Julie McNamara November 6 and 7 2014 Have you ever heard Yours is not to reason why Just invert and multiply In contrast to Children who are successful at making sense of mathematics are those who believe that mathematics makes sense ID: 322225
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Slide1
Beyond Invert and Multiply: Making Sense of Fraction Computation
Julie McNamara
November
6 and 7
, 2014Slide2
Have you ever heard….
Yours is not to reason why,
Just invert and multiply!Slide3
In contrast to…..
“Children who are successful at making sense of mathematics are those who believe that mathematics makes sense.”
-Lauren
ResnickSlide4
CCSS Standards for Mathematical Practice
Make sense of problems and persevere in solving the
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others
.
Model with mathematics.
Use appropriate
tools
strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning. Slide5
NCTM 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. Slide6
Brendan, Grade 4Slide7
Fractions as numbers…
“
In mathematics, do whatever it takes to help you learn something, provided you do not lose sight of what you are supposed to learn
. In the case of fractions, it means you may use any pictorial image you want to process your thoughts on fractions, but
at the end, you should be able to formulate logical arguments in terms of the original definition of a fraction as a point on the number line.
”
-Wu
, 2002, p. 13 Slide8
Rod Relations
Using your Cuisenaire rods, find as many fractional relationships as you can.
For example:
1 orange = 2 yellows, so 1 yellow
=
½ orangeSlide9
Making Sense of Fraction AdditionSlide10
Whole Number Addition and Subtraction Strategies
Decomposing
/recomposing
Associative property
Commutative
property
RenamingSlide11
Use the Cuisenaire Rods to solve:
1
2
1
2
b
rown rod +
b
rown rod
1
4
1
4
b
rown rod +
brown rod
1
2
1
4
b
rown rod +
brown rod Slide12
Addition with Cuisenaire Rods, V1 and V2
Version 1:
All problems use brown rod as the whole
May need to rename one addend
Version 2:
Problems use different rods as the whole
May need to rename both addendsSlide13
Get to the Whole!
Decomposing
and recomposing fractions to “get to the whole” when adding and subtracting.
3
4
3
4
+Slide14
:
Will’s Strategy
Video from
Beyond
Invert & Multiply
(in press)
3
4
3
4
+Slide15
:
Belen’s Strategy
3
4
3
4
+
Video from
Beyond Invert & Multiply
(in press)Slide16
:
Malia’s
Strategy
3
5
4
5
+
Video from
Beyond
Invert & Multiply
(in press)Slide17
Student workSlide18
Student workSlide19
Student workSlide20
Making Sense of Fraction MultiplicationSlide21
Connecting Multiplication to AdditionSlide22
Using repeated addition to solve
1
2
x
6
Video from
Beyond
Invert & Multiply
(in press)Slide23
Connecting to Whole
Number MultiplicationSlide24
Farmer Liz
Fruit
Vegetable
sSlide25
Farmer Liz
Fruit
Vegetable
sSlide26
Farmer Liz
Farmer Liz wants
to further partition the two halves of her
field such that --
half
of the fruit section will be planted with fruit trees and half with fruit
bushes
half
of the vegetable section will be planted with vegetables that grow above ground and half with vegetables that grow below
ground Slide27
Farmer Liz
Fruit trees
Fruit bushes
Above ground vegetables
Below ground vegetablesSlide28
Farmer Liz
What fraction of Farmer Liz’s field is planted with fruit trees?
What fraction of Farmer Liz’s field is planted with fruit
bushes
?
What fraction of Farmer Liz’s field is planted with
above ground vegetables?
What fraction of Farmer Liz’s field is planted with
below
ground vegetables? Slide29
Multiplying fractions
How does the “field” show that
=
1
2
1
2
x
1
4
?Slide30
Farmer Bruce
Farmer Bruce is
going to plant half of his field with flowers and leave the other half unplanted to use as a pasture.
He plans to use 1/3 of the flower half for roses, 1/3 for daisies, and 1/3 for geraniums
.
What
fraction of the field will be planted with roses?
What fraction of the field will be planted with daisies?
What fraction of the field will be planted with geraniums? Slide31
Farmer Bruce
Farmer Bruce decided
that he really didn’t like geraniums and instead planted 2/3 of the flower section with daisies and kept 1/3 for roses, how much of the field
is planted
with daisies? Slide32
Tell Me All You Can
Before
coming up with an exact answer, consider what you know about the answer as a means of getting a sense of the “neighborhood” of the answer. Slide33
Tell Me All You Can
The answer will be less than ________ because _________.
The answer will be greater than ________ because _________.
The answer will be between ________ and ________ because _________.Slide34
What do you know about ?
1
2
x
2
6
Video from
Beyond
Invert & Multiply
(in press)Slide35
What do you know about ?
1
2
x
4
5
Video from
Beyond
Invert & Multiply
(in press)Slide36
Making Sense of Fraction DivisionSlide37
Two types of division situations:
Quotative
(also called measurement division):
Size of group is known; number of groups is unknown
6 ÷ 2: How many 2 ’s are in 6?
Partitive
:
Number of groups is known; how many in each group is unknown
6 ÷ 2:
Split 6 into 2 groups
6 is 2 of what?Slide38
Quotative Division
•
6 ÷ 2
: How many 2 ’s are in 6Slide39
How Long? How Far? Part 1Slide40
Video from
Beyond
Invert & Multiply
(in press)Slide41
Quotative DivisionSlide42
Reasoning about 1 ÷
1
6Slide43
Reasoning about 2 ÷
1
6Slide44
Reasoning about 10 ÷
1
3Slide45
Reasoning about 6 ÷
3
4Slide46
Partitive Division
6
÷
3:
Split 6 into
3
groups
6 is 3 of what?Slide47
How Long? How Far? Part 2
Beach Clean-Up (2 people)
Distance
Each person cleans
8 miles
4 miles
4 miles
2 miles
2 miles
1 mile
1 mile
½ mile
½ mile
?Slide48
How Long? How Far? Part 2
1
2
÷
3
1
6
÷
2
3
4
÷
3Slide49
Do you always have to
invert and multiply?
Your friend tells you she doesn’t understand why your teacher makes you invert and multiply to divide fractions. She says you can just divide across the numerators and denominators to get your answer. She shows you the two examples below to prove her point:
What do you think of her idea?
Is she right?
If so, why? If not, why not?
49Slide50
What about in this case?Slide51
Write a word problem that can be solved by the expression below: Slide52
CCSS Number
and Operations - Fractions
3.NF: Develop understanding of fractions as numbers.
4.NF: Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
5.NF: Use equivalent fractions as a strategy to add and subtract fractions.
5.NF: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Slide53
Fractions as numbers…
“
In mathematics, do whatever it takes to help you learn something, provided you do not lose sight of what you are supposed to learn
. In the case of fractions, it means you may use any pictorial image you want to process your thoughts on fractions, but
at the end, you should be able to formulate logical arguments in terms of the original definition of a fraction as a point on the number line.
”
-Wu
, 2002, p. 13 Slide54
Remember….
“Children who are successful at making sense of mathematics are those who believe that mathematics makes sense.”
-
Lauren
ResnickSlide55
Coming Soon!Slide56
Thank you!!!
juliemcmath@gmail.com