Mixed Numbers and Improper Fractions
Mood Rings, Part 1
ACTIVITY
1.6
SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/
Retell, Vocabulary Organizer
My Notes
Tyrell and four of his friends from West Middle School went
to a craft fair and they all decided to buy mood rings. When
discussing their ring sizes with the ring maker, they learned about
anthropometry. Did you know that the average length around a
1 inches and the average length
woman’s ring finger is about 2 ___
16
9 inches?
around a man’s ring finger is about 2 ___
16
Before they can buy their rings, the friends must first measure
their ring fingers.
Anthropometry is the study of
the measurement of the human
body in terms of its dimensions.
People who design things for
others to use, such as bracelets
or rings, have to take typical body
measurements into account.
Jose
1 in.
2 __
4
2
Nisha
3 in.
1__
4
#1
#2
#3–4
Ema
3 in.
2 ___
16
© 2010 College Board. All rights reserved.
Remember, a mixed number is
the sum of a whole number and a
proper fraction.
2
1
3
1 inches? Count the __
1 inches on
1 inches are in 2 __
b. How many __
2
2
2
the diagram and write your answer as an improper fraction.
0
2
1
MATH TERMS
A positive improper fraction has
a value greater than or equal to
1. The numerator is greater than
or equal to the denominator.
3
__5 in.
2
c. Another way to see this is to use
a model. Name the shaded parts
of this diagram two ways.
1
2 __
2
Mixed Number: ______
(wholes + part)
1 The purpose of this question
is to engage students in the
context of the activity by making
a personal connection. Students
may need a review mini-lesson on
1 inch.
measuring to the nearest ___
16
2 Vocabulary Organizer
(a, b) This begins a series of four
questions that serve as a review
of writing mixed numbers and
improper fractions using pictures,
number lines, and abstract rules.
__5
2
Improper Fraction: ______
(total parts)
Unit 1 • Number Concepts
033-038_SB_MS1_1-6_SE.indd 33
© 2010 College Board. All rights reserved.
#5
#13–14
#6–11
#12
Paragraph
Summarize/Paraphrase/Retell
a. Look at this measuring tape to determine Bob’s finger size.
Write your answer as a mixed number in the table above.
0
Materials
Chunking the Activity
2. This table shows the finger sizes of the five friends.
1 in.
2 __
• Converting between mixed
numbers and improper fractions
• Comparing and ordering mixed
numbers and improper fractions
• Measuring tapes (at least 1 per
pair of students)
• Fraction circles/bars (optional)
Answers may vary. Sample answer: Ring sizes should vary
3 inches.
1 to 2 __
roughly from 1 __
2
4
Bob
Mixed Numbers and
Improper Fractions
Activity Focus
CONNECT TO SCIENCE
1. How big is your ring finger? To measure your ring finger wrap
a measuring tape snugly around the base of your finger. Record
1 of an inch.
your measurement to the nearest ___
16
Tyrell
7 in.
1__
8
ACTIVITY 1.6 Investigative
33
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1 Inch
MINI-LESSON: Measuring to the Nearest ___
16
Give each student a long strip of paper. Have students draw a line at
each end of their strips and label the left side 0 and the right side 1,
1.
then fold their strips in half and at the top label this crease, __
2
Students fold their strips in half again and label each crease now with _14_,
_2_, just below the label for _1_, and _3_. Have students do this two more
4
2
1 4’s. Encourage students to use different
times in order to create _18_’s and ___
16
sized lines when labeling the creases. Compare the strips students create
to a standard ruler. Let students practice measuring objects to the
1 units of their “rulers.”
nearest _12_, _14_, _18_, and ___
16
Unit 1 • Number Concepts
33
ACTIVITY 1.6 Continued
ACTIVITY 1.6
continued
Mixed Numbers and Improper Fractions
Mood Rings: Part 1
2 (continued) Think/Pair/
Share, Look for a Pattern (f),
Quickwrite (f) Stress the relationships between the parts so that
students understand why the
“trick,” or algorithm, works, and
do not simply memorize a process.
Students may need a vocabulary
reminder on the following terms:
quotient
3r1
__
_1
dividend
divisor 2 7
-6
remainder
1
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Look for a Pattern, Quickwrite, Create Representations
My Notes
d. Complete this description of how to find the improper
fraction for the number of shaded halves:
2 shaded
2 whole shaded circles with ____
There are ____
4 shaded halves. Plus, there
halves in each. This gives me ____
5
1 shaded half in the last circle, making a total of ____
is ____
shaded halves.
e. Both the mixed number and the improper fraction in Part c
describe the same number. Write an equation that shows the
two are equal.
5
1 = __
2 __
2 2
f. Without drawing a model, describe a method to use the
digits from the mixed number to find its equivalent improper
fraction. (Hint: Consider your description in Part d.)
3 Create Representations
Answers may vary. Sample answer: Multiply the denominator
by the whole number to get 2 × 2 = 4, then add the numera5.
tor to get 4 + 1 = 5. Put 5 over the denominator to get __
2
3. Now convert Nisha’s finger size to an improper fraction.
a. Use circles:
__7
4
0
1
© 2010 College Board. All rights reserved.
b. Use a number line:
2
__7
4
c. Use the method you discovered in Part e of Question 2. Did
you get the same answer as you did in Parts a and b?
__7 ; yes
4
4. Use your method from Question 2e to convert the other friends’
sizes to improper fractions.
a. Tyrell
b. Ema
c. Jose
15
___
8
35
___
16
__9
4
34 SpringBoard® Mathematics with Meaning™ Level 1
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033-038_SB_MS1_1-6_SE.indd 34
34 SpringBoard® Mathematics with Meaning™ Level 1
Mixed Numbers and Improper Fractions
ACTIVITY 1.6
Mood Rings: Part 1
continued
ACTIVITY 1.6 Continued
5 Identify a Subtask,
Debriefing
SUGGESTED LEARNING STRATEGIES: Identify a Subtask, Quickwrite
My Notes
5. You can use two methods to change from an improper fraction
to a mixed number: drawing a model or dividing.
9 . Then write a mixed number to represent
a. Draw a model of __
4
the whole and fractional parts of your model.
Suggested Assignment
CHECK YOUR UNDERSTANDING
p. 38, #1
WRITING MATH
A fraction is a way of writing a
division problem:
9
__
= 9 ÷ 4.
4
UNIT 1 PRACTICE
p. 59, #40–41
1
2 __
4
6 Quickwrite Students are
b. Divide 9 by 4. Express the remainder as a fraction to get a
mixed number.
expected to use prior knowledge
and compare the whole numbers.
1
9 ÷ 4 = 2 __
4
7 Quickwrite Students compare
The five friends want to compare finger sizes to see whose ring
finger is the largest and whose is the smallest.
the unit fractions using prior
knowledge—the smaller the
denominator, the bigger the
unit fraction.
6. Nisha and Jose compare their finger sizes. Write an inequality
symbol in the circle to make the statement true. Explain your
thinking.
Nisha
3
1__
4
<
Jose
1
2__
4
8 Quickwrite Students use
knowledge from Activity 1.5
to solve this problem. Make a
quick informal assessment using
anecdotal records to identify
students in need of reteaching for
comparing fractions.
© 2010 College Board. All rights reserved.
Explanations may vary. Sample answer: Jose; the whole
3 < 2 __
1.
numbers are different so I compare them: 1 < 2, so 1 __
4
4
7. Next, Bob and Jose compare. Which of them has the larger ring
finger? Explain your thinking.
Bob
1
2 __
2
>
Jose
1
2 __
4
Encourage students to share
and compare their different
approaches.
Explanations may vary. Sample answer: Bob; the whole
1 and __
1 are
numbers are the same so I compare the fractions. __
2
4
1
1.
unit fractions, and halves are bigger than fourths, so 2 __ > 2 __
2
4
3 and 1 __
7 . Whose
8. Then Nisha and Tyrell compare their sizes of 1 __
4
8
finger is larger? Express your answer as an inequality. Explain
your thinking.
3 . Students’ explanations may
7 > 1 __
Tyrell’s finger is larger: 1 __
8
4
vary depending on the process they choose to arrive at their
answers.
Unit 1 • Number Concepts
12/16/09 5:44:53 PM
© 2010 College Board. All rights reserved.
PM
033-038_SB_MS1_1-6_SE.indd 35
35
Unit 1 • Number Concepts
35
9 Think/Pair/Share,
Quickwrite (b) Students should
write names and sizes. They then
compare Ema to Jose and see
4 would be equal to _1_ which
that ___
4
16
is Jose’s fraction, so Ema’s size is
3
.
smaller, as her fraction is ___
16
Students know Ema’s finger size
is bigger than Tyrell’s though
because her mixed number has a
whole number of 2.
ACTIVITY 1.6
continued
Mixed Numbers and Improper Fractions
Mood Rings: Part 1
SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/
Pair/Share, Create Representations, Debriefing
My Notes
9. You have compared the sizes of four friends, two at a time.
a. Use what you have learned to order the friends by their ring
finger sizes, going from smallest to largest.
Nisha, Tyrell, Jose, Bob
b. Where does Ema fit into this order? Explain.
Ema fits in between Tyrell and Jose.
10. Ema tells her friends that they could order their fractions all
at once by using common denominators. Show how this can
be done. Change the mixed numbers to improper fractions with
0 The purpose of this question is
for students to discover that they
can use common denominators to
compare multiple fractions at the
same time.
15 = ___
30 ; Bob: 2 __
5 = ___
40 ;
7 = ___
1 = __
16 as the denominator. Tyrell: 1 __
8
8
16
2 2 16
9 = ___
36 ; Nisha: 1__
3 = __
28 ; Ema: 2 ___
3 = ___
35 ; then
1 = __
7 = ___
Jose: 2 __
4 4 16
4 4 16
16 16
compare the numerators to find that the order from smallest to
largest is Nisha, Tyrell, Ema, Jose, and Bob.
a Create Representations,
Debriefing Students practice
ordering on a number line as
another way of ordering. Be sure
students are plotting points and
not shading the line.
b Create Representations,
Quickwrite, Think/Pair/
Share This question is meant
to give students the opportunity
to think about a variety of
possibilities and decide how they
can use what they already know
to answer the questions.
11. Order the friends’ ring finger sizes from least to greatest on
the number line below. Label each point with each person’s
initial as shown for Bob.
N T
1
CONNECT TO AP
Looking at the pattern of points
on a number line can help you
decide if the numbers are getting
close to a particular number.
Discovering such patterns is a
fundamental concept in advanced
math courses.
EJ
B
2
3
© 2010 College Board. All rights reserved.
ACTIVITY 1.6 Continued
12. You now know different ways to order measurements. How
would you order the measurements of several other friends?
Describe your method.
Karen
Ahmed
Ileana
Keisha
Hunter
5
1__
7
__
5
2 ___
4
__
14
___
8
4
16
2
8
Answers may vary. Sample answer: First change all numbers to
either mixed numbers or improper fractions, and then compare
them as done earlier.
The finger sizes are easy to order since the fractions are all
1 ’s, or ___
1 ’s, __
1 ’s, like the increments on a ruler. However,
1 ’s, __
in __
2 4 8
16
98 , 1 ___
23 , and 2 ___
15 , 2 ____
2 , do not have
many numbers, such as 1___
31 100 44
71
denominators that are as simple to compare.
36 SpringBoard® Mathematics with Meaning™ Level 1
Connect to AP
Students will need to be able to graph and order fractions on a number
line or in the coordinate plane when they study sequences and series,
especially, those that converge to a real number. Consider the sequence
3 __
9
99
1 , __
2 , __
__
, 4 , …, ___
, … ____
, …. Have your class graph these fractions on a
2 3 4 5
10
100
number line. They can then observe that the points are getting closer and
closer to the number 1. However, no number in the sequence will ever
equal 1 or be greater than 1. An activity like this will set the stage for
future understanding of limits, a fundamental concept in calculus.
36 SpringBoard® Mathematics with Meaning™ Level 1
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0
© 2010 College Board. All rights reserved.
033-038_SB_MS1_1-6_SE.indd 36
Mixed Numbers and Improper Fractions
ACTIVITY 1.6
Mood Rings: Part 1
continued
SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/
Retell, Create Representations, Quickwrite, Debriefing, Self
Revision/Peer Revision
To order numbers like these, think about how close each fraction
1 , 1, and so on. For example,
is to benchmark numbers, such as 0, __
2
15
15
1
___
__
1 is a little less than 1 , because half of 32 is 16. Thus, place 1___
31
2
31
1
__
just before the 1 mark.
2
1 23
44
1
15
1 31
22
71
R
Read
and Think Aloud is
a good teacher strategy
wh reading the informato use while
tion before Question 13.
TEACHER TO
TEACHER
My Notes
MATH TERMS
Benchmark numbers are
numbers used as points of
comparison when estimating.
Paragraph
Summarize/Paraphrase/Retell
2 98
100
2
C Create Representations
98
Students reason that 2____
is just
23 100
___
a little less than 3, 1 44 is a little
greater than 1_12_ because _12_ of 44
2 is a little greater
is 22, and 2___
71
than 2 but less than 2_12_.
3
23 , and 2 ___
98 , 1 ___
2 on
13. Continue using benchmarks to order 2 ____
100 44
71
the number line above.
14. Summarize your findings on comparing and ordering mixed
numbers and improper fractions by discussing the following
cases.
d Quickwrite, Debriefing, Self
Revision/Peer Revision Students
form generalizations for
comparing and ordering different
combinations of mixed numbers
and improper fractions.
3 and 3 ___
1
a. Whole numbers are different: 2 ___
19
27
Answers may vary. Sample answer: Just compare the
whole numbers. The bigger the whole number, the bigger
the mixed number.
Suggested Assignment
7
2 and 5 __
b. Whole numbers are the same: 5 __
3
9
© 2010 College Board. All rights reserved.
ACTIVITY 1.6 Continued
CHECK YOUR UNDERSTANDING
p. 38, #2–7
Answers may vary. Sample answer: Compare the fractions
by either writing equivalent fractions or using crossmultiplication.
UNIT 1 PRACTICE
p. 59, #42–44
32 and ___
27
c. Both are improper fractions: ___
5
4
Answers may vary. Sample answer: Compare them the
same way you compare common fractions.
13
3 and ___
d. One mixed number and one improper fraction: 6 __
5
2
Answers may vary. Sample answer: Either change the
improper fraction to a mixed number or the mixed number
to an improper fraction, then solve using the methods
discussed earlier.
Unit 1 • Number Concepts
12/16/09 5:45:01 PM
© 2010 College Board. All rights reserved.
PM
033-038_SB_MS1_1-6_SE.indd 37
37
Unit 1 • Number Concepts
37
ACTIVITY 1.6 Continued
ACTIVITY 1.6
continued
Mixed Numbers and Improper Fractions
Mood Rings: Part 1
Answer Key
5
b. 5__
9
2. Kendra, Miley, Bryson, Wyatt
3. Juan
4. See answer to #4 below.
9
47
5a. 8___
> ___
11
6
53
3
b. ___
< 7__
7
5
6
7 , 3 ___
11 , ___
24 , 4 __
6. __
7
2
20 5
7. Answers may vary. Sample
answer: Fractions can be
expressed as proper fractions,
improper fractions, or mixed
numbers. Comparing mixed
numbers and improper
fractions is like comparing
proper fractions because you
can find equivalent fractions
if you need to and then
use common denominators
or common numerators to
compare; comparing mixed
numbers and improper
fractions is different from
comparing proper fractions
because with mixed numbers
sometimes you can just
compare the whole number
parts and sometimes you have
to convert the mixed number
to an improper fraction.
CHECK YOUR UNDERSTANDING
Write your answers
answers on
on notebook
notebook paper.
paper.Show your work.
4. Order each of the following numbers
Show your work.
by placing them on a number line. Use
benchmark numbers to determine their
1. Convert the improper fraction to a mixed
placement.
number and the mixed number to an
10 , and __
8
2 , ___
11 , 2 __
1___
20 5 7
3
5. Compare.
improper fraction.
50
4
a. 7 __
b. ___
7
9
2. Mr. White’s students are playing a game.
He gives each student in a group a fraction
to help them decide the order in which
they will play. The person with the largest
fraction goes first, and so on. The table
below shows the fractions for the students
in one group.
Wyatt
3
1__
4
Kendra
8
2 ___
11
Miley
19
___
7
9 and ___
3
53 and 7 __
47
b. ___
a. 8 ___
7
5
11
6
6. Order from least to greatest:
6 , ___
7 , 3 ___
24 , __
11
4 __
7 5 2 20
7. MATHEMATICAL Describe the three forms
R E F L E C T I O N in which fractions can be
written. Explain how to compare the three
forms. Give examples using different
forms of fractions.
Bryson
9
__
5
List the order in which they will take their
turns.
© 2010 College Board. All rights reserved.
53
1a. ___
7
3. Kim and Juan are measuring their wrists to
5 in.
purchase watches. Kim’s measures 5 __
8
3
__
and Juan’s measures 5 in. Who has the
4
larger wrist?
38 SpringBoard® Mathematics with Meaning™ Level 1
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4.
1
38 SpringBoard® Mathematics with Meaning™ Level 1
10 111
7 20
2
22
5
8
3
3
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033-038_SB_MS1_1-6_SE.indd 38