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Introduction

NUMBER GRID COURSEWORK

10 X 10 NUMBER GRID

2 x 2 BOX

1213
2223

22 x 13 = 286
12 x 23 = 276
286 – 276 =
10

8990
99100

99 x 90 = 8910
89 x 100 = 8900
8910 – 8900 =
10

7273
8283

82 x 73 = 5986
83 x 72 = 5976
5986 – 5976 =
10

6667
7677

76 x 67 = 5092
77 x 66 = 5082

5092 – 5082 = 10

I have noticed that the outcome is always 10.

2 x 2 BOX ALGEBRA

aa+1
a+10a+11

(a + 10)(a + 1) = a² + a + 10a + 10 = a² + 11a + 10
a(a + 11) = a² + 11a
(a² + 11a + 10) – (a² + 11a) =
10

3 x 3 BOX

68        69        70
78        79        80
88        89        90

88 x 70 = 6160
90 x 68 = 6120
6160 – 6120 =
40

18        19        20
28        29        30
38        39        40

38 x 20 = 760
18 x 40 = 720
760 – 720 =
40

3 x 3 BOX ALGEBRA

a        a+1        a+2
a+10        a+11        a+12
a+20        a+21        a+22

(a + 20)(a + 2) = a² + 2a + 20a + 40 = a² + 22a + 40
a(a + 22) = a² + 22a
(a² + 22a + 40) – (a² + 22a) =
40

4 X 4 BOX

31        32        33        34
41        42        43        44
51        52        53        54
61        62        63        64

61 x 34 = 2074
64 x 31 = 1984
2074 – 1984 =
90

1        2        3        4
11        12        13        14
21        22        23        24
31        32        33        34

31 x 4 = 124
34 x 1 = 34
124 – 34 =
90

4 X 4 BOX ALGEBRA

a        a+1        a+2        a+3
a+10        a+11        a+12        a+13
a+20        a+21        a+22        a+23
a+30        a+31        a+32        a+33

(a + 30)(a + 3) = a² + 3a + 30a + 90 = a² + 33a + 90
a(a + 33) = a² + 33a
(a² + 33a + 90) – (a² + 33a) =
90

5 X 5 BOX

56        57        58        59        60
66        67        68        69        70
76        77        78        79        80
86        87        88        89        90
96        97        98        99        100

96 x 60 = 5760
100 x 56 = 5600
5760 – 5600 =
160

6        7        8        9        10
16        17        18        19        20
26        27        28        29        30
36        37        38        39        40
46        47        48        49        50

46 x 10 = 460
50 x 6 = 300
460 – 300 =
160

10 X 10 NUMBER GRID RESULTS SO FAR

 BOX SIZE RESULT 2 x 2 10 3 x 3 40 4 x 4 90 5 x 5 160

I’ve spotted a pattern. I predict that the result for the 6 x 6 box will be 250. This is because each time the box size increases the result goes up 20 more than it went up last time.

6 X 6 BOX

1        2        3        4        5        6
11        12        13        14        15        16
21        22        23        24        25        26
31        32        33        34        35        36
41        42        43        44        45        46
51        52        53        54        55        56

51 x 6 = 306

Middle

I have noticed a relationship within my results. The result, which is after a result, is the box sizes squared number, multiplied by 10(as shown colour co-ordinated within the final result table). Therefore, the formula for a 10 x 10 number grid with multiples of 1 and heights and widths of n is…

(n – 1)(n – 1) x 10

To further my investigation, I’m going to change the shape, which contains the numbers being multiplied. I am going to use a rectangle, which has a width of n and a height of 2.

10 X 10 NUMBER GRID 2

2 X 3 BOX

51        52        53
61        62        63

61 x 53 = 3233
63 x 51 = 3213
3233 – 3213 =
20

32        33        34
42        43        44

42 x 34 = 1428
44 x 32 = 1408
1428 – 1408 =
20

2 X 3 BOX ALGEBRA

a        a+1        a+2
a+10        a+11        a+12

(a + 10)(a + 2) = a² + 10a + 2a + 20 = a² + 12a + 20
a(a + 12) = a²  + 12a
(a² + 12 a + 20) – (a² + 12a) =
20

2 X 4 BOX

1        2        3        4

11        12        13        14

11 x 4 = 44

14 x 1 = 14
44 – 14 =
30

11
12        13        14
21        22        23        24

21 x 14 = 294
24 x 11 = 264
294 – 264 =
30

2 X 4 BOX ALGEBRA

a        a+1        a+2        a+3
a+10        a+11        a+12        a+13

(a + 10)(a + 3) = a² + 10a + 3a + 30 = a² + 13a + 30
a(a + 13) = a² + 13a
(a² + 13a + 30) – (a² + 13a) =
30

2 X 5 BOX

16        17        18        19        20
26        27        28        29        30

26 x 20 = 520
30 x 16 = 480
520 – 480 =
40

31        32        33        34        35
41        42        43        44        45

41 x 35 = 1435
45 x 31 = 1395
1435 – 1395 =
40

2 X 5 BOX ALGEBRA

a        a+1        a+2        a+3        a+4
a+10        a+11        a+12        a+13        a+14

(a + 10)(a + 4)

Conclusion

6 x 6 box will be 200 because…

(6 – 1)(6 – 1) x 8 = 5 x 5 = 25 x 8 =
200

6 X 6 BOX

1        2        3        4        5        6
9        10        11        12        13        14
17        18        19        20        21        22
25        26        27        28        29        30
33        34        36        37        38        39
41        42        43        44        45        46

41 x 6 = 246
1 x 46 = 46
246 – 46 =
200

11        12        13        14        15        16
19        20        21        22        23        24
27        28        29        30        31        32
35        36        37        38        39        40
43        44        45        46        47        48
51        52        53        54        55        56

16 x 51 = 816
56 x 11 = 616
816 – 616 =
200

My prediction was correct. However I will make sure by predicting the
7x7 box. I predict that the result be will 288 because…

(7 – 1)(7 – 1) = 6 x 6 = 36 x 8 =
288

7 X 7 BOX

1        2        3        4        5        6        7
9        10        11        12        13        14        15
17        18        19        20        21        22        23
25        26        27        28        29        30        31
33        34        35        36        37        38        39
41        42        43        44        45        46        47
49        50        51        52        53        54        55

7 x 49 = 343
55 x 1 = 55
343 – 55 =
288

10        11        12        13        14        15        16
18        19        20        21        22        23        24
26        27        28        29        30        31        32
34        35        36        37        38        39        40
42        43        44        45        46        47        48
50        51        52        53        54        55        56
58        59        60        61        62        63        64

16 x 58 = 928
10 x 64 = 640
928 – 640 =
288

My second prediction was also correct.

8 X 8 NUMBER GRID RESULTS
 BOX SIZE RESULT 2 x 2 8 3 x 3 32 4 x 4 72 5 x 5 128 6 x 6 200 7 x 7 288

8 X 8 NUMBER GRID CONCLUSION

The formula for an 8x8 number grid with multiples of 1 and heights and widths of n is…

(n – 1)(n – 1) x 8

OVERALL NUMBER GRID COURSEWORK CONCLUSION

My 4 formulas are:

10 x 10 grid, multiples of 1, width of n, height of n = (n – 1)(n – 1) x 10
10 x 10 grid, multiples of 1, width of n, height of 2 = (h – 1)(w – 1) x 10
10 x 10 grid, multiples of 2, width of n, height of 2 = (l – 1)(w – 1) x 2² x 10
8 x 8 grid, multiples of 1, width of n, height of n =  (n – 1)(n – 1) x 8

I have found and recognised the similarities in each formula to product my general formula for a number grid with any number sizes grid, any multiple and any width and height.

(l – 1)(w – 1) x m2 x g

l = length
w = width
m = multiple
g = grid size
n = nth term

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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