# Number Grid Investigation.

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Introduction

Number Grid Investigation Introduction: The coursework task is to investigate the patterns generated from using rules in a square grid. The grid provides a structured approach to learning number relationships. 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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 In accordance with the task there are four steps to follow: 1. A box is drawn round four numbers [see grid above]. 2. Find the product of the top left number and the bottom right number. 3. Do the same with the top right number and the bottom left number. 4. Calculate the difference between these numbers. From the example in the above grid: [45 x 56 = 2520 - 46 x 55 = 2530] I find that the product difference for the diagonal square [2x2] is 10. I am going to start by investigating as to whether or not the location of the 2x2 square on the grid is significant. A. 81 x 92 = 7452 - 82 x 91 = 7462. Product difference is 10. B. 9 x 20 = 180 - 10 x 19 = 190. Product difference is 10. C. 12 x 23 = 276 - 13 x 22 = 286. Product difference is 10. From the worked examples A, B & C I find that the product difference is 10. Taking these results into account, I predict for any 2x2 square the result will always be 10. ...read more.

Middle

Once again this is the size of the grid I am working with, and also it is the constant difference from a 2x2 square. 3x3 Results A. ( 7x 21 ) - ( 5x 23 ) = 32 B. ( 20x 34) - ( 18x 36) = 32 C. ( 40x 54 ) - ( 38x 56) = 32 Algebraic table for 3x3: N N+2 N+16 N+18 Equation: (N + 2) ( N + 16) - N ( N + 18) N� +2N +16N +32 - N� + 18N + 18N + 32 - 18N = 32 4x4 Results A. ( 4x 25 ) - ( 1x 28 ) = 72 B. ( 24x 45) - ( 21x 48 ) = 72 C. ( 36x 57 ) - ( 33x 60) = 72 Algebraic table for 4x4: N N+3 N+24 N+ 27 Equation: ( N + 3 ) ( N + 24) - N ( N + 27) N� + 3N + 24N + 72 - N� + 27N +27N + 72 - 27N = 72 Table 4. Results for 8x8 grid. Square selection size Product difference Difference between each difference p.d. Increase 2 x 2 8 3 x 3 32 24 16 4 x 4 72 40 16 5 x 5 128 56 16 6 x 6 200 72 16 7 x 7 288 88 16 By investigating further and looking at my results for T.4. I have established there is a formula for an nth term for any square selection size within an 8x8 grid. 1st 2nd 3rd 4th (2x2) (3x3) (4x4) (5x5) 8 32 72 128 24 40 56 16 16 Nth term Product difference p.d. Increase The formula for finding any term in any size from the 8x8 grid is 8n� E.g. * 2nd term - 8 x 2� = 32 (correct, see T.4. 3x3) * 4th term - 8 x 4� = 128 (correct, see T.4. ...read more.

Conclusion

3 x D = 20n + 20 I now have a formula for a 3 x D rectangle. E.g. 3 x 7 - 5th term in the sequence - 20 x 5 + 20 = 120. 7 x 21 - 1 x 27 = 120. My formula of 20n + 20 is correct. From this I can begin to see the product differences are increasing by 10`s, and also the formulas are going up by 10`s. I can predict that the formula for finding a 4 x? rectangle will be 30 n + 30, but I will have to test this to see if this is true. A. 4x 3 = 60 B. 4 x 4 = 90 C. 4 x 5 = 120 My prediction was correct. You need to multiply the nth term by 30 this time, and then add 30 which is the difference between each difference. I can now construct a formula for a 4 x D rectangle: 4 x D = 30 n + 30. E.g. 4 x 6 - 4th term in the sequence - 30 x 4 + 30 = 150 6 x 31 - 1 x 36 = 150 M y formula of 30n + 30 is correct. These workings out and formulas give me a rule for any sized rectangle, providing I know the width. Possibly there could be a rule for find any width size rectangle with an unknown sized length? Conclusion In this project I have found that number grids are an extremely powerful tool for a wide range of maths concepts such as, number patterns, problem solving and investigation. I have successfully predicted what can come next in square selection sized differences, nth terms and algebraic equations that simplify expressions. There are many different rules formulas within grid sizes and square or rectangle selections. If I were to extend this project I would possibly change the original rules with regards to diagonal multiplication and subtraction, to see if there are any different patterns, or I would change the grid sizes and rectangle sizes or perhaps try different shapes. ...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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