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Secondary Magazine - Issue 119: Building Bridges

Created on 05 February 2015 by ncetm_administrator
Updated on 26 February 2015 by ncetm_administrator

Building Bridges

In the last issue, we considered how the area model for multiplication helps pupils visualise and understand multiplicative invariance: that A × B = (kA) × (B÷K). The area model is one of the best examples of a bridge that can stretch all the way from KS1 to KS5, and so for this issue we thought we’d pull together some uses of it, and invite you to tell us others – if you have pictures you can tweet @NCETMsecondary or send to info@ncetm.org.uk, we’ll share them in the next issue.

Most children will, in KS1, use arrays of concrete objects to represent multiplication, for example arranging 15 counters into 3 rows of 5 (and 5 rows of 3). The extension to the area model – often called the grid method – in KS2 comes naturally, perhaps thinking about a farmer with a field that is 16m by 37m who wants to grow four crops – one of which, of course, is rhubarb! Dividing the field into four smaller sub-fields enables the calculation of 16 × 37 to be broken into four easier calculations:

 × 30 7 10 300 70 6 180 42 592

Part of the power of this visualisation is that it can indeed be visualised, and pupils’ mental multiplication of two digit numbers can be practised and improved – passers-by of many of my KS3 lessons will have heard the slightly odd instructions “right, close your eyes, think, ok, top left, top right, bottom left, bottom right, what have you got?” Handheld whiteboards might well be useful at first.

The grid also explains the steps of column multiplication. This formal algorithm is certainly more efficient with larger numbers and must be taught and practised to ensure procedural fluency – pupils mustn’t be left only able to use the grid method – but it doesn’t develop conceptual understanding as well as thinking about fields (of rhubarb!) does.

Once pupils are confident with negative numbers, the efficiency of the grid method can be refined: to evaluate 21 × 19, ask pupils to compare

 × 20 1 10 200 10 9 180 9 399

and

 × 20 1 20 400 2 -1 -20 -1 399

This enables pupils to make and justify conjectures equivalent to the difference of two squares without having to use algebraic symbols, if they would obfuscate the reasoning.

The area model can be adapted throughout KS2 to 5:

 4⁄5 × 2⁄3

The grid shows why the algorithm is to multiply the numerators and the denominators.

• 223 × 514
 × 5 1⁄4 2 10 1⁄2 2⁄3 31⁄3 1⁄6 14

This is certainly not as efficient as the algorithm “make the fractions top-heavy and then multiply the numerators and the denominators”, but the connection it makes to pupils’ earlier understanding of multiplication is clear, and thus develops their secure conceptual understanding.

• Expanding brackets
 × 3x 2 x 3x2 2x -5 -15x -10 3x2 - 13x - 10

in particular

 × x 6 x x2 6x 6 6x 36 x2 + 12x + 36

and

 × x 6 x x2 6x -6 -6x -36 x2 - 36

and then rich challenges such as (a + b + c)2

• Pupils can teach themselves to factorise quadratics, from a well-designed sequence of “guess and check” intelligent practice questions such as
 × x ? x x2 ? ? ? 5 x2 + 6x + 5

then

 × x ? x x2 ? ? ? -8 x2 + 2x - 8

then

 × ?x ? x 3x2 ? ? ? ? 3x2 + 15x

then

 × x ? x x2 ? ? ? ? x2 - 25

then

 × ?x ? ?x 9x2 ? ? ? ? 9x2 - 4

then

 × ?x ? x 3x2 ? ? ? ? 3x2 - 8x - 3

then

 × ?x ? ?x ? ? ? ? ? 4x2 + 7x - 15

then

 × ?x ? ?x ? ? ? ? ? 4x2 + 12x + 9

and so on.

• And, in KS5,
 × 5 2j 3 15 6j -4j -20j -8j2 23 - 14j

and

 × ? ? ? a2 ? ? ? ? a2 + b2

and a personal favourite: factorising x4 + 4y4 (hint, this farmer grows 9 crops).

The area model is, therefore, a Humber-spanning bridge that takes pupils all the way from their first experiences of positive integer multiplication to their exploration of the complex plane in Further Pure Maths: a route well worth their following.

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