Memorising Vs Collective Summation | 4x, x6, x7, x8, x12 & x 13

My Gr 2 headmaster in the village would let us out for recess if only we recite our multiplication tables. Those were the times. Here, is a strategy and if used properly can be very effective too.

I'd like to called it 'Collective Summation'. It uses the idea of patterns and simple multiplication & addition.

Students can easily recall the first 5 multiples of 4

4 x 1 = 4
4 x 2 = 8......I will us this to show the strategy
4 x 3 = 12
4 x 4 = 16
4 x 5 = 20

Strategy 1
Key: remember the last multiple and add 4.

4 x 2 = 8

then 4 x 3 = 12 = 4 + 8
        4 x 4 = 16 = 4 + 12
        4 x 5 = 20 = 4 + 16

Instead of responsive recall of multiples of 4, what you are doing is actually working with what you know and adding a 4 to the subsequent number to get the next.

This strategy can be applied to hard-to-recall multiples in 6, 7, 8 and 13 times tables.

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Strategy 2

Using square numbers:

4 = 2 x 2
9 = 3 x 3
16 = 4 x 4
25 = 5 x 5
36 = 6 x 6
49 = 7 x 7
64 = 8 x 8
81 = 9 x 9
100 = 10 x 10
121 = 11 x 11
144 = 12 x 12
169 = 13 x 13

If you know, you can easily work out the multiples, when these numbers are doubled.
2 x 2 = 4            2 x 4 = 8 = 4 + 4

3 x 3 = 9            3 x 6 = 18 = 9 + 9
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16 = 4 x 4          4 x 8 = 32 = 16 + 16
25 = 5 x 5          5 x 10 = 50 = 25+ 25

36 = 6 x 6           6 x 12  = 72 = 36 + 36
49 = 7 x 7           7 x 14 = 58= 49 + 49

Got the idea? Now, try these

64 = 8 x 8              8 x 16 =
81 = 9 x 9              9 x 18 =

100 = 10 x 10              10 x 20 =
121 = 11 x 11              11 x 11 =

144 = 12 x 12              12 x 24 =
169 = 13 x 13               13 x 26 =

Teacher's Note: You may have a technique you have used in class over time - this is what I personally find useful. It can give your students confidence in you, too. (You know what I mean when you cannot do 12 x 24 in front of your student :))

Students' Note: You will find this useful when doing calculation involving 12 x 12, 12 x 24, 13 x 13 and 13x 26. It can save you lots of exam time.

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Perimeter | Finding a missing length

The diagram shows a rectangle and a square.

The perimeter of the rectangle is the same as the perimeter of the square.

Work out the length of one side of the rectangle.

Solution.........perimeter, p, is the sum of lengths around a shape


rectangle, p = 8+8+2+2 = 20 cm

square, s = 20/4 = 5 cm ....4 is the number of equal sides in a square

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Students note
- showing a clear working out will result in awarding of full marks
- key to getting this question right is knowing how to find perimeter and 4 sides of a square are equal.

Teachers note
- students should know the formulae p = 2( l + w) and p = 4s, for rectangle and square respectively. But, I think the best strategy is to show students the concept, not formula, as the concept can be used in both regular and irregular shapes. Remembering to use formulae can create confusion in contextual exam setting if a student can not recall them easily.
 

Perimeter | Adding the lengths by collecting like terms

In the diagram, all measurements are given in centimetres.

All angles are right angles.

Show that the perimeter of the shape can be written as 2(3x + 5).

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Solution

perimeter = side 1 + side 2 + side 3 + ...... (two sides are missing)

                = 2x + x + 3 + 2 + horizontal missing side + vertical missing side

                = 2x + x + 3 + 2 + 2x + 3 + x + 2

                = 6x + 10

                = 2(3 + 10) ......you've got to factorise to show

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Note: to get 4 full marks for this question, student must show (by calculation).

Prime Number, Square Number, Sequence

Here are 5 rows of numbers.

 

Row A

2

4

6

8

10

12

14

16

 

 

Row B

3

5

7

9

11

13

15

17

 

 

Row C

2

3

5

7

11

13

17

 

 

Row D

1

2

5

10

20

50

100

200

 

 

Row E

1

2

4

8

16

32

64

 

 

All the numbers are even in one of the rows.

 

(a)   Which row?

 

……………

 

(1)

 

(b)   The numbers in Row C are the first 7 prime numbers written in order of sizes.

 

 

 

Write the next prime number.

 

……………

(1)

 

(c)   Write down a square number from Row D.

 

 

 

……………

 

(1)

 

The numbers in Row E are the first 7 numbers of a sequence.

 

 

 

(d)   Work out the next number in the sequence.

 

……………

 

(1)

 

(Total 4 marks)

Answers

 

(a) Row A

(b) 19

(c) 100 (can also be 1)

(d) 128 ( each number doubles, nth term =  2n)

 

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Students note:

- knowing your square numbers (say the first 10) will help with finding answers to square roots, e.g. square root of 100 is 10

 

- there are two ways of finding nth term in a sequence: finding the right pattern and or working our the rule.

 

Teachers note:

It can be helpful to use the same test format when setting test