Equivalent Fractions
Objective To guide the development and use of a rule for
generating equivalent fractions.
g
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ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Identify fractional parts of regions. [Number and Numeration Goal 2]
• Name equivalent fractions. [Number and Numeration Goal 5]
• Use a rule for generating equivalent
fractions. [Number and Numeration Goal 5]
• Develop a rule for generating
equivalent fractions. [Patterns, Functions, and Algebra Goal 1]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Fraction Match
Student Reference Book, p. 243
Math Masters, pp. 389 and 473–476
Students practice identifying
equivalent names for fractions.
Fraction and Mixed-Number
Addition and Subtraction
Math Journal 2, p. 200B
pattern blocks
Students add and subtract fractions
and mixed numbers.
Key Activities
Math Boxes 7 7
Students use examples of equivalent
fractions to develop a rule for finding
equivalent fractions.
Math Journal 2, p. 202
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Informing Instruction See page 605.
Ongoing Assessment:
Recognizing Student Achievement
Study Link 7 7
Math Masters, p. 223
Students practice and maintain skills
through Study Link activities.
Use Math Masters, page 225. Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Identifying Equivalent Fractions on the
Fraction Number-Line Poster
Math Masters, p. 388 or 389
per group: Fraction Number-Line Poster
(Math Masters, pp. 204 and 205),
straightedge
Students use a Fraction Number-Line Poster
to identify equivalent fractions.
ENRICHMENT
Investigating Egyptian Fractions
Math Masters, p. 224
Students investigate how early Egyptians
represented a fraction as the sum of
unit fractions.
EXTRA PRACTICE
Completing Name-Collection Boxes
Math Masters, p. 397
Students complete name-collection boxes
for fractions.
[Number and Numeration Goal 5]
Key Vocabulary
equivalent fractions Equivalent
Fractions Rule
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, pp. 1, 17, 79, and 165
Students practice finding equivalent fractions.
Materials
Math Journal 2, pp. 201, 342, and 343
Study Link 7 6
Math Masters, p. 225
calculator colored chalk slate pattern
blocks (optional)
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 141, 142
Lesson 7 7
603
Mathematical Practices
SMP2, SMP3, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
4.OA.4, 4.NF.1, 4.NF.2, 4.NF.3a, 4.NF.3c, 4.NF.3d
Mental Math and Reflexes
Math Message
Have students name all the factor pairs for numbers less than 100.
Suggestions:
Complete journal page 201.
6 1 and 6; 2 and 3
4 1 and 4; 2 and 2
5 1 and 5
50 1 and 50; 2 and 25; 5 and 10
52 1 and 52; 2 and 26; 4 and 13
72 1 and 72; 2 and 36; 3 and 24;
4 and 18; 6 and 12; 8 and 9
12 1 and 12; 2 and 6; 3 and 4
15 1 and 15; 3 and 5
21 1 and 21; 3 and 7
Study Link 7 6
Follow-Up
Have small groups compare answers.
Ask volunteers to draw additional
representations of the fractions in
Problems 1–4.
1 Teaching the Lesson
Math Message Follow-Up
(Math Journal 2, p. 201)
WHOLE-CLASS
DISCUSSION
ELL
Ask students to examine the squares they colored on journal
page 201. Point out that the three fractions they wrote for each
problem all name the same fractional part of the square. Such
fractions are called equivalent fractions. To support English
language learners, have students write equivalent fractions next
to the examples in the journals.
Students should notice that whenever the total number of equal
parts is doubled (or quadrupled), the number of colored parts is
also doubled (or quadrupled), but the fractional part represented
by the colored parts does not change.
Student Page
Date
77
Many Names for Fractions
Color the squares and write the missing numerators.
1.
Tell students that in this lesson they will develop a rule for finding
equivalent fractions.
Time
LESSON
1
Color _
2 of each large square.
Whole
49
Developing a Rule for
square
Finding Equivalent Fractions
1
is colored.
2
2.
is colored.
4
4
is colored.
4
2
(Math Journal 2, p. 201)
is colored.
is colored.
8
4
In each problem on journal page 201, the numerator and
denominator of the first fraction are each multiplied by 2 to
obtain the second fraction. They are each multiplied by 4 to
obtain the third fraction.
is colored.
16
To support English language learners, write the following on
the board.
3
Color _
4 of each large square.
3
is colored.
4
6
is colored.
8
12 is colored.
16
604
1∗2
_
2∗2
Problem 1:
= _24
1∗4
_
2∗4
= _48
Problem 2:
= _28
1∗4
_
4∗4
4
=_
16
1∗2
_
4∗2
Math Journal 2, p. 201
185-218_EMCS_S_MJ2_G4_U07_576426.indd 201
ELL
8
1
Color _
4 of each large square.
1
3.
2
WHOLE-CLASS
DISCUSSION
3/31/11 3:39 PM
Unit 7 Fractions and Their Uses; Chance and Probability
Student Page
Date
Problem 3:
3∗2
_
4∗2
3∗4
_
4∗4
= _68
2
_
2
Time
Equivalent Names for Fractions
12
=_
16
Fraction
Equivalent Fractions
Decimal
Percent
0
0%
1
100%
Decimal
Percent
ᎏ0ᎏ
2
4
_
4
ᎏ1ᎏ
2
Write and with colored chalk to emphasize that the numerator
and denominator were multiplied by the same number.
2 3
ᎏᎏ, ᎏᎏ
4 6
ᎏ2ᎏ
2
ᎏ1ᎏ
3
The Equivalent Fractions Rule can be used to rename any
fraction: If the numerator and denominator of a fraction are
multiplied by the same nonzero number, the result is a fraction
that is equivalent to the original fraction.
ᎏ2ᎏ
3
ᎏ1ᎏ
4
ᎏ3ᎏ
4
ᎏ1ᎏ
5
ᎏ2ᎏ
5
ᎏ3ᎏ
5
Adjusting the Activity
ᎏ4ᎏ
5
Present a more abstract rationale for this rule:
ᎏ5ᎏ
6
ᎏ1ᎏ
6
If any number is multiplied by 1, the product is the number you started with.
A fraction with the same numerator and denominator, such as _44 , is
ᎏ1ᎏ
8
ᎏ3ᎏ
8
ᎏ5ᎏ
8
equivalent to 1.
ᎏ7ᎏ
8
Multiplying the numerator and denominator of a fraction by the same number
(not 0) is the same as multiplying the fraction by 1. So, the product is
equivalent to the original fraction.
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
Generating Equivalent Fractions
Math Journal 2, p. 342
V I S U A L
PARTNER
ACTIVITY
(Math Journal 2, pp. 342 and 343; Math Masters, p. 225)
Have students turn to the Equivalent Names for Fractions
table on journal page 342. Ask them to write 10 fractions that
are equivalent to _13 . Ask students to explain how they know these
fractions are all equivalent to _13 . Encourage them to use pattern
blocks or drawings to support their reasoning.
Have students look for patterns in fractions that are equivalent
to _13 . Point out how these patterns relate to the Equivalent
Fractions Rule.
Student Page
Date
Time
Equivalent Names for Fractions
Fraction
Ongoing Assessment: Informing Instruction
Watch for students who note that not every pair of equivalent fractions can be
found by multiplying (or dividing) by the same whole number. For example:
1
__
3 ∗ 13
4
=_
8
1
__
6 ∗ 13
In this example, the numerator and denominator are both multiplied by the mixed
1
number 1_3 .
Equivalent Fractions
continued
ᎏ1ᎏ
9
ᎏ2ᎏ
9
ᎏ4ᎏ
9
ᎏ5ᎏ
9
ᎏ7ᎏ
9
ᎏ8ᎏ
9
ᎏ1ᎏ
10
ᎏ3ᎏ
10
ᎏ7ᎏ
10
ᎏ9ᎏ
10
ᎏ1ᎏ
12
Working in pairs, students use the Equivalent Fractions Rule to
find three equivalent fractions for each of the remaining fractions
in the table.
ᎏ5ᎏ
12
ᎏ7ᎏ
12
11
ᎏ
ᎏ
12
Ask students to explain how to use a calculator to find equivalent
fractions. Sample answer: Enter a fraction. Multiply it by any
fraction whose numerator and denominator are the same.
Math Journal 2, p. 343
Lesson 7 7
605
Name
Date
LESSON
Time
An Equivalent Fractions Rule
77
Margot says the value of a fraction does not change if you do the same thing to
the numerator and denominator. Margot says that she added 2 to the numerator
3
1
and the denominator in _4 and got _6 .
1+2
_
_3
4+2 = 6
When students complete their work on journal pages 342 and 343
ask them to solve the problem on Math Masters, page 225 on
their own.
3
1
Therefore, she says that _4 = _6 . How could you explain or show Margot that she is wrong?
Sample answer: _14 does not equal _36 , because
_3 equals _1 . You can
multiply or divide the
6
2
numerator and denominator by the same
number and not change the value of the fraction,
but you cannot just add or subtract the same
number from the numerator and denominator.
Math Masters, page 225
Ongoing Assessment:
Recognizing Student Achievement
Math Masters
Page 225
Use Math Masters, page 225 to assess students’ understanding of
equivalent fractions. Students are making adequate progress if they are
able to draw a picture or use the Equivalent Fractions Rule to demonstrate that
3
1
_
_
4 ≠ (is not equal to) 6 . Some students may rename the fractions as decimals
and show that 0.25 ≠ 0.5.
[Number and Numeration Goal 5]
Adjusting the Activity
Have a table of equivalent fractions
available, such as Math Journal 2, pages 342
and 343 or Student Reference Book, page 51.
203-246_EMCS_B_MM_G4_U07_576965.indd 225
1/25/11 9:58 AM
AUDITORY
KINESTHETIC
TACTILE
VISUAL
2 Ongoing Learning & Practice
Playing Fraction Match
SMALL-GROUP
ACTIVITY
(Student Reference Book, p. 243; Math Masters,
pp. 389 and 473–476)
Students play Fraction Match to practice naming equivalent
fractions. When students have finished playing the game, ask
them to select two fractions that match. On an Exit Slip, have
them explain how they know the fractions are equivalent.
Fraction and Mixed-Number
INDEPENDENT
ACTIVITY
Addition and Subtraction
Student Page
(Math Journal 2, p. 200B)
Games
Fraction Match
Materials 䊐 1 deck of Fraction Match Cards
(Math Masters, pp. 473–476)
Players
2 to 4
Skill
Recognizing equivalent fractions
Students add and subtract fractions and mixed numbers.
Encourage students to use pattern blocks or a method of their
choice to solve the problems.
Object of the game To match all of your cards and have none left.
Directions
1. Shuffle the deck and deal 7 cards to each player. Place the
remaining cards facedown on the table. Turn over the top
card and place it beside the deck. This is the target card.
If a WILD card is drawn, return it to the deck and continue
drawing until the first target card is a fraction.
2. Players take turns trying to match the target card with
a card from their hand in one of 3 possible ways:
Math Boxes 7 7
(Math Journal 2, p. 202)
♦ a card with an equivalent fraction
♦ a card with a like denominator
♦ a WILD card.
2
ᎏᎏ
3
INDEPENDENT
ACTIVITY
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 7-5. The skill in Problem 6
previews Unit 8 content.
is the target card. It can be matched with:
ᎏ8ᎏ , or
♦ an equivalent fraction card such as ᎏ46ᎏ , ᎏ69ᎏ , or 12
♦ a like denominator card such as ᎏ03ᎏ , ᎏ13ᎏ , or ᎏ33ᎏ , or
♦ a WILD card. The player names any fraction (with a denominator
of 2, 3, 4, 5, 6, 8, 9, 10, or 12) that is equivalent to the target card.
ᎏ8ᎏ . The player may not
The player can match ᎏ23ᎏ by saying ᎏ46ᎏ , ᎏ69ᎏ , or 12
match ᎏ2ᎏ by saying ᎏ2ᎏ .
3
3
3. If a match is made, the player’s matching card is placed on
top of the pile and becomes the new target card. It is now the
next player’s turn. When a WILD card is played, the next
player uses the fraction just stated for the new target card.
4. If no match can be made, the player takes 1 card from the
deck. If the card drawn matches the target card, it may be
played. If not, the player keeps the card and the turn ends.
Writing/Reasoning Have students write a response to the
following: How did you determine the number of squares
you needed to circle in Problem 1? Sample answer: There
are 24 total squares. I divided them into 8 equal groups with 3
squares in each group. Then I circled 3 of the groups.
5. The game is over when one of the players runs out of cards,
when there are no cards left in the Fraction Match deck, or
time runs out. The player with the fewest cards wins.
Student Reference Book, p. 243
606
Unit 7 Fractions and Their Uses; Chance and Probability
Student Page
Study Link 7 7
INDEPENDENT
ACTIVITY
(Math Masters, p. 223)
Date
Time
LESSON
Fraction and Mixed-Number Addition and Subtraction
77
Add and subtract. Use pattern blocks to help you.
Home Connection Students identify the missing
numerator or denominator of equivalent fractions to
complete name-collection boxes.
3.
5.
7.
9.
11.
13.
3 Differentiation Options
1
__
4 , or 2
1
__
4
1
2
__
12
– __
– __ =
12 12 12
2
1
__
5
, or 5__
3 2
1
2
4_4 – _4 + 1_4 = 4
3
1
__
__
, or 1 2
2
2 _
1
_
_
2
+2–2=
2
2
1__
3
4
2
1
9
(2 _9 + _9 ) – (_9 + 1_9 ) =
3
__
6 1
4 2
8
( _8 – _8 ) – ( _8 – _8 ) =
5
__
2
__
1.
1
2 1
_
+ _4 – _4 =
4
8
4
2.
3 _
2
4
_
– + _8 =
8 8
4.
7_6 + 1_6 – 1_6 =
6.
4
2
2
_
+ _8 + 3_8 =
8
8.
3 2
4 1
( _8 – _8 ) + ( _8 – _8 ) =
10.
12.
5
3
2
__
7__
6 , or 7 3
4
8
3__
8 , or 4
4
__
1
__
8 , or 2
10
5
5__, or 5_6_
3
11
1
1
+ __
– __ ) = 12
(10 __
) – (5 __
12
12
12 12
5
__
, or 1
3
1
2 1
( _5 + _5 ) + ( _5 – _5 ) = 5
Paulo, Regina, and Ted picked a bucket of apples on the
3
field trip to the apple orchard. Paulo took _
16 of the apples,
1
6
__
Regina took _
16 of the apples, and Ted took 16 of the apples.
They decided to give the rest of the apples to the teacher.
Regina
Who took the most apples?
6
__
3
__
16 , or 8
What fraction of the apples did their teacher get?
How do you know?
6
3
10
1
__
__
__
Sample answer: I added __
16 + 16 + 16 and I got 16 . Since I
16
10
__
,
I
subtracted
know that the total number of apples was __
16
16
6
from that and I got __
.
That’s
how
many
the
teacher
gets.
16
SMALL-GROUP
ACTIVITY
READINESS
Identifying Equivalent Fractions
5–15 Min
14.
5
Julie was making a quilt. She had _8 yard of fabric.
7
4
She bought another _8 yard of fabric. She gave _8 yard
of the fabric to her friend.
8
__
on the Fraction Number-Line
Poster
8 , or 1
How many yards of fabric does she have left?
(Math Masters, pp. 204, 205, and 388 or 389)
yard
Math Journal 2, p. 200B
185-218_EMCS_S_MJ2_G4_U07_576426.indd 200B
3/3/11 12:39 PM
To explore equivalent fractions using a number-line model, have
students use a straightedge to vertically line up fractions on the
Fraction Number-Line Poster (see the optional Readiness activity
in Lesson 7-1) that are equivalent to _14 , _13 , _12 , _23 , and so on. Ask
students to record the results of their exploration in a Math Log or
on an Exit Slip.
1 Whole
0
1
Halves
0
2
1
2
2
2
Student Page
Fourths
Date
0
4
1
4
2
4
3
4
4
4
77
1.
1
8
2
8
3
8
4
8
Math Boxes
Eighths
0
8
Time
LESSON
5
8
6
8
7
8
8
8
3
1
_
Circle _
8 of all the squares. Mark Xs on 6
of all the squares.
2.
Sample answer:
Thirds
Insert parentheses to make these
number sentences true.
a.
0
3
1
3
2
3
(
)
2 ∗ 3 + 10 = 26
)
(24 - 5)∗ 2 = 38
d. 12 + 24 = 3 ∗(6 + 6)
b.
(
12 = 6 ∗ 6 - 4
c.
3
3
Sixths
59
0
6
1
6
2
6
3
6
4
6
5
6
6
6
3.
Plot and label each point on the
coordinate grid.
A (0,2)
B (4,0)
C
5
1
D (5,5)
A straightedge highlights equivalent fractions.
0
0
T
E
A
2
1
2
3
B
4
obtuse
This angle is an
(acute or obtuse) angle.
144
A bag contains
5
6
1
3
6.
green blocks,
red blocks,
blue block, and
yellow blocks.
You put your hand in the bag and, without
looking, pull out a block. About what
fraction of the time would you expect to
get a blue block?
1
__
15
P
O
5
E (5,3)
5.
Draw and label a 125° angle.
Sample answer:
D
4
3
C (1,5)
150
4.
92 93
143
If 1 inch on a map represents 40 miles,
then how many inches represent
10 miles? Fill in the circle next to the
best answer.
A
2 in.
B
1 in.
_
4
C
1 in.
_
2
D
4 in.
145
45
Math Journal 2, p. 202
185-218_EMCS_S_MJ2_G4_U07_576426.indd 202
1/27/11 10:51 AM
Lesson 7 7
607
Study Link Master
Name
Date
STUDY LINK
Time
ENRICHMENT
Fraction Name-Collection Boxes
77
In each name-collection box:
49 50
Write the missing number in each fraction so that the fraction belongs
in the box. Write one more fraction that can go in the box.
_1
1.
_2
2.
2
4.
4
6
3
4
9
12
5
12
5
10
18
20
10
20
10
20
30
40
(Math Masters, p. 224)
To apply students’ understanding of fraction addition and
equivalent fractions, have students investigate how early
Egyptians represented a fraction as the sum of unit fractions.
9
8
25
18
12
100
Answers
vary.
Answers
vary.
Answers
vary.
Make up your own
a.
name-collection box
problems like the ones
above. Ask a friend to
solve your problems.
Check your friend’s work.
To solve Problems 5 and 6, students need to divide the rectangle
into more regions than indicated by the denominator of
the fraction.
b.
NOTE Egyptians also used the fraction _23 .
Answers vary.
py g
g
p
15–30 Min
Fractions
_1
3.
3
2
Investigating Egyptian
SMALL-GROUP
ACTIVITY
EXTRA PRACTICE
Practice
5.
23 R3
= 95 4
6.
19
57 ÷ 3 =
7.
42
= 882 21
Completing Name-Collection
INDEPENDENT
ACTIVITY
5–15 Min
Boxes
Math Masters, p. 223
203-246_EMCS_B_MM_G4_U07_576965.indd 223
1/25/11 9:58 AM
(Math Masters, p. 397)
To provide practice generating equivalent names for fractions,
have students complete name-collection boxes. Encourage students
to complete the boxes with equivalent fractions and mathematical
expressions that include fractions.
Use Math Masters, page 397 to create problems to meet the needs
of individual students or have students create and solve their
own problems.
EXTRA PRACTICE
5-Minute Math
Teaching Master
Name
Date
LESSON
Time
Egyptian Fractions
77
Ancient Egyptians only used fractions with 1 in the numerator. These are called
3
4
unit fractions. They wrote non-unit fractions, such as _4 and _9 , as sums of unit
fractions. They did not use the same unit fraction more than once in a sum.
55 57
4
1
1
_
= _3 + _9
9
1
2
1
4
1
3
1
9
Use drawings and what you know about equivalent fractions to help you find the
Egyptian form of each fraction.
_3
1. 8
=
_1 + _1
4
8
5
_
2. 12
224
7
_
3. 10
1
4
1
8
=
_1 + _1
5
2
=
1
_1 + _
3
12
*
1
2
3
_
5. 5
=
1
3
=
1
5
1
_1 + _
2
10
1
2
_5
4. 6
1
_1 + _1
2
3
1
2
6.
1
10
1
3
1
_1 + _
2
14
4
_
7 =
1
2
1
14
Math Masters, p. 224
203-246_EMCS_B_MM_G4_U07_576965.indd 224
608
5–15 Min
To offer students more experience with equivalent fractions,
see 5-Minute Math, pages 1, 17, 79, and 165.
Examples:
1
1
_3 = _
+_
4
2
4
SMALL-GROUP
ACTIVITY
1/25/11 9:58 AM
Unit 7 Fractions and Their Uses; Chance and Probability