mctm 2016 (final)
TRANSCRIPT
Turning Down the VolumeFractions
Dr. Patrick SullivanMissouri State University
https://goo.gl/wKGEWS
2 0.2
20023
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
23
?
• “2 over 3”
• ”2 thirds”
• 2 “copies” or “groups of”
• “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 *
• 2
How do we “represent” it?
Strategy #1: Notation
Which fraction is greater?
As loud as it gets!
Reasoning Strategies
• Whole Number Reasoning-–”they are equal”(especially when comparing and )
• 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 of a whole pound of cheese and her
friend Sally has 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 + 9 = 13 = + =
Addition “Simplified”
1 half + 3 eighths
“There are 4 eighths in 1 half”
4 eighths + 3 eighths = 7 eighths
5÷ 13
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”?
OR
There are 5 whole Twizzlers and each person is going to receive of a whole Twizzler? Using all of the Twizzlers, how many people can you give of a whole Twizzler?
Strategy #2: Context
There are 5 whole Twizzlers and each person is going to receive of a whole Twizzler? Using all of the Twizzlers, how many people can you give 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 of a whole peanut butter and jelly sandwich. How many students can you feed?
Problem 2
You have 24 peanut butter and jelly sandwiches and each student will eat 2 peanut butter and jelly sandwiches. How many students can you feed?
Making the Connection
𝟖÷𝟐𝟑
• 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
• … - = 0
• = 12
• 24(1) – 2(1) – 2(1) …. – 2(1) = 0
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 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?
Problem 3
Problem 4
Making the Connection𝟓∗𝟐𝟑
• 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 are the multiplicative unit.
A Closer Look
++++ = 10
5 * = 10
10 is how many whole?
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 of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive?
* 15 (Each box represents 1 whole cup of sugar)
is the multiplier and 15 is multiplicative unit
as the multiplier implies an action on the multiplicative unit—”partition multiplicative unit into 3 equal groups and take 2 of those groups”
* 15 = 2 * = 2 * 5 = 10
Moment of Reflection
Think about how the fraction is treated in each problem.
• Each batch of cookies requires 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 of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive?
“Tame Way”
Joann has of a whole cup of sugar. She wants to give 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?
* 15 sixteenth (Each box represents 1 sixteenth of whole cup of sugar)
is the multiplier and 15 sixteenths, 15is multiplicative unit
as the multiplier implies an action on the multiplicative unit—”partition multiplicative unit into 3 equal groups and take 2 of those groups”
* 15 sixteenths = 2 * = 2 * 5 sixteenths = 10 sixteenths
OR * 15 = 2 * = 2 * 5 = 10
Moment of Reflection
Think about how the fractions and are interpreted in the problem you just completed.
Joann has of a whole cup of sugar. She wants to give of what she has to her friend Ann so Ann can make some cookies. How much sugar will Ann receive?
Your Challenge
*
“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
* 15 sixteenths =2 sixteenths = 2 * (5 sixteenths) = 10 sixteenths
Modeling *
15 Sixteenths
15 Sixteenths partitioned into thirds creating 45 Forty-eighths
=
Modeling *
Review
*
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
*8 nineteenths = 3 * nineteenths = 6 nineteenths =
Session Reflection
https://goo.gl/NQTxmU
Going Further
• Jamie has of a whole foot of string and wants to create pieces of string that are of a foot long. How many pieces of string that are 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
You have 12 peanut butter and jelly sandwiches 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 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 of a pound of sugar. She wants to give 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 ÷ 0.01
• 0.06 ÷ 0.02
