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• Level: GCSE
• Subject: Maths
• Word count: 1541

# Maths number grid

Extracts from this document...

Introduction

Number Grid Coursework

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 9 x 20 = 180 21 22 23 24 25 26 27 28 29 30 19 x 10 = 190 31 32 33 34 35 36 37 38 39 40 d = 10 41 42 43 44 45 46 47 48 49 40 51 52 53 54 55 56 57 58 59 60 89 x 100 = 8900 61 62 63 64 65 66 67 68 69 70 99 x 90  = 8910 71 72 73 74 75 76 77 78 79 80 d  = 10 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

I have noticed that there is a difference of ten between the answers when multiplying the diagonally opposite corners of a 2 x 2 grid.  I have also noticed that the 2 x 2 grids I have used are the corners of the 100 square grid.  I will try using a 2 x 2 grid located more towards the centre of  the 100 square grid to see if it is  just the corner 2x2 grids that give a difference of 10.

34 x 45 = 1530

44 x 35 = 1540

d  = 10

As you can see by my calculations it is not just a 2 x 2 grid in the corner that gives a difference of 10 when the diagonally opposite corners are multiplied. The position of the 2 x 2 grid on a 100 square grid does not change the difference.

I am now going to investigate if the difference is the same in a 2 x 3 grid when multiplying diagonally opposite corners.

2 x 3 grid

 63 64 65 63 x 75 = 4725 73 74 75 65 x 73 = 4745

d  = 20

 39 39 40 38 x 50 = 1900 48 49 50 48 x 40 = 1920

d  = 20

 11 12 13 11 x 23 = 253 21 22 23 21 x 13 = 273

d  =  20

Middle

From my results, I have discovered that if you multiply diagonally opposite corners of a 2 x 4 box grid, taken from any position in a 100 square grid, the difference between the two answers will be 30.

2 high boxes

 Height Width Difference 2 2 10 2 3 20 2 4 30 2 5 40

As you can see in my table, I feel I could now use my findings to predict the difference for a 2 x 5 grid.  As I have already predicted, the difference will be 40.

2 x 5 grid

 1 2 3 4 5 1 x 15 = 15 11 12 13 14 15 11 x 5  = 55

d  = 40

My prediction was correct in saying that the difference in a 2 x 5 grid would be 40.

I have noticed that it does not make any difference to the outcome wherever the grid is taken from in a 100 square grid.

From now on I only need to try one box of each size as I know that position has no effect on the difference.

2 x w Algebra (Any width)

 n n+w n+w-1 n+10 n+10+w-1 (n+10+w-1) = n+w+9

(n)  (n+w+9) = n2 + nw +9n

(n+10) (n+w-1) = n2 + nw – 1n +10n + 10w -10

(n2 +nw+9n) – (n2+nw+9n+10w-10) = d

10w-10= d

Conclusion

 n n+w-1 g(h-1)+n g(h-1)+n+(w-1)

gh-g+n                               gh-g+n+gw-1

n(gh-g+n+w-1)

=ngh-ng+(n2)+nw-n

(n+w-1)(gh-g+n)

=ngh-ng+(n2)+wgh-wg+wn-gh+g-n

d=(ngh-ng+(n2)+nw-n)-(ngh-ng+(n2)+wgh-wg+wn-gh+g-n)

d=wgh-wg-gh+g

I will now demonstrate how my formula works:

If g=10 h=4 w=3 n=5:

d= wgh-wg-gh+g

d=(3X10X4)-(3x10)-(10x4)+10

d=120-30-40+10

d=60

As you can see below, this is the answer I got for a 4x3 grid.

 5 6 7 5 x 37 = 185 15 16 17 7 x 35 = 245 25 26 27 d = 60 35 36 37

By Philip Redpath

## Algebra for h x w box on a 10x10 grid

 n n+w-1 10(h-1)+n 10(h-1)+n+(w-1)

10h-10+n                             10h-10+n+w-1

n(10h-10+n+w-1)

=10hn-10n+(n2)+wn-n

(n+w-1)(10h-10+n)

=10hn-10n+(n2)+10wh-10w+nw-10h+10-n

d=(10hn-10n+(n2)+wn-n)-( 10hn-10n+(n2)+10wh-10w+nw-10h+10-n)

d=10wh-10w-10h+10

I will now demonstrate how my formula works:

h=3     w=2     n=6

d=10wh-10w-10h+10

d=(10x2x3)-(10x2)-(10x3)+10

d=60-20-30+10

d=20

As you can see below, this is the same answer I got for a 3x2 grid:

 3 4 3 x 24 = 72 13 14 23 x 4 = 92 23 24 d = 20

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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