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Introduction

Number Grid Coursework My task is to investigate a 2x2 box on a 100 square I will take a 2x2 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference. Test 1 54 55 54 x 65= 3510 64 65 55 x 64= 3520 3520-3510= 10coff ff" . "r se" . ff . "ff" . "w or". ff . " " . ff . "k inff foff " . ff . "! DIFFERENCE = 10 Test 2 5 6 5 x 16= 80 15 16 6 x 15= 90 90-80= 10 DIFFERENCE = 10 Test 3 18 19 18 x 29= 522 28 29 19 x 28= 532 532-522= 10 DIFFERENCE = 10 Prediction I predict that in a two by two square the difference will always be 10 Proof 83 84 83 x 94= 7802coaf af" . "r se" . af . "af" . "w or". af . " " . af . "k inaf foaf " . af . ". 93 94 84 x 93= 7812 7812-7802= 10 DIFFERENCE = 10 Algebraic Explanation I will assign a letter to the first number in the 2 x 2 square, n. The next number to the right will therefore be n+1 The number directly below it will then be n+10 The number diagonally across from it will be n+11 I will then times the corners together, like In did on the above examples. Top Left hand corner x bottom right hand corner = n(n+11) = n� + 11n Top right hand corner x bottom left hand corner = n� +1n+10n+10 n� +11n+10 (n� +11n+10) - (n� + 11n) = 10 Therefore the difference between the corners multiplied together will always be 10. Expanding the Task I now feel it will be interesting to look at a 3x3 number square on a 100 grid. ...read more.

Middle

75 76 4088 - 4028 = 60 DIFFERENCE = 60 Test 2 67 68 69 70 67 x 90 = 6030 77 78 79 80 70 x 87 = 6090 87 88 89 90 6090 - 6030 = 60 DIFFERENCE = 60 Test 3 31 32 33 34 31 x 54 = 1674 41 42 43 44 34 x 51 = 1734 51 52 53 54 1734 - 1674 = 60 DIFFERENCE = 60 Prediction I predict that in a 3 x 4 rectangle the difference will always be 60 Algebra I will assign a letter to the first number in the 3x4 rectangle, n. The right hand top corner will therefore be n+3 The left hand bottom corner will then be n+20 The corner diagonally across from it will be n+23 I will then times the corners together, like I did on the above examples. Top Left hand corner x bottom right hand corner = n(n+23) = n� + 23n Top right hand corner x bottom left hand corner = (n+20)(n+3) = n�+60+20n+3n ` = n2+60+23n (n�+60+23n) - (n� + 23n) = 60 Therefore the difference between the corners multiplied together will always be 60. Expanding the Task I now feel it will be interesting to look at a 4x5 rectangle on a 100 grid. I will take a 4x5 rectangle on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference. 66 67 68 69 70 66 x 100 = 6660 76 77 78 79 80 70 x 96 = 6720 86 87 88 89 90 6720 - 6600 = 120 96 97 98 99 100 DIFFERENCE = 120 41 42 43 44 45 41 x 75 = 3075 51 52 53 54 55 45 x 71 = 3195 61 62 63 64 65 3195 - 3075 = 120 71 72 73 74 75 DIFFERENCE = 120 36 37 38 39 40 36 x 70 = 2520 sabkaurb, please do not redistribute this project. ...read more.

Conclusion

When n is the number in the top left hand corner, L is the length and w is the width. The top left hand corner is n, so therefore the top right hand corner will be n + the length of the rectangle minus 1 (as shown), this is because you move along the length of the box, but as the top left hand corner (n) takes up one column you minus 1. The bottom right hand corner of the rectangle will have plus n + 9 because as you move down vertically the number increases by 9 each time. This will be multiplied by the width of the rectangle minus 1, n+7(w-1). This is because you move down the width of the rectangle, however the top left hand corner (n) takes up 1 row so you minus 1. To find the bottom right hand corner we have to keep the n+9(w-1) of the left hand bottom corner and then add L-1 (see diagram below for visual explanation of the algebra). I will then multiply the corners out, as I do on my investigation. Left hand top corner x bottom right corner n x [n + 9 (w-1) + (L-1)] = n2 + 9n(w-1) + n(L-1) Right hand top corner x bottom left corner [n+(L-1)] [n+7(w-1)] = n2 + 9n(w-1) + n(L-1) + 9(L-1)(w-1) Find the difference between the two multiplied corners [n2 + 9n(w-1) + n(L-1) + 9(L-1)(w-1)] - [n2 + 7n(w-1) + n(L-1)] = 9(L-1)(w-1) I will now try this formula out with 3 examples on a 49 square, each with different sized rectangles, to prove whether it is correct 2x3 sized rectangle Formula 9(L-1)(w-1) = 9(3-1)(2-1) = 9 x 2 x 1 = 18 Therefore in theory the difference between the two corners multiplied together in a 2 x 3 rectangle on a 81 square grid will be 18. Example 37 38 39 37 x 48 = 1776 46 47 48 39 x 46 = 1794 1794 - 1776 = 18 DIFFERENCE = 18 The formula worked for this example 3x4 sized rectangle Formula 9(L-1)(w-1) = 9(3-1)(4-1) ...read more.

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