same amount are equivalent.
2.
SMARTER What do you notice about how
the numerators changed for the shaded part as
you folded the paper? _ The_ numer_ ators g_ ot lar_ ger.
What does this tell you about the change in the
number of parts? _ The _ numb_ er of p_ arts in_ creas_ ed.
How did the denominators change for the shaded
part as you folded? __Th_ ey do_ ubled_ . _
What does this tell you about the change in the size of the parts? _ The s_ ize of_ the p_ arts go_ t sma_ ller.
N MMaakkee CCoonnnneeccttiioonnss
HOannds
You can use a number line to find equivalent fractions.
Find a fraction equivalent to 2_3.
Materials fraction strips
MTaatlkh
MATHEMATICAL PRACTICES 2
Use Reasoning Explain how the number of sixths in a distance on the number line is related to the number of thirds in the same distance.
0
1
6
6
2 6
3 6
4 6
5 6
6 6
The number of sixths is
double the number of thirds.
0
1
2
3
3
3
3
3
STEP 1 Draw a point on the number line to represent the distance 2_3.
STEP 2 Use fraction strips to divide the number line into sixths. At the end of each strip, draw a mark on the number line and label the marks to show sixths.
STEP 3 Identify the fraction that names the same point as 2_3. _ 4_6
So,
_2_ 3
=
_46__.
540
Advanced Learners
Logical / Mathematical Individual / Partners
? Present this problem to students:
There were three sandwiches of the same size.
Mwaasrkcuattein1_2toof8tehqeufiarlsptioencees..TAhme yseactoen4d
sandwich pieces. The
third sandwich was cut into 6 equal pieces. Lola ate
3 pieces. How could you cut another sandwich into
equal pieces so that David could eat an equivalent
amount?
? Ask students to draw a picture to solve the problem.
? Have students write to explain how they solved the problem and label their drawings to illustrate their explanations. Check students' work.
? Houghton Mifflin Harcourt Publishing Company
Draw Conclusions
Have students read the Math Idea before completing the exercises. Discuss the definition for the term equivalent and have students provide examples of equivalent numbers.
Have students refer to their folded papers to explain the answer for Exercise 1 and to extend their thinking to complete Exercise 2.
SMARTER
?
How
do
you
know
that
2 _
8
equals
1_4?
Explain.
Possible explanation: the two eighths parts are the
same
size
as
the
1 _ 4
part.
? t3_6haenndu1_2mabreereoqfupivaarltesn, at nfrdacthtieonsisz.eCsoomfptharee
parts, in each fraction.
they will be smaller than the
p3_6awrtilflohra1_2v.e
more
parts
and
Make Connections HOannds
Review with students how they used fraction strips in Chapter 8 to divide a number line into equal parts.
? How can you identify equivalent fractions on a number line? Equivalent fractions name the
same point on the number line.
? Use your number line to find another pair
of equivalent fractions. Explain how you
found
your
answer.
Possible
answer:
1 _ 3
and
2 _ 6
;
I used thirds and sixths to name the same distance
from 0 on the number line.
MTaatlkh
Use Math Talk to focus on students' understanding of
equivalent fractions on a number line.
? What number do you multiply by 3 to get 6? 2
? How many sixths equal one-third? 2
COMMON ERRORS
Error Students may think that fractions with the same numerator are equivalent.
Example In Exercise 1, students think
that
_ 1
8
=
_41 .
Springboard to Learning Have students
outline
1 _
4
students
students
part on their paper. Then, instruct tpooisnhtatdoeetahcehtw1_8 opa1_8rtpaanrtds.cHouanvet
"1 eighth, two eighths."
Lesson 9.6 540
3 EXPLAIN
Share and Show
MATH
BOARD
HOannds
The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.
Discuss with students how drawing lines in Exercises 1 and 2 is like tracing the folds from the paper-folding investigation.
Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.
Quick Check
3 2
RtI
1
If
a student misses the checked
exercises
Then Differentiate Instruction with ? Reteach 9.6
? Personal Math Trainer 3.NF.A.3a
? Rtl Tier 1 Activity (online)
4 ELABORATE
Problem Solving ? Applications
MP6 Attend to precision. Exercise 5 focuses on fractions equal to 1. ? Describe how all fractions that name
a whole will be alike. The numerator and
denominator will be the same number, because the numerator will name all the parts in the whole.
SMARTER
Personal Math Trainer Be sure to assign this problem to students in the Personal Math Trainer. It features a video to help them model and answer the problem. Students need to be able to identify equivalent fractions on the number line. They should understand that each mark on the number line that has a fraction both above and below it represents an equivalence between those two fractions. Students who do not correctly select the first choice may not understand that both of the fractions represent one whole. Other errors may be clarified if students discuss how the number line shows thirds as well as sixths.
541 Chapter 9
? Houghton Mifflin Harcourt Publishing Company
Name
SShhaarree aanndd SShhow
MATH
BOARD
Shade the model. Then divide the pieces to find the
equivalent fraction. Possible shadings are shown.
1.
2.
1 _ 4
=
_2__ 8
HOannds
2 _ 3
=
_4__ 6
Use the number line to find the equivalent fraction.
3. 0 1 2 3 4 5 6 6666666
0
1
2
2
2
2
1 _ 2
=
_3__ 6
4. 0 1 2 3 4 5 6 7 8 888888888 0 1 234 44444
3 _ 4
=
_6__ 8
OOqqnnaakkddllRRnnkkuuhhmmff??@@ooookkhhbb``sshhnnmmrr
5. MATHEMATICAL PRACTICE
6
Explain
why
2 _ 2
=
1.
Write another fraction that is equal
to 1. Draw to justify your answer.
Possible
explanation:
2 _ 2
=
1
because
the numerator names both parts in the
whole;
possible
answer:
8 _ 8
.
Drawings may vary.
2__ 2
1
8__ 8
Personal Math Trainer
6.
SMARTER For numbers 6a?6d, select True or False
to tell whether the fractions are equivalent.
6a.
6 _ 6
and
3 _ 3
6b.
4 _ 6
and
1 _ 3
6c.
2 _ 3
and
3 _ 6
6d.
1 _ 3
and
2 _ 6
True True True True
False False False False
01 23456 6666666
0
1
2
3
3
3
3
3
Chapter 9 ? Lesson 6 541
................
................
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