This rectangle has a differentce of 40. from all these we can see that my prediction was right.
I will now do a formula for this:
For rectangles with 2 rows high and 3 columns wide the difference is always 20
For rectangles with 2 rows high and 4 columns wide the difference is always 30
For rectangles with 2 rows high and 5 columns wide the difference is always 40
So the number of columns wide – 1 × 10 gives the answer
So 10 × (Width – 1) is the formula
Or 10 × (W – 1)
Now I am going to do a rectangle this is 3 squares down. My prediction is that the number will be ×5 e.g. 45, 35 or 25. the first rectangle I will do is 3 2.
9a 11 12 13
21 22 23
31 32 33
b. 11 × 33 = 363
3 × 13 = 403
c. 403 – 363 = 40
This rectangle has a differentce of 40. As you can see, I did not get my prediction right. I do not see any pattern yet but this was only my first rectangle using 3 down. I will now do another 3 × 3 rectangle.
10a. 77 78 79
87 88 89
97 98 99
b. 77 × 99 = 7623
97 × 79 = 7663
c. 7663 – 7623 = 40
This rectangle also has a differentce of 40. This is the second last rectangle I will do with this size, the next size I will do will be 3×4.
11a. 34 35 36 37
44 45 46 47
54 55 56 57
b. 34 × 57 = 1938
54 × 37 = 1998
c. 1998 – 1938 = 60
This rectangle has a differentce of 60. I will do one more rectangle this size before I change size.
12a. 15 16 17 18
25 26 27 28
35 36 37 38
b. 15 × 38 = 570
35 × 18 = 630
c. 630 – 570 = 60.
This rectangle also has a differentce of 60. From this rectangle I have made a new prediction that they will go up in 20’s. So the next size I do will be 3 × 5 and should be 80.
13a. 5 6 7 8 9
- 16 17 18 19
25 26 27 28 29
b. 5 × 29 = 145
25 × 9 = 225
c. 225 – 145 = 80
This rectangle has a different of 80. this means that my partition was rite that they do go up in 20’s. and to further rove this point I will do one more size of 3 down, which will be 6 across (3×6). But first I will do another rectangle with the size of
3 × 5.
14a. 74 75 76 77 78
84 85 86 87 88
94 95 96 97 98
b. 74 × 98 = 7252
94 × 78 = 7332
c. 7332 – 7252 = 80
This rectangle has a different of 80. I am now going to use a rectangle that is 3 × 6. To prove, that my prediction is right.
15a. 5 6 7 8 9 10
15 16 17 18 19 20
25 26 27 28 29 30
b. 5 × 30 = 150
25 × 10 = 250
c. 250 – 150 = 100
This rectangle has a different of 100. this proves that my perdition was right. I am now going to do this size rectangle once more. Then I will move on to a rectangle that is 4 down, and I will start at 2 across (4 × 2).
16. 61 62 63 64 65 66
71 72 73 74 75 76
81 82 83 84 85 86
b.61 × 86 = 5246
81 × 66 = 5346
c. 5346 – 5246 = 100
This rectangle has a different of 100 also.
The pattern for rectangles with 3 rows is:
For a rectangle with 3 rows high and 3 columns wide the difference is always 40
For a rectangle with 3 rows high and 4 columns wide the difference is always 60
For a rectangle with 3 rows high and 5 columns wide the difference is always 80
For a rectangle with 3 rows high and 6 columns wide the difference is always 100
I will now do the 4 × 2 rectangle. My perdition for this is that it will go up in 30’s. e.g. 30, 60, 90.
17a. 69 70
79 80
89 90
90 100
b. 69 × 100 = 6900
99 × 70 = 9630
c. 6930 – 6900 = 30.
This rectangle has a different of 30, this means that the first part of my perdition is right but I will do another rectangle this size (4 × 2).
18a. 25 26
- 36
45 46
55 56
b. 25 × 56 = 1400
55 × 26 = 1430
c. 1430 – 1400 = 30
This rectangle has a different of 30. I m now going to do a rectangle that is the size of 4 × 3. these rectangles should be 60.
19a. 7 8 9
17 18 19
27 28 29
37 38 39
b. 7 × 39 = 273
37 × 9 = 333
c. 333 – 273 = 60
This rectangle, as I prettied has a different of 60, I will do one more rectangle this size before I make it bigger.
20a. 53 54 55
63 64 65
73 74 75
83 84 85
b. 53 × 85 = 4505
83 × 55 = 4565
c.4565 – 4505 = 60
This rectangle has a different of 60. now I am going to make the rectangle bigger by one across (4×4) this should be 90.
21a. 1 2 3 4
11 12 13 14
21 22 23 24
31 32 33 34
b. 1 × 34 = 34
31 × 4 = 124
c. 124- 34 = 90
This rectangle has a different of 60.
The pattern for rectangles with 4 rows high is:
For a rectangle with 4 rows high and 2 columns wide the difference is always 30
For a rectangle with 4 rows high and 3 columns wide the difference is always 60
For a rectangle with 4 rows high and 4 columns wide the difference is always 90
Now I am going to do rectangles with 5 down e.g. 5 × 2. And I will start and 5 × 2.
22a. 27 28
37 38
47 48
57 58
67 68
b. 27 × 68 = 1836
67 × 28 = 1876
c. 1876 – 1836 = 40
This rectangle has different of 40. From this I have predicted that they will go up in 40s for 5 down. E.g. for 5×3 it should be 80 and for 5 × 4 it should be 120 and so on. I will do one more rectangle of this size before I move on and make the size of the rectangle bigger.
23a. 8 9
18 19
28 29
38 39
48 49
b. 8 × 49 = 392
48 × 9 = 432
c. 432 – 392 = 40
This rectangle has a different of 40. Now I make the rectangle bigger across by one. So the next rectangle I will do, will be 5×3, this should have a different of 80.
24a. 33 34 35
43 44 45
53 54 55
63 64 65
73 74 75
b. 33 × 75 = 2475
73 × 35 = 2555
c. 2555 – 2475 = 80
This rectangle has a different of 80 just as my perdition said it should. I will now do one more rectangle of this size before I make it bigger.
25a. 54 55 56
64 65 66
74 75 76
84 75 76
94 95 96
b. 54 × 96 = 5184
94 × 56 = 5264
c.5264 – 5184 = 80
This rectangle has a different of 80. I will now change the size to make it
5 × 4. This had a different of 120.
26a. 16 17 18 19
26 27 28 29
36 37 38 39
46 47 48 49
56 57 58 59
b. 16 × 59 = 944
56 × 19 = 1064
c. 1064 – 944 = 120
Just as my perdition said, this rectangle has a different of 120.
For rectangles with 5 rows high and 2 columns wide the difference is always 40
For rectangles with 5 rows high and 3 columns wide the difference is always 80
For rectangles with 5 rows high and 4 columns wide the difference is always 120
I am going to make a table now showing the results of my work and extending the pattern.
I am now going to use the algebra that we were given in class on some random rectangles to check the to see if my algebra is right. The first rectangle I am going to do is a 5 × 4.
(X + 40) (X+3)
X (X+43)
=X2 + 3X + 40 × 30
=X2 + 43 × + 30
(X + 43X)
X2 + 43X+30
-(X2 + 13X)
=30
This algebra does not work, we know this because it shows that a rectangle that is 5×4 has a different of 30, but we know from the table that this is untrue and should be 120.
As I have worked out algebra, I will use my other version from now on. My version of the algebra uses the formula
10 × (W – 1)
So for 2 rows and 2 columns the answer would be
10 × (2 – 1) = 10
This only works for 2 rows down and must be adapted for rectangles with more then 2 rows down. e.g. with 2 columns:
With 3 rows down, it should be 20
With 4 rows down, it should be 30
And so on.
So the formula to work these out would be
(10 × Number of Rows) – 10
We can simplify this to
10 × (Number of Rows – 1)
Or 10 × (H – 1)
We must join these two formulas together to get one formula
Which are:
10 × (W – 1)
And
10 × (H – 1)
This gives the formula:
10 × (H – 1) × (W – 1)
Both are multiplied by the same 10.
I will now chose random rectangles and use my formula, then check them against the table I have done.
Size 3 × 3
71 72 73
81 82 83
91 92 93
10 (H-1) × (W-1)
10 (3-1) × ( 3-1)
(10×2) × 2 = 40
Using the formula I have worked out that this rectangle has a different of 40. looking back to the table and to the original way to do this, this size also had a different of 40.
Now using my formula I will try a rectangle the size of 4 × 5
55 56 57 58 59
65 66 67 68 69
75 76 77 78 79
85 86 87 88 89
10 (H-1) × (W-1)
10 (4-1) × (5-1)
(10×3) ×4 = 120
Using the formula the rectangle has a different of 120. and using the table you can see that this is rights for a rectangle this size.
Now using my formula I will try a rectangle that has a size of 10×10
1 /10 10
/10 /10
91 / 10 100
10 (H-1) × (W-1)
10 (10-1) × (10-1)
(10×9) ×9= 810
The different of this rectangle is 810. If you did it the long way, you would find that it would also have a different of 810.
I will now do some graphs to so the relation between the different size rectangles.