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Number Grid.

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Introduction

GCSE Mathematics Coursework Number Grid Prepared by: Scott Bayfield This investigation relates to the product differences in different sized areas within a 10 x 10 number grid. For example: A 2 x 2 grid has been selected inside of the above grid. The objective of the investigation is to find the product of the bottom right and the top left numbers in the selected area. To calculate the product I must: Multiply the number in the bottom right cell with the number in the top left cell. The product difference is 10. This figure remains the same no matter where I carry out the above sum when used on a 2 x 2 grid within a 10x10 matrix. For example: Example 1 (75 x 86) - (76 x 85) = 10 Example 2 (1 x 12) - (2 x 11) = 10 Example 3 (17 x 28) - (18 x 27) = 10 These examples prove that the answer remains the same regardless of where we complete the sum with in the grid. ...read more.

Middle

term formula. The formula that I found effective was this: Total grid size x NTh term 2 = product difference i.e. NTh. term 1st 2nd 3rd 4th 5th Grid size 2 x 2 3 x 3 4 x 4 5 x 5 6 x 6 NTh. term calculation 10n2 Product difference 10 40 90 160 250 Note that the NTh. term is always 1 less than the grid size To see if this formula would work with a different size number grid I used the same formula as before only this time worked from a 10 x 8 matrix. Within the grid I tried the same variations as above i.e. 2 x 2, 3 x 3 etc. and discovered that the results remained exactly the same. I continued to investigation further, only this time I experimented using the 8 x 8 grid that can be seen below: When calculating the product from the above selection I encountered different results than found using the 10 x 10 matrix. ...read more.

Conclusion

I experimented with three individual 3 x 2 grids to prove my results remained constant. Example 1 (12 x 24) - (14 x 22) = 20 Example 2 (44 x 56) - (46 x 54) = 20 Example 3 (88 x 100) - (90 x 98) = 20 I found that the product differences remained the same however, in an effort to once again, further the investigation I carried out several more individual tests, increasing the size of the internal grid on each occasion. My results were as follows: Grid Size Product Difference 3 x 2 20 4 x 3 60 5 x 4 120 6 x 5 200 7 x 6 300 And again, I worked out a formula to find the product difference. Formula: Grid size x internal grid length -1 x internal grid height -1 or: 10(L-1)(H-1) NTh. term 1st 2nd 3rd 4th 5th Grid size 3 x 2 4 x 3 5 x 4 6 x 5 7 x 6 NTh. term calculation 10(L-1)(H-1) Product difference 8 32 72 128 200 Note that the NTh. term is always 1 less than the grid height ...read more.

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