Transcript How many halves are in 1?

TUNING UP FRACTIONS
LINDA WEST
[email protected]
SMARTTRAINING, LLC
©
BIG IDEAS IN MATHEMATICS - COMPETENCIES
– Visualization
– Making Connections (looking for patterns in
order to generalize)
– Communication
Problem Solving
Number Sense
• Based on the Theories of:
– Jerome Bruner
– Zoltan Dienes
– Richard Skemp
©
NUMERATOR
• An adjective
• The counting number
• Tell how many
• The “numberator”
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DENOMINATOR
• A noun
• A label
• Tells what kind of unit
• The “deNAMEnator”
• An Ordinal number
©
What makes these types
of numbers so difficult for
students?
©
•
•
•
•
Notation confusing
“bigger” numbers indicate “smaller amount”
Same number indicates different amount
Multiplication sometimes yields smaller
result while division sometimes yields a
larger one
• The “whole” must always be held in the mind
• Fractions, decimals and percents
traditionally taught as disparate topics
©
FRACTIONS
Fractions
Quantity
Proportion
Percentage is exclusively used for Proportion
Decimal is exclusively used for Quantity
©
Whole vs. Fractional
Number Sense
Whole Numbers
 Counting units
 RELATIONSHIPS TO
OTHER NUMBERS
• More/Less
• 5 & 10 benchmarks
• Part/Whole relationships
Fractional Numbers
 COUNTING UNITS
 RELATIONSHIPS TO
OTHER NUMBERS
•
•
•
•
More /Less
0, ½, 1 Benchmarks
Part /Whole Relationships
Equivalency
©
Whole Numbers
Fractional Numbers
• EQUAL UNITS BASED
ON 10’S WITH
RELATIONSHIPS TO
EACH OTHER
• WHOLE UNIVERSE
OF “UNITS” THAT
ARE LESS THAN 1,
BUT ARE RELATED
TO EACH OTHER!
• CONNECT
CONSTRUCTS WITH
ABSTRACT
NOTATION AND
OPERATIONS
• CONNECT
CONSTRUCTS WITH
ABSTRACT
NOTATION AND
OPERATIONS
©
Keys to Understanding
• Concepts and Problem Solving before
Rules and Drill
• Connections before Calculations
©
The Dangerous Rush to Rules
• None of the rules help students think about
the meaning
• Rules give students no means of assessing
whether an answer is reasonable
• Surface mastery of rules is quickly lost
• Algorithm rules do not immediately apply to
every situation
• Incredibly defeating for students
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2 ÷ =?
3
4
1
2
Please:
1. Solve.
2. Draw a picture.
3. Write a word problem.
©
12 ÷ 4 = ____
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Our real question is:
How many halves are
contained in 2 ¾?
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WOULD ANYONE LIKE TO SHARE THEIR
WORD PROBLEM?
EXAMPLE:
Jacqui ran 2 ¾ miles.
This was ½
𝟑
A recipe
calls that
for 2 she
cupsruns
of flour.
The
the
distance
each
𝟒
only What
size measuring
cup available
day.
is the total
distanceis a
𝟏
cupJacqui
measure.
How
many
measuring
that
runs
each
day?
𝟐
cups are required to complete the
recipe?
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NOW TRY THIS ONE
2½÷⅓
•
•
•
Use some of the pattern blocks on your table to solve this problem.
Verify with algorithm.
Write a word problem that can be solved using this equation.
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2 ½ ÷ 1/3
2 ½ ÷ 2/3
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FUN WITH FRACTIONS
Using digit tiles 1 – 9, once only in each equation,
how many equations can you create?
+
=
©
HOW DO WE DEVELOP VISUALIZATION
SKILLS WITH FRACTIONS?
•
Take out the yellow hexagon. Cover it with as many different pattern
blocks as you can until the entire hexagon is filled.
•
How can we express what we have done concretely in a number
sentence?
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HOW DO WE USE
Logic
To explain
The fraction division algorithm?
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𝟏
=
𝟐
6÷
____
WHAT DOES THAT MEAN?
Let’s go to an easier problem:
6 ÷ 3 = ____ What does that mean?
• How can I read that differently?
• How many threes are in six?
• 6 is 3 of what number?
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𝟏
=
𝟐
6÷
____
WHAT DOES THAT MEAN?
• Who can read the equation differently?
• How many ½’s are in 6?
• 6 is ½ of what number?
©
DEVELOPING THE CONCEPT
• 6 ÷ 6 = ____
• 6 ÷ 3 = ____
• 6 ÷ 2 = ____ Does anyone see a pattern?
• 6 ÷ 1 = ____
• 6 ÷ 𝟏𝟐 = ____
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DO YOU SEE A PATTERN?
•
•
•
•
How many halves are in 1?
How many halves are in 2?
How many halves are in 3?
Let’s focus on thirds:
– How many thirds are in 1?
– How many thirds are in 2?
– How many thirds are in 3?
– How many thirds are in 4?
(2)
(4)
(6)
(3)
(6)
(9)
(12)
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DO YOU SEE A PATTERN?
• LET’S FOCUS ON FOURTHS:
– How many fourths are in 1? (4)
– How many fourths are in 2? (8)
– How many fourths are in 3? (12)
– How many fourths are in 4? (16)
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DO YOU SEE A PATTERN?
• LET’S SWITCH IT UP A LITTLE BIT:
– How many halves are in 1?
– How many thirds are in 1?
– How many fourths are in 1?
– How many fifths are in 1?
– How many halves are in 2?
– How many thirds are in 2? etc., etc.
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AREA MODEL TO REAL LIFE
We need multiple embodiments.
(Zhang 2012)
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DIVISION BY A FRACTION
• 1
𝟏
÷ 𝟔=
1
0
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USING BAR MODELS WITH FRACTIONS
Divide 3 by 2/3
2/3
There are four 2/3’s. and another half of a 2/3 in 3. So there are
four-and-a-half 2/3’s in 3.
3 ÷ 2/3 = 3 x 3/2 = ?
©
3 DIVIDED BY
𝟐
𝟑
• What is your word problem?
©
A man travelled from Town X to Town Y in three days. He covered
1
1
2
of the
journey of the first day and 4 of the remainder of the journey on the
second day. What fraction of the total journey did he cover on the last
day?
He travelled
of the total journey on the last day.
©
𝟏
𝟐
𝟏
𝟑
You painted of a room. of the area painted
was green. What fraction of the room was
green?
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A PICTURE IS WORTH …
• Find ¼ of 48.
• Find ¾ of 48.
• 1/3 of a number
is 48. Find the number.
• 4/5 of a number
is 48. Find the number.
©
•
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Disagree
0
1
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Agree
2
Strongly
Agree
3
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