Focus on Math Concepts
Lesson 13
Part 1: Introduction
MAFS
4.NF.1.1
Understand Equivalent Fractions
What’s really going on when fractions are equivalent?
Equivalent fractions name the same part of a whole.
Think about how you could explain to a third grader why ​  5  ​  and ​ 1 ​are equivalent.
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You could shade area models to show ​  5  ​  and ​ 1 ​ .
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Both models are the same size. Both show the same
amount shaded, so ​  5  ​  and ​ 1 ​are equivalent fractions.
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Think Equivalent fractions show a fraction a different way.
Fractions can be written many different
ways by changing the number of equal parts in the whole.
Start with a rectangle divided into 2 equal
parts. Shade one part to show ​ 1 ​ .
1
2
Underline the part
that explains how to
write a fraction a
different way.
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Divide the same rectangle into 4 equal parts.
There are 2 times as many parts and 2 times
as many parts shaded. Now 2 out of 4 equal
parts are shaded.
2
4
But, your rectangle still shows ​ 1 ​ shaded.
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Divide the original rectangle into 8 equal
parts. There are 4 times as many parts and
4
8
4 times as many parts shaded. Now 4 out of
8 equal parts are shaded. Again, your
rectangle shows ​ 1 ​ shaded.
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So, ​ 1 ​ , ​ 2 ​ , and ​  4 ​are all equivalent fractions, since they name the same part of a whole.
2 ··
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L13: Understand Equivalent Fractions
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Part 1: Introduction
Lesson 13
Think Every fraction has many equivalent fractions.
You can start with any fraction and change the way the whole is divided to get an
equivalent fraction.
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3
This model is divided into 3 equal parts.
The shaded section shows the fraction ​ 1 ​ .
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​  2 ​has 2 times as many parts shaded and
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6
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2 times as many equal parts.
​  4  ​ has 4 times as many equal parts and
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4 times as many parts shaded as ​ 1 ​ .
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All three models have the same shaded area. So, ​ 1 ​ , ​ 2 ​ , and ​  4  ​ are equivalent.
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1
You can also multiply the numerator and denominator of ​   ​by the same number.
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2 times as many equal parts and 2 times as many parts
shaded: ​ 1 ​ 3 ​ 2 ​ 5 ​ 2 ​ .
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2
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Think of 2 times
as many as 3 2
4 times as many equal parts and 4 times as many parts
shaded: ​ 1 ​ 3 ​ 4 ​ 5 ​  4  ​  .
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Reflect
1 Explain how you can find equivalent fractions.
L13: Understand Equivalent Fractions
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Part 2: Guided Instruction
Lesson 13
Explore It
Dividing models is one way to think about equivalent fractions.
2 The model shows ​ 1 ​ . How many equal parts make up the
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whole? Draw 2 more lines on the circle to make 8 equal parts.
3 Compare the 4 equal parts to the 8 equal parts. How many times as many parts
are there now? Now how many parts are shaded? Why are there two times as many parts shaded as there were in the ​ 1 ​ model?
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Use the model above to answer problems 4 and 5.
4 If 3 of the original 4 parts were shaded, how many of the 8 parts would be
shaded? 5 If all 8 parts were shaded, how many of the original 4 parts would be shaded?
Now try these two problems.
6 Draw a model to show ​ 2 ​and then
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7 This model shows ​  30  ​  . If the model
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···
divide it into a different number of
had only 10 equal parts, how many
parts to find an equivalent fraction.
would be shaded? 122
L13: Understand Equivalent Fractions
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Part 2: Guided Instruction
Lesson 13
Talk About It
Solve the problems below as a group.
8 Write the equivalent fractions from problems 2 and 3. Multiply both the numerator and denominator of ​ 1 ​by the same number to get ​ 2 ​ .
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What number did you use? Why does this make sense?
What happens if you divide both the numerator and the denominator in ​ 2 ​by 2?
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9 To find an equivalent fraction to ​ 6 ​ , Beth divided by 2 to get 4 in the denominator.
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What should Beth do to find the numerator? What are the equivalent fractions?
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10 Fill in the missing numbers to find an equivalent fraction to ​   ​ .
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10
2
5
____
____
​   ​ 3 ​     ​ 5 ​    ​ 
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Try It Another Way
Work with your group to model equivalent fractions.
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11 Shade the model to show ​   ​ . Then show 10 equal parts and write an equivalent
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fraction.
1
12 Shade the model to show ​   ​ . Then show 12 equal parts and write an equivalent
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fraction.
L13: Understand Equivalent Fractions
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Part 3: Guided Practice
Lesson 13
Connect It
Talk through these problems as a class, then write your answers below.
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13 Compare: Use different methods to find two fractions that are equivalent to ​   .​
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14 Illustrate: Explain why you can multiply both the numerator and denominator by
the same number to make an equivalent fraction. Draw a model to show an
example.
15 Choose: Think about the cooking problem below.
Fia needs ​ 3 ​ of a cup of brown sugar. She only has a ​ 1 ​ -cup measuring cup and
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3
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1
a ​   ​ -cup measuring cup. Which should she use, and why?
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L13: Understand Equivalent Fractions
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Part 4: Performance Task
Lesson 13
Put It Together
16 Use what you have learned to complete this task.
A Draw a model to show the fraction ​  6  ​ and two equivalent fractions.
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BHow can you use multiplication and division to check your equivalent fractions
in Part A? Why does this work?
L13: Understand Equivalent Fractions
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Develop Skills and Strategies
Lesson 16
Part 1: Introduction
MAFS
4.NF.2.3a
4.NF.2.3d
Add and Subtract Fractions
In Lesson 15, you learned that adding fractions is a lot like adding whole
numbers. Take a look at this problem.
Lynn, Paco, and Todd split a pack of 12 baseball cards. Lynn got 4 cards, Paco
got 3 cards, and Todd got the rest of the cards. What fraction of the pack did
Todd get?
Explore It
Use the math you already know to solve the problem.
How many cards did Lynn and Paco get altogether? How many cards did Todd get? There are 12 cards in the pack. What fraction represents the whole pack of cards?
If Lynn got 4 cards out of 12, that means she got ​  4  ​ of the pack. If Paco got 3 cards
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out of 12, what fraction of the pack did he get? What fraction of the pack did Lynn and Paco get altogether? Explain how you could find the fraction of the pack that Todd got.
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L16: Add and Subtract Fractions
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Part 1: Introduction
Lesson 16
Find Out More
We often use fractions in real life. Sometimes they refer to parts of a set of objects,
like the baseball card problem. In that problem, the “whole” is the pack, and 12 cards
means there are 12 parts of the whole.
Each person got baseball cards from the same pack, so each fraction refers to the
same whole. When you add or subtract baseball cards, the whole will stay the same
because the cards are all from the same pack of 12.
4
12
3
12
5
12
Fractions in real life can also refer to equal parts of a whole object. Lynn, Paco, and
Todd might share a pizza cut into 8 slices. The “whole” is the pizza, and 8 slices means
there are 8 equal parts of the whole. Even if a person takes away 1 slice or 3 slices
from the pizza, the whole will stay the same.
Reflect
1 Describe another example of a set of objects or a whole object divided into
fractions.
L16: Add and Subtract Fractions
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Part 2: Modeled Instruction
Lesson 16
Read the problem below. Then explore different ways to understand it.
Josie and Margo made 10 clay pots in art class. Josie painted ​ 3  ​ of the pots.
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4
Margo painted ​    ​ of the pots. What fraction of the clay pots did they paint?
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Picture It
You can use models to help understand the problem.
The following model shows the pots. Each pot is ​  1  ​ of the total number of pots.
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Josie painted 3 pots, and Margo painted 4 pots. They painted a total of 7 pots.
J
J
J
1
J
J
M
M
3 tenths
J
M
M
5
4 tenths
J
J
J
M
M
M
M
7 tenths
Model It
You can also use a number line to help understand the problem.
The following number line is divided into tenths, with a point at ​  3  ​  .
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0
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Start at ​  3  ​ and count 4 tenths to the right to add ​  4  .​  
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L16: Add and Subtract Fractions
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Part 2: Guided Instruction
Lesson 16
Connect It
Now you will solve the problem from the previous page using equations.
2 How do you know that each pot is ​  1  ​ of the total number of pots?
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3 What do the numerators, 3 and 4, tell you? 4 How many clay pots did Josie and Margo paint altogether? 5 Write equations to show what fraction of the clay pots Josie and Margo painted
altogether.
Use words:
3 tenths
Use fractions:​  3  ​  
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1
4 tenths
1​  4  ​  
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tenths
5
  ​ 
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5​ 
6 Explain how you add fractions with the same denominator.
Try It
Use what you just learned to solve these problems. Show your work on a
separate sheet of paper.
7 Lita and Otis are helping their mom clean the house. Lita cleaned ​ 1 ​of the rooms.
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Otis cleaned ​ 1 ​of the rooms. What fraction of the rooms did Lita and Otis clean
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altogether? 8 Mark’s string is ​ 1 ​of a meter long. Bob’s string is ​ 3 ​of a meter long. How long are
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the two strings combined? of a meter
L16: Add and Subtract Fractions
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Part 3: Modeled Instruction
Lesson 16
Read the problem below. Then explore different ways to understand it.
Alberto’s water bottle had ​ 5 ​of a liter in it. He drank ​ 4 ​of a liter. What fraction
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of the bottle still has water in it?
Picture It
You can use models to help understand the problem.
The following model shows the water bottle divided into 6 equal parts. Each part
is ​ 1 ​ of a liter. Five shaded parts show how much water is in the bottle.
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Alberto drank 4 parts of the water in the bottle, so take away 4 shaded parts of the
bottle. There is 1 part of the bottle left with water in it.
2
5
5 sixths
4 sixths
1 sixth
Model It
You can use a number line to help understand the problem.
The following number line is divided into sixths, with a point at ​ 5 ​ .
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1
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0
2
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1
Start at ​ 5 ​and count 4 sixths to the left to subtract ​ 4 .​
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0
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L16: Add and Subtract Fractions
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Part 3: Guided Instruction
Lesson 16
Connect It
Now you will solve the problem from the previous page using equations.
9 How do you know that each part is ​ 1 ​of a liter?
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10 What do the numerators, 5 and 4, tell you? 11 How many parts of water are left in the bottle after Alberto drank 4 parts? 12 Write equations to show what fraction of the bottle has water left in it.
Use words:
5 sixths
Use fractions:​ 5 ​
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2
4 sixths
2​  4 ​
sixth
5
5​     ​
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13 Explain how you subtract fractions with the same denominator.
Try It
Use what you just learned to solve these problems. Show your work on a
separate sheet of paper.
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2
14 Mrs. Kirk had ​   ​of a carton of eggs. She used ​   ​of the carton to make breakfast.
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What fraction of the carton of eggs does Mrs. Kirk have left? 8
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15 Carmen had ​    ​ of the yard left to mow. She mowed ​    ​ of the yard. What fraction
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of the yard is left to mow? L16: Add and Subtract Fractions
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Part 4: Guided Practice
Lesson 16
Study the model below. Then solve problems 16–18.
Student Model
The student used labels
and “jump” arrows to
show each part of the
hike on a number line. It
is just like adding whole
numbers!
Jessica hiked ​ 2 ​mile on a trail before she stopped to get a drink
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of water. After her drink, Jessica hiked another ​ 2 ​mile. How far
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did Jessica hike in all?
Look at how you could show your work using a number line.
before drink
0
Pair/Share
How else could you
solve this problem?
What fraction represents
the whole fruit
smoothie?
2
5
1
5
after drink
2
5
2
5
3
5
4
5
1
​ 4 ​ mile
5
Solution: ··
1
16 Ruth made a fruit smoothie. She drank ​   ​of it. What fraction of the
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fruit smoothie is left?
Show your work.
Pair/Share
How did you and your
partner decide what
fraction to start with?
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Solution: L16: Add and Subtract Fractions
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Part 4: Guided Practice
Lesson 16
3
17 Mr. Chang has a bunch of balloons. ​    ​ of the balloons are red.
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2
​    ​ of the balloons are blue. What fraction of the balloons are
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neither red nor blue?
I think that there are at
least two different steps
to solve this problem.
Show your work.
Pair/Share
How is this problem
different from the
others you’ve seen in
this lesson?
Solution: 1
2
18 Emily ate ​   ​of a bag of carrots. Nick ate ​   ​of the bag of carrots.
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What fraction of the bag of carrots did Emily and Nick eat
altogether? Circle the letter of the correct answer.
A​  1 ​
To find the fraction of
the bag Emily and Nick
ate altogether, should
you add or subtract?
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B​  1 ​
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C​  3 ​
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D​  3  ​ 
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Rob chose D as the correct answer. How did he get that answer?
L16: Add and Subtract Fractions
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Pair/Share
Does Rob’s answer
make sense?
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Part 5: MAFS Practice
Lesson 16
Solve the problems.
1 Liang bought some cloth. He used }
​ 5 ​  of a yard for a school project. He has }
​ 2 ​  of a yard
8
left. How much cloth did Liang buy?
8
3 ​  of a yard
A​ }
8
7  ​ of a yard
B​ }}
16
7 ​  of a yard
C​ }
8
8 ​  of a yard
D​ }
8
2 Carmela cut a cake into 12 equal-sized pieces. She ate }}
​  2  ​ of the cake, and her brother
12
3
ate }}
​    ​ of the cake. What fraction of the cake is left?
12
1  ​ of the cake
A​ }}
12
5  ​ of the cake
B​ }}
12
7  ​ of the cake
C​ }}
12
12 ​  of the cake
D​ }}
12
3 Lee’s muffin mix calls for }
​ 2 ​  cup of milk, }
​ 1 ​  cup of oil, and }
​ 1 ​  cup of sugar. How much
3
3
more milk than oil does she need for the muffin mix?
3
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L16: Add and Subtract Fractions
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Part 5: MAFS Practice
Lesson 16
4 Lucy and Margot are mowing the lawn. They divided the lawn into 8 equal sections.
Lucy mowed 2 sections and Margot mowed 4 sections. Which model can be used to
find the total fraction of the lawn they mowed? Circle the letter of all that apply.
A
B
C
D
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8
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8
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0
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5 In all, Cole and Max picked }}
​  9  ​ of a bucket of blueberries. Cole picked }}
​  3  ​ of the
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10
bucket of blueberries. What fraction of the bucket of blueberries did Max pick?
Show your work.
Answer Max picked of the bucket of blueberries.
6 A pizza is cut into 8 equal slices. Together, Regan and Juanita will eat }
​ 5 ​  of the pizza.
What is one way the girls could eat the pizza?
8
Show your work.
Answer Regan could eat of the pizza, and
Juanita could eat of the pizza.
Self Check Go back and see what you can check off on the Self Check on page 119.
L16: Add and Subtract Fractions
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