Generalising from number properties to algebra
description
Transcript of Generalising from number properties to algebra
CENTRE FOR
EDUCATIONAL DEVELOPMENT
Generalising from
number properties to algebra
A
Adviser in Numeracy, Mathematics and NCEA
CENTRE FOR EDUCATIONAL DEVELOPMENT
Materials
Teaching progression (adapted from Pierre – Kieren)
Knowledge
Images
CENTRE FOR EDUCATIONAL DEVELOPMENT
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
CENTRE FOR EDUCATIONAL DEVELOPMENT
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
CENTRE FOR EDUCATIONAL DEVELOPMENT
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
CENTRE FOR EDUCATIONAL DEVELOPMENT
One bracket with addition
Start with 3 rows of 7 counters
CENTRE FOR EDUCATIONAL DEVELOPMENT
CENTRE FOR EDUCATIONAL DEVELOPMENT
Discuss how this might be written
2173 =×Focus on
CENTRE FOR EDUCATIONAL DEVELOPMENT
Place a straw between two columns
What does it now show?
Record it as 532373 ×+×=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
How else can you place the straw to show the same thing?
23535323 ×+×=×+×Discuss what this shows:
CENTRE FOR EDUCATIONAL DEVELOPMENT
How else can you place the straw to show something different?
433373 ×+×=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
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×
CENTRE FOR EDUCATIONAL DEVELOPMENT
What about placing the straw along the row?
717273 ×+×=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
Repeat using different numbers with one straw.
Progress to using grids to show the same thing.
CENTRE FOR EDUCATIONAL DEVELOPMENT
73×
CENTRE FOR EDUCATIONAL DEVELOPMENT
532373 ×+×=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
Generalise to number properties
bnanban
baba
×+×=+××+×=+××+×=+×
)(33)(3
5323)52(3
CENTRE FOR EDUCATIONAL DEVELOPMENT
717273 ×+×=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
Generalise to number properties
nbnanba
baba
×+×=×+×+×=×+
×+×=×+
)(777)(
71727)12(
CENTRE FOR EDUCATIONAL DEVELOPMENT
Numbers greater than 10…
?135 =×
CENTRE FOR EDUCATIONAL DEVELOPMENT
?135 =×
?135 =×
CENTRE FOR EDUCATIONAL DEVELOPMENT
?135 =×
35105)310(5135 ×+×=+×=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
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
CENTRE FOR EDUCATIONAL DEVELOPMENT
Investigate
Two brackets with addition
One bracket with subtraction
Two brackets with subtraction
CENTRE FOR EDUCATIONAL DEVELOPMENT
Two brackets with addition
?1213 =×
CENTRE FOR EDUCATIONAL DEVELOPMENT
?1213 =×
CENTRE FOR EDUCATIONAL DEVELOPMENT
)23()103(
)210()1010(
)210()310(1213
×+×+×+×=+×+=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
One bracket with subtraction
13103
)110(393
×−×=−×=×
CENTRE FOR EDUCATIONAL DEVELOPMENT
What about these?
?1623 =×
?619 =×
?4729 =×
?247129 =×
CENTRE FOR
EDUCATIONAL DEVELOPMENT
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?
CENTRE FOR
EDUCATIONAL DEVELOPMENT