# Number Grids

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Introduction

Number Grids

Introduction

I have a10x10 grid. The task is to select various size squares and rectangles, I will then be multiplying the opposite corners together then subtracting the results to get the difference between the two answers. I hope to find patterns and rules in the results, if I can find patterns I will use algebra to describe them. My aim is to find the general rule for any size rectangle on any size grid.

Plan

- investigate different squares on a 10x10 grid.
- Find an algebraic rule.
- Repeat for rectangles on a 10x10 grid.
- Adapt the rule for squares to fit rectangles.
- Change the size of the grid
- Find an overall rule for any rectangle on any grid.

25 26 25 x 36 = 900

35 36 26 x 35 = 910

910 – 900 = 10

The difference between the two answers is 10.

1 2 1 x 12 = 12

11 12 2 x 11 = 22

22 – 12 = 10

Middle

The difference between the two answers is 90.

1 2 3 4 1 x 34 = 34

11 12 13 14 4 x 31 = 124

21 22 23 24 124 – 34 = 90

31 32 33 34

The difference between the two answers is 90.

67 68 69 70 67 x 100 = 6700

77 78 79 80 70 x 97 = 6790

87 88 89 90 6790 – 6700 = 90

97 98 99 100

The difference between the two answers is 90.

Now that I have my results from each of the different size squares I done, I will put them into the table.

Size of Square | The difference |

2 x 2 3 x 3 4 x 4 | 10 40 90 |

I now need to look for a pattern which is within the results.

After looking at the results I notice that if I ignore the noughts on the results, for example 1, 4, 9 . . . . . these numbers are all square numbers.

For the nth term, squaring must be involved.

When I square the number in the first column, I get the answer in the second column but on the line below. So instead of squaring n, I need to square (n – 1).

Conclusion

2 – 1 = 1 1 x 5 = 5

6 – 1 = 5

5 x 10 = 50 using the formula I predict the the difference will be 50.

10(2 -1)(6 -1) = 10 x 5 = 50

This proves my formula works.

I have a theory that the reason I have to times by 10 is because I am usining a 10 x 10 grid. So I am going to test this theory by changing the size of the grid to a 5 x 5 grid, to see if the number I have to multiply by is 5.

I am going to test this theory on a 2 x 3 rectangle.

7 8

12 13

17 18

By using my previous formula I am going to guess what the difference is going to be.

2 – 1 = 1

3 – 1 = 2

1 x 2 = 2

2 x 5 = 5. I predict the number 5 will be the difference.

7 x 18 = 126

8 x 17 = 136

136 – 126 = 10 this provesmy theory was correct.

I have noticed the number you times it by is the size of the grid, so therefore to work out any rectangle on any size grid the formula would be:

g(n-1)(m-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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