# Investigating the relationship between the T-totals and the T-number.

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Introduction

Mathematics

G.C.S.E.

Course Work

T-Totals

By: Bharatjit Basuta

10kw

07/04/2004

Part One: I’ll be investigating the relationship between the T-totals and the T-number.

To show the relationship between T-totals and the T-numbers I will use a nine by nine grid to explain. There is a shape in the grid called the T-shape. This is shown below highlighted in the colour red.

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The number 20 at the bottom of the T-shape, this is called the T-number. All the other numbers highlighted in the T-shape are called the T-total.

For this T-shape

T-number = 20

T-total = 37

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For this T-shape

T-number = 21

T-total = 42

After viewing the information that every time the T-number goes up one the T-total goes up by five. So in this case the ratio between the T-number and the T-total is 1:5. This can help me because when I want to translate a T-shape that is in another position. For instance when I the T-shape here.

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For red T-shape

T-number = 20

T-total = 1+2+3+11+20 = 37

For Orange T-shape

T-number = 74

T-total = 55+56+57+65+74 = 307

I all ready know the answer to the red T-shape from the previous work that I did. To work out the orange T-shape I will have to work out the difference of the T-number. In this case it is 54 (74-20). Then I will times the 54 by 5 because the T-total rises by 5 ever time the T-number goes up by one.

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5x78-63 = T-total

5x78-63 = 327

Checking…

59+60+61+69+78 =327

As show the formula works because I have found the relationship between T-totals and T-numbers for the.

Part Two: I’ll be using grids of different sizes and then translate the T-shape in to different positions. Then ill investigate the relationship between the T-total, T-number and the grid size.

Here I am doing what I did in part one. This time ill be finding out more about the grid sizes and what they are capable of doing.

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For this T-shape (11x11 Grid)

T-number = 24

T-total = 1+2+3+13+24 = 43

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For this T-shape (9x9 Grid)

T-number = 20

T-total = 1+2+3+11+20 = 37

Even though T-shape looks to be in the same place the T-total and the T-number have risen. The T-number has risen by four and the T-total has risen by six. If I use the same rules I made in the part one I will get a new formula for the new size grid.

Using the long method

24-1 = 23

24-2 = 22

24-3 = 21

24-13 = 11

TOTAL = 77

Shorter method

7* 11 = 77

Multiply by eleven because that is the grid size

New formula

5T-number-77 = T-total

5x24-77 = 43

The same formula works with only changing the last number in the formula. Now I will try the same method on a smaller grid size to make sure that my method doesn’t only work when the grid size gets bigger.

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T-number = 10

T-total = 1+2+3+6+10 = 22

7x4 = 28

Multiply by four because that is the grid size

5T-number-28 = T-total

5x10-28 = 22

By changing the size of the grids I‘ve learned that there is an overall formula for any size grid to find the T-total. This formula is 5T-number-7G-number. I found this formula out by combining the formula which I used to find the difference of the T-number with the formula that I’ve been using with the other grids. This formula is shown below.

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Conclusion

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12-1 =11

12-10= 2

12-19= -7

12-11 = 1

TOTAL = 7

Formula

5T-number – 7 = T-total

5x12 - 7= 53

Checking…

T-number = 12

T-total = 1 +10 +19 +11 +12 = 53

This formula has worked. Now if I rotate the t-shape 180 degrees, the same will happen, as what happened when the T-shape when it was turned 180 degrees from its first original position. To prove this it will be shown below.

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5T-number + 7 = T-total

5x70 + 7 = 357

Checking…

T-number = 70

T-total = 70+71+72+63+81 = 357

Now that I have worked out all the formulas for the position in the normal sized T-shape. No I’m going to enlarging the T-shape. I will double the T-shape. The new shape is shown below on the 9x9 grid. I have added the numbers together in the squares of the T-shape. This leaves me with my original T-shape but with larger numbers in the grid.

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176-24(1+2+10+11) = 152

176-32(3+4+12+13) = 144

176-40(5+6+14+15) = 136

176-104(21+22+30+31) = 72

TOTAL= 504

I have the rest of the formula. The formula is identical apart from the number we minus or plus.

Formula

5T-number –504 = T-total

5x176-504 = 376

I have proven that the formula works

Conclusion

In conclusion I have learned that in this project I have found out many ways in which to solve the problem I have with the T-shapes being in many different positions with different many sizes of grids. The way I have made the calculations of this project less difficult is was by creating a many main formulas that change for all the different circumstances.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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