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Introduction

Maths Coursework

Number Grids

## Chapter One

For the first part of my maths G.C.S.E coursework I have been provided with a 10x10 number grid, which is numbered 1 to 100.  I have been instructed to find the product of the top left number and the bottom right number and the same with the top right and bottom left number within selected squares used.

I am going to use this grid to examine at random various sizes of squares and rectangles.  My objective here is to establish a trend to identify an overall formula.

I am firstly going to examine a series of 2x2 squares the primary one I will look at has been outlined in the number grid I was provided with, I will then select alternative 2x2 squares at random from the grid.

1.                                             13x22 - 12x23   286 - 276 Difference = 10

58x67- 57x68

1.  3886 - 3876 Difference = 10

1.                                            35x44 – 34x45

1540 - 1530 Difference = 10

By looking at the three 2x2 squares chosen above it is possible to see each time that there is a difference of 10.  So in conclusion to this I can say that any further investigations using 2x2 squares will always result in a difference of 10.

Looking at my results and the number grid at this stage, I feel I can suggest that the reason I may get the same result of 10 each time is one of two small theories, my first one being that it is possible to see that the selected 2x2 squares have a difference of 10 between the bottom and the top line of each set of numbers, before even using multiplication.  And the second piece of theory for this difference could be that the grid I am using is a 10x10 number grid.

Middle

I feel unable to make a prediction of a defined difference at this point; I hope to make able to predict some type of trend or pattern further on.

9x25 – 5x29  225 –145

Difference = 80  56x72 – 52x76 4032 –3952 Difference = 80

As can be seen my defined difference for any 5x3 rectangle gives me an answer of 80.  I am going to use algebra to ensure my answer is accurate.

(s+4)(s+20) – s(s+24)

s(s+20)+4(s+20) – s  -24s

s  +20s +4s+80 – s  -24s

=80

I have now calculated the answer for my 3x2 rectangles and came to a difference of 20 and an answer for my 5x3 rectangles and came to a defined difference of 80. Still I do not see any major patterns forming and the only trend I can establish is that my answers are still multiples of 10. I will continue my investigation by increasing to a larger rectangle.

Furthering my investigation

I am now going to look at 6x4 rectangles, I would hope after this I will be able to establish some type of pattern to help me reach my aim of finding an overall formula for any rectangle that could be investigated. 30x55 – 25x60

1650 –1500 Difference =150

66x91 – 61x96

6006 – 5856 Difference = 150

By looking at the above calculations it can be assumed that they are correct, I will use algebra to prove this.

(s+5)(s+30) – s(s+35)

s(s+30) +5(s+30) – s  - 35s

s  +30s+5s+150 – s  -35s

=150

I find it very difficult to see any major trend, this is because I am randomly selecting various sizes of rectangles and therefore they do not come in any particular order and the sizes do not carry on from the one before.

## Furthering my investigation

Conclusion

 r x Result Multiples of 9 2 x2 9 9 x 1 9 x 1 3 x3 36 9 x4 9 x 2 4 x4 81 9 x 9 9 x 3 5 x5 144 9 x 16 9 x 4

I can now draw upon the conclusion that the formula for this sequence of rhombuses  is (10-1)(r-1) .

Chapter 8

To conduct a thorough investigation I felt it was important to examine the affects that may occur if I was to reflect he shape of the rhombus within the 10x10 number grid. To ensure validity I will examine the same sizes of rhombuses that I used in my prior investigation in chapter seven, only this time all the rhombuses that I use will be this shape:-

13x23 – 12x24

299 – 288

Difference = 11 55x75 – 53x77

4125 – 4081

Difference = 44

I will now show the algebra to this 3x3 rhombus to show that this calculation is correct.

(r+2)(r+22)-r(r+24)

r(r+22)+2(r+22)-r –24r

r +22r+2r+44-r –24r

= 44

I have distinguished that these answers are multiples of 11, therefore I will predict that the next answer will too be a multiple of 11.

24x54 – 21x57 1296 – 1197

Difference = 99

Again I can see that my prediction was correct. 36x76 – 32x80

2736 – 2560

Difference = 176

I will show the algebra for any 5x5 rhombus within a 10x10 number grid, this will show that the defined difference will always be 176.

(r+4)(r+44)-r(r+48)

r(r+44)+4(r+44)-r –48r

r +44r+4r+176-r –48r

=176

I will now place my findings from this investigation into a table to help find the end formula for any given rhombus within a 10x10 number grid.

 r x Results Multiples of 11 2 x 2 11 11 x 1 11 x 1 3 x 3 44 11 x 4 11 x 2 4 x 4 99 11 x 9 11 x 3 5 x 5 176 11 x 16 11 x 4

In conclusion of this chapter I can now revel that the overall formula for any

shaped rhombus is (10+1)(r-1) .

Conclusion for chapter 7 & 8

Overall this investigation into the sizes and shapes of rhombuses has provided me with evidence to suggest that the determination on the formula will depend on the actual shape of the rhombus ie.    Will always be a –1    whereas   Will always be a +1

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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