Primary School Focus || Maths Resource: Multiplication 2

Maths Resource: Multiplication 2Author(s): Ian Sugarman and Don StewardSource: Mathematics in School, Vol. 23, No. 1, Primary School Focus (Jan., 1994), pp. 29-34Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215074 .

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Multiplication 2

lan Sugarman, Shropshire Mathematics Centre Don Steward, MEDIAN

In MiS 22.4 we promoted the idea that it can be very helpful for pupils to associate multiplication ideas with rectangular arrays of squares or dots. Such an appreciation can be extended to larger arrays, allowing the construction of mental images helpful to the development of common-sense strategies for solving long multiplication sums. Multibase structured equipment can be a valuable aid to this understanding.

Draw some large rectangles on a sheet of paper, some "squarish" and others elongated. The rectangles should have whole number centimetre dimensions, e.g. 16 x 12, 18 x 12, 15 x 13 etc. Pupils can be asked to use multibase ten blocks (hundreds, tens and units) to ascertain which rectangle has the largest area. By placing first as many "flats" as possible, then as many "longs" as possible and finally counting the number of unit cubes needed to complete a covering of the shape, the rectangles' areas can be easily established.

Reinforcement of this idea, in a slightly different context, is provided in MRes'. Here the intention is to solve the multiplication problem presented, by drawing a line which partitions the dots into lines of 10. There will then be a certain number of tens and a single digits multiplication fact:

to 5

*******o*** !15 8

-eeee oooo o XS 80

0**************

oooo *

oo x = o

*************** 120

MRes2 extends this skill to double digit multiplication sums, such as 15 x 13. Bearing in mind the task of covering the rectangle with the multibase blocks, pupils should be able to adapt this work to partition these arrays of dots to show the boundaries of a hundred square (10 x 10), the ten rods and the remaining rectangular array of units.

(continued on page 34)

Mathematics in School, January 1994 29



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30 Mathematics in School, January 1994



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MRes 200 @




In General, looking at digits.

AB means a two-digit number. EFG would be a three-digit number.

3EF has 3 hundreds, E tens and F units. Try to say as much as you can about these:

* If A x B = C what can you say?

* If AB x C = DE what can you say?

* If AB x CD = EFG what can you say?

* If AB x CD = EFG1 what can you say about B and D?



* look at some other end digits and decide upon possibilities.

* If AB x CD = 3EF what can you say about A and C?

* Try to find a quick way of squaring a two digit number that ends in 5. Use a diagram to explain why your method works.

MRes 201 @




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(continued from page 29)

An appropriate partition of the 15 x 13 rectangle might be:

lo x 10o 10o0 5x (c = 50 3x to = 30

3x5 - IS iA C

to 10 00 0 0 0 0 0 000

3000*000000 0*00

********** **** ********** **** ********** ****

00***0 *0 0* ***

to .. 0

********** ****

ooo0ooo 5

Arrays like the ones included at the bottom of the page allow opportunities to explore a variety of partitionings to arrive at the total number of dots: e.g.

ioo((l too 6 xo, 2 . ... .. . ... ..... .... - 1 r**r*6ir r -13--- b = 4% ....-- "- '- "":''"" ..... LI

a ~o. 0000.00

0 o..oo.... o .7 9(4. = 6 0.. .... .............

0000 00 * a 0 0 0000 906696060006006666 .~

X ~ o Qooo ooo~ - -- oooo--- oooo- --- --7 - .~ ~ - oo o ,- ooe .oooo...oeo-,-eo..oo2

The grids in MRes3 provide a means of pictorially representing any multiplication sum up to 20 x 20. For any such sum, e.g. 17 x 14, the boundary of the rectangle is drawn. Initially, coloured pens can be used to highlight the four partitioned rectangles. This can later be developed to recording the totals within each section:

-AF ' - IALI ] [

v i I :

,

I .L4LL ifffLA

Practice should establish that a rough diagram is all that is required:

30 3

1 30 0 S

7 20 to2.1

In MRes4 there are a variety of problems in which the pupils are asked to establish statements about particular arrangements of digits within long multiplication sums. There is a particular focus on the properties of last digits. We envisage that calculators would be used for this sheet. 0




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Primary School Focus || Maths Resource: Multiplication 2