Algebraic Generalisation

Achievement Objectives

Level 3

• Record and interpret additive and simple multiplicative strategies, using words, diagrams and symbols with an understanding of equality

• Generalise the properties of addition and subtraction with whole numbers

• Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns

Achievement Objectives

Level 4

• Form and solve simple equations• Generalise the properties of multiplication and division with

whole numbers• Use graphs, tables and rules to describe linear relationships

found in number and spatial patterns

Achievement Objectives

Level 5

• Form and solve simple linear and quadratic equations• Generalise the properties of operations with fractional

numbers and integers• Relate graphs, tables and rules to linear and simple quadratic

relationships found in number and spatial patterns

Progressions

1: move from symbols to linear equations and on to quadratic equations

2: generalise add/sub for integers, mult/div for integers and then both for fractions

3: use and understand patterns, through linear relationships to quadratic relationships. Its all about rules, tables and graphs.

A pet hate, and an important one

Algebra;

Letters represent numbers not objects….they represent the number of objects.

Sometimes we confuse students by placing an emphasis on objects not number…..dont!

Introduce letters for unknown numbers please.

Numeracy

Work through these three progressions using:• Strategies and knowledge at and just in front

of the appropriate stage• The numeracy teaching model: materials,

imaging, abstract• Activities – NZ Maths, Figure it out, etc

Algebra in the strands….• Statistics

– Modelling formulas

• Geometry– Coaster patterns– Match patterns

• Measurement– Area formulas

• Number

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Algebraic Generalisation

•Explain the strategy

•Give another example

•Generalise

Example 1

Task 1

9 + 9 + 9 + 5 + 5 + 5 = 3 9 + 3 5

6 + 4 + 6 + 4 + 6 + 4 = 3 (6 + 4) 9 + 9 + 9 – 5 – 5 – 5 = 3 9 – 3 5 6 – 4 + 6 – 4 + 6 – 4 = 3 (6 – 4)

Task 215 + 16 = 15 + 15 +1

= 2 15 + 1

19 + 20 = 20 + 20 – 1 = 2 20 – 1

9 + 10 + 11 = 9 + (9+1) + (9+2) = 3 9 + 3

9 + 10 + 11 = (10–1) + 10 + (10+1) = 3 10

9 + 10 + 11 = (11–2) + (11–1) + 11 = 3 11 – 3

Task 3

12 13 = 12 12 + 12 1 =122 + 12

13 12 = 13 13 – 13 1 =132 – 13

Task Four

7 32 = 7 30 + 7 2 7 39 = 7 40 – 7 1

Task Five

32 42 = 30 40 + 2 40 + 30 2 + 2 2

32 48 = 30 50 + 2 50 + 30 -2 + 2 -2

39 42 = 40 40 + -1 40 + 40 2 + -1 2

39 49 = 40 50 + -1 50 + 40 -1 + -1 -1

Task Six9 9 9 9 9 9 9 = 97

92 95 = (9 9) (9 9 9 9 9) = 97

97 = 9 9 9 9 9 9 995 9 9 9 9 9 = 92

(94)3 = 94 94 94

= 912

Using this resource…

• Diagnostic Task• Starter / Plenary• Challenge…

Next…..patterning

Handouts.

Work through these and place them at the appropriate numeracy stage/curriculum level.

• https://www.ncetm.org.uk/resources/10848