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Introduction

Maths Coursework Part 1 Introduction I am to conduct an investigation involving a number grid. The first part of the investigation will involve me: � Drawing a box around various random numbers on a number grid i.e. 2 x 2 numbers, 3 x 3 numbers, 4 x 4 numbers. � I will then find the product of the top left number and the bottom right number in the boxes by multiplying them. � I will then do the same with the top right and bottom left numbers. � I am then going to calculate the difference between these products 10x10 Grid � 2x2 Boxes Box 1 52 53 X X+1 62 63 X+10 X+11 [image001.gif] [image002.gif] 52 x 63 = 3276 x (x + 11) x2 + 11x [image003.gif] 62 x 53 = 3286 (x + 1) (x + 10) x2 + 11x + 10 = (x2 + 11x + 10) - (x2 + 11x) = 10 3286 - 3276 = 10 The difference between the two numbers is 10 Box 2 81 82 91 92 [image004.gif] 81 x 92 = 7452 91 x 82 = 7462 7462 - 7452 = 10 The difference between the two numbers is 10 Box 3 69 70 79 80 [image005.gif] 69 x 80 = 5520 79 x 70 = 5530 5530 - 5520 = 10 The difference between the two numbers is 10 � 3x3 Boxes Box 1 8 9 10 X X+1 X+2 18 19 20 X+10 X+11 X+12 28 29 30 X+20 X+21 X+22 [image006.gif] [image007.gif] 8 x 30 = 240 x (x + 22) x2 + 22x [image008.gif] 28 x 10 = 280 (x + 2) (x + 20) x2 + 20x + 2x + 40 = (x2 + 22x + 40) - (x2 + 22x) = 40 280 - 240 = 40 The difference between the two numbers is 40 Box 2 34 35 36 44 45 46 54 55 56 [image009.gif] 34 x 56 = 1904 54 x 36 = 1944 ...read more.

Middle

- (x2 + 23x) = 45 160 - 115 = 45 The difference between the two numbers is 45 Box 2 32 33 34 35 37 38 39 40 42 43 44 45 47 48 49 50 [image027.gif] 32 x 50 = 1600 47 x 35 = 1645 1645 - 1600 = 45 The difference between the two numbers is 45 Box 3 44 45 46 47 49 50 51 52 54 55 56 57 59 60 61 62 [image028.gif] 44 x 62 = 2728 59 x 47 = 2773 2773 - 2728 = 45 The difference between the two numbers is 45 Evaluation This formula is working with the same pattern as before using square numbers only instead of multiplying by 10 it is now being multiplied by 5 using the same size of grid. For example: When multiplying by 10 the pattern was going up by 20 [image029.gif] 2 x 2 = 10 (+ 30) [image030.gif] [image031.gif] 3 x 3 = 40 (+ 50) 4 x 4 = 90 (+ 70) 5 x 5 = 160 When multiplying by 5 the pattern was going up by 10 [image032.gif] 2 x 2 = 5 (+15) [image033.gif] [image034.gif] 3 x 3 = 20 (+25) 4 x 4 = 45 (+35) So if I was to do a 5 x 5 table the result would be: 5 x 5 = 80 (45 + 35) Part 2 For the second part of this investigation I plan to predict the difference between numbers in a different sized number grid. When doing the grid before which was a 10 x 10 grid the numbers turned out: 2 x 2 = 10 (+ 30) 3 x 3 = 40 (+ 50) 4 x 4 = 90 (+ 70) 5 x 5 = 160 The formula given from these numbers turned out to be: Square number x 10 i.e. 3 x 3 = 9 x 10 = 90 which was the 4 x 4 box difference between numbers. ...read more.

Conclusion

The pattern for the previous two is as follows: 3 x 2 = 20 (+ 40) 4 x 3 = 60 (+ 60) I predict the pattern is going up by 20 each time and that the 5 x 4 grid will be 5 x 4 = 120 � 5 x 4 boxes 1 2 3 4 5 X X+1 X+2 X+3 X+4 11 12 13 14 15 X+10 X+11 X+12 X+13 X+14 21 22 23 24 25 X+20 X+21 X+22 X+23 X+24 31 32 33 34 35 X+30 X+31 X+32 X+33 X+34 [image049.gif] [image015.gif] 1 x 35 = 35 x (x + 35) x2 + 35x [image001.gif] 31 x 5 = 155 (x + 4) (x + 31) x2 + 30x + x + 31 = (x2 + 35x + 31) - (x2 + 35x) = 120 155 - 35 = 120 The difference between the two numbers is 120 Box 2 13 14 15 16 17 23 24 25 26 27 33 34 35 36 37 43 44 45 46 47 [image050.gif] 13 x 47 = 611 43 x 17 = 731 731 - 611 = 120 The difference between the two numbers is 120 Box 3 35 36 37 38 39 45 46 47 48 49 55 56 57 58 59 65 66 67 68 69 [image051.gif] 35 x 69 = 2415 65 x 39 = 2535 2535 - 2415 = 120 The difference between the two numbers is 120 Now that I have completed the grid I can see that my prediction was correct and that whether you do a square or rectangle box the formula will always have a similar pattern. For example: Overall summarised table of results Grid size Box size Difference 10 x 10 2 x 2 10 3 x 3 40 4 x 4 90 5 x 5 160 5 x 5 2 x 2 5 3 x 3 20 4 x 4 45 7 x 7 2 x 2 7 3 x 3 28 4 x 4 63 10 x 10 3 x 2 20 4 x 3 60 5 x 4 120 ...read more.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.