100 Number Grid

Step 1.                x (x + 44)

Step 2.                (x + 4)(x + 40)          

Step 3.                (x2 + 44x + 160) – (x2 - 44x)                Difference = 160

Sequence Prediction

The results from each square investigation show a sequence forming.  I will demonstrate this in the following table:

I will continue this investigation by finding out the product difference of a 6 x 6 square along with a 7 x 7 square.  I predict that the product difference of a 6 x 6 square will be 250, and that a 7 x 7 square will be 360.  

6 x 6 Square

1. 1 x 56 = 56                6 x 51 = 306                Product difference = 250
2. 33 x 88 = 2904        38 x 83 = 3154        Product difference = 250

Algebraic Equation = (x2 + 66x + 250) – (x2 – 66x)        Difference = 250

7 x 7 Square

1. 4 x 70 = 280                10 x 64 = 640                Product difference = 360
2. 32 x 98 = 3136        38 x 92 = 3496        Product difference = 360

Algebraic Equation = (x2 + 77x + 360) – (x2 – 77x)        Difference = 360

The calculations show that my prediction was correct. Using this sequence pattern, I will calculate the 8th and 9th term of the number grid:

Looking closely at this table, I can see a formula for finding the nth term (n).  Multiplying 10 by n2 will provide the product difference for any nth term in a 10 x 10 number grid.

Nth Term = 10n2.

3rd Term:        10 x 32 = 90

6th Term:        10 x 62 = 360

9th Term:        10 x 92 = 810

Further Investigation

The investigation of various sized squares in a 10 x 10 number grid has provided me with an algebraic and nth term formula.  I am going to investigate further using different sized number grids, following the same rules of the 10 x 10 number grid, for example: 8 x 8 and 9 x 9 number grids.  My prediction is that by multiplying the nth term by the size of the grid, the product difference will be calculated.  For example:  finding the 3rd term in a 8 x 8 grid, Formula = 8n2, and finding the 5th term in a 9 x 9 grid, Formula = 9n2.

8 x 8 Number Grid

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.

 

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.

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.

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.

Step 1.                x (x + 36)

Step 2.                (x + 4)(x + 32)          

Step 3.                (x2 + 36x + 128) – (x2 - 36x)                Difference = 128

Sequence Prediction

The results from each square investigation show a sequence forming.  I will demonstrate this in the following table:

Looking closely at this table, I can see a formula for finding the nth term (n).  Multiplying 8 by n2 will provide the product difference for any nth term in an 8 x 8 number grid.

Nth Term = 8n2.

3rd Term:        8 x 32 = 72

6th Term:        8 x 62 = 288

7th Term:        8 x 92 = 810

9 x 9 Number Grid

2 x 2 Square

1. 1 x 11 = 11                2 x 10 = 20                Product difference = 9
2. 24 x 34 = 816                25 x 33 = 825                Product difference = 9
3. 47 x 57 = 2679        48 x 56 = 2688        Product difference = 9

I predict that for any 2 x 2 square in a 9 x 9 number grid, the product difference will always be 9.  I will prove this theory by using an algebraic equation.

 

Step 1.                x (x + 10)

Step 2.                (x + 1)(x + 9)          

Step 3.                (x2 + 10x + 9) – (x2 - 10x)                Difference = 9

3 x 3 Square

1. 1 x 21 = 21                3 x 19 = 57                Product difference = 36
2. 29 x 49 = 1421        31 x 47 = 1457         Product difference = 36
3. 51 x 71 = 3621        53 x 69 = 3657        Product difference = 36

I predict that for any 3 x 3 square in a 9 x 9 number grid, the product difference will always be 36.  I will prove this theory by using an algebraic equation.

Step 1.                x (x + 20)

Step 2.                (x + 2)(x + 18)          

Step 3.                (x2 + 20x + 36) – (x2 - 20x)                Difference = 36

4 x 4 Square

1. 4 x 34  = 136                7 x 31 = 217                Product difference = 81
2. 24 x 54 = 1296        27 x 51 = 1377        Product difference = 81
3. 37 x 67 = 2479        40 x 64 = 2560        Product difference = 81

I predict that for any 4 x 4 square in a 9 x 9 number grid, the product difference will always be 81.  I will prove this theory by using an algebraic equation.

Step 1.                x (x + 30)

Step 2.                (x + 3)(x + 27)          

Step 3.                (x2 + 30x + 81) – (x2 - 30x)                Difference = 81

5 x 5 Square

1. 4 x 44  = 176                8 x 40 = 320                Product difference = 144
2. 11 x 51= 561                15 x 47 = 705                Product difference = 144
3. 39 x 79 = 3081        43 x 75 = 3225        Product difference = 144

I predict that for any 5 x 5 square in a 9 x 9 number grid, the product difference will always be 144.  I will prove this theory by using an algebraic equation.

Step 1.                x (x + 40)

Step 2.                (x + 4)(x + 36)          

Step 3.                (x2 + 40x + 144) – (x2 - 40x)                Difference = 144

Sequence Prediction

The results from each square investigation show a sequence forming.  I will demonstrate this in the following table:

Looking closely at this table, I can see a formula for finding the nth term (n).  Multiplying 9 by n2 will provide the product difference for any nth term in a 9 x 9 number grid.

Nth Term = 9n2.

3rd Term:        9 x 32 = 81

5th Term:        9 x 52 = 225

8th Term:        9 x 82 = 810

Summary

The results from this investigation show that for any size number grid there is a working formula to find out the product difference of any sized square.  I conclude that the formula for finding out the product difference in a 10 x 10 square is 10n2, a 9 x 9 square is 9n2 and an 8 x 8 square is 8n2.  These results suggest that for any size number grid, a formula for finding out the product difference will be the size of the grid, multiplied by the nth term/square size and then squared.  For example:

Grid size =                         G

Nth Term/Square Size =        N

Formula =                         GN2

Further Investigation

I am going to carry on with this investigation, using the same rules as before but instead of examining squares, I will examine rectangles.  I will begin this investigation using a 10 x 10 number grid.

2 x 3 Rectangle

1. 1 x 22 = 22                2 x 21 = 42                Product Difference = 20
2. 13 x 34 = 442                14 x 33 = 462                Product Difference = 20
3. 48 x 69 = 3312        49 x 68 = 3332        Product Difference = 20

2 x 4 Rectangle

1. 2 x 33 = 66                3 x 32 = 96                Product Difference = 30
2. 33 x 64 = 2112        34 x 63 = 2142        Product Difference = 30
3. 69 x 100 = 6900        70 x 99 = 6930        Product Difference = 30

2 x 5 Rectangle

1. 4 x 45 = 180                5 x 44 = 220                Product Difference = 40
2. 46 x 87 = 4002        47 x 86 = 4042        Product Difference = 40
3. 51 x 92 = 4692        52 x 91 = 4732        Product Difference = 40

These results suggest that for every rectangle in the 2 x X series the product difference will increase in size by 10.  I will test this theory by investigating two more rectangle sizes of 2 x 6 and 2 x 7.  I predict that the product difference of a rectangle size of 2 x 6 will be 50 and that a 2 x 7 rectangle will be 60.

2 x 6 Rectangle

1. 2 x 53 = 106                3 x 52 = 156                Product difference = 50
2. 35 x 86 = 3010        36 x 85 = 3060        Product difference = 50

2 x 7 Rectangle

1. 16 x 77 = 1232        17 x 76 = 1292        Product difference = 60
2. 39 x 100 = 3900        40 x 99 = 3960        Product difference = 60

My prediction is correct. This provides me with two working formulas for working out the nth term (N) and product difference of any 2 x X rectangle in a 10 x 10 number grid.  The formula for working out (N) is:

(N + 1) 10

3rd Term =         (3 + 1) x 10 = 40

5th Term =         (5 + 1) x 10) = 60

The formula for working out the product difference is:

Length (L) (L – 1) 10

2 x 5 Rectangle = (5 – 1) x 10 = 40

2 x 8 Rectangle = (8 – 1) x 10 = 70

I will now investigate this further, using rectangles of various sizes.  For example: 3 x X series and 4 x X series.

3 x 4 Rectangle

1. 3 x 35 = 105                5 x 33 = 165                Product Difference = 60
2. 15 x 47 = 705                17 x 45 = 765                Product Difference = 60
3. 57 x 89 = 5073        59 x 87 = 5133        Product Difference = 60

3 x 5 Rectangle

1. 4 x 46 = 184                6 x 44 = 264                 Product Difference = 80
2. 42 x 84 = 3528        44 x 82 = 3608        Product Difference = 80
3. 58 x 100 = 5800        60 x 98 = 5880        Product Difference = 80

3 x 6 Rectangle

1. 4 x 56 =224                6 x 54 = 324                Product Difference = 100
2. 44 x 96 = 4224        46 x 94 = 4324        Product Difference = 100
3. 27 x 79 = 2133        29 x 77 = 2233        Product Difference = 100

These results suggest that for every rectangle in the 3 x X series the product difference will increase in size by 20.  I will prove this theory firstly by showing a table with further results, followed by a sequence formula.

My prediction is correct. This provides me a working formula for working out the nth term (N) and the product difference of any 3 x X rectangle in a 10 x 10 number grid. The formula for working out (N) is:

(N + 1) 20 + 20

3rd Term =         (3 + 1) x 20 + 20 = 100

5th Term =         (5 + 1) x 20 + 20 = 140

The formula for working out the product difference is:

Length (L) (L – 1) 20

3 x 6 Rectangle = (6 – 1) x 20 = 100

3 x 9 Rectangle = (9 – 1) x 20 = 160

By examining these results I predict that in the 4 x X series the product difference will be 30.  

My prediction is correct.  I will prove this by showing calculations of these examples.

4 x 5 Rectangle

1. 1 x 44 = 44                4 x 41 = 164                Product Difference = 120

4 x 6 Rectangle

1. 3 x 56 = 168                6 x 53 = 318                Product Difference = 150

4 x 7 Rectangle

1. 34 x 97 = 3298        37 x 94 = 3478        Product Difference = 180

4 x 8 Rectangle

1. 26 x 99 = 2574        29 x 96 = 2784        Product Difference = 210

These results provide me with a formula for working out (N) in this series:

(N + 1) 30 + 30 +30

4th Term =         (4 + 1) x 30 + 30 + 30 = 210

6th Term =         (6 + 1) x 30 + 30 + 30 = 270

The formula for working out the product difference for any rectangles in this series will be:

 (L – 1) 30

4 x 6 Rectangle = (6 – 1) x 30 = 150

4 x 9 Rectangle = (9 – 1) x 30 = 240

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.