Comparing and Ordering Fractions
Analyzing Elections
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Summarize/Paraphrase/Retell, Create Representations,
Vocabulary Organizer
ACTIVITY
1.5
My Notes
Every West Middle School homeroom must elect a student council
representative. Since Mr. Fare’s homeroom students do not know
each other yet, he has asked interested students to volunteer. Andy,
Betty, Carla, and Deon decide to volunteer.
To simulate a regular election, each of the 23 students in his
homeroom will roll a number cube to vote. A 1 is a vote for Andy.
A 2 is a vote for Betty. A 3 is a vote for Carla. A 4 is a vote for Deon.
If 5 or 6 is rolled, the student continues to roll until 1, 2, 3, or 4
is rolled.
1. Work together to simulate this election.
a. In your group, roll a number cube until you have 23 votes.
Organize your data in this table.
Andy (1)
Betty (2)
Carla (3)
Deon (4)
Total
Votes
© 2010 College Board. All rights reserved.
b. Who did your group elect as the homeroom representative?
2. List the names of the candidates in order of most to least
number of votes. Next to each name, write the number of votes
he or she received.
MATH TERMS
3. What fraction of the total votes did each candidate receive?
Write the fractions in order from greatest to least.
The number of votes each
candidate received can be
written as a fraction or as a
ratio of the number of votes
received to the total number of
votes. These ratios are called
rational numbers.
Unit 1 • Number Concepts
25
ACTIVITY 1.5
Comparing and Ordering Fractions
continued
Analyzing Elections
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Quickwrite
My Notes
5 of the
4. In the election in Mr. Fare’s homeroom, Andy received ___
23
7 of the total, Carla received ___
8 of
total votes, Betty received ___
23
23
3 of the total. Who was elected?
the total, and Deon received ___
23
The 300 students at West Middle School held a traditional election
for student council officers. Eden, Frank, Gabrielle, and Hernando
ran for president.
3 of the votes,
4 of the votes, Frank received ___
5. Eden received ___
15 ___
10
2 of
Gabrielle received 1 of the votes, and Hernando received __
5
30
the votes. Why is it more difficult to decide who won this election than it was for the election in Question 4?
6. What common denominator do all the fractions in Question 4
share?
The LCD is simply the LCM for two
or more different denominators.
The LCM of 6 and 8 is 24, so you
use 24 as the LCD to write
1 and __
1.
equivalent fractions for __
8
6
26
7. You can draw a model to compare fractions. Use this method to
3 of the votes to Hernando’s __
2 of the votes.
compare Frank’s ___
5
10
a. What is the least common denominator, or LCD, of these
two fractions? (Hint: Look for the least common multiple, or
LCM, of 5 and 10.)
SpringBoard® Mathematics with Meaning™ Level 1
© 2010 College Board. All rights reserved.
To make it easier to compare the results from this election, you
can rewrite these fractions as equivalent fractions with a common
denominator.
Comparing and Ordering Fractions
ACTIVITY 1.5
continued
Analyzing Elections
SUGGESTED LEARNING STRATEGIES: Create Representations,
Quickwrite, Self Revision/Peer Revision, Think Aloud, Summarize/
Paraphrase/Retell, Vocabulary Organizer
My Notes
b. Draw a rectangle in the My Notes space. Then divide it into
the number of equal parts you found in Part a.
2.
c. Shade your rectangle to represent __
5
2.
d. Write an equivalent fraction for __
5
2 to write an inequality
e. Use the equivalent fraction for __
5
comparing the votes for Frank and Hernando. Who received
more votes?
WRITING MATH
8. Next compare the votes for Eden and Gabrielle.
a. Can you use 10 as the common denominator to compare
their votes? Explain your reasoning.
The symbols <, >, ≤, and ≥ are
inequality symbols. Remember,
each symbol opens towards the
greater number and points to
the smaller number: 5 > 1.
© 2010 College Board. All rights reserved.
b. One way to compare all four students’ votes is to find how
many of the 300 votes each candidate received. Would you
want to draw a model to do this? Why or why not?
MATH TERMS
You can use the Property of One to find equivalent fractions.
3,
2 , __
When you use the Property of One, you multiply a fraction by __
2
4 , and so on. This is the same as multiplying the fraction by the 3
__
4
3 , __
2 , __
4 , describes the number 1 in
number 1. Each of the fractions, __
2 3 4
a different way.
1 , you
To use the Property of One to find an equivalent fraction for __
2
multiply this way.
3 = _____
1 × 3 = __
3
1 = __
1 × 1 = __
1 × __
__
2 2
2 3 2×3 6
1 are each
Notice that the numerator and denominator of __
2
multiplied by 3.
The Property of One for
fractions states that if the
numerator and the denominator
of a fraction are multiplied by
the same number, its value is
not changed.
Unit 1 • Number Concepts
27
ACTIVITY 1.5
Comparing and Ordering Fractions
continued
Analyzing Elections
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Simplify the Problem, Quickwrite, Self Revision/Peer Revision
My Notes
9. Use the Property of One to rename all four fractions to find the
fraction of the 300 total votes each student received.
a. Eden
b. Frank
c. Gabrielle
d. Hernando
4 =
___
15
3 =
___
10
1 =
___
30
2=
__
5
10. Compare the renamed fractions. Then list the original
fractions from least to greatest.
Now explore some ideas about common denominators.
12. List other common denominators that could be used to write
equivalent fractions for comparing the presidential election
votes at West Middle School.
13. Choose one of the common denominators you listed in
Question 12 that you think may be easier to work with than
300 to compare the fractions. Explain your choice.
14. Change each fraction from Question 9 to an equivalent
fraction with the denominator you chose in Question 13.
28
SpringBoard® Mathematics with Meaning™ Level 1
© 2010 College Board. All rights reserved.
11. You changed each fraction to an equivalent fraction with a
common denominator of 300. Why did this make it easier to
compare the fractions of the total votes for each candidate?
Comparing and Ordering Fractions
ACTIVITY 1.5
continued
Analyzing Elections
SUGGESTED LEARNING STRATEGIES: Create Representations,
Look for a Pattern
My Notes
15. Use this table to organize your data for the election results.
Candidate
Fraction of
Votes
Eden
4
___
Frank
Gabrielle
Hernando
Fraction
Fraction
Final Rank in
(Denominator of 300) (Denominator You Chose) Election (1st–4th)
15
3
___
10
1
___
30
2
__
5
16. Explain how you determined the final ranking.
Two ways to compare fractions are to rewrite the fractions using a
common denominator or to use cross products.
You do not have to find the LCD
to write equivalent fractions.
You can always find a common
denominator by multiplying the
denominators of the fractions.
© 2010 College Board. All rights reserved.
EXAMPLE 1
5.
4 and ___
Compare __
9
11
Using a common denominator:
5
4 ? ___
__
Step 1:
9 × 11 = 99
Step 2:
Step 3:
Multiply the denominators to find a
common denominator.
Write equivalent fractions.
Compare the fractions.
9 11
5 = ___
45
4 = ___
44 and ___
__
9 99
11 99
45 , so __
5
44 < ___
4 < ___
___
99 99
9 11
Using cross products:
Step 1:
Compare the products found by
multiplying the numerator of one
fraction by the denominator of the
other fraction.
4 × 11 = 44
4
__
9
5 × 9 = 45
5
___
11
5
4 < ___
44 < 45, so __
9 11
TRY THESE A
3.
2 and __
a. Compare __
7
9
5 and ___
7.
b. Compare __
9
13
Unit 1 • Number Concepts
29
ACTIVITY 1.5
Comparing and Ordering Fractions
continued
Analyzing Elections
SUGGESTED LEARNING STRATEGIES: Identify a Subtask,
Simplify the Problem, Create Representations
My Notes
As president of the Student Council, Hernando wants to speak with
all the student groups about their concerns. The guidance counselor
gave Hernando the following data:
8
• ___
15 of the students take part in music.
1 of the students are in the art club.
• __
6
16 of the students participate in sports.
• ___
33
4 of the students are in academic clubs.
• __
9
Hernando decides to speak first with the groups that have the most
participants. To do so he must order these fractions. He knows that
a common denominator for them would be very large, so he asks
his math teacher, Ms. Germain, if there is an easier way to order
the fractions.
17. Ms. Germain decides to explain the concept with less
complicated fractions. She starts by asking Hernando to
represent each of these unit fractions.
1
3
1
4
1
2
1
5
b. She tells Hernando that he can also use number lines to
compare the fractions. Graph each fraction on the number
lines below.
1
3
1
4
30
0
0
SpringBoard® Mathematics with Meaning™ Level 1
1
1
1
2
1
5
0
1
0
1
© 2010 College Board. All rights reserved.
a. Shade each rectangle to show the fraction.
Comparing and Ordering Fractions
ACTIVITY 1.5
continued
Analyzing Elections
SUGGESTED LEARNING STRATEGIES: Quickwrite, Self Revision/
Peer Revision, Questioning the Text, Identify a Subtask, Create
Representations
My Notes
1 , __
1,
c. Use your work from Parts a and b to order the fractions __
4
3
1 , and __
1 from greatest to least.
__
5
2
d. Each of the four fractions you just ordered has the
same numerator. Tell Hernando how he can use just the
denominators to order the fractions.
4 , __
4 from
4 , ___
4 , and ___
e. Use mental math to order the fractions __
5 11 7
25
greatest to least.
Hernando can see that the fractions he wants to order do not have
either a common numerator or a common denominator.
You may recall that mental math
is working a problem in your
head without writing it on paper.
© 2010 College Board. All rights reserved.
18. He thinks that it will be easier to find a common numerator
for them rather than a common denominator.
8 , __
1,
a. What is the least common numerator of the fractions ___
15 6
16 , and __
4?
___
33
9
b. Change each of the fractions above to an equivalent fraction
with the common numerator found in Part a.
c. Order the fractions in Part b from least to greatest using the
number line below.
0
1
d. In what order will Hernando talk with the student groups?
Unit 1 • Number Concepts
31
ACTIVITY 1.5
Comparing and Ordering Fractions
continued
Analyzing Elections
CHECK YOUR UNDERSTANDING
Write your answers
answers on
on notebook
notebook paper.
paper.Show your work.
6. Your school is holding a mock election for
Show your work.
president. 250 students vote.
1. A jar is filled with 70 centimeter cubes.
There are 15 red, 9 green, 21 yellow,
20 purple, and 5 orange. Write the
fractions for each color in order from
least to greatest.
2. Draw and shade rectangles and then order
the fractions from greatest to least.
10 of the total votes.
Candidate 1 receives ___
50
9 of the total votes.
Candidate 2 receives ___
25
4 of the total votes.
Candidate 3 receives ___
10
5 of the total votes.
Candidate 4 receives ____
125
Rank the candidates by the number of
votes each received, from least to greatest.
4
1
__
2
7
__
8
5.
7 and __
3. Consider the fractions __
9
6
a. What is the LCD for these fractions?
b. Use the LCD you just found and the
Property of One to write equivalent
7 and __
5.
fractions for __
9
6
c. Which fraction is greater?
4. What is the difference between an LCD
and an LCM?
5. Two students are playing a game with
fraction cards. Each player lays a card
down and whoever has the greater amount
wins the two cards. Who wins this pair?
7. The table below shows the fraction of
students who voted for each after-school
activity. Use mental math to order the
activities from most popular to least
popular. Explain your thinking.
Computer
Games
1
__
2
Read
1
___
12
Watch
TV
1
__
6
Play
Sports
1
__
4
8. Use common numerators to compare the
weekly growth of the plant. In which week
did the plant grow the most? Explain how
you reached your conclusion.
Week
1
Player 1
Player 2
2
11
14
3
4
3
Growth
(in.)
3
___
11
6
__
7
12
___
13
9. MATHEMATICAL Describe the steps for
R E F L E C T I O N comparing and ordering
fractions with unlike denominators.
32
SpringBoard® Mathematics with Meaning™ Level 1
© 2010 College Board. All rights reserved.
3
__