MCTM 2016 (Final)

Turning Down the Volume
Fractions
Dr. Patrick Sullivan
Missouri State University
patricksullivan@missouristate.edu
https://goo.gl/wKGEWS

2 ones 2 tenths
2 hundreds2 thirds

Goals
• Establish a clear connection between operations with
whole numbers and fractions
• Engage in contextual problems
• Reason/sense-making of operations involving fractions
• Reflect on “your” thinking as you do the problems
• Reflect on important language

Mathematics Teaching Practices
• Establish mathematics goals to focus learning.
• Implement tasks that promote reasoning and
problem solving.
• Use and connect mathematical representations.
• Facilitate meaningful mathematical discourse.
• Pose purposeful questions.
• Build procedural fluency from conceptual
understanding.
• Support productive struggle in learning mathematics.
• Elicit and use evidence of student thinking.

Volume Lowering Strategies
• Strategy #1: Notation/Comparison
• Strategy #2: Context
• Strategy #3: Connect to Whole Number Problems

Tensions
• “Their” models vs. “Our” models
• Concrete vs. Abstract
• Notation and Language
• Fractions as part-whole, scalar, operator, ratio

How do we “interpret”
2
3
?
• “2 over 3”
• ”2 thirds”
• 2 “copies” or “groups of”
1
3
• “multiply by 2 and divided by 3”
• ”2 hits in 3 total at-bats”
• ”2 parts out of 3 total parts”

How do we “write” it?
• 2 thirds
• 2 *
1
3
• 2
1
3
•
2
3

Strategy #1: Notation
Which fraction is greater?
1
3
1
6
As loud as it gets!

Reasoning Strategies
• Whole Number Reasoning-–”they are equal”
(especially when comparing
5
6
and
7
8
)
• Gap Reasoning—”one-third is 2 from the
whole and one-sixth is 5 from the whole so
one-third is greater”

Strategy #1: Notation
Which fraction is greater?
1 third 1 sixth
Turning down the volume a little!

Strategy #2 (Context)
• Drew ate 1 sixth of a whole Twizzler. Brad ate 1 third
of a whole Twizzler. Who ate more?
• Group A is going to equally share 1 Twizzler with 6
people. Group B is going to equally share 1 Twizzler
with 3 people. The people in which group will get
more get more Twizzler? How much will each a
person get in each group?
Turning the Volume Way Down!

Strategy #3 (Connect to Whole Numbers)

Addition
• Jenny has 4 whole apples and her friend Sue
has 9 whole apples. Between the two of them
how many total whole apples do they have?
• Jenny has
1
3
of a whole pound of cheese and
her friend Sally has
3
4
of a whole pound of
cheese. Between the two of them how much
cheese do they have?

Addition with Whole Numbers
4 groups of 1 apple + 9 groups of 1 apple = 13 groups of 1 apple
4 whole apples + 9 whole apples = 13 whole apples

Addition with Fractions
1 third + 3 fourths = ????
1 copy of 1 third + 3 copies of 1 fourths = ????

4 twelfths + 9 twelfths = 13 twelfths
4 groups of 1 twelfth + 9 groups of 1 twelfth = 13
groups of 1 twelfth
4
1
12
+ 9
1
12
= 13
1
12
=
4
12
+
9
12
=
13
12

Addition “Simplified”
1
2
+
3
8
1 half + 3 eighths
“There are 4 eighths in 1 half”
4 eighths + 3 eighths = 7 eighths

5 ÷
1
3
What does the expression mean?

Thinking about “Division”
• Lily has 12 Starbursts. She is going to give them
away to her 3 friends. How many Starbursts will
each friend receive?
• Gwen has 12 Starbursts. She is going to give 3
Starbursts to as many friends as she can. How many
friend will receive 3 Starbursts?
What are the similarities/differences in the structure
of the two problems?

Which is “louder”?
5 ÷
1
3
OR
There are 5 whole Twizzlers and each person is
going to receive
1
3
of a whole Twizzler? Using all of
the Twizzlers, how many people can you give
1
3
of a
whole Twizzler?

Strategy #2: Context
There are 5 whole Twizzlers and each person is going to
receive
1
3
of a whole Twizzler? Using all of the Twizzlers,
how many people can you give
1
3
of a whole Twizzler?

Problem Prompts
• Using only pictorial representations
determine the answer to each problem.
• Create a numerical expression that models
your pictorial representation.

Problem 1
You have 8 peanut butter and jelly sandwiches
and each student will eat
2
3
of a whole peanut
butter and jelly sandwich. How many students
can you feed?

Problem 2
and each student will eat 2 peanut butter and
jelly sandwiches. How many students can you
feed?

Making the Connection
𝟖 ÷
𝟐
𝟑
• Question: How many groups
of 2 thirds can I make from
24 thirds?
• Answer: 12 groups of 2
thirds of a whole.
𝟐𝟒 ÷ 𝟐
• Question: How many groups
of 2 whole can I make from
24 whole?
• Answer: 12 groups of 2
whole.

A Closer Look
• 8 whole sandwiches became 24
thirds 8 = 24
1
3
• 24
1
3
− 2
1
3
− 2
1
3
… - 2
1
3
= 0
• 24
1
3
÷ 2
1
3
= 12
• 24(1) – 2(1) – 2(1) …. – 2(1) = 0
• 24(1) ÷ 2(1) = 12

Thinking about Multiplication
Lily is making gift boxes of cookies. Each gift box
will hold 5 cookies. If she wants to make 4 gift
boxes how many cookies will she need to make?
4 * 5 = 20
(multiplier) * (multiplicative unit) = product

Different Scenarios
• Scenario #1: Multiplier is whole number;
Multiplicative Unit is a fraction
• Scenario #2: Multiplier is a fraction; Multiplicative
Unit is a whole number
• Scenario #3: Multiplier and Multiplicative Unit are
both fractions

Problem 3
Each batch of cookies requires
2
3
of a whole cup
of sugar. How much sugar will be needed to
make 5 batches of cookies?

Problem 4
Each batch of cookies requires 2 whole cups of
sugar. How much sugar will be needed to make 5
batches of cookies?

Making the Connection
𝟓 ∗
𝟐
𝟑
• Question: How much is 5
groups of 2 thirds?
• Answer: 10 thirds
5*2
• Question: How much is 5
groups of 2 whole?
• Answer: 10 whole
5 is the multiplier,
2
3
and 2 are the multiplicative unit.

A Closer Look
2
1
3
+2
1
3
+2
1
3
+2
1
3
+2
1
3
= 10
1
3
5 * 2
1
3
= 10
1
3
10
1
3
is how many whole? 3
1
3
2(1) + 2(1) + 2(1) + 2(1) + 2(1) = 10
5 * 2 = 10

Problem 5 (The “Twist”)
Joann has 15 cups of sugar. She wants to give
2
3
of what she has to her friend Ann so Ann can
make some cookies. How much sugar will Ann
receive?

2
3
* 15 (Each box represents 1 whole cup of sugar)
2
3
is the multiplier and 15 is multiplicative unit
2
3
as the multiplier implies an action on the multiplicative unit—
”partition multiplicative unit into 3 equal groups and take 2 of those
groups”
2
3
* 15 = 2 *
1
3
∗ 15 = 2 * 5 = 10

Moment of Reflection
Think about how the fraction
2
3
is treated in each
problem.
• Each batch of cookies requires
2
3
of a whole cup of
sugar. How much sugar will be needed to make 5
batches of cookies?
• Joann has 15 cups of sugar. She wants to give
2
3
of
what she has to her friend Ann so Ann can make
some cookies. How much sugar will Ann receive?

“Tame Way”
Joann has
15
16
of a whole cup of sugar. She wants
to give
2
3
of what she has to her friend Ann so
Ann can make some cookies. How much sugar
will Ann receive?
How is this problem similar/different in
structure to the problem you just solved?

2
3
* 15 sixteenth (Each box represents 1 sixteenth of whole cup of sugar)
2
3
is the multiplier and 15 sixteenths, 15
1
16
, is multiplicative unit
2
3
as the multiplier implies an action on the multiplicative unit—”partition
multiplicative unit into 3 equal groups and take 2 of those groups”
2
3
* 15 sixteenths = 2 *
1
3
∗ 15 sixteenths = 2 * 5 sixteenths = 10 sixteenths
OR
2
3
* 15
1
16
= 2 *
1
3
∗ 15
1
16
= 2 * 5
1
16
= 10
1
16

Moment of Reflection
Think about how the fractions
2
3
and
15
16
are
interpreted in the problem you just completed.
Joann has
15
16
of a whole cup of sugar. She wants
to give
2
3
of what she has to her friend Ann so
Ann can make some cookies. How much sugar
will Ann receive?

Your Challenge
2
3
*
15
16
“Why can’t we just multiply straight across?”
Would the answer be less than or greater than 1?

15 Sixteenths
15 Sixteenths partitioned into three equal units
2
3
* 15 sixteenths =2
1
3
∗ 15 sixteenths = 2 * (5 sixteenths) = 10 sixteenths
Modeling
𝟐
𝟑
*
𝟏𝟓
𝟏𝟔

15 Sixteenths
15 Sixteenths partitioned into thirds creating 45 Forty-eighths
2
3
∗ 45 forty − eighths = 2
1
3
∗ 45 forty − eighths = 2 15 forty − eighths = 30 forty − eighths =
𝟑𝟎
𝟒𝟖
Modeling
𝟐
𝟑
*
𝟏𝟓
𝟏𝟔

Review
3
4
*
8
19
Multiplier Multiplicative Unit
Interpret fraction as an operator
Partition multiplicative unit into
4 equal groups and “use” 3 of
those groups.
Interpret fraction as a scalar
multiple.
8 nineteenths or 8 copies of
1
19
3
4
*8 nineteenths = 3 *
1
4
∗ 8 nineteenths = 6 nineteenths =
6
19

Session Reflection
https://goo.gl/NQTxm
U

Going Further
• Jamie has
4
5
of a whole foot of string and wants to
create pieces of string that are
1
3
of a foot long.
How many pieces of string that are
1
3
of a foot
long can she create? (Be sure to include partial
pieces.)
• Jamie has 12 feet of string and wants to create
pieces that are 5 feet long. How many of those
size pieces can she create?

Problem 11
and each student will eat 0.6 of a peanut butter
and jelly sandwich. How many students can you
feed?

Connecting to Problem 3
Joann has 5 cups of sugar. She wants to give
2
3
of
what she has to her friend Ann so Ann can make
some cookies. How much of a whole pound of
sugar will Anne receive?

Going Further
Joann has
4
5
of a pound of sugar. She wants to
give
2
3
of what she has to her friend Ann so Ann
can make some cookies. How much sugar will
Ann receive?

Going Even Further
• Each batch of cookies requires 0.4 of a cup of
sugar. How much sugar will be needed to
make 5 batches of cookies?
• Joann has 0.8 of a cup of sugar. She wants to
give 0.2 of what she has to her friend Ann so
Ann can make some cookies. How much of a
whole cup of sugar will Ann receive?

“Division” is “Division”
• 6 ÷ 2
• 6 ÷
1
4
• 6 ÷ 0.01
• 0.06 ÷ 0.02

Contact Information
• Dr. Patrick Sullivan
(patricksullivan@missouristate.edu)

MCTM 2016 (Final)

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