Transcript Ticket sales 1

Developing Multiplicative
Reasoning
Mike Askew
King’s College London
Comparing
5 kg
8 kg
Smarties
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There are two jars.
One contains 25 smarties
The other contains 75 smarties
The smarties are poured into one jar
How many smarties in the jar?
Water
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There are two jars.
One contains water at 25o C
The other contains water at 75o C
The water is poured into one jar
What is the temperature of water in the
jar now?
What fraction (1)?
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2/5
3/5
2 1/2
1 1/2
2/3
1 2/3
What fraction (2)?
MaTCH - School culture
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Lack of belief in children’s ability to think
and be creative
‘Empty vessels’ model of teaching
Low expectations, especially about
children’s language
Development emphases
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Teaching methods:
Use of discussion and cooperative work
Children explaining methods
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Teaching approaches:
Developing tools for thinking
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Addition through empty number line
Multiplication through arrays and areas
Artifacts and tools
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Artifacts:
symbols, concrete materials, diagrams,
words
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Tools:
artifacts linked to mental representation
Interpreting children’s thinking
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Artifacts – empty number line, arrays,
areas – introduced by teacher
Tool use – evidence for this from
children’s talk and recording?
Artifacts into tools
Multiplicative reasoning
Multiplication arises from
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Repeated sets (3 tables, 4 at each)
Scale factors (3 times as long as)
Rectangular areas (3 rows of 4 chairs)
Cartesian product (3 breads, 4 fillings)
Division arises from:
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Measurement/grouping (quotition)
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12 children at tables of 4
Sharing (partition)
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12 children at 4 tables
Teaching approach
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‘Ban’ on teaching multiplication as
repeated addition
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Use of arrays and areas line to model
strategies
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Later use of ratio tables for linking with
division
Early work: describing arrays
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Why arrays?
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Links with 10 x 10 multiplication grid
Less obvious link with addition
Cartesian product can be modelled on
an array (bread x fillings)
Extension of one dimensional number
line into two dimensions
Tool for thinking with
Torn grids (1)
Torn grids (2)
35 x 23
30
20
3
5
35 x 23
30
5
20
600
100
3
90
15
89 x 75
80
70
5
9
7 x 32
26 x 16
42 x 36
32 x 69
Reduced scale
15 x 14 - child’s extension
9 x 22
34 x 9
Orientation
Left waiting
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650 people want to travel into Athens by
train
There are 11 carriages
Each carriage carries 45 people
How many people are left waiting for the
next train?
Left waiting 1
Left waiting 2
Left waiting 3
Turnstiles
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There are 42 turnstiles at the North East
end of the stadium
285 (210) people go through each
turnstile.
How many people entered the stadium
by the North East entrance?
Turnstiles 1
Turnstiles 2
Grandmother’s sweets
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Grandmother arrives with a bag of 24
sweets.
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There are three children at home who
decide to share the sweets equally.
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Just about to eat them when another
child arrives.
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And then the twins arrive.
24 shared between 3
24 shared between 4
24 shared between 6
24 shared between 3 and 4
24 shared between 6
How many numbers?
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Have have 6 apples in one bag and four
in another. If I put them together, how
many apples do I have?
I have 4 bags each containing 6 apples.
How apples do I have?
Ratio tables
B a gs
1
7
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I am putting apples
into bags. There are
six apples in each
bag. I fill seven
bags. How many
apples is that?
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Multiplication
Apples
6
?
Ratio tables
B a gs
1
?
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I am putting apples
into bags. There are
six apples in each
bag. I have 42
apples. How many
bags can I fill?
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Division as repeated
subtraction/grouping
Apples
6
42
Ratio tables
B a gs
1
7
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I am putting apples
into bags. There are
seven bags. I have
42 apples. If I put
the same number of
apples in each bag,
how many is that?
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Division as sharing
Apples
?
42
Marathon
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Sam is running the London marathon
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Every two miles she drinks 100 ml of
water
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How much does she drink over the 26
miles?