# Number Grid

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Introduction

Number Grid Coursework The grid I will using is a 10 x 10 grid, numbers ranging from 1 to 100. Grid Size = 10 x 10 Box size = 2 x 2 Task 1: I will take a 2 x 2 box on the 100 numbered grid and I will find the product of the top left number and the bottom right number of this box, and will do the same with the top right and bottom left numbers of this box. I will then look at the relationship between the two products, by finding the difference. 12 13 22 23 12 x 23 = 276 13 x 22 = 286 286-276 = 10 Difference = 10 Now I want to find out whether or not the difference of these two products is constant throughout the number grid, therefore I will randomly move the box somewhere else: 78 79 88 89 78 x 89 = 6942 79 x 88 = 6952 6952-6942 = 10 Difference = 10 I want to further test out the experiment: 55 56 65 66 55 x 66 = 3630 56 x 65 = 3640 3640-3630 = 10 Difference = 10 I will now represent all the numbers in the box with a letter, therefore the first letter in the box (left- top corner) will be a: a b c d Table of results: a ad bc bc-ad 12 276 286 10 78 6942 6952 10 55 3630 3640 10 39 1950 1960 10 7 126 136 10 From the results of this experiment, I have come to realize that there is a distinct pattern in the results. From the results, this suggests that the difference between the products of ad and bc is always 10. One can also see that if a is increased then so does ad and bc even though the difference between the two is not affected. ...read more.

Middle

The reason 10 was added was because it was the size of the grid, 10 x 10, and also I thought that it would make sense to add it onto the part of the formula with the brackets as all the differences are in the 10 times table. Hence, due to this I could make a formula: w = width h = length 10(h-1) Therefore, I decided to test the formula by substituting w with 3 from the 3 x 3 box: y= difference 10(3-1) = y y= 20 However, for some reason this formula did not work for the 3 x 3 box, because the difference was supposed to be 40, but I decided to try it with a 4 x 4 anyway: 10(4-1) = y Y = 30 This showed that the formula was incorrect and I needed to change something in the formula, as I needed to get difference to equal 90 not 30 for a 4 x 4 box. I realized that for the 3 x 3 box, I had to multiply the 10 by 4 in order to get 40 and in the 4 x 4 box, I needed to multiply the 10 by a 9 to get 90. I noticed that the numbers are the square of the side of square minus 1. I discovered that if I squared the brackets then both 3 x 3 and 4 x 4 boxes would get their differences correct, as 40 and 90. 3 x 3 box: 10(3-1)�= y 10 x 2�= y y= 40 4 x 4 box: 10(4-1)�= y 10 x 3�= y y=90 Therefore, I proved that the general formula for the difference of bc and ad in a 10 x 10 squared grid was: 10(h-1)� Task 5: I wanted to see what change would occur if I changed the grid size. The formula for the 10 x 10 grid was 10(h-1)�, I predict that the grid size for a 5 x 5 grid will be 5(h-1)� and ...read more.

Conclusion

Hence, due to this I could make a formula: w = width h = length 10(h-1) Therefore, I decided to test the formula by substituting w with 3 from the 3 x 3 box: y= difference 10(3-1) = y y= 20 However, for some reason this formula did not work for the 3 x 3 box, because the difference was supposed to be 40, but I decided to try it with a 4 x 4 anyway: 10(4-1) = y Y = 30 This showed that the formula was incorrect and I needed to change something in the formula, as I needed to get difference to equal 90 not 30 for a 4 x 4 box. I realized that for the 3 x 3 box, I had to multiply the 10 by 4 in order to get 40 and in the 4 x 4 box, I needed to multiply the 10 by a 9 to get 90. I noticed that the numbers are the square of the side of square minus 1. I discovered that if I squared the brackets then both 3 x 3 and 4 x 4 boxes would get their differences correct, as 40 and 90. 3 x 3 box: 10(3-1)�= y 10 x 2�= y y= 40 4 x 4 box: 10(4-1)�= y 10 x 3�= y y=90 Therefore, I proved that the general formula for the difference of bc and ad in a 10 x 10 squared grid was: 10(h-1)� Task 5: I wanted to see what change would occur if I changed the grid size. The formula for the 10 x 10 grid was 10(h-1)�, I predict that the grid size for a 5 x 5 grid will be 5(h-1)� and for a 8 x 8 grid, it will be 8(h-1)�: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ...read more.

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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