Generalising from number properties to algebra

31
CENTRE FOR EDUCATIONAL DEVELOPMENT Generalising from number properties to algebra A Adviser in Numeracy, Mathematics and NCEA

description

Generalising from number properties to algebra. A Adviser in Numeracy, Mathematics and NCEA. Teaching progression (adapted from Pierre – Kieren). Materials. Images. Knowledge. A Teaching Progression. Start by: Using materials, diagrams to illustrate and solve the problem Progress to: - PowerPoint PPT Presentation

Transcript of Generalising from number properties to algebra

Page 1: Generalising from  number properties to algebra

CENTRE FOR

EDUCATIONAL DEVELOPMENT

Generalising from

number properties to algebra

A

Adviser in Numeracy, Mathematics and NCEA

Page 2: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Materials

Teaching progression (adapted from Pierre – Kieren)

Knowledge

Images

Page 3: Generalising from  number properties to algebra

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

Page 4: Generalising from  number properties to algebra

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

Page 5: Generalising from  number properties to algebra

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

Page 6: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

One bracket with addition

Start with 3 rows of 7 counters

Page 7: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Page 8: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Discuss how this might be written

2173 =×Focus on

Page 9: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Place a straw between two columns

What does it now show?

Record it as 532373 ×+×=×

Page 10: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

How else can you place the straw to show the same thing?

23535323 ×+×=×+×Discuss what this shows:

Page 11: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

How else can you place the straw to show something different?

433373 ×+×=×

Page 12: Generalising from  number properties to algebra

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×

Page 13: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

What about placing the straw along the row?

717273 ×+×=×

Page 14: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Repeat using different numbers with one straw.

Progress to using grids to show the same thing.

Page 15: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

                   

                   

                   

                   

                   

                   

                   

                   

                   

                   

73×

Page 16: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

                   

                   

                   

                   

                   

                   

                   

                   

                   

                   

532373 ×+×=×

Page 17: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Generalise to number properties

bnanban

baba

×+×=+××+×=+××+×=+×

)(33)(3

5323)52(3

Page 18: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

                   

                   

                   

                   

                   

                   

                   

                   

                   

                   

717273 ×+×=×

Page 19: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Generalise to number properties

nbnanba

baba

×+×=×+×+×=×+

×+×=×+

)(777)(

71727)12(

Page 20: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Numbers greater than 10…

?135 =×

Page 21: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

?135 =×

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

?135 =×

Page 22: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

?135 =×

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

35105)310(5135 ×+×=+×=×

Page 23: Generalising from  number properties to algebra

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

Page 24: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Investigate

Two brackets with addition

One bracket with subtraction

Two brackets with subtraction

Page 25: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

Two brackets with addition

?1213 =×

Page 26: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

?1213 =×                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

Page 27: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

                                       

)23()103(

)210()1010(

)210()310(1213

×+×+×+×=+×+=×

Page 28: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

One bracket with subtraction                   

                   

                   

                   

                   

                   

                   

                   

                   

                   

13103

)110(393

×−×=−×=×

Page 29: Generalising from  number properties to algebra

CENTRE FOR EDUCATIONAL DEVELOPMENT

What about these?

?1623 =×

?619 =×

?4729 =×

?247129 =×

Page 30: Generalising from  number properties to algebra

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?

Page 31: Generalising from  number properties to algebra

CENTRE FOR

EDUCATIONAL DEVELOPMENT