# Maths Coursework on T-Shapes

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Maths Coursework on T-Shapes

In this maths coursework I will be investigating T-shapes, T-shapes exist in grids, like the 9x9 grid above. The blue highlighted shape is a the simplest T-shape in the grid. The number at the bottom of the T-shape is called the T-number; in this case the T-number is 20. The sum of all the numbers in a T-shape is called the T-number. This T-number is 37, 1+2+3+11+20=37. In the first part of the investigation into T-shapes I will try to find a formula that finds the T-total using only the T-number. Now lets recap on the basic facts about T-shapes.

- The number at the bottom of the T-shape is called the T-number
- The total all of the numbers in a t-shape is called the T-total.

In this investigation I will first find a formula to find the T-total using the T-number in a 9x9 grid, I will then use differing grid sizes to see what an affect this has on the formula. Then using this information I will find a formula, which will be true to all grid sizes. Then I will experiment with rotations and reflections on differing grid sizes to see what an affect it has on the formula. Then a Conclusion about T-shapes.

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To find the formula, which will show the T-total with the T-number, I will first make a note of some of the T-totals in this grid.

- 1+2+3+11+20=37
- 2+3+4+12+21=42
- 3+4+5+13+22=47
- 4+5+6+14+23=52

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I haven’t chosen T-shapes close to each other in case of the eventuality that the formula would only work on the first line. First I will work out if the formula works on the blue T-shape.

5N-63; 21x5=105 105-63=42 2++3+4+12+21=42.

The formula works for the blue T-shape, if the formula works for the other highlighted T-shapes in this grid I will have found the formula for the 9x9 grid.

Red T-shape; 5N-63 43x5=215 215-63=152 24+25+26+34+43=152

Green T-shape; 5N-63 47x5=235 235-63=172 28+29+30+38+47=172

Yellow T-shape; 5n-63 79x5=395 395-63=332 60+61+62+70+79= 332

All the formulae were correct; the formula for the 9x9 grid is 5N-63. Now I will find the formula of other grid sizes and use that information to find a formula, which fits all the grids. I will begin with a 10x10 grid, as it is only one step up from a 9x9 grid and the formula shouldn’t be that different.

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I will apply the same method to this grid, I will find the difference between each T-total and apply that to the beginning of the formula, then times the T-numbers by this amount and subtract the T-totals from the multiplied t-numbers.

I will begin by finding out the t-totals of the first five t-shapes on the grid e.g. the ones that begin with 1 through to 5.

1st T-shape= 1+2+3+12+22=40

2nd T-shape= 2+3+4+13+23=45

3rd T-shape=3+4+5+14+24=50

4th T-shape=4+5+6+15+25=55

5th T-shape=5+6+7+16+26=60

Again the T-totals go up in 5, so therefore the beginning of the formula would be 5N like the previous formula. Now that we have found that the difference between each T-total is 5 we need to times the T-numbers by this amount. I will chose the same T-shapes, which I used previously to find the formula.

1st T-shape= 22x5= 110

2nd T-shape= 23x5= 115

3rd T-shape= 24x5= 120

4th T-shape= 25x5= 125

5th T-shape= 26x5= 130

Now we subtract the multiplied T-numbers from the T-totals to find out the end part of the formula.

1st T-shape=110-40=70

2nd T-shape=115-45=70

3rd T-shape=120-50=70

4th T-shape=125-55=70

5th T-shape=130-60=70.

So the formula for the 10x10 grid should be 5N-70. I will now test it on the T-shape below.

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69x5= 345 345-70=275 48+49+50+59+69=275

The formula was correct. I will now find out the formula of a grid smaller than 9x9 to see what affect lessening the grid size has on the formula.

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Conclusion

I will first find out the formula of the red T-shape because, the formula for the blue T-shape was already worked out on page 4. I will work out the answer by multiplying the T-number by five and subtracting the T-total from it.

41x5=205 34+41+42+43+52=212 212-205=7 5N-7

The formula for this T-shape is 5N-7. However to see if it works on other T-shapes, I will test it on the one below.

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16x5=80 80+7=87 9+16+17+18+27=87

The formula works. Now I will experiment on the next, green T-shape again by multiplying the T-number by five and subtracting it from the T-total.

41x5=205 41+50+58+59+60=268 268-205=63

Again I will test the formula 5N-63 on the T-shape below.

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57x5=285 74-75+76+66+57=348 348-285=63.

The formula is correct, and interestingly it is the exact opposite of the blue T-shapes, which was 5N+63. I therefore predict that the yellow T—shape will be the same as the red T-shapes, except if the red was an addition the yellow would be a subtraction, and visa versa. Since the red formula is 5N-7 the yellow should be 5N+7. I will now test this theory.

41x5=205 30+39+40+41+48=198 205-198=7

It is correct, therefore anything when rotated 180 degrees is the same but if it was an addition it would become a subtraction, and if it was a subtraction, it would become an addition. However I still have to test it on the T-shape below.

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39x5=195 28+37+38+39+46=188 195-188=7

It is correct proving the theory works.

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

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