# Investigate different shapes in different sized number grids.

Thomas Shaw

Introduction

In this piece of course work I will investigate different shapes in different sized number grids. The shapes I will look at will be the square, the rectangle, the rhombus and the parallelogram. I will investigate these shapes in two different number grids, one a 10x10 grid and the other a 5x5 grid. I will draw these shapes on the number grids at random points and take the corner numbers and then multiply the opposite corners. From these results I will attempt to work out a formula for how the size of shape affects the result taken from the table.

I will take the number from the bottom left corner of the shape and the top right corner of the shape and multiply these numbers together. This set of numbers will be represented by the colour RED. I will then take the top left corner and the bottom right corner and multiply these two numbers together. This set of numbers will be represented by the colour BLUE. I will then subtract the RED result from the BLUE which will give me the number I need to work out my formula. The number gained from the subtraction will be called the difference.

All my results will be taken from either a 5x5 grid like this:

Or a 10x10 grid like this:

From the four shapes I am using I will do different sizes of each in my paper, the sizes used are as follows:

Square

2x2

3x3

4x4

5x5

Once I have done this I will work out the formula and work out what 6x6 would be then do 6x6 to show that it is correct. NOTE: the 6x6 test will only apply for the 10x10 grid, for the 5x5 grid I will do up to 4x4 then work out the formula and find out what 5x5 will be.

Rectangle

3x2

4x3

5x4

Once this is completed I will work out the formula for the rectangle and then work out 6x5. Once again the 5x5 grid will go up to 4x3 and then work out the formula to 5x4.

Rhombus

2x2

3x3

4x4

I will then work out the rhombus formula and work out what 5x5 is then do 5x5 to show that the formula is correct.

Squares - 5x5 Grid

2x2

2 x 6 = 12

1 x 7 = 7

Difference = 5

18 x 14 = 252

13 x 19 = 247

Difference = 5

21 x 17 = 357

16 x 22 = 352

Difference = 5

Formula for 2x2 on 5x5

A = (X + 5) (X + 1)

= X² + X + 5X + 5

= X² + 6X + 5

B = X (X + 6)

= X² + 6X

Difference = A – B

= (X² + 6X + 5) – (X² + 6X)

= 5

3x3

17 x 9 = 153

7 x 19 = 133

Difference = 20

23 x 15 = 345

13 x 25 = 225

Difference = 20

16 x 8 = 128

6 x 18 = 108

Difference = 20

Formula

A = (X + 10) (X + 2)

= X² +2X + 10X + 20

= X² + 12X + 20

B = X (X + 12)

= X² + 12X

Difference = A – B

= (X² + 12X +20) – (X² + 12X)

=20

4x4 Square

17 x 5 = 85

2 x 20 = 40

Difference = 45

22 x 10 = 220

7 x 25 = 175

Difference = 45

21 x 9 = 189

6 x 24 = 144

Difference = 45

Formula

A = (X + 15) (X + 3)

= X² +3X + 15X + 45

= X² + 18X + 45

B = X (X + 18)

= X² + 18X

Difference = A – B

= (X² + 18X + 45) – (X² + 18X)

= 45

5x5 grid - square formula

These are the differences from my different squares on the 5x5 grid:

If we look at the difference’s we notice that all of them are multiple’s of 5 so we can predict that the formula involves the number 5. Also, the grid is 5x5 so if G = Grid Size (Which is 5 NOT 25). Therefore if D = Difference and G = Grid Size we can make the following formula:

D=(Square size - 1)² x G

This table will show the formula in action to show how it produces the formula:

If this Formula is correct it should allow us to work out what the difference of a 5x5 square is:

D = (5-1)² x 5

16 x 5

80

If the formula I have worked out is correct then the difference for my 5x5 square should be 80. I will now do the only 5x5 square I can do to show it is correct.

5x5

21 x 5 = 105

1 x 25 = 25

Difference = 80

The Difference I predicted ...