40 40
80x94=7520
90x84=7560
40
2x6 Rectangle
Prediction
I predict that when I multiply a 2x6 rectangle the opposite corners will have a difference of 50.
52x67=2484 1x16=16
62x57=2534 11x6=66
50 50
77x92=7084
87x82=7134
50
Pattern Difference
2x2 10
2x3 20
2x4 30
2x5 40
2x6 50
A Graph to Show the Difference within the Rectangle 2n
For any size rectangle
N
m
2x3
X(X+12) (X+2) (X+10)
X² +12X X(X+10) +2(X+10)
X² +12X X² +10X+2X+20
X² +12X X² +12X+20
X² +12X
- X² +12X+20
20
This shows that for any 2x3 rectangle the difference will be 20.
I shall now move on and investigate a 3x3 Rectangle (square) and more rectangles in the same way.
3x3 Rectangle (square)
54x76=4104 71x93=6603
74x56=4144 91x73=6643
- 40
Prediction
I predict that when I
10x32=320 multiply a 3x3 30x12=360 rectangle (square) the
Opposite corners will
Have a difference of 40.
3x4 Rectangle
42x65=2730 17x40=680
62x45=2790 37x20=740
60 60
68x91=6188
88x91=6248
60
Again I have noticed a pattern occurring each time the width increases by 1 the difference increases by 20.
Prediction
Using the theory I predict that when I multiply a 3x4 rectangle the opposite corners will have a difference of 60.
3x5 Rectangle
Prediction
I predict that when I multiply a 3x5 rectangle the opposite corners will have a difference of 80.
1x25=25 73x97=7081
21x5=105 93x77=7161
- 80
40x64=2560
60x44=2640
80
3x6 Rectangle
Prediction
I predict that when I multiply a 3x6 rectangle the opposite corners will have a difference of 100.
18x43=774 61x86=5246
38x23=874 81x66=5346
- 100
8x33=264
28x13=364
100
Pattern Difference
3x2 20
3x3 40
3x4 60
3x5 80
3x6 100
A Graph to Show the Difference within the Rectangle 3n
3x5
X(X+24) (X+4) X+20)
X² +24X X(X+20) +4(X+20)
X² +24X X² +20X+4X+80
X² +24X X² +24X+80
X² +24X
-X² +24X+80
80
I shall now move on and investigate 4x4 Rectangle (square) and more Rectangles in the same way.
4x4 Rectangle (square)
56x89=4984 19x52=988
86x59=5074 49x22=1078
- 90
1x34=34
31x4=124
90
Prediction
I predict that when I multiply a 4x4 rectangle the opposite corners will have a difference of 90.
4x5 Rectangle
6x40=240 44x78=3432
36x10=360 74x48=3552
- 120
66x100=6600
96x70=6720
120
I have noticed a pattern occurring each time the width increases, by 30 by 1.
Prediction
Using the theory I predict that when I multiply a 4x5 rectangle the opposite corners will have a difference of 120.
4x6 Rectangle
Prediction
I predict that when I multiply a 4x6 rectangle the opposite corners will have a difference of 150.
64x99=6336 28x63=1764
94x69=6486 58x33=1914
150 150
32x67=2144
62x37=2294
150
Pattern Difference
4x4 90
4x5 120
4x6 150
A Graph to Show the Difference within the Rectangle 4n
4x6
X(X+35) (X+5) (X+30)
X² +35X X(X+30) +5(X+30)
X² +35X X² +30X+5X+150
X² +35X X² +35X+150
X² +35X
-X² +35X+150
150
From my results on the 2x….. 3x….. And the 4x…..I will now predict the difference in a 5x Rectangle.
Pattern Difference
5x2 40
5x3 80
5x4 120
5x5 160
5x6 200
I will now test this theory by choosing one to see if my prediction is correct.
5x6 Rectangle
1x46=46
41x6=246
200
Looking at this I can tell that my prediction is correct.
I will now predict the difference in a 8x Rectangle.
Pattern Difference
8x2 70
8x3 140
8x4 210
8x5 280
8x6 350
I will do the same as before and test the theory by choosing one to see if my prediction
8x3 Rectangle
66x93=6138
86x73=6278
140
Looking at this I can tell that my prediction is correct.
I have found the rule for any size rectangle. I think that I can write down the expression to show this, that does not require the actual figures for the length.
(X + L – 1 ) x (X + (G(H – 1))) = XxX + X(G + X(H - 1)) + L – 1 (G (H - 1))
X x (X + (G(H – 1 ))) + L – 1 ) = XxX + X(G+(G + X(H – 1)) + X L – 1
To Test This Expression
L-1 (G (H-1))
Lets substitute the value of X for 1 in a 2x2 Rectangle on a 10x10 grid and check that it calculates the square values correctly.
And now I will use the above expression to find the difference (which should be 10)
L-1 (G (H-1))
2-1 (10 (2-1))
1 x (10 x 1)
1 x 10 =10
I will now check it with a 3x4 Rectangle on a 10x10 grid. I know that the difference should be 60 from previous working outs.
L-1 (G (H-1))
4-1 x (10 (3-1))
3 x (10 (2))
3 x 20 = 60
L-1 (G (H-1)) is the universal rule with the brackets in different positions:
L-1(G (H-1)) = L-1 x (Grid Width) x (H-1) = Difference
Evaluation
I have completed this piece of coursework and succeeded in my aims I set out to do. I have found the rule to work out the difference for any Rectangle of any size on any grid. I have done this by progressive investigation and the use of algebraic methods.
I have taken the coursework as far as I can in the amount of time I had been allotted. I am happy with the amount of work that I have done on this bit of coursework and can’t think of anything I could of carried out better or improved upon.
