Generalising from number properties to algebra Anne Lawrence Adviser in Numeracy, Mathematics and NCEA CENTRE FOR EDUCATIONAL DEVELOPMENT
Teaching progression (adapted from Pierre Kieren) Materials Images Knowledge CENTRE FOR EDUCATIONAL DEVELOPMENT 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 CENTRE FOR EDUCATIONAL DEVELOPMENT 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 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 CENTRE FOR EDUCATIONAL DEVELOPMENT 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
CENTRE FOR EDUCATIONAL DEVELOPMENT CENTRE FOR EDUCATIONAL DEVELOPMENT Discuss how this might be written Focus on
3 7 21 CENTRE FOR EDUCATIONAL DEVELOPMENT Place a straw between two columns What does it now show?
Record it as 3 7 3 2 3 5 CENTRE FOR EDUCATIONAL DEVELOPMENT How else can you place the straw to show the same thing?
Discuss what this shows: 3 2 3 5 3 5 3 2 CENTRE FOR EDUCATIONAL DEVELOPMENT How else can you place the straw to show something different?
3 7 3 3 3 4 CENTRE FOR EDUCATIONAL DEVELOPMENT How many different ways of placing the straw can you find?
How many different ways can you find of writing 37 ? Record
them all. FOR EDUCATIONAL DEVELOPMENT CENTRE Can you find a pattern? What about placing the straw along the row?
3 7 2 7 17 CENTRE FOR EDUCATIONAL DEVELOPMENT Repeat using different numbers with one
straw. Progress to using grids to show the same thing. CENTRE FOR EDUCATIONAL DEVELOPMENT
37 CENTRE FOR EDUCATIONAL DEVELOPMENT CENTRE FOR EDUCATIONAL DEVELOPMENT 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 CENTRE FOR EDUCATIONAL DEVELOPMENT
CENTRE FOR EDUCATIONAL DEVELOPMENT 3 7 2 7 17 Generalise to number properties (2 1) 7 2 7 17
(a b) 7 a 7 b 7 (a b) n a n b n CENTRE FOR EDUCATIONAL DEVELOPMENT Numbers greater than 10 5 13 ?
CENTRE FOR EDUCATIONAL DEVELOPMENT 5 13 ? 5 13 ? CENTRE FOR EDUCATIONAL DEVELOPMENT
5 13 5 (10 3) 5 10 5 3 5 13 ? CENTRE FOR EDUCATIONAL DEVELOPMENT 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 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 13 12 ? CENTRE FOR EDUCATIONAL DEVELOPMENT
13 12 ? CENTRE FOR EDUCATIONAL DEVELOPMENT 13 12 (10 3) (10 2) (10 10) (10 2) (3 10) (3 2)
CENTRE FOR EDUCATIONAL DEVELOPMENT One bracket with subtraction 3 9 3 (10 1) 3 10 3 1 CENTRE FOR EDUCATIONAL DEVELOPMENT
What about these? 19 6 ? 23 16 ? 29 47 ? 129 247 ? 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 CENTRE FOR EDUCATIONAL DEVELOPMENT