The number 7 comes up when using each grid size because of the amount of squares each number is away from the T. The T shape below will make this clearer:-
From the investigations I have done so far I have found the information for my formula and need to assemble it and make sure it works. The formula I put together, using all the relevant information, is 5x – y.
(The x variable stands for the T number and the y variable stands for 7 x grid size).
I tried this formula with the following grid sizes: 8 x 8, 11 x 11 and 7 x 7. When I applied my formula it worked.
1+2+3+10+18 = 34 1+2+3+13+24 = 43 1+2+3+9+16 = 31
5 x 18 – 7 x 8 = 34 5 x 24 – 7 x 11 = 43 5 x 16 – 7 x 7 = 31
I am correct I am correct I am correct
Stage Two – Translating
The formula I have found to find any T number of an upright T in any grid size, will now be used at the beginning of the formula for translating the T shape. This is because I still want to find out the T total.
Moving to the right: As each T total increases by 5 as it is moved along, I will have to times 5 by the amount of squares I will be moving along. This is another variable and the letter a will stand for moving across. The formula now becomes
5x – y +5a.
I will test this on a 9 x 9 grid. My original T number is 20 and the translated T will have the grid number 25. I will also try my formula with the original T number being 65 and the translated T will have the T number 69.
WORKING OUT AND RESULTS
My formula has worked.
Moving to the left: The same approach is taken as moving the T shape to the left as moving the T shape to the right except for a minor adjustment. When moving the T shape to the left, instead of adding 5 x amount of squares moved across, a minus is used. This is because the T number is decreasing when the T is moved to the left. This makes the formula 5x – y – 5a. I will now test this with a 9 x 9 grid starting with the original T number being 25 and the translated T number will be 20. The T totals for both of these T shapes are in the table above so do not have to be written again.
WORKING OUT AND RESULTS
My formula has worked.
Moving the T shape down: for this I will be using a 8 x 8 grid but my formula will have a variable grid size. As the T shape moves down one square, 8 is added as that is the amount of squares moved. Therefore, to move down more than one square, 5 (as there are 5 squares in the T) will have to be multiplied by the grid size and the amount of squares moving down. From this I have found that the formula for moving the T shape down is 5x – y + 5bz (z is the variable for the grid width and b is the variable for the vertical movement).
Moving the T shape up: the same formula is used for moving the T shape up as moving the T shape down but with a minor adjustment. This is because the T shape does not increase, but decreases. The adjustment made is that 5bz is not added but subtracted. I will now test these two formulae on the same 8 x 8 grid.
WORKING OUT AND RESULTS
My formula works and I am correct.
Working out the movement of vertical and horizontal in one formula: To move the T shape vertically and horizontally in one formula you must combine the four formulae that have been worked out previously. The formulae that need to be focused on are translating the T shape up and down, and translating it right and left.
Moving down and to the right: the formula for moving down is 5x – y + 5zb and the formula for moving right is 5x – y + 5a. As there is already 5x – y in both of the formulae this part of the formula does not be to be repeated but obviously needs to be in the final formula. The 5zb part of the first formula calculates the movement moving down and the 5a calculates the movement of moving to the right. These are the two movements that are made in the process so are vital and must be in the formula. Therefore, when the three parts of each formula or combined the formula 5x – y +5zb + 5a is created. I will now test this formula on a 10 x 10 grid.
WORKING OUT AND RESULTS
Formula: 5x – y + 5zb + 5a
Substituting in the relevant information: (5 x 32) – 70 + (5 x 10 x 5) + (5 x 5)
(I have added brackets into the equation to make it clear where numbers has substituted the variables)
Simplified: 160 – 70 + 125 + 25
Answer: 365
Checking: 66 + 67 + 68 + 77 + 87
Answer: 365
My formula is correct and can be used in any grid size as it has a grid size variable.
Moving down and to the left: moving down and to the left consists of the same method of combining the two formulae but the formula for moving to left is different to moving to the right. The slight difference is that as the numbers are decreasing, the +5a becomes –5a because of the direction. The 5zb part of the formula still remains the same as the T shape is still moving in the downward direction. The formula has now become 5x – y + 5zb – 5a. I will now test this formula out on a 9 x 9 grid to show my variation in grid sizes and to prove that the grid size does not effect my result as there is a grid width variable.
(My grid is on the next page)
WORKING OUT AND RESULTS
Formula: 5x – y + 5zb – 5a
Substituting in the relevant information: (5 x 35) – 63 + (5 x 9 x 5) – (5 x 6)
Simplified: 175 – 63 + 225 – 30
Answer: 307
Checking: 55 + 56 + 57 + 65 + 74
Answer: 307
My formula has worked for this example.
Moving up and right: with this next stage I will be incorporating the same formula into this new formula as I did with the previous one. I will be using the formula 5x – y + 5a in my formula. The second part to the formula is the formula for moving the T shape up. This formula is 5x – y – 5zb. This seems the same as the formula for the downward movement, but there is a minus instead of an addition sign. This is because the numbers decrease as they move upward. The formula for this movement becomes 5x – y – 5zb + 5a when the two formulae are combined. I will now test this formula on an 8 x 8 grid.
WORKING OUT AND RESULTS
Formula: 5x – y – 5zb + 5a
Substituting in the relevant information: (5 x 42) – 56 – (5 x 8 x 3) + (5 x 5)
Simplified: 210 – 56 – 120 + 25
Answer: 59
Checking: 6 + 7 + 8 + 15 + 23
Answer: 59
I am correct with my example.
Moving up and left: moving up and left is using the same formula as moving up and right but with one adjustment. As we are decreasing our T total by moving it to the left, the additions sign next to the 5a changes to a minus sign. The formula has now become 5x – y – 5zb – 5a. I will now test this on an 11 x 11 grid.
WORKING OUT AND RESULTS
Formula: 5x – y – 5zb – 5a
Substituting in the relevant information: (5 x 109) – 77 – (5 x 11 x 6) – (5 x 8)
Simplified: 545 – 77 – 330 – 40
Answer: 98
Checking: 12 + 13 + 14 + 24 + 35
Answer: 98
My formula works and I am correct.
Stage Three – Rotation
The first step I will take to start of my rotation investigation on T totals is to draw a 10 x 10 grid and draw on an upright T shape and then rotate it 90 degrees. I will investigate the relationship between them and draw more of these types of diagrams to make the relationship clearer.
I will now subtract each pair of T shapes from each other and see what results I get which will depend on what I do next in my investigation.
T number 22: 117 – 40 = 77
T number 77: 392 – 315 = 77
I have noticed from these numbers that the difference between the two numbers is 77. The number 77 is the grid width add 7.
A way to look at this situation is to replace the rotated T number with x and change the rest of the numbers appropriately. This would give me the following on a 9 x 9 grid: -
When all the numbers add up they make the number seven. This shows that the formula must b 5x + 7. I will now test my formula on a rotated T shape of 90 degrees on a 9 x 9 grid.
(The grid is on the next page)
WORKING OUT AND RESULTS
Formula: 5x + 7
Substituting in the relevant information: 5 x 40 + 7
Answer: 207
Checking: 33 + 42 + 51 + 41 + 40
Answer: 207
Another example
Substituting in the relevant information: 5 x 64 + 7
Answer = 327
Checking: 64 + 65 + 66 + 57 + 75
Answer: 327
My formula is correct.
Rotating 270 degrees: rotating the T shape 90 degrees is adding to the T total of the new rotated shape. When the T shape is rotated 270 degrees the t total is decreased. The T shape is still in the same sideways position but is on the opposite side. As the formula for the rotation of 90 degrees is 5x +7, I am assuming that rotating the shape 270 degrees would mean that the formula would be 5x – 7.
A way to find out if my assumption is true is to use the same method as I used before. I will substitute the T number of the rotated shape with x and apply this to all the other numbers in the T shape. The following is what I came up with: -
The numbers in this T shape add up to –7 proving my assumption correct. I will now test this formula on a 10 x 10 grid.
(The grid is on the next page)
WORKING OUT AND RESULTS
All using the formula 5x - 7
Now I will see is the formula I used to work out the T totals is correct and gets me to the right answer.
The answers I got from the formula match the answers I got from checking them individually. My formula is correct.
Rotation of 180 degrees: to start with finding out the formula for rotating the T shape 180 degrees, I will draw some T shapes on a 9 x 9 grid and work out the relationship between all of the T shapes.
I can not see the connection between these numbers so I will try a different strategy. I will replace the T number with a x and adjust the rest of the numbers. Once I have done that I will try and figure out the connection and relationship. I will try this with a 9 x 9 grid and the T number will be 62.
A general way to look at this for any size grid is by adding in the grid variable. This is the following: -
The z variables in this T shape add up to 7. In the formula for a rotation of 180 degrees there must be 7 x grid size. As I want to find out the T total, 5x must be in the formula as well. When I combined the two, I came up with 5x + 7z. I will now test this formula on a 10 x 10 grid and see if my formula makes sense. I will use 3 examples of T shapes to make sure that thy work for all parts of the grid.
WORKING OUT AND RESULTS
Formula: 5x + 7z
Substituting the relevant information: 52 x 5 + 7 x 10
Answer: 330
Checking: 52+62+72+71+73
Answer: 330
This is correct
Substituting in the relevant information: 5 x 35 + 7 x 10
Answer: 245
Checking: 35+45+55+54+56
Answer: 245
I am correct
Substituting in the relevant information: 5 x 77 + 7 x 10
Answer: 455
Checking: 77+87+97+96+98
Answer: 455
I am correct and my formula seems to work.
---------------------
In conclusion of my investigation on T totals, I am happy with the results I was able to obtain and I am pleased with the ways I went about my investigation. If I had had more time along the way I would have been able to achieve more challenging aspects of this task. I would have attempted enlargement and reflection. From my investigation I gained the following information:
- How to find the T total with any T number in any size grid
- How to find the T total of a T by moving it up, down, right and left
- I have found how to move the T over a certain amount of squares and up and down a certain amount of squares and finding the T total.
- How to find the T total of a T shape when it is at 90 degrees, 180 degrees and 270 degrees.
The formulae I have found are the following:
- 5x – y
- 5x – y + 5a, 5x – y – 5a, 5x – y + 5zb and 5x – y – 5zb
- 5x – y + 5zb + 5a, 5x – y + 5zb – 5a, 5x – y – 5zb + 5a, and 5x – y – 5zb – 5a.
Of the work I have done and completed, I am pleased that I have understood it all and that I have been able to follow my plan and not become stuck on where to go next. I feel as if I have planned my investigation well and that it was very useful throughout the whole of the T totals assignment.