27 multiplied by 49 equals 1323
Subtract 1323 from 1363 equals 40.
I choose another section, this time I choose,
75 multiplied by 57 equals 4275
55 multiplied by 77 equals 4235
Subtract 4235 from 4275 equals 40.
I am starting to think that there may be a general formula for any square on a 10x10 number grid. I will do one more different size square on my 10x10 square grid just to make sure. I will do a 4x4 square grid. Please find below a diagram of what to do in regards to a 4x4 square on a 10x10 grid and how to do it.
As you can see I got a result of 90. I choose another 4x4 section on my 10x10 number grid. I choose the section,
44 multiplied by 17 equals 748
14 multiplied by 47 equals 658
Subtract 658 from 748 equals 90.
I almost certainly believed there must be a general formula for any size square on a 10x10 number grid, however, I wanted to make sure, so I carried on with one more 4x4 square on a 10x10 grid. I choose the section,
32 multiplied by 5 equals 160
2 multiplied by 35 equals 70
Subtract 70 from 160 equals 90.
I will now try to create a formula for any size square on a 10x10 number grid using algebra.
A=
W=side of square
(A+10(W-1)) (A+(W-1)) – A(A+11(W-1))
= A2 + A(W-1) + 10A(W-1)+10(W-1)2 - A2 -11A(W-1)
= 10(W-1) 2
The formula for a any size square on a 10x10 number grid is
(A+10(W-1)) (A+(W-1)) – A(A+11(W-1)).
I am now going to carry on with my investigation and find out if there is a general formula for rectangles on a 10x10 grid.
I am going to start with a 3x2 rectangle. Please find below a diagram telling you how to work the difference out.
As you can see my answer came out as 20. I choose another 3x2 rectangle on my 10x10 number grid and came out with the section underneath.
26 multiplied by 18 equals 468
16 multiplied by 28 equals 448
Subtract 448 from 468 equals 20.
The answer, was again, 20. I tried once more to see if there is a pattern. I choose the section,
67 multiplied by 59 equals 3953
57 multiplied by 69 equals 3933
Subtract 3933 from 3953 equals 20.
As you can see, my answer came out as 20. I now wanted to find a formula for any 3x2 rectangle on a 10x10 number grid.
A=1
((A+10)(A+2)) – (A(A+12))
= A2 + 2A + 10A + 20 – A2 + 12A
= 20
My formula for a 3x2 rectangle on a 10x10 number grid is,
((A+10)(A+2)) – (A(A+12))
I wanted to pursue my rectangles investigation further. I pursued this, first with a 4x3 rectangle on a 10x10 number grid. Please find below a diagram of how to work out the difference of a 4x3 rectangle on a 10x10 number grid.
As you can see my answer came out as 60. I repeated this twice more. For my first choice, I choose the section,
66 multiplied by 49 equals 3234
46 multiplied by 69 equals 3174
Subtract 3174 from 3234 equals 60.
My second choice was,
26 multiplied by 9 equals 234
6 multiplied by 29 equals 174
Subtract 174 from 234 equals 60.
The answer was 60. I believed there was a general formula for a any size rectangle on a 10x10 grid. But, I wanted to make sure of this, so I carried on my investigation with rectangles with a 5x4 rectangle on a number grid. Please find below a diagram showing what I did to find the difference between the sum of the corners of a 5x4 rectangle on a 10x10 number grid.
As you can tell from the previous page, my answer was 120. I further chose 2 more 5x4 sections from the table. My first choice section was,
91 multiplied by 65 equals 5915
61 multiplied by 95 equals 5795
Subtract 5795 from 5915 equals 120.
My answer was 120. My next choice section was,
96 multiplied by 70 equals 6720
66 multiplied by 100 equals 6600
Subtract 6600 from 6720 equals 120.
Again my answer was 120. I was now confident enough to expand my investigation and try to discover a general rule for any size rectangle on a 10x10 square number grid using algebra.
I have noticed that the (height-1)(length-1)x10 = the difference for any size rectangle on a 10x10 square number grid. So if,
H=height
X=length
The formula would be, 10(H-1)(X-1)
Now I have discovered the formula for a any size rectangle on a 10x10 square number grid. I wanted to further my investigation. I wanted to know the general formula for any size rectangles, on, an any size number grid. Then hopefully, I will be able to expand my algebra further and create a general rule for a any size object on an any size grid.
Using the formula from my previous work on any size rectangles on a 10x10 number grid, I believe I can foretell the formula for any size rectangles on a any size square number grid.
The formula for any size rectangle on a 10x10 number grid is
10(H-1)(X-1)
I believe the number here, in this case being 10, is the number for the amount of squares on one side of the number grid. If I changed that number for an algebraic number meaning the grid size, I believe I can come up with the formula I want.
H=height
X=length
G=grid size
G(H-1) (X-1)
To prove this work, I will place numbers in the formula.
Height =6
Length =9
Grid size =17
17(6-1) (9-1)
= 17x5x8
= 680
Please find a diagram of what I have just done and how I did it on the following page.
This proves that my formula is correct for any rectangle on any square number grid. I now want to reach the core objective of my investigation and find a general rule that will work out the difference between the sums of the corners, of any box, on any size square number grid. With the knowledge I have gained throughout this investigation I am able to work out:
G= grid size
X= length
H= height
G(X-1)(H-1)
To prove this works, I will create a 13x13 square number grid. On that number grid, I will select a square, being 6x6. I will then calculate the differences between the sums of the corners and check to see if my general formula adds up. Please find below a diagram of what I have done in regards to the last paragraph.
G=13
H=6
X=6
13(6-1) (6-1)
=13 X 5 X 5
= 325.
This proves my formula does work.
During my investigation, I have managed to:
• Find a general formula to work out the difference between the sums of the corners of an any size rectangle on an any size square grid.
• Find a general formula to work out an any size box on an any size square number grid.
I have also been able to prove these formulas.
If I were to carry on my investigation, I could expand it by:
• Changing the shape of the number grid (i.e. into a triangle)
• Changing the shapes I use within the number grid (i.e. again, a number grid.
