Post on 07-Jan-2016
description
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Generalising from
number properties to algebra
A
Adviser in Numeracy, Mathematics and NCEA
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Materials
Teaching progression (adapted from Pierre – Kieren)
Knowledge
Images
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A Teaching ProgressionStart by:
Using materials, diagrams to illustrate and solve the problem
Progress to:Developing mental images to help
solve the problemExtend to:
Working abstractly with the number property
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To reinforce and consolidateMove back and forth between:
Using materials, diagrams to illustrate and solve the problem
and:Developing mental images to help
solve the problemand:
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 conference of the mathematics education research group of Australasia Darwin 1995).
2. http://www.blackdouglas.com.au
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One bracket with addition
Start with 3 rows of 7 counters
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Discuss how this might be written
2173 =×Focus on
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Place a straw between two columns
What does it now show?
Record it as 532373 ×+×=×
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How else can you place the straw to show the same thing?
23535323 ×+×=×+×Discuss what this shows:
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How else can you place the straw to show something different?
433373 ×+×=×
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How many different ways of placing the straw can you find?
How many different ways can you find of writing ?
Record them all.
Can you find a pattern?
73×
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What about placing the straw along the row?
717273 ×+×=×
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Repeat using different numbers with one straw.
Progress to using grids to show the same thing.
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73×
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532373 ×+×=×
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Generalise to number properties
bnanban
baba
×+×=+××+×=+××+×=+×
)(33)(3
5323)52(3
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717273 ×+×=×
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Generalise to number properties
nbnanba
baba
×+×=×+×+×=×+
×+×=×+
)(777)(
71727)12(
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Numbers greater than 10…
?135 =×
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?135 =×
?135 =×
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?135 =×
35105)310(5135 ×+×=+×=×
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A suggested progression
Start with rows of counters in columnsUse a straw to generate different number
propertiesRepeat for different numbersGeneralise 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
?1213 =×
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?1213 =×
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)23()103(
)210()1010(
)210()310(1213
×+×+×+×=+×+=×
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One bracket with subtraction
13103
)110(393
×−×=−×=×
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What about these?
?1623 =×
?619 =×
?4729 =×
?247129 =×
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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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