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Texas
4 Mathematics
STAAR
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Instruction
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Lesso
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Student Book
Sample Lessons
Includes two sample lessons:
Lesson 10: Compare Fractions
Lesson 11: Understand Fraction Addition and Subtraction
Table of Contents
Unit 1: Number and Operations, Part 1
STAAR Reporting Categories 1 and 2
TEKS
Lesson 1 Understand Place Value
2
4(2)(A)♦, 4(2)(B)✶
Lesson 2 Compare and Order Whole Numbers
8
4(2)(B)✶, 4(2)(C)♦
Lesson 3 Round Whole Numbers
16
4(2)(D)♦
Lesson 4 Add and Subtract Whole Numbers
24
4(4)(A)✶
Lesson 5 Multiply by One-Digit Numbers
34
4(4)(B)♦, 4(4)(D)♦
Lesson 6 Multiply by Two-Digit Numbers
46
4(4)(C)✶, 4(4)(D)✶
4(4)(H)✶
Lesson 7 Divide Whole Numbers
54
4(4)(E)♦, 4(4)(F)♦
4(4)(H)✶
STAAR Practice
64
Unit 2: Number and Operations, Part 2
STAAR Reporting Categories 1 and 2
Lesson 8 Understand Fractions
66
4(3)(A)♦, 4(3)(B)♦
Lesson 9 Understand Equivalent Fractions
72
4(3)(C)♦
Lesson 10 Compare Fractions
78
4(3)(D)✶
Lesson 11 Understand Fraction Addition and Subtraction
88
4(3)(E)✶
Lesson 12 Add and Subtract Fractions
94
4(3)(E)✶
Lesson 13 Add and Subtract Mixed Numbers
106
STAAR Practice
118
4(3)(E)✶, 4(3)(F)♦
✶ = STAAR Readiness Standard
♦ = STAAR Supporting Standard
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iii
Table of Contents
Unit 3: Number and Operations, Part 3
STAAR Reporting Categories 1 and 2
TEKS
Lesson 14 Understand Decimals . . . . . . . . . . . . . . . . . . . . . . 1204(2)(B)✶, 4(2)(E)♦
Lesson 15 Relate Decimals and Fractions . . . . . . . . . . . . . . . . .1264(2)(G)✶, 4(2)(H)♦
4(3)(G)♦
Lesson 16 Compare and Order Decimals . . . . . . . . . . . . . . . . .1364(2)(F)♦, 4(2)(E)♦
Lesson 17 Add and Subtract Decimals . . . . . . . . . . . . . . . . . . 1484(4)(A)✶
STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Unit 4: A
lgebraic Reasoning, Number and Operations, Part 4
STAAR Reporting Categories 2 and 3
Lesson 18 Estimation and Problem Solving . . . . . . . . . . . . . . . 1584(4)(G)♦
Lesson 19 Model Multi-Step Problems . . . . . . . . . . . . . . . . . . 1684(4)(H)✶, 4(5)(A)✶
Lesson 20 Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . 1764(5)(B)✶
Lesson 21 Perimeter and Area . . . . . . . . . . . . . . . . . . . . . . . .1864(5)(D)✶
STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
✶ = STAAR Readiness Standard
♦ = STAAR Supporting Standard
iv
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Table of Contents
Unit 5: Geometry and Measurement
STAAR Reporting Category 3
TEKS
Lesson 22 Points, Lines, Rays, Angles . . . . . . . . . . . . . . . . . . . 196
4(6)(A)♦, 4(6)(C)♦
Lesson 23 Classify Two-Dimensional Figures . . . . . . . . . . . . . . 208
4(6)(D)✶, 4(6)(A)♦
4(6)(C)♦
Lesson 24 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4(6)(B)♦
Lesson 25 Measure and Draw Angles . . . . . . . . . . . . . . . . . . . 230
4(7)(C)✶, 4(7)(D)♦
Lesson 26 Add and Subtract With Angles . . . . . . . . . . . . . . . . 240
4(7)(E)♦
Lesson 27 Convert Measurements . . . . . . . . . . . . . . . . . . . . 250
4(8)(A)♦, 4(8)(B)♦
Lesson 28 Time and Money . . . . . . . . . . . . . . . . . . . . . . . . . 260
4(8)(C)✶
Lesson 29 Length, Liquid Volume, and Mass . . . . . . . . . . . . . . 270
4(8)(C)✶
STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
Unit 6: Data Analysis and Personal Financial Literacy
STAAR Reporting Category 4
Lesson 30 Represent Data . . . . . . . . . . . . . . . . . . . . . . . . . . 290
4(9)(A)✶
Lesson 31 Use Data to Solve Problems . . . . . . . . . . . . . . . . . . 298
4(9)(B)♦
Lesson 32 Fixed and Variable Expenses . . . . . . . . . . . . . . . . . 306
4(10)(A)♦
Lesson 33 Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
4(10)(B)♦
Lesson 34 Financial Institutions . . . . . . . . . . . . . . . . . . . . . . 318
4(10)(E)♦
STAAR Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
✶ = STAAR Readiness Standard
♦ = STAAR Supporting Standard
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v
Develop Skills and Strategies
Lesson 10
Part 1: Introduction
TEKS
4.3.D
Compare Fractions
In the past, you learned to compare fractions using models. Take a look at this
problem.
Adriana ate 2 of a granola bar and June ate 2 of a same-size granola bar. Which
4
··
5
··
girl ate more granola bar?
Adriana
June
Explore It
Use the math you already know to solve the problem.
How many equal pieces of granola bar did Adriana eat?
How many equal pieces of granola bar did June eat?
Since both girls ate the same number of pieces, what can you look at to find out
who ate more?
What does the size of the denominator tell you about the size of the pieces of
granola bar?
Who ate more? Explain why.
78
L10: Compare Fractions
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Part 1: Introduction
Lesson 10
Find Out More
Deciding who ate more of the granola bar means comparing the fractions 2 and 2 .
4
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2
4
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. 2
2
5
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2 is greater than 2 .
5
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4
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5
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5
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, 2
4
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2 is less than 2 .
4
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5
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What if June’s granola bar was larger than Adriana’s? Would the comparison make
sense?
To compare fractions, you must use the same-size whole.
You can also use equivalent fractions to compare fractions. Look for numbers that
you can multiply by the denominators so that the fractions end up with the same
denominators.
2 3 4 5 8 and 2 3 5 5 10
4 ··
20
4 ··
5 ··
20
··
··
8 , 10 , so 2 , 2
20 ··
20
5 ··
4
··
··
5
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Reflect
1 Explain how you can tell which fraction is greater, 2 or 3 .
5 ··
10
··
L10: Compare Fractions
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79
Part 2: Modeled Instruction
Lesson 10
Read the problem below. Then explore different ways to understand it.
A grasshopper weighs about 2 of an ounce. A beetle weighs 8 of an ounce.
100
···
Which weighs more?
10
··
Picture It
You can use models to help compare fractions.
The following model shows the weights of the grasshopper and beetle.
Grasshopper
Beetle
Solve It
You can use a common denominator to help you solve the problem.
It is hard to compare two fractions with different numerators and different
denominators. You can write an equivalent fraction for one or both of the fractions so
they have a common denominator. Fractions with the same denominator are divided
into the same number of equal parts. If fractions have the same denominator, you
can just compare the numerators.
8 and 2 .
Compare 10
100
··
···
First, look at the denominator, 10. Can you multiply 10 by any number to get 100? Yes,
10 3 10 5 100.
Find a fraction equivalent to 8 that has a denominator of 100:
8 3 10 5 80
10
10 ···
100
·· ··
10
··
Compare the numerators of 80 and 2 :
100
···
100
···
80 . 2
So, 80 . 2 .
100
···
80
100
···
L10: Compare Fractions
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Part 2: Guided Instruction
Lesson 10
Connect It
Now you will solve the problem from the previous page by finding a common
numerator.
2 What is an equivalent fraction for 2 that has a numerator of 8?
100
···
3 One model is divided into 400 equal parts
and the other is divided into 10 equal parts.
Which has smaller parts?
4 Shade 8 pieces of each model.
5 Which model has a greater area shaded?
6 Which fraction is greater, 8 or 8 ?
400 ··
10
···
7 Look at the denominators of 8 and 8 . When two fractions have the same
400
10
···
··
numerator and different denominators, how do you know which one is greater?
Explain.
Try It
Use what you just learned to solve these problems.
8 Mel’s tomato plant is 8 of a foot tall. Her pepper plant is 3 of a foot tall. Compare
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4
··
··
the heights of the plants using a symbol.
9 Compare the fractions 3 and 5 using a symbol.
5
10
··
··
L10: Compare Fractions
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81
Part 3: Modeled Instruction
Lesson 10
Read the problem below. Then explore different ways to use benchmarks to
compare fractions.
Jasmine’s swimming lesson lasts for 2 of an hour. It takes her 1 of an hour to do
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6
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her homework. Will Jasmine spend more time on her homework or at her
swimming lesson?
Model It
You can use a number line to help you compare fractions.
The number line shows where the fractions 2 and 1 are compared to 0 and 1.
3
··
1
6
6
··
2
3
1
2
0
1
The number line shows that 1 is closer to 0 than 2 is, and that 2 is closer to 1 than 1 is.
This means that 2 . 1 .
3 ··
6
··
6
··
3
··
3
··
6
··
Solve It
You can use a benchmark fraction to solve the problem.
Another way to compare fractions is by using the fraction 1 as a benchmark.
2
··
Look at the number line. It shows that 1 is less than 1 and 2 is greater than 1 .
So, 1 , 2 and 2 . 1 .
6 ··
3
3 ··
6
··
··
6
··
2
··
3
··
2
··
Jasmine will spend more time at her swimming lesson than on homework.
82
L10: Compare Fractions
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Part 3: Guided Instruction
Lesson 10
Connect It
Now you will solve a similar problem using 1 as a benchmark.
Think about these two fractions: 11 and 7
10
··
8
··
10 Which fraction is greater than 1?
11 Which fraction is less than 1?
12 Which fraction is greater? Explain why.
13 Fill in the blank with the correct symbol to show the comparison.
11
10
··
7
8
··
14 Explain how you can use benchmarks to compare fractions.
Try It
Use what you just learned to solve these problems.
15 Fill in the blank. Explain how you found your answer.
16 Nathan walked
5
10
··
3
4
··
10 of a mile. Sarah walked 19 of a mile. Who walked a greater
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··
10
··
distance? Explain.
L10: Compare Fractions
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83
Part 4: Guided Practice
Lesson 10
Study the model below. Then solve problems 17–19.
Student Model
It is important that both
measurements use the
same unit!
Becker catches a fish that is 3 of a yard long. To keep the fish,
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1
it has to be longer than of a yard. Can Becker keep his fish?
3
··
Look at how you could show your work using a number line.
1
2
3
4
5
6
7
8
9
10 11
0 12 12 12 12 12 12 12 12 12 12 12 1
1
3
Pair/Share
How else could you
solve this problem?
Which strategy for
comparing do you think
works best with these
fractions?
Solution:
2
3
Since 3 is less than 1 , Becker can’t keep his fish.
12
···
3
··
17 Myron and Jane are working on the same set of homework
problems. Myron has finished 7 of the problems and Jane has
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2
finished of the problems. Who has finished more of the
3
··
homework?
Show your work.
Pair/Share
How did you and your
partner choose what
strategy to use to solve
the problem?
84
Solution:
L10: Compare Fractions
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Part 4: Guided Practice
18 Compare the fractions
3 and 7 using the benchmark fraction 1 .
12
2
··
··
10
··
Show your work.
Lesson 10
You already know about
how big 1 is!
2
··
Pair/Share
Draw a model to check
your answer.
Solution:
19 Janelle walked
3 of a mile. Pedro walked 6 of a mile. Which
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6
··
statement shows how to find the greater fraction? Circle the letter
There are several ways to
compare fractions!
of the correct answer.
A 3 5 6 and 6 , 6
6
··
B
C
D
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12
··
10
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3 5 6 and 6 . 6
12
12 ··
10
··
··
6 5 3 and 3 , 3
10 ··
5
5 ··
6
··
··
3 , 1 and 6 . 1
6 ··
2
10 ··
2
··
··
6
··
Tina chose B as the correct answer. How did she get that answer?
Pair/Share
How can you find the
answer using a
benchmark fraction?
L10: Compare Fractions
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85
Part 5: TEKS Practice
Lesson 10
Solve the problems.
1
2
2 cup of raisins and 3 cup of almonds to make trail mix. Which statement
Grant needs }
}
4
3
can be used to find out if there are more raisins or almonds in the mix?
8
2
4
3
4
2
6
3
6
2
6
3
6
9
B
5 } and } 5 }
}
4
5
3
6
C
5 } and } 5 }}
}
4
3
9
12
D
5 } and } 5 }
}
4
7
3
9
Tell whether each sentence is True or False.
b.
c.
d.
e.
2, 6
15
···
True
False
7 .7
8
··
True
False
1.3
8
··
True
False
254
6
··
True
False
30 5 3
50
···
True
False
5
··
10
···
2
··
4
··
500
····
Fill in the blank with one of the symbols shown to compare 5 and 5 .
10
···
,
5
}}
10
86
3
5 }} and } 5 }}
}
4
3
12
12
a.
3
2
A
5
8
··
.
5
}
8
L10: Compare Fractions
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Part 5: TEKS Practice
4
Lesson 10
5 of an hour. He spent
Sam’s music teacher told him to practice his trombone for }}
10
2 of an hour practicing. Did he practice long enough?
}
6
Show your work.
Answer Sam
5
practice long enough.
Olivia and Eleanor each made the same amount of lemonade to sell at a lemonade
stand. Olivia poured all of her lemonade into 10 equal glasses. Eleanor poured all of
her lemonade into 5 equal glasses. Olivia sold 7 glasses of lemonade and Eleanor sold
2 glasses. Which girl sold a greater fraction of her lemonade? Compare the fractions
using a symbol.
Show your work.
sold a greater fraction of her lemonade.
Answer
6
9 of her
Rachel and Sierra are selling boxes of fruit as a fundraiser. Rachel has sold }}
10
5 of her boxes. Which girl has sold a greater fraction
boxes of fruit and Sierra has sold }
8
of her boxes of fruit? Draw a model to show your answer.
Show your work.
Answer
has sold a greater fraction of her boxes of fruit.
L10: Compare Fractions
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87
Focus on Math Concepts
Lesson 11
Part 1: Introduction
TEKS
4.3.E
Understand Fraction Addition and Subtraction
What’s really going on when we add numbers?
Adding means joining or putting things together.
Think about how you could explain adding 2 1 3 to a first grader. You could start
at 2, count on 3 more, and see where you end up: 2 . . . 3 . . . 4 . . . 5.
Or, you could put a segment with a length of 2 and a segment with a length of 3 next
to each other on a number line to show 2 1 3.
1
0
1
1
1
2
1
1
3
4
5
6
7
8
9
10
When you add 2 1 3, you are putting ones together.
Think
Adding fractions means joining or putting together parts of the
same whole.
You can put a segment with a length of 2 and a segment
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3
with a length of next to each other to show 2 1 3 .
4
4 ··
4
··
··
1
4
0
4
1
4
1
4
1
4
2
4
1
4
3
4
0
Underline the
sentence that
explains what adding
fractions means.
1
4
4
4
1
5
4
6
4
7
4
8
4
2
When you add 2 1 3 , you are putting one-fourths together.
4
··
88
4
··
L11: Understand Fraction Addition and Subtraction
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Part 1: Introduction
Lesson 11
Think
Subtracting means separating or taking away.
Look at the whole
numbers. Now look at
the numerators of the
fractions. I think I see a
connection.
On a number line, you can start with a segment of length
5 and take away a segment of length 2 to show 5 2 2.
1
1
0
1
1
2
1
1
1
1
3
4
5
6
7
8
9
10
When you subtract 5 2 2, you are taking away ones.
You can show subtracting fractions on a number line. Start with a segment of
length 5 and take away a segment of length 2 to show 5 2 2 .
4
··
4
··
1
4
0
4
1
4
1
4
1
4
1
4
1
4
2
4
3
4
4
··
1
4
1
4
4
4
0
4
··
1
5
4
6
4
7
4
8
4
2
When you subtract 5 2 2 , you are taking away one-fourths.
4
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4
··
Now you’ll have a chance to think more about how adding or subtracting fractions is
like adding or subtracting whole numbers. You may find that using number lines or
area models can help you explain your thinking.
Reflect
1 Use your own words to describe what you just learned about adding and
subtracting fractions.
L11: Understand Fraction Addition and Subtraction
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89
Part 2: Guided Instruction
Lesson 11
Explore It
Counting on and using a number line are two ways to think about adding
fractions.
2 Count by fourths to fill in the blanks: 1 , 2 ,
4 ··
4
··
, 4, 5,
4 ··
4
··
,
,
,
Now label the number line.
0
1
4
2
4
4
4
5
4
10
4
3 Count by fifths to fill in the blanks: 1 , 2 ,
5 ··
5
··
,
,
Now label the number line.
0
1
5
2
5
6
5
Use the number lines above to answer numbers 4 and 5.
4 What is 1 more than 6 ?
4
4
··
··
5 What is 1 more than 3 ?
5
5
··
··
Now try these two problems.
6 Label the number line below and use it to show 2 1 1 .
4 ··
4
··
7 Label the number line below and use it to show 3 1 1 .
4 ··
4
··
90
L11: Understand Fraction Addition and Subtraction
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Part 2: Guided Instruction
Lesson 11
Talk About It
Solve the problems below as a group.
8 Look at your answers to problems 2 and 3. How is counting by fractions the same
as counting with whole numbers?
How is it different?
9 Label the number line below and use it to show 7 2 2 .
8 ··
8
··
5 1
10 Label the number line below and use it to show 2 .
6 ··
6
··
Try It Another Way
Work with your group to use the area models to show adding or subtracting
fractions.
1 2
11 Show 1 .
8 ··
8
··
L11: Understand Fraction Addition and Subtraction
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6 2 2.
12 Show
10 ··
10
··
91
Part 3: Guided Practice
Lesson 11
Connect It
Talk through these problems as a class, then write your answers below.
2 1
13 Compare: Draw two different models to show 2 .
3 ··
3
··
14 Explain: Rob had a large pizza and
a small pizza. He cut each pizza into
fourths. He took one fourth from each
pizza and used the following problem
to show their sum: ··14 1 ··14 5 ··24 .
What did Rob do wrong?
15 Demonstrate: Think about how you would add three whole numbers. You add
two of the numbers first, and then add the third to that sum. You add three
fractions the same way.
1 1 3 1 4.
Use the number line and area model below to show ··
10 ··
10 ··
10
92
L11: Understand Fraction Addition and Subtraction
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Part 4: Performance Task
Lesson 11
Put It Together
16 Use what you have learned to complete this task.
Jen has 4 of a kilogram of dog food. Luis has 3 of a kilogram of dog food.
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··
A large dog eats 2 of a kilogram in one meal.
10
··
10
··
A Write two different questions about this problem that involve adding or
subtracting fractions.
i
ii
B Choose one of your questions to answer. Circle the question you chose.
Show how to find the answer using a number line and an area model.
L11: Understand Fraction Addition and Subtraction
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93
Teacher Resource Book
Sample Lessons
Includes two sample lessons:
Lesson 10: Compare Fractions
Lesson 11: Understand Fraction Addition and Subtraction
Develop Skills and Strategies
Lesson 10
(Student Book pages 78–87)
Compare Fractions
Lesson objectives
the Learning Progression
• Use symbols (., ,, 5) to compare fractions with the
same denominator and different numerators.
In Grade 3, students used models to compare two
• Recognize that fractions with different denominators
and the same numerators represent different values.
denominator by reasoning about their size. In Grade 4,
• Use benchmark fractions to compare fractions.
compare two fractions with different numerators and
• Recognize that you can only compare two fractions
when both refer to the same whole.
different denominators. Emphasis is placed on
Prerequisite skiLLs
the two fractions have the same size wholes. Students
• Represent fractions with denominators 2, 3, 4, 6, or
8 using a number line or visual models.
compare fractions by creating common numerators or
• Identify, create, and explain equivalent fractions.
fractions with the same numerator or the same
they extend their understanding of fractions to
understanding that a comparison only makes sense if
use models (e.g., fraction bars, area models, etc.) to
denominators. Students also learn to use benchmark
• Express whole numbers as fractions.
fractions 1 e.g., 1 2 to compare fractions. They record
• Compare fractions whose numerators or
denominators are the same.
work focuses on visual models and benchmark
vocabuLary
formally address fraction comparison in later grades,
There is no new vocabulary. Review the following
key terms.
but they will later apply their understanding of fraction
compare: to decide if one number is greater than, less
than, or equal to another number
when they compare fractional quantities in their lives.
2
··
comparisons using the ., ,, and 5 symbols. Students’
fractions, rather than an algorithm. Students will not
comparison when they learn to compare decimals and
greater than (.): a comparison of two numbers that
says one has greater value than the other
less than (,): a comparison of two numbers that says
one has less value than the other
teks Focus
4.3.D Compare two fractions with different numerators and different denominators and represent the comparison using the symbols
., 5, or ,. Readiness Standard
MatheMaticaL Process standards (MPs): 4.1.A, 4.1.C, 4.1.D, 4.1.E, 4.1.F (See page A9 for full text. Also see MPS Tips in the lesson.)
140
L10: Compare Fractions
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Part 1: Introduction
Lesson 10
At A GlAnce
Develop Skills and Strategies
Students explore a fraction comparison problem
involving fractions that have the same numerator but
different denominators. They use an approach they
already know. They explain that the fractions have the
same number of pieces but the fraction with the greater
denominator has smaller pieces.
Lesson 10
Part 1: Introduction
TEKS
4.3.D
Compare Fractions
In the past, you learned to compare fractions using models. Take a look at this
problem.
Adriana ate 2 of a granola bar and June ate 2 of a same-size granola bar. Which
4
··
5
··
girl ate more granola bar?
Adriana
Step By Step
• Tell students that this page shows a way to compare
fractions using a visual model.
• Have students read the problem at the top of the page.
Explore It
Use the math you already know to solve the problem.
• Work through Explore It as a class.
How many equal pieces of granola bar did June eat?
the model alone as justification for saying that
Adriana ate more. Encourage them to use the model
mathematically (4.1.E) by reasoning about the
number of pieces and sizes of the pieces. Discuss
how the number of pieces in each square relates to
the denominators and how the number of shaded
pieces relates to the numerators.
ell Support
Have students identify the comparison word in
this problem [more]. Connect this word to the
mathematical term greater than. Have students think
of other words that might be used to compare
quantities (e.g., bigger, larger, longer, taller, most,
etc.). Do the same with less than (e.g., less, fewer,
smaller, shorter, least, etc.).
L10: Compare Fractions
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2
Since both girls ate the same number of pieces, what can you look at to find out
who ate more? the size of the pieces
What does the size of the denominator tell you about the size of the pieces of
granola bar? The greater denominator means there are more and smaller
• Have students explain the model. (See Mathematical
Discourse, below.)
MpS tip: Some students may be tempted to use
2
How many equal pieces of granola bar did Adriana eat?
• Have students describe the problem and state what
needs to be done mathematically (compare the
fractions).
• Ask student pairs or groups to explain their answers
for the last two bullets. Look for understanding that
a larger denominator means the whole is broken into
more pieces, which means each piece is smaller. Ask,
Would you rather share your favorite treat with
3 classmates or with 1 classmate? Why? Students
should apply this reasoning to explain that Adriana’s
2 pieces were bigger than June’s.
June
pieces.
Who ate more? Explain why. Both girls ate the same number of pieces.
Adriana’s 2 pieces are larger than June’s 2 pieces. So, Adriana ate more.
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Mathematical Discourse
• What does the model tell you?
The two rectangles show equal sized wholes.
One has 4 equal parts; the other has 5 equal
parts. Two parts are shaded in each. It looks
like more space is shaded in the rectangle with
4 parts.
• Why do we look at the denominators to find out who
ate more?
When numerators are the same, the number of
pieces is the same. So we compare sizes of the
pieces. The denominator tells how many pieces.
The more pieces, the smaller they are.
• How can you use the model to find out who ate more?
We see more shaded area in Adriana’s granola
bar. June’s granola bar has more pieces, so they
are smaller than Adriana’s. Therefore, June ate
less than Adriana did.
141
Part 1: Introduction
Lesson 10
At A GlAnce
Students use symbols to compare fractions. Then they
reflect on the importance of comparing fractions from
same-size wholes.
Part 1: Introduction
Lesson 10
Find Out More
Deciding who ate more of the granola bar means comparing the fractions 2 and 2 .
4
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5
··
Step By Step
• Read Find Out More as a class.
• Explain that you can show either fraction first in the
comparison; 2 . 2 is the same as 2 , 2 . Make sure
4
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5
··
5
··
4
··
students understand what . and , mean and how
2 . 2
4
5
··
··
2 is greater than 2 .
4
5
··
··
2 , 2
5
4
··
··
2 is less than 2 .
5
4
··
··
What if June’s granola bar was larger than Adriana’s? Would the comparison make
sense?
the direction of the sign shows the comparison.
• Discuss the question about the sizes of the granola
bars. Have students explain their thinking.
• Students answer Reflect on their own. Consider
having students share their ideas with a partner.
Then discuss as a group. Reinforce the idea that the
comparison doesn’t make sense unless the wholes are
the same size. Point out that often students will see
fractions in a problem with no diagram.
To compare fractions, you must use the same-size whole.
You can also use equivalent fractions to compare fractions. Look for numbers that
you can multiply by the denominators so that the fractions end up with the same
denominators.
2 3 4 5 8 and 2 3 5 5 10
4 ··
20
4 ··
5 ··
20
··
··
8 , 10 , so 2 , 2
20 ··
20
5 ··
4
··
··
5
··
Reflect
1 Explain how you can tell which fraction is greater, 2 or 3 .
5 ··
10
··
Possible answer: Multiply the numerator and denominator by 2 to find an
equivalent fraction in tenths: 4 . Since 4 . 3 , 2 . 3 .
10
···
Hands-On Activity
10
···
10 ··
5
···
10
···
79
L10: Compare Fractions
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Investigate the importance of comparing
fractions from same-size wholes.
Materials: 8 squares of paper of four different sizes
with two of each size, cards, markers
• Have students divide different squares into thirds,
fourths, fifths, sixths, and tenths.
• Have students shade 2–5 parts in each square and
write each resulting fraction on a card.
• Set out the squares so that two different-sized
squares and two same-sized squares are paired.
• Students discuss whether or not each pair of
fractions can be compared, and why. Fractions of
different-sized wholes cannot be compared, even
if they have the same numerators or same
denominators. Fractions of the same-sized wholes
can be compared; students have not yet learned
how to compare fractions of same-sized wholes
that have different numerators and different
denominators, but they will learn to do that in
this lesson.
142
Real-World connection
compare fractions in everyday recipes.
Materials: recipes, measuring cups and measuring
spoons, sand or rice, bowls
Provide students with a variety of recipes that have
fractions of cups, tablespoons, and teaspoons in
their ingredients. Have students read through the
recipes and write down the fractions they see and
arrange them by unit. Students write all the fractions
of cups in one group, the fractions of tablespoons in
another group, and the fractions of teaspoons in a
third group. Have them use visual models and ., ,,
and 5 to compare the fractions in each group. If
time allows, show students the measuring cups and
spoons and have them measure the fractional
quantities using sand or rice to compare amounts.
L10: Compare Fractions
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Part 2: Modeled Instruction
Lesson 10
At A GlAnce
Part 2: Modeled Instruction
Students use a model to study a problem involving
fraction comparison. They solve the problem using
common denominators.
Lesson 10
Read the problem below. Then explore different ways to understand it.
A grasshopper weighs about 2 of an ounce. A beetle weighs 8 of an ounce.
100
···
Which weighs more?
SteP By SteP
10
··
Picture It
• Read the problem at the top of the page as a class.
You can use models to help compare fractions.
The following model shows the weights of the grasshopper and beetle.
• Have students identify this as fraction comparison.
Grasshopper
Beetle
• Ask students to describe the models in Picture It and
explain how to use these to solve the problem.
• Point out that the fractions have different
denominators. Show this on the models.
Solve It
You can use a common denominator to help you solve the problem.
It is hard to compare two fractions with different numerators and different
denominators. You can write an equivalent fraction for one or both of the fractions so
they have a common denominator. Fractions with the same denominator are divided
into the same number of equal parts. If fractions have the same denominator, you
can just compare the numerators.
• Read Solve It as a class. Have students identify a
common denominator. [hundredths]
8 and 2 .
Compare 10
100
··
MPS tip: Students must apply knowledge of
···
First, look at the denominator, 10. Can you multiply 10 by any number to get 100? Yes,
10 3 10 5 100.
Find a fraction equivalent to 8 that has a denominator of 100:
equivalent fractions to solve this problem.
8 3 10 5 80
10 ···
100
··
10
··
10
··
Encourage them to analyze mathematical
Compare the numerators of 80 and 2 :
100
···
as they search for a viable common denominator.
Ask, How could you rewrite 2 as tenths? [You can’t
100
···
80 . 2
relationships to connect mathematical ideas (4.1.F)
So, 80 . 2 .
100
···
80
100
···
L10: Compare Fractions
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100
···
because you would have to divide 2 by 10.] How
could you rewrite 8 as hundredths? [Multiply both
10
··
numbers by 10 to get 80 hundredths.]
Mathematical Discourse
Visual Model
compare the fractions using a number line.
• After completing the page, present another model.
• Draw two number lines from 0 to 1, one above the
other. On one number line, mark and label tenths.
On the other number line, mark and label every
10 hundredths: 10, 20, etc. Mark 10 hundredths
between each tenth.
• Guide students to recognize that the number lines
are the same length.
• Have a volunteer mark 2 on the hundredths
100
···
number line. Have another volunteer mark 8 .
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··
• Students compare the two values.
2 , 8 or 8 . 2 .
3 ···
100
10
10
100 4
··
···
···
L10: Compare Fractions
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• What do you notice about the two models?
One model has 10 parts and the other has
100 parts. The models are the same size.
Different numbers of parts are shaded.
• Why are the models divided into different numbers
of parts?
The grasshopper’s weight is given in
hundredths, so that model is divided into 100.
The beetle’s weight is given in tenths, so that
model is divided into 10.
• How can you compare these fractions?
Make both fractions have a common
denominator so that both wholes have the same
number of parts. Using the common
denominator, write an equivalent fraction for
the beetle’s weight.
143
Part 2: Guided Instruction
Lesson 10
At A GlAnce
Students revisit the problem on page 80. They use
common numerators, along with a visual model, to
compare the fractions and solve the problem.
SteP By SteP
• Tell students that Connect It refers to the problem on
page 80.
• Work through problems 2–7 with students. Students
should understand that, when the numerators are the
same, the same number of pieces are shaded in each
fraction. Then they need to look at the total number
and size of the pieces in the whole to determine
which fraction has the greater portion shaded.
• Students work with a partner to solve the Try It
problems. Circulate and support students’ work.
• Have students explain how they found the solutions
and discuss their explanations with the class.
MPS tip: Connect It problem 7 may challenge
Part 2: Guided Instruction
Lesson 10
Connect It
Now you will solve the problem from the previous page by finding a common
numerator.
2 What is an equivalent fraction for 2 that has a numerator of 8?
100
···
8
400
····
3 One model is divided into 400 equal parts
and the other is divided into 10 equal parts.
Which has smaller parts?
the one divided into 400 equal parts
4 Shade 8 pieces of each model.
5 Which model has a greater area shaded?
6 Which fraction is greater, 8 or 8 ?
400 ··
10
···
the one divided into 10 pieces
8
10
···
7 Look at the denominators of 8 and 8 . When two fractions have the same
400
10
···
··
numerator and different denominators, how do you know which one is greater?
Explain. The fraction with the smaller denominator has bigger parts, so it
is greater. The numerators show the same number of parts, but the parts
are different sizes.
Try It
Use what you just learned to solve these problems.
8 Mel’s tomato plant is 8 of a foot tall. Her pepper plant is 3 of a foot tall. Compare
12
4
··
··
the heights of the plants using a symbol.
8 , 3 or 3 . 8
4
4 ···
12
··
··
12
···
9 Compare the fractions 3 and 5 using a symbol. 3
5
10
··
··
. 5 or 5 , 3
5 ···
10
10 ··
5
··
···
students because it can be confusing that the
greater number actually means smaller parts.
81
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Encourage students to create and use
representations (4.1.E) as they work on this
problem. Ask, What does a numerator of 400 mean?
If students struggle, draw a different example on
the board. Use a simpler model showing 2
5
··
compared to 2 . Elicit that the fraction with the
10
··
smaller denominator always has bigger parts.
tRy It SolutIonS
8 Solution: 8 , 3 or 3 . 8 ; Students may use
12
··
4
··
4
··
12
··
common numerators. 24 is a common numerator
(3 3 8). 24 5 8 and 24 5 3 . 36ths are smaller than
36
··
12
··
32nds, so 8 , 3 .
12
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32
··
4
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4
··
ERROR ALERT: Students who wrote 8 . 3 may
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··
4
··
have reasoned that since 12 . 4 and 8 . 3, the
fraction 8 must be greater. Have students draw
12
··
fraction bar models to illustrate that 3 . 8 . Then
4
··
12
··
help them find common numerators and solve.
9 Solution: 3 . 5 or 5 , 3 ; Students may recognize
5
10
10
5
··
··
··
··
5
1
3
1
that 5 and . , so 3 . 5 .
10
2
5
2
5
10
··
··
··
··
··
··
144
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Part 3: Modeled Instruction
Lesson 10
At A GlAnce
Part 3: Modeled Instruction
Students explore a problem involving fraction
comparison. A number line and a benchmark fraction
help students understand the quantities being
compared.
Lesson 10
Read the problem below. Then explore different ways to use benchmarks to
compare fractions.
Jasmine’s swimming lesson lasts for 2 of an hour. It takes her 1 of an hour to do
3
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6
··
her homework. Will Jasmine spend more time on her homework or at her
swimming lesson?
SteP By SteP
Model It
• Read the problem at the top of the page as a class.
You can use a number line to help you compare fractions.
The number line shows where the fractions 2 and 1 are compared to 0 and 1.
3
··
• Discuss the meaning of the problem.
1
6
• Read Model It as a class. Have students describe the
features of the number line and explain where the
fractions from the problem are located.
6
··
2
3
1
2
0
1
The number line shows that 1 is closer to 0 than 2 is, and that 2 is closer to 1 than 1 is.
6
3
3
6
··
··
··
··
This means that 2 . 1 .
3 ··
6
··
• Read Solve It as a class. Have a volunteer describe the
Solve It
meaning of 1 . [It is midway between 0 and 1, so it is
You can use a benchmark fraction to solve the problem.
easy to find.]
Look at the number line. It shows that 1 is less than 1 and 2 is greater than 1 .
Another way to compare fractions is by using the fraction 1 as a benchmark.
2
··
2
··
6
··
So, 1 , 2 and 2 . 1 .
6
··
• Work through the comparisons and guide students
to the solution.
3
··
3
··
2
··
3
··
2
··
6
··
Jasmine will spend more time at her swimming lesson than on homework.
• Remind students to relate the mathematical answer
to the problem context.
ell Support
82
Review the term benchmark fraction. The term
benchmark originally came from surveying, or
measuring land. Surveyors made a cut into stone to
help measure the height of the land the same way
every time. These cuts were called benchmarks
because they served as a “bench” for the surveyor’s
leveling tools. We use benchmark fractions as a
reference point for comparing other fractions.
Hands-On Activity
compare fractions to 1 .
2
··
Materials: Number line from 0 to 1, cards with a
variety of fractions written on them (denominators
of 3, 4, 5, 6, 8, 10, 12, 100)
• Have students label 1 on the number line.
2
··
L10: Compare Fractions
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Mathematical Discourse
• How do you know that 1 is less than 1 ?
6
2
··
··
Sixths are smaller than halves.
• How does a number line help you compare fractions?
It helps you see which fractions are closer to 0
and which are closer to 1. You can mark off
different sized pieces for easy comparison.
• How does a benchmark fraction, such as 1 , help you
2
··
compare fractions?
• Give students cards with a variety of fractions.
You know how large 1 is, so it is a useful
• Students take a card and place it on the number
reference point. It is often easier to compare a
line between 0 and 1 or between 1 and 1.
2
··
2
··
• Ask students to explain each fraction’s placement.
Discuss any fractions they are not sure about.
L10: Compare Fractions
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2
··
fraction to 1 than to another fraction. By
2
··
comparing two fractions to 1 , you can often see
2
··
how they compare to each other.
145
Part 3: Guided Instruction
Lesson 10
At A GlAnce
Part 3: Guided Instruction
Students compare two fractions using 1 as a
benchmark. Then they solve problems using
benchmark fractions or other methods.
Lesson 10
Connect It
Now you will solve a similar problem using 1 as a benchmark.
Think about these two fractions: 11 and 7
10
··
SteP By SteP
11 Which fraction is less than 1?
• Introduce the idea that 1 is not the only fraction that
2
··
10
···
10
··
10
···
8
··
11
10
··
7
.
8
··
14 Explain how you can use benchmarks to compare fractions.
10
··
You can compare both fractions to the same number to see which one is
more than 10 . Also, 8 is the same as 1, and 7 is less
greater than, less than, or equal to that benchmark. The fraction that is
8
··
greater than the benchmark is greater than the one that is less than or
equal to the benchmark.
• Have students share and explain their answers to
problem 14. Encourage students to ask each other
questions to clarify the reasoning.
Try It
Use what you just learned to solve these problems.
15 Fill in the blank. Explain how you found your answer.
5
10
··
I used 1 as a benchmark. 5 5 1 and 3 . 1 . So, 3 . 5 .
2
··
10
···
16 Nathan walked
MPS tip: Help students use benchmark fractions
strategically (4.1.C). When students discuss their
comparison, guide them to consider what
benchmark fraction will be most helpful in solving
a given problem. For example, in problems 10–13, 1
is a useful benchmark because both fractions in the
problem are near 1 on a number line.
8
··
13 Fill in the blank with the correct symbol to show the comparison.
They should see that 10 is the same as 1, and 11 is
8
··
7
8
··
Since 11 is greater than 1 and 7 is less than 1, 11 must be greater than 7 .
• Walk through problems 10–13 with students.
10
··
10
···
12 Which fraction is greater? Explain why.
students can use as a benchmark fraction.
than 8 .
8
··
8
··
11
10 Which fraction is greater than 1?
2
··
4
··
2
··
4
··
,
3
4
··
10
···
10 of a mile. Sarah walked 19 of a mile. Who walked a greater
20
··
10
··
distance? Explain.
10 out of 10 is equal to 1. 19 out of 20 is 19 , which is less than 1. Nathan
20
···
walked a greater distance than Sarah.
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L10: Compare Fractions
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tRy It SolutIonS
15 Solution: 5 , 3 ; Students may use 1 as
10
··
4
··
2
··
a benchmark to determine that 5 5 1 and 3 . 1 .
Therefore, 5 , 3 .
10
··
10
··
2
··
4
··
2
··
4
··
16 Solution: Nathan; Students may realize that
10 is the same as 1 and 19 is less than 1.
20
··
Therefore, 10 is a greater distance.
10
··
ERROR ALERT: Students who chose 19 may have
20
··
10
··
reasoned that 19 parts is more than 10 parts.
Remind students that they also need to take into
account the total number of parts that each mile is
divided into (the denominators of the fractions).
146
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Part 4: Guided Practice
Lesson 10
Part 4: Guided Practice
Lesson 10
Study the model below. Then solve problems 17–19.
Student Model
It is important that both
measurements use the
same unit!
Part 4: Guided Practice
18 Compare the fractions
Lesson 10
3 and 7 using the benchmark fraction 1 .
12
2
··
··
10
··
Show your work.
You already know about
how big 1 is!
2
··
Becker catches a fish that is 3 of a yard long. To keep the fish,
12
··
Possible answer: 3 , 1 and 7 . 1 so 3 , 7
10 ··
2
12 ··
2
10 ···
12
···
···
···
it has to be longer than 1 of a yard. Can Becker keep his fish?
3
··
Look at how you could show your work using a number line.
Pair/Share
1
2
3
4
5
6
7
8
9
10 11
0 12 12 12 12 12 12 12 12 12 12 12 1
1
3
Pair/Share
How else could you
solve this problem?
Which strategy for
comparing do you think
works best with these
fractions?
Solution:
2
3
19 Janelle walked
Since 3 is less than 1 , Becker can’t keep his fish.
12
···
3
··
problems. Myron has finished 7 of the problems and Jane has
9
··
finished 2 of the problems. Who has finished more of the
3
··
homework?
Show your work.
9
··
Pair/Share
How did you and your
partner choose what
strategy to use to solve
the problem?
84
9 ··
9
··
There are several ways to
compare fractions!
A 3 5 6 and 6 , 6
6
··
B
C
D
12
12 ··
10
··
··
3 5 6 and 6 . 6
12
12 ··
10
··
··
6 5 3 and 3 , 3
10 ··
5
5 ··
6
··
··
3 , 1 and 6 . 1
6 ··
2
10 ··
2
··
··
6
··
Tina chose B as the correct answer. How did she get that answer?
Possible answer: Tina found an equivalent fraction, but
Possible answer: 2 5 6
Since 7 . 6 , 7 . 2
3 of a mile. Pedro walked 6 of a mile. Which
10
··
6
··
statement shows how to find the greater fraction? Circle the letter
of the correct answer.
17 Myron and Jane are working on the same set of homework
3
··
Draw a model to check
your answer.
3 , 7 or 7 . 3
10 ···
12
12 ···
10
···
Solution: ···
9
··
compared them incorrectly. She thought that 6 is greater
3
··
12
···
than 6 because 12 is greater than 10.
10
···
Pair/Share
How can you find the
answer using a
benchmark fraction?
Solution: Myron has finished more of the homework.
L10: Compare Fractions
85
L10: Compare Fractions
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©Curriculum Associates, LLC
Copying is not permitted.
At A GlAnce
SolutionS
Students study an example that uses a number line and
equivalent fractions to compare two fractions. Then
they solve word problems involving fraction
comparison using a variety of methods.
Ex Since 3 , 1 , Becker can’t keep his fish.
12
··
3
··
The number line shows twelfths and thirds.
17 Solution: Myron finished more of the homework;
Multiply the numerator and denominator of 2 by
SteP By SteP
3 so both fractions have a denominator of 9.
• Ask students to solve the problems individually.
Circulate and provide support.
(DOK 2)
• Watch for students who struggle with the reasoning
required for using benchmark fractions or finding
common numerators.
• When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
18 Solution: 3 , 7 or 7 . 3 ; Compare the fractions
10
··
12
··
12
··
10
··
to 1 . Use common denominators or common
2
··
numerators. 3 , 1 , 7 . 1 . (DOK 2)
10
··
2 ··
12
··
2
··
19 Solution: A; Use common numerators and then look
at the denominators to compare.
Explain to students why the other two answer
choices are not correct:
C is not correct because fifths are greater than
sixths.
D is not correct because 3 5 1 . (DOK 3)
6
··
L10: Compare Fractions
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3
··
2
··
147
Part 5: TEKS Practice
Lesson 10
Part 5: TEKS Practice
Lesson 10
Solve the problems.
1
2
4
2 cup of raisins and 3 cup of almonds to make trail mix. Which statement
Grant needs }
}
4
3
can be used to find out if there are more raisins or almonds in the mix?
2
8
3
2
4
3
4
2
6
3
6
5 }} and } 5 }}
}
4
3
12
12
B
5 } and } 5 }
}
4
5
3
6
C
5 } and } 5 }}
}
4
3
9
12
D
2 5 6 and 3 5 6
}
}
}
}
4
7
3
9
Show your work.
2 , 1 and 5 5 1 , so 2 , 5
Possible answer: }
}}
}
6 }
6 }}
2
10 }
2
10
Answer Sam
5
Tell whether each sentence is True or False.
2, 6
15
···
b.
7 .7
10
8
···
··
c.
1.3
2
8
··
··
d.
254
4
6
··
··
e.
True
5
··
True
30 5 3
50
···
500
····
3 False
3 False
3 True
False
True
3 False
3 True
False
Lesson 10
5 of an hour. He spent
Sam’s music teacher told him to practice his trombone for }}
10
2 of an hour practicing. Did he practice long enough?
}
6
9
A
a.
3
Part 5: TEKS Practice
practice long enough.
Olivia and Eleanor each made the same amount of lemonade to sell at a lemonade
stand. Olivia poured all of her lemonade into 10 equal glasses. Eleanor poured all of
her lemonade into 5 equal glasses. Olivia sold 7 glasses of lemonade and Eleanor sold
2 glasses. Which girl sold a greater fraction of her lemonade? Compare the fractions
using a symbol.
Possible answer: Olivia sold 7 of her lemonade and Eleanor
10
···
Show your work.
sold 2 of hers. 2 5 4 ; 7 . 4 , so Olivia sold more.
5
5 ···
10 ···
10 ···
10
··
··
Answer
6
did not
Olivia
sold a greater fraction of her lemonade.
9 of her
Rachel and Sierra are selling boxes of fruit as a fundraiser. Rachel has sold }}
10
5 of her boxes. Which girl has sold a greater fraction
boxes of fruit and Sierra has sold }
8
of her boxes of fruit? Draw a model to show your answer.
Fill in the blank with one of the symbols shown to compare 5 and 5 .
10
···
,
5
}}
10
,
5
8
··
Show your work.
.
9
10
5
}
8
5
8
9 has a bigger area shaded, so it is greater than 5 .
10
8
···
··
Rachel
Answer
has sold a greater fraction of her boxes of fruit.
86
L10: Compare Fractions
87
L10: Compare Fractions
©Curriculum Associates, LLC
Copying is not permitted.
AT A GlAncE
Students compare fractions to solve word problems that
might appear on a mathematics test.
SoluTionS
1 Solution: A; Find a common denominator: 12.
Multiply numerator and denominator of 2 by 4 and
3
··
numerator and denominator of 3 by 3. Then
4
··
numerators can be compared. (DOK 2)
©Curriculum Associates, LLC
Copying is not permitted.
4 Solution: did not; Compare the fractions using
the benchmark fraction 1 . 5 5 1 and 2 , 1 .
So, 2 , 5 . (DOK 1)
6
··
2 ··
10
··
2
··
6
··
2
··
10
··
5 Solution: Olivia; Compare the fractions 7 and 2 .
10
··
5
··
Students may find a common denominator and
write 2 as tenths. (DOK 1)
5
··
6 Solution: Rachel; See possible student work above.
Compare the shaded parts of the bar models.
(DOK 1)
2 Solution: a. False; b. False; c. True; d. False; e. True
(DOK 1)
3 Solution: ,; 5 of 10 equal parts is a smaller amount
than 5 of 8 equal parts. (DOK 1)
148
L10: Compare Fractions
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Differentiated Instruction
Lesson 10
Assessment and Remediation
• Ask students to compare 7 and 4 and to show a visual model and explain their work.
10
5
··
··
• For students who are struggling, use the chart below to guide remediation.
• After providing remediation, check students’ understanding. Ask students to compare 3 and 5 .
8
12
··
··
• If a student is still having difficulty, use STAAR Ready Instruction, Level 3, Lesson 18.
If the error is . . .
Students may . . .
To remediate . . .
7 is greater than
not understand the
reason for comparing
fractions with the
same number of
pieces (numerators).
Have students draw same-size bar models of 7 and 4 . Point out that the
10
5
··
··
size of the pieces is not the same, so you can’t compare numerators. Ask
students to come up with a common numerator (such as 28) so that the
number of pieces is the same and write equivalent fractions for 7 and 4
10
5
··
··
so they can compare.
not understand the
reason for comparing
fractions with
same-size pieces
(denominators).
Have students draw same-size bar models of 7 and 4 . Ask if the number
10
5
··
··
of shaded pieces is the same (no) and explain that therefore you can’t
compare based on the size of the pieces. Ask students to come up with a
common denominator (such as 10) so that the size of the pieces is the
same and write equivalent fractions for 7 and 4 so they can compare.
not understand
when it is
appropriate to use a
benchmark fraction.
Explain that when both fractions are greater than (or less than) the
benchmark fraction, you don’t have enough information to compare.
Have students make a number line from 0 to 1. Help them mark and label
tenths and fifths. Have students locate 7 and 4 on the number line to
10
5
··
··
make the comparison.
10
··
4 because 7 . 4
5
··
7 is greater than
10
··
4 because 10 . 5
5
··
7 5 4 because
5
··
7 . 1 and 4 . 1
10 ··
2
5 ··
2
··
··
10
··
10
··
5
··
Hands-On Activity
Challenge Activity
Draw models to compare fractions.
Compare three or more fractions.
Materials: 1-cm grid paper, scissors, markers
or pencils
Materials: Fractions written on cards
Have students work with a partner. Provide students
with 1-centimeter grid paper. Instruct each student in
the pair to draw and cut out two 3-by-4 arrays. Have
them use the arrays to show halves, thirds, fourths,
sixths and twelfths. Have each student color part of
each of their models and write a fraction to show the
shaded part. Have students compare the fractions
using ., ,, or 5. Have students repeat by drawing
and cutting out two 2-by-8 arrays. They should use
the array to model and compare halves, fourths,
eighths, and sixteenths. Ask students if they can use
these models to compare eighths and twelfths. [No,
because the models are different sizes.] Have students
turn over one of their 2-by-8 arrays and draw 12
equal parts. Ask students if they could use this array
and another of their 2-by-8 arrays to compare eighths
and twelfths [yes], and have them do so.
L10: Compare Fractions
©Curriculum Associates, LLC
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Give a pair or small group of students a pile of cards
with fractions written on them. Students set out
three, four, or even five fractions and place them in
order from least to greatest. The strategy is to choose
one fraction and then compare it to another fraction.
Then choose a third fraction and compare it to each
of the already ordered fractions. Then choose a
fourth fraction and compare to each, and so on. To
make the comparisons, students may draw number
lines or visual models or compare to benchmark
fractions using what they know about equivalent
fractions.
149
Focus on Math Concepts
Lesson 11
(Student Book pages 88–93)
Understand Fraction Addition and Subtraction
Lesson objectives
the Learning Progression
• Understand addition as joining parts.
One goal of the Texas Essential Knowledge and Skills
for Mathematics is to develop a deeper understanding
of fractions by using a progression of concepts from
simple to complex. This lesson prepares students for
the conceptual shift involved in progressing from
adding and subtracting whole numbers to adding and
subtracting fractions. Students are guided to think of
operations with fractions as very much like operations
with whole numbers.
• Understand subtraction as separating parts.
• Extend their understanding of addition and
subtraction of whole numbers to addition and
subtraction of fractions.
• Use fraction models to add and subtract fractions
with like denominators.
Prerequisite skiLLs
Students see that you can count with unit fractions
In order to be proficient with the concepts in this
lesson, students should:
just as you count with whole numbers. And because
• Know addition and subtraction basic facts.
do arithmetic with them. If you walked 2 of a
• Understand the meaning of fractions.
mile (2 fifths) yesterday and 4 of a mile (4 fifths) today,
• Identify numerators and denominators.
• Write whole numbers as fractions.
vocabuLary
There is no new vocabulary. Review the following
key terms.
numerator: the top number in a fraction; it tells the
number of equal parts that are being described
you can count with unit fractions, you can also
5
··
5
··
altogether you walked 6 of a mile (6 fifths; because
5
··
2 things plus 4 more of those things is 6 of those
things).
Students use the meaning of fractions and the
meanings of addition and subtraction that were built in
earlier grades to understand why the procedures for
adding and subtracting fractions make sense.
denominator: the bottom number in a fraction; it tells
the total number of equal parts in the whole
teks Focus
4.3.E Represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build
to the number line and properties of operations. Readiness Standard
MatheMaticaL Process stanDarDs (MPs): 4.1.A, 4.1.C, 4.1.D, 4.1.E, 4.1.F, 4.1.G (See page A9 for full text. Also eee MPS Tips in
the lesson.)
150
L11: Understand Fraction Addition and Subtraction
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Part 1: Introduction
Lesson 11
At A GlAnce
Focus on Math Concepts
Students explore the idea that adding fractions is not
essentially different from adding whole numbers.
Lesson 11
4.3.E
What’s really going on when we add numbers?
expression 2 1 3.
4
··
Adding means joining or putting things together.
Think about how you could explain adding 2 1 3 to a first grader. You could start
at 2, count on 3 more, and see where you end up: 2 . . . 3 . . . 4 . . . 5.
Step By Step
Or, you could put a segment with a length of 2 and a segment with a length of 3 next
to each other on a number line to show 2 1 3.
• Introduce the Question at the top of the page.
1
• Help students relate the number line diagram to the
sum 2 1 3.
0
1
3
4
5
6
7
8
9
10
You can put a segment with a length of 24 and a segment
··
Underline the
sentence that
explains what adding
fractions means.
with a length of 3 next to each other to show 2 1 3 .
4
··
1
4
made up of 5 one-fourths.
0
4
• If students need additional support with locating
1
4
1
4
4
··
1
4
2
4
1
4
3
4
0
4
··
1
4
4
4
1
5
4
6
4
7
4
8
4
2
When you add 2 1 3 , you are putting one-fourths together.
fractions on a number line, have them build a
4
··
88
4
··
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
Copying is not permitted.
Mathematical Discourse
To extend students’ understanding of decomposing
fractions, follow these steps:
• Draw and label a number line on the board from
0 to 2 like the one on the page showing fourths.
• Ask students to think of two different fractions
that you could put together that would give you
the same sum as adding 2 and 3 .
4
··
2
1
Adding fractions means joining or putting together parts of the
same whole.
number 5 is made up of 5 ones, the number 5 is
4
··
concept extension
1
1
Think
• Guide students to recognize that just as the
number line by putting 1 fraction strips end-to-end,
4
··
creating a concrete model to show 2 1 3 .
4
4
··
··
1
When you add 2 1 3, you are putting ones together.
• Read Think with students. Reinforce the idea that
fractions are numbers.
4
··
• Have a volunteer go to the board to show the
two fractions on the number line. 3 1 and 4 in
4
··
either order 4
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
TEKS
Understand Fraction Addition and Subtraction
A number line diagram gives meaning to the
4
··
Part 1: Introduction
Copying is not permitted.
4
··
• How would you explain adding in your own words?
Responses should include phrases such as
“join” or “put together.”
• How is adding fractions like adding whole numbers?
Students may mention that, in both cases, you
are putting things together.
• Can you think of another way to explain adding
fractions?
Students may suggest that you can count on
with fractions just like you count on with
whole numbers.
151
Part 1: Introduction
Lesson 11
At A GlAnce
Students explore the idea that subtracting fractions is
not essentially different from subtracting whole
numbers. A number line diagram gives meaning to the
expression 5 2 2 .
4
··
Part 1: Introduction
Think
Subtracting means separating or taking away.
• Read Think with students.
• Discuss how the number line represents the problem
5 2 2. Show how to subtract on the number line.
(start at 5 and count back 2)
• Ask a volunteer to explain how to use the number
line to find 5 2 2 . Provide 1 fraction strips for
4
4
4
··
··
··
students who need more support.
• Have students read and reply to the Reflect directive.
Look at the whole
numbers. Now look at
the numerators of the
fractions. I think I see a
connection.
On a number line, you can start with a segment of length
5 and take away a segment of length 2 to show 5 2 2.
4
··
Step By Step
Lesson 11
1
1
0
1
1
2
1
1
1
1
3
4
5
6
7
8
9
10
When you subtract 5 2 2, you are taking away ones.
You can show subtracting fractions on a number line. Start with a segment of
length 5 and take away a segment of length 2 to show 5 2 2 .
4
··
4
··
1
4
0
4
1
4
1
4
1
4
1
4
1
4
2
4
3
4
4
4
0
4
··
4
··
1
4
1
4
1
5
4
6
4
7
4
8
4
2
When you subtract 5 2 2 , you are taking away one-fourths.
4
··
4
··
Now you’ll have a chance to think more about how adding or subtracting fractions is
like adding or subtracting whole numbers. You may find that using number lines or
area models can help you explain your thinking.
Reflect
Visual Model
• Tell students that you will use a number line to
show 5 2 3 .
8
··
8
··
1 Use your own words to describe what you just learned about adding and
subtracting fractions. Possible answer: I learned that adding and subtracting
fractions is just like adding and subtracting whole numbers. When the
denominators are the same, you can just add or subtract the numerators.
89
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
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• Draw a number line from 0 to 1 on the board.
• Ask students for ideas on how to divide the line so
that you can use it to help you solve the problem.
• Have students explain why dividing the line into
eighths makes sense.
• Label 0 and 1 on the line and have students
provide labels for the other marks as you move
your finger along the line.
• Ask a volunteer to show how to find the answer to
the problem using the number line.
MpS tip: In the Visual Model activity, students
are asked to create and use a representation and
explain why dividing the line into eighths makes
sense. (4.1.E)
152
Mathematical Discourse
• How would you explain subtracting in your
own words?
Listen for phrases such as “take apart” or
“take away.”
• How is subtracting fractions like subtracting whole
numbers?
Students may note that subtracting means
taking away. It doesn’t matter what kinds of
numbers you’re subtracting.
• Do you see a connection between the whole numbers
and the numerators of the fractions on this page?
Students may mention that the whole numbers
and the numerators of the fractions are the
same numbers, and to answer both problems
you subtract 2 from 5.
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
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Part 2: Guided Instruction
Lesson 11
At A GlAnce
Part 2: Guided Instruction
Students use number lines to answer questions,
reinforcing the understanding that fractions are
numbers.
Explore It
Counting on and using a number line are two ways to think about adding
fractions.
2 Count by fourths to fill in the blanks: 1 , 2 ,
4 ··
4
··
Step By Step
3
6
7
8
9
, 4 , 5 , ··
4 , ··
4 , ··
4 , ··
4
4 ··
4
··
4
··
Now label the number line.
• Tell students that they will have time to work
individually on the Explore It problems on this page
and then share their responses in groups. You may
choose to work through the first problem together as
a class.
0
5
··
• Take note of students who are still having difficulty
and wait to see if their understanding progresses as
they work in their groups during the next part of
the lesson.
STUDENT MISCONCEPTION ALERT: Some
students may think that a fraction is always less
than 1. If this misconception persists, use fraction
strips to demonstrate fractions less than, equal to,
and greater than 1. Then, encourage students to use
the fraction strips to show and name other fractions
greater than 1.
L11: Understand Fraction Addition and Subtraction
Copying is not permitted.
2
4
3
4
··
4
4
5
4
6
4
··
7
4
··
8
4
··
9
4
··
10
4
4
5
5 , ··
5
, ··
Now label the number line.
0
1
5
2
5
3
5
··
4
5
··
5
5
··
6
5
Use the number lines above to answer numbers 4 and 5.
7
4 What is 1 more than 6 ?
4
4
··
··
4
··
5 What is 1 more than 3 ?
5
5
··
··
5
··
4
Now try these two problems.
6 Label the number line below and use it to show 2 1 1 .
4 ··
4
··
• If students need more support, suggest that they
count out loud to help them fill in the missing
numbers in problems 2 and 3.
• To help students answer problem 4, have them put
their finger on 6 on the number line, then count on
4
··
by 1 . Similarly, to answer problem 5, have them put
4
··
their finger on 3 on the number line and count on
5
··
by 1 .
1
4
3
3 Count by fifths to fill in the blanks: 1 , 2 , ··
5
5 ··
5
··
• As students work individually, circulate among them.
This is an opportunity to assess student
understanding and address student misconceptions.
Use the Mathematical Discourse questions to engage
student thinking.
©Curriculum Associates, LLC
Lesson 11
0
4
1
4
2
4
3
4
4
4
5
4
6
4
7
4
8
4
7 Label the number line below and use it to show 3 1 1 .
4 ··
4
··
0
4
90
1
4
2
4
3
4
4
4
5
4
6
4
7
4
8
4
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
Copying is not permitted.
Mathematical Discourse
• In which direction on the number line do you move
when adding? Explain.
Responses might include the fact that adding
means joining so you will be getting segments
that are longer or an answer farther to the right
than the number you started with.
• For problem 5, will the answer change if you find
3 more than 1 ? Explain.
5
··
5
··
Listen for responses that demonstrate an
understanding that you can add two numbers
in any order and get the same sum.
153
Part 2: Guided Instruction
Lesson 11
At A GlAnce
Students use number lines to show subtracting
fractions. Then they use models to show adding and
subtracting fractions.
Part 2: Guided Instruction
Lesson 11
Talk About It
Solve the problems below as a group.
8 Look at your answers to problems 2 and 3. How is counting by fractions the same
Step By Step
• Organize students in pairs or groups. You may
choose to work through the first Talk About It
problem together as a class.
• Walk around to each group, listen to, and join in
on discussions at different points. Use the
Mathematical Discourse questions to help support or
extend students’ thinking.
as counting with whole numbers? Possible answer: When you count with
whole numbers, you count by ones. When you count with fractions, the
numerator counts by ones as long as the denominators are the same.
How is it different?
Possible answer: When you count by fractions, you are counting by parts.
9 Label the number line below and use it to show 7 2 2 .
8 ··
8
··
0
• When sharing ideas about problems 9 and 10, be sure
to emphasize that when labeling the number line,
numerators count on by ones, but the denominator
remains the same.
2
8
3
8
4
8
5
8
6
8
7
8
8
8
9
8
10
8
11
8
12
8
11
6
12
6
5 1
10 Label the number line below and use it to show 2 .
6 ··
6
··
MpS tip: During this time, you may choose to ask
a particular group to prepare to share their thinking
or solution. Encourage students to justify their
thinking using precise mathematical language.
(4.1.G)
1
8
0
1
6
2
6
3
6
4
6
5
6
6
6
7
6
8
6
9
6
10
6
Try It Another Way
Work with your group to use the area models to show adding or subtracting
fractions.
1 2
11 Show 1 .
8 ··
8
··
6 2 2.
12 Show
10 ··
10
··
91
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
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• Direct the group’s attention to Try It Another Way.
Have a volunteer from each group come to the board
to draw the group’s solutions to problems 11 and 12.
Hands-On Activity
Use fraction strips to subtract fractions.
Materials: strips of paper, markers, scissors
• Model how to fold the strip of paper in half, in
half again, and in half a third time. Tell students
to unfold the strips and use a marker to show the
8 equal sections.
• Direct students to cut out each section. Ask them
to name the fraction that represents each section.
3 ··18 4 Have them label each section.
• Write 7 2 5 on the board. Have students use their
8
8
··
··
strips to show that the difference is 2 .
8
··
154
Mathematical Discourse
• What is another name for 8 ? 12 ? Explain your
8 ··
6
··
thinking.
Students should recognize that eight 1 pieces
8
··
make up 1 whole and that twelve 1 pieces make
6
··
up 2 wholes.
• Can you think of another way to show finding a
difference on a number line?
Students may mention adding up to subtract.
For example, to find 7 2 2 , you might start
8
··
8
··
at 2 and think, “What do I need to add to
8
··
get to 7 ?”
8
··
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
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Part 3: Guided Practice
Lesson 11
At A GlAnce
Part 3: Guided Practice
Students demonstrate their understanding of adding
and subtracting fractions as they talk through three
problems.
Lesson 11
Connect It
Talk through these problems as a class, then write your answers below.
2 1
13 Compare: Draw two different models to show 2 .
3 ··
3
··
Step By Step
Possible answers:
• Discuss each Connect It problem as a class using the
discussion points outlined below.
1
3
0
2
3
3
3
14 Explain: Rob had a large pizza and
a small pizza. He cut each pizza into
fourths. He took one fourth from each
Compare:
pizza and used the following problem
to show their sum: ··14 1 ··14 5 ··24 .
• You may choose to have students work in pairs to
encourage sharing ideas. Each partner draws a
different model.
What did Rob do wrong?
Possible answer: Rob’s addition is correct, but he cannot add one fourth of
the large pizza and one fourth of the small pizza in this way because the
wholes are not the same.
• For a quick and easy assessment, have students draw
their models on small whiteboards or paper and hold
them up. Choose several pairs to explain their
models to the class.
• Use the following to lead the class discussion:
Explain how you knew the number of parts to draw
in the whole.
How did you show subtraction in your model?
How are the models the same? How are they
different?
15 Demonstrate: Think about how you would add three whole numbers. You add
two of the numbers first, and then add the third to that sum. You add three
fractions the same way.
1 1 3 1 4.
Use the number line and area model below to show ··
10 ··
10 ··
10
0
92
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
Copying is not permitted.
Demonstrate:
Explain:
• The second problem focuses on the importance of
the whole and the fact that you cannot add or
subtract fractions unless they refer to the same
whole.
• Read the problem together as a class. Ask students to
continue to work in pairs to discuss and write their
responses about what Rob did wrong.
• Begin the discussion by asking questions, such as:
What fraction describes a slice of the larger pizza?
3 ··14 4 What fraction describes a slice of the
smaller pizza? 3 1 4
4
··
Are both 1 s the same size? [no] Why not? [the whole
4
··
• This discussion gives students an opportunity to
think about problems that involve adding three
fractions.
MpS tip: Ask students to show how to use a
number line as a tool to model the sum of three
whole numbers. (4.1.C)
• Discuss how you can add three (or more) fractions in
the same way as adding whole numbers as long as
you are talking about the same type of fractions.
Have students explain how they used the models to
show the sum.
• Remind students to start at 0 when labeling the
number line.
pizzas are not the same size]
Why doesn’t it make sense to add these two fractions?
[the wholes are not the same]
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
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155
Part 4: Performance Task
Lesson 11
At A GlAnce
Part 4: Performance Task
Students write two questions that can be answered
using some or all of the given information about the
problem situation. Then they answer one of the
questions.
Lesson 11
Put It Together
16 Use what you have learned to complete this task.
Jen has 4 of a kilogram of dog food. Luis has 3 of a kilogram of dog food.
10
10
··
··
A large dog eats 2 of a kilogram in one meal.
10
··
Step By Step
A Write two different questions about this problem that involve adding or
subtracting fractions.
• Direct students to complete the Put It Together task
on their own.
i Possible answer: How much dog food do Jen and Luis have altogether?
ii Possible answer: How much more dog food does Jen have than Luis?
• Explain to students that the questions they write do
not have to use all of the given information.
B Choose one of your questions to answer. Circle the question you chose.
Show how to find the answer using a number line and an area model.
Possible answers:
• As students work on their own, walk around to
assess their progress and understanding, to answer
their questions, and to give additional support, if
needed.
0
• If time permits, have students share one of their
questions with a partner and show how to find
the answer to their partner’s question using a
visual model.
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ScorinG ruBricS
L11: Understand Fraction Addition and Subtraction
See student facsimile page for possible student answers.
A
points expectations
2
156
The response demonstrates the student’s
mathematical understanding of adding and
subtracting fractions. Both questions can be
answered using the information given in
the problem.
1
An effort was made to accomplish the task.
The response demonstrates some evidence
of verbal and mathematical reasoning, but
the student’s questions may contain some
misunderstandings.
0
There is no response or the response shows
little or no understanding of the task.
©Curriculum Associates, LLC
B
Copying is not permitted.
93
points expectations
2
Both a number line and an area model are
correctly drawn and labeled to show the
solution to the problem.
1
Only one model is correctly drawn and
labeled or the models drawn may contain
minor errors. Evidence in the response
demonstrates that with feedback, the
student can revise the work to accomplish
the task.
0
There are no models drawn or the models
show no evidence of providing visual
support for solving the problem.
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
Copying is not permitted.
Differentiated Instruction
Lesson 11
Intervention Activity
On-Level Activity
Use fraction strips to model adding and
subtracting fractions.
Find a sum greater than one.
Guide students through the following steps in
finding the sum of two fractions whose sum is
greater than one. You may have students work in
groups and present their results to the class.
Materials: fraction strips
Write an addition expression on the board, such as
2 1 3 . Have students lay 1 fraction strips end-to-end
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1. Have students draw a picture of two pizzas of
equal size, both cut into 8 equal pieces. Tell
students that two slices of one pizza and five
slices of the other pizza were eaten. Ask students
to write a fraction to represent the leftover slices.
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to show the sum. Ask them to tell you how many
1 s there are in all. Continue with similar problems.
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Include expressions whose sums are greater than
one, such as 3 1 2 .
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2. Ask students to imagine replacing missing slices
of one pizza with remaining slices from the
other pizza. Have students draw another pizza
to represent this situation and write a fraction
to represent the leftover slices.
Write a subtraction expression on the board, such
as 5 2 2 . Have students lay 1 fraction strips end-to6
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end to show 5. Then have them “take away” 2 . Ask
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them to tell you how many 1 s are left. Continue with
3. Ask students to write a number sentence that
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similar problems. Be sure to provide expressions that
represents the total amount of pizza left.
include fractions greater than one, such as 6 2 3 .
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3 ··68 1 ··38 5 ··98 4
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4. Ask students to write a fraction equivalent to 9
that makes it easier to see how many whole
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pizzas are left over. 3 11 4
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Challenge Activity
Write a question for the answer given.
Write the following problem on the board: The answer is 7 . What could the question be?
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Encourage students to think about both addition and subtraction. Provide number lines, area models, or
fraction strips for support as necessary.
Note the methods students use. Do they just guess, work out their problem, check to see if it’s correct, and then
adjust their responses if necessary? Do they use a visual model or do they work symbolically?
If time permits, give students (or pairs or groups) practice with similar problems. You might ask them to write
two questions for each answer you supply, one using addition and one using subtraction.
L11: Understand Fraction Addition and Subtraction
©Curriculum Associates, LLC
Copying is not permitted.
157