Transcript The Numeracy Professional Development Project in Secondary

Generalising from
number properties to algebra
Anne Lawrence
Adviser in Numeracy,
Mathematics and NCEA
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Teaching progression
(adapted from Pierre – Kieren)
Materials
Images
Knowledge
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A Teaching Progression
Start by:
Using materials, diagrams to illustrate
and solve the problem
Progress to:
Developing mental images to help
solve the problem
Extend to:
Working abstractly with the number
property
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To reinforce and consolidate
Move back and forth between:
Using materials, diagrams to illustrate
and solve the problem
and:
Developing mental images to help
solve the problem
and:
Working abstractly with the number
property
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Discovering algebraic ‘rules’ for
expanding
Based on an array model for multiplication
Ideas adapted from
1. Cyril Quinlan. Analysing teaching/learning strategies for
algebra. P 459-464. MERGA 18 (Eighteenth annual
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conference
of the mathematics education research group of
Australasia Darwin 1995).
2. http://www.blackdouglas.com.au
One bracket with addition
Start with 3 rows of 7 counters
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Discuss how this might be written
Focus on
3 7  21
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Place a straw between two columns
What does it now show?
Record it as
3 7  3 2  3 5
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How else can you place the straw to show
the same thing?
Discuss what this shows:
3 2  3 5  3 5  3 2
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How else can you place the straw to show
something different?
3 7  3 3  3 4
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 How many different ways of placing the straw
can you find?
 How many different ways can you find of writing
3 7 ?
 Record them all.
 Can you find a pattern?
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What about placing the straw along the
row?
3 7  2  7 1 7
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Repeat using different numbers with one
straw.
Progress to using grids to show the same
thing.
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3 7
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3 7  3 2  3 5
Generalise to number properties
3  (2  5)  3  2  3  5
3  ( a  b)  3  a  3  b
n  ( a  b)  n  a  n  b
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3 7  2  7 1 7
Generalise to number properties
(2  1)  7  2  7  1 7
( a  b)  7  a  7  b  7
( a  b)  n  a  n  b  n
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Numbers greater than 10…
5 13  ?
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5 13  ?
5 13  ?
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5  13  5  (10  3)  5  10  5  3
5 13  ?
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A suggested progression
Start with rows of counters in columns
Use a straw to generate different number
properties
Repeat for different numbers
Generalise number properties with words
Extend from counters to grids or arrays
Generalise properties using symbols
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Investigate
Two brackets with addition
One bracket with subtraction
Two brackets with subtraction
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Two brackets with addition
1312  ?
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1312  ?
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13  12  (10  3)  (10  2)
 (10  10)  (10  2)
 (3  10)  (3  2)
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One bracket with subtraction
3  9  3  (10  1)
 3 10  3 1
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What about these?
19  6  ?
23 16  ?
29  47  ?
129  247  ?
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Questions to consider…
Is the use of counters necessary?
Do students need to cut out grids or is
shading of rectangles sufficient?
How important is recording?
What is the best way of leading into
the use of symbols?
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