1*28 = 28
8*21 = 168
= 168-28
Product = 140
Therefore the product equals 140
31*62 = 1922
32*61 = 1952
= 1952-1922
Product = 30
1*32 = 32
2*31 = 62
= 62-32
Product =30
Therefore the product equals 30
65*99 = 6435
69*95 = 6555
= 6555-6435
Product =120
2*36 = 72
6*32 = 192
= 192-72
Product = 120
Therefore the product equals 120
61*96 = 5856
66*91 = 6006
= 6006-5856
Product = 150
54*80 = 3600
50*75 = 3750
= 370-3600
Product = 150
Therefore the product Equals 150
1*38 = 38
8*31 = 248
= 248-38
Product = 210
32*60 = 1380
30*53 = 1590
= 1590-1380
Product = 210
Therefore the product equals 210
22*64 = 1408
24*62 = 1488
= 1488-1408
Product = 80
57*99 = 5643
59*97 = 5723
= 5723-5643
Product = 80
Therefore the product equals 80
1*44 = 44
4*41 = 164
= 164- 44
Product = 120
53*96 = 5088
56*93 = 5208
= 5208-5088
Product = 120
Therefore the product equals 120
1*46 = 46
6*41 = 246
= 246-46
Product = 200
33*78 = 2574
38*73 = 2774
= 2774-2574
Product = 200
Therefore the product equals
2*50 =100
10*42 = 420
= 420-100
Product = 320
52*100 = 5200
60*92 = 5520
= 5520-5200
Product = 320
Therefore the product equals 320
1*53 = 53
3*51 = 153
= 153-53
Product = 100
48*100 = 4800
50*98 = 4900
= 4900-4800
Product = 100
Therefore the product equals 100
1*54 = 54
4*51 = 204
= 204-54
Product = 150
13*66 = 858
16*63 = 1008
= 1008-858
Product = 150
Therefore the product equals 150
1*57 = 57
7*51 = 357
= 357- 57
Product = 300
32*98 = 3136
38*92 = 3496
= 3496-3136
Product = 300
Therefore the product equals 300
1*59 = 59
9*51 = 459
= 459-59
Product = 400
2*60 = 120
10*52 = 520
= 520-120
Product = 400
Therefore the product equals 400
3*63 = 192
4*63 = 252
= 252- 192
Product = 60
15*76 = 1140
16*75 = 1200
= 1200-1140
Product = 60
Therefore the product equals 60
22*84 = 1848
24*82 = 1968
= 1968-1848
Product = 120
1*63 = 63
3*61 = 183
= 183-63
Product = 120
Therefore the product equals 120
1*65 = 65
5*65 = 305
= 305-65
Product = 240
14*78 = 1092
18*74 = 1332
= 1332-1092
Product = 240
Therefore the product equals 240
2*69 = 138
9*62 = 558
= 558-138
Product = 420
33*100 = 330
40*93 = 3720
= 3720-3300
Product = 420
Therefore the product equals 420
1*72 = 72
2*71 = 142
= 142-72
Product = 70
19*90 = 1710
20*89 = 1780
= 1780- 1710
Product = 70
Therefore the product equals 70
2*74 = 148
4*72 = 288
= 288-148
Product = 140
25*97 = 2425
27*95 = 2565
= 2565-2425
Product = 140
Therefore the product equals 140
1*74 = 74
4*71 = 284
= 284-71
Product = 210
26*99 = 2574
29*96 = 2784
= 2784-2574
Product = 210
Therefore the product equals 210
2*78 = 156
8*72 = 576
= 576-156
Product = 420
13*89 = 1157
19*83 = 1577
= 1577-1157
Product = 420
Therefore the product equals 420
19*100 = 1900
20*99 = 1980
= 1980-1900
Product = 80
1*82 = 82
2*81 = 162
= 162-82
Product = 80
Therefore the product equals 80
13*96 = 1248
16*93 = 1488
= 1488-1248
Product = 240
16*99 = 1584
19*66 = 1824
= 1824-1584
Product = 240
Therefore the product equals 240
2*89 = 178
9*82 = 738
= 738-178
Product = 560
13*100 = 1300
20*93 = 1860
= 1860-1300
Product = 560
Therefore the product equals 560.
2.2 Identifying a pattern
In this section I will be identifying the patterns which from the solutions of the previous sections. One of the key patterns in the previous section we have identified, is that regardless of the values of a certain rectangle the product will always be the same, for example a 2*3 rectangle will have the same product regardless of the values within, this is shown in the previous section.
These are a summary of the results from the previous section, as we can see the values are increasing by each column.
Fig 1
Increasing by 10 ------------------
Increasing by 20-------------------
Increasing by 30 ------------------
Increasing by 40 ------------------
Increasing by 50-------------------
Increasing by 60 ------------------
Increasing by 70 ------------------
Increasing by 80 ------------------
Increasing by 90 ------------------
The values in red are the ones that I have predicted. I came up with these predictions due to following the pattern. I am now going to prove my predicted answers are correct in the same format, as we have done previously in section 2.1.
1*33 = 33
3*31 = 93
= 93-33
Product = 60
58*90 = 5220
60*88 = 5280
= 5280-5220
Product = 60
Therefore the product equals 60, as predicted on figure 1
1*58 = 58
8*51 = 408
408-58 = 350
42*99 = 4158
49*92 = 4508
= 4508-4158
Product = 350
Therefore the product equals 350, as predicted on figure 1.
- These examples prove my predictions are correct.
Y
Figure 2.
X
The reason I have labelled the grid x and y is because it suggests both sides are not equal. This will be explained in the next section.
2.3 Generating a formula from the previous section we have identified patterns occurring
A rectangle is defined by x*y where x does not equal ( ) y. In order to obtain the incremental value we multiply the x by 10 and subtracting 10 away. Therefore the equation is:
(X*10)-10 } Can be written as 10x-10
e.g. (3*10)-10
=20
Therefore from figure 1 and 2, by looking at our chosen x value (3) we can tell that the row is increasing by 20
If we multiply the previous equation we have generated by y and subtract the incremental value as given in the previous equation we can workout any product of any size rectangle. The full equation now is:
((X*10)-10)*Y) – ((X*10)-10)
We are going to prove this equation with different size rectangles to workout the product. I will check the product with a table (figure 2)
Rectangle = x*y
= 2*3
((X*10)-10)*Y)) – ((X*10)
((2*10)-10)*3) – ((2*10)-10)
(20-10)*3) – (20-16)
(10*3)-10
30-10
= 20
This product answers match these of figures 1 and 2. So I have found out a formula to answer any square in the grid
2.4 Further Maths
In this section I will introduce mathematical terms, which have not been taught, within our syllabus. The notation we are going to introduce is:
This will tell us the parameters and conditions, which we have set in this equation such as, the definition of a rectangle. The symbol actually denotes the sum of the equation.
Therefore, the final equation, with our new mathematical notation is:
(X, Y)
((X*10)-10)*Y) – ((X*10)-10)
I = 2
X does not = Y
3.0 Conclusion
Firstly I started of by finding out a pattern that was occurring. Then in section 2.3, I expanded on the pattern that was occurring, by generating a formula from the patterns I had identified. I had then proven my formulae by giving an example. To further heighten my marks, I researched into textbooks to expand on the formulae, which I have not been taught in this syllabus. From this I have learnt to further expand on my equations using the notation and I am confident this was a successful Investigation.
