# Number Grids Coursework

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Introduction

Number Grids Coursework Introduction In the following piece of coursework, I intend to investigate taking a square of numbers from a 10 x 10 grid, multiplying the opposite corners and then finding the difference between the two products. I was first asked to take a 2 x 2 square from a 10 x 10 grid, multiply the opposite corners and then find the difference. This is the result I received; 2x2 squares 15 16 25 26 Square 1 15 x 26 = 390 16 x 25 = 400 Difference = 400 - 390 = 10 Here I found that the difference was 10, to find out if the difference was the same for every 2 x 2 square, I decided to test another 3 squares, and here are the results I received; Square 2 18 19 28 29 18 x 29 = 522 19 x 28 = 532 Difference = 532 - 522 = 10 Square 3 32 33 42 43 32 x 43 = 1376 33 x 42 = 1386 Difference = 1386 - 1376 = 10 35 36 45 46 Square 4 35 x 46 = 1610 36 x 45 = 1620 Difference = 1620 - 1610 = 10 After calculating the differences between the products of four 2 x 2 square grids, I can conclude that the difference is always 10, and I predict that the difference will always remain 10 for 2 x 2 square grids. I then decided to look at a 3 x 3 square on a 10 x 10 grid, to see if the difference would be 10, or anything different, and here is the result; 32 33 34 42 43 44 52 53 54 3 x 3 squares Square ...read more.

Middle

s This is the 2 x 2 square so far: To get the top-right corner I am going to look at a 2 x 2 square taken from a grid: 15 16 is one more than 15 25 26 From this I have seen that the top right corner is one more than the top left corner. This is the same in any other 2 x 2 square. So using the top left corner 's' I can say that the top right corner is s + 1. I now have the two to expressions s S + 1 Again to find like the top right corner, I am going to have to look at some example of a 2 x 2 square to find the bottom two corners expressions. 15 16 is one more than 15 25 is ten more than 15 26 is eleven more than 15 From this I have seen that the bottom left corner is ten more than the top left corner, so I can say that the bottom left corner is s + 15, and from the bottom right corner I can see that it is eleven more than the top left corner, so I can say that the bottom right corner is s + 11, and now I have a four complete expressions for a 2 x 2 square, which now looks like this; S S + 1 S + 11 S + 11 Now I just have to multiply the corners, and find the differences like this; s( s + 11) = s� + 11s (s + 1)(s + 10) = s� + 10s + s + 10 = s� + 11s + 10 Difference = (s� + 11s + 10) ...read more.

Conclusion

+ (J - 1) Now I will just multiply the corners, and then find the expression for the difference like so; S (s + g(J - 1) + (J - 1) = s� + gs (J - 1) + s (J - 1) (S + (J - 1) ) (S + g(J - 1) = s� + gs( J - 1) + s( J - 1) + g( J - 1) � Difference = (s� + gs (J - 1) + s (J - 1) ) - (s� + gs( J - 1) + s( J - 1) + g( J - 1) �) = g(J - 1) � Now I have an algebraic expression that can be used on any sized square in any sized grid, to work out the difference of the answers, when the corners were multiplied. To prove that this equation is correct I am going to take a 3 x 3 square from an 8 x 8 grid, using the equation first. 1 2 3 9 10 11 17 18 19 Difference = 8( 3 - 1) � = 8 x 2� = 8 x 4 = 32 So if my equation is correct then the difference will always we 32 of a 3 x 3 square taken from an 8 x 8 grid. 1 2 3 9 10 11 17 18 19 1 x 19 = 19 3 x 17 = 51 Difference = 51 - 19 = 32 So the difference is 32 of a 3 x 3 square taken from an 8 x 8 grid, which proves that my equation was correct, and it can be used on any sized square in any sized grid, to work out the difference of the answers, when the two opposite corners are multiplied. Number Grids Coursework Saira Javed 10C 1 ...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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