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# 100 Number Grid

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Introduction

Name:                 Hazel Goodman

Coursework 1:   Number Grid

Number Grid

 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

Introduction

The aim of this piece of coursework is to investigate whether a pattern or sequence occurs when using the following rules:

• A box is drawn round four numbers using the above number grid.
• Find the product of the top left number and the bottom right number in this box.
• Do the same with the top right and bottom left numbers.
• Calculate the difference between these products.

To begin, an investigation using a 2 x 2 square will take place.  This will be followed by further investigations, using varied sizes of squares and rectangles.  Furthermore, a prediction of a formula for each number grid size will be included.

2 x 2 Square

1. 1 x 12 = 12                2 x 11 = 22                Product difference = 10
2. 22 x 33 = 726                23 x 32 = 736                Product difference = 10
3. 67 x 78 = 5226        68 x 77 = 5236        Product difference = 10
4. 89 x 100 = 8900        90 x 99 = 8910        Product difference = 10

The results indicate that for every 2 x 2 square, the product difference is 10.  Using this theory, I predict that every 2 x 2 square will have a product difference of 10.   I will continue this investigation to determine whether this prediction is correct.

1. 36 x 47 = 1692        37 x 46 = 1702        Product difference = 10
2. 49 x 60 = 2940        50 x 59 = 2950        Product difference = 10
3. 74 x 85 = 6290        75 x 84 = 6300        Product difference = 10
4. 85 x 96 = 8160        86 x 95 = 8170        Product difference = 10

By continuing this investigation, my prediction proves to be correct.  As a result, an algebraic formula can be used to work out the product difference. This is as follows:

 X X + 1 X + 10 X + 11

Step 1.                x (x + 11)

Step 2.                (x + 1)(x + 10)

Step 3.                (x2+ 11x + 10) – (x2- 11x)                Difference = 10

3 x 3 Square

1. 3 x 25 = 75                5 x 23 = 115                Product difference = 40
2. 14 x 36 = 504                16 x 34 = 544                Product difference = 40

Middle

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64

2 x 2 Square

1. 1 x 10 = 10                2 x 9 = 18                Product difference = 8
2. 3 x 12 = 36                4 x 11 = 44                Product difference = 8
3. 36 x 45 = 1620        37 x 44 = 1628        Product difference = 8

I predict that for any 2 x 2 square in an 8 x 8 number grid, the product difference will always be 8.  I will prove this theory by using an algebraic equation.

 X X + 1 X + 8 X + 9

Step 1.                x (x + 9)

Step 2.                (x + 1)(x + 8)

Step 3.                (x2+ 9x + 8) – (x2- 9x)                Difference = 8

3 x 3 Square

1. 1 x 19 = 19                3 x 17 = 51                Product difference = 32
2. 35 x 53 = 1855        37 x 51 = 1887         Product difference = 32
3. 46 x 64 = 2944        48 x 62 = 2976        Product difference = 32

I predict that for any 3 x 3 square in an 8 x 8 number grid, the product difference will always be 32.  I will prove this theory by using an algebraic equation.

 X X + 1 X + 2 X + 8 X + 9 X + 10 X + 16 X + 17 X + 18

Step 1.                x (x + 18)

Step 2.                (x + 2)(x + 16)

Step 3.                (x2+ 18x + 32) – (x2- 18x)                Difference = 32

4 x 4 Square

1. 4 x 31  = 124                7 x 28 = 196                Product difference = 72
2. 26 x 53 = 1378        29 x 50 = 1450        Product difference = 72
3. 37 x 64 = 2368        40 x 61 = 2440        Product difference = 72

I predict that for any 4 x 4 square in an 8 x 8 number grid, the product difference will always be 72.  I will prove this theory by using an algebraic equation.

 X X + 1 X + 2 X + 3 X + 8 X + 9 X + 10 X + 11 X + 16 X + 17 X + 18 X + 19 X + 24 X + 25 X + 26 X + 27

Step 1.                x (x + 27)

Step 2.                (x + 3)(x + 24)

Step 3.                (x2+ 27x + 72) – (x2- 27x)                Difference = 72

5 x 5 Square

1. 4 x 40  = 160                8 x 36 = 288                Product difference = 128
2. 11 x 47 = 517                15 x 43 = 645                Product difference = 128
3. 27 x 63 = 1701        31 x 59 = 1829        Product difference = 128

I predict that for any 5 x 5 square in an 8 x 8 number grid, the product difference will always be 128.  I will prove this theory by using an algebraic equation.

 X X + 1 X + 2 X + 3 X + 4 X + 8 X + 9 X + 10 X + 11

Conclusion

Further Investigation

The results from these investigations have provided me with working formulas for both the Nth Term and product differences of squares and rectangles in the 100 number grid.  I predict that there is a formula for finding the product difference for any size rectangle and number grid.  My investigations have shown that for every rectangle (L – 1) occurs, I now need to work out the remaining part of the rule.  By examining my previous investigations it shows that if I minus 1 from the width and multiply it by 10 it will provide me with the sequence difference.  I predict that the (x 10) rule is linked to the grid size of 10 x 10.  From this I can start building a formula using the following expressions:

Grid (G)

Length (L)

Width (W)

Differerence (D)

Therefore if (L-1) works, and so does (W-1) 10, I can compose a formula for any size rectangle:

Formula = (L-1) (W-1) G = D

Conclusion

The results from the 100 number grid investigation have proven that my predictions have been correct throughout this study.  I have found working formulas for finding the Nth Term (N) and the product difference (D) of both squares and rectangles in a 100 number grid.

To find (N) of a square = GN2

To find (D) of a rectangle = (L-1) (W-1) G

As well as the above formulas I have been consistent in finding algebraic equations for each square and rectangle in a number grid.

If I were to carry this investigation further, I would change the rules with regard to multiplication.  I would investigate using addition and subtraction or try using other shapes, such as a stair pattern.

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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