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  • Level: GCSE
  • Subject: Maths
  • Word count: 2022

T-totals, T-numbers and T-shapes.

Extracts from this document...

Introduction

Introduction

Recently, we have been set a task to investigate about T-totals, T-numbers and T-shapes. T-shapes consist of 5 numbers; these 5 numbers will form a T-total when added together. A T-number is the bottom number of a T-shape.

This piece of coursework will consist of 3 parts. We will have to try and investigate what the relationship is between the T-total, T-number and the T-shape in the time provided to complete this piece of coursework. Various methods will have to be used to find the formula; these various methods will consist of things such as algebra, trial and improvement, generalization etc. Different grid sizes would be used in order to find the “Master Equation” of T-totals, T-numbers and T-shapes.

An example of a T-shape and what it consists of.

x

x

x

x

x

This is a T-shape.

When all the numbers are added up together it would form a T-total.

X+X+X+X+X=T-total.

Along this row is the T-number.

Part 1

Using T-shapes, T-totals and T-numbers I am going to investigate the relationship between the T-total and the T-number. To do this I need a formula relating the T-total and T-number.

I am going to use a 9x9 grid

...read more.

Middle

Firstly I am going to use an 8x8 grid, reason being that I am going to start off with small grids then work up to larger grids.

8x8 Grid

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Below I have chosen my examples of T-shapes that I am going to experiment with.


1

2

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10

18

        The T-total of the numbers in this T-shape is:

        1+2+3+10+18=34.

        The T-number is 18.

2

3

4

11

19

        The T-total of the numbers in this T-shape is:

             2+3+4+11+19=39.

             The T-number is 19.

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12

20

        The T-total of the numbers in this T-shape is:

3+4+5+12+20=44.

The T-number is 20.

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5

6

13

21

        The T-total of the numbers in this T-shape is:

        4+5+6++13+21=49.

        The T-number is 21.

5

6

7

14

22

The T-total of the numbers in this T-shape is:

5+6+7+14+22=54.

The T-number is 22.

6

7

8

15

23

The T-total of the numbers in this T-shape is:

6+&+8+15+23=59.

The T-number is 23.


Using my examples I have worked out that a pattern is starting to form.

T-number

T-total

18

34

19

39

20

44

21

49

22

54

23

59

52

204

Predictionimage00.png

Every time a T-number goes up by 1, The T-total goes up by 5.

35

36

37

44

52

        The T-total of the numbers in this T-shape is:

        35+36+37+44+52=204.

        The T-number is 52.                Correct

The equation for an 8x8 grid is 5N-56.

N-17

N-16

N-15

N-8

N

N+ (N-8)

...read more.

Conclusion

I have proved this above.

Expressing result algebraically:

N-19

N-20

N-21

N-10

N

            T-total= N+ (N-10) + (N-20) + (N-21) + (N-19)=5N-70

Proof that my equation does work.

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12

22

            T-total=40

            40=5x-70

            (5x22)=110

             40=110-70

             110-70=40

Master equation for Part 2:

5x-(grid size x7) =T-total

Proof of Master equation for grid size 8

46

47

48

55

63

        5x63-(8x7) =259

        46+47+48+55+63=259        Correct.

Proof of Master equation for grid size 8

71

72

73

82

92

                                5x92-(10x7) =390

                                71+72+73+82+92=390                Correct.

Part 3

Now instead of just shifting the directions of the T-shape I am going to rotate them, by this I mean turning the T-shape upside down and moving it in opposite directions. But I am still going to have to investigate the relationships between the T-total, the T-numbers, the grid size and the Transformations.

I am going o start off with grid size 11.

11x11 Grid

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        Below I have chosen my examples of T-shapes that I am going to experiment with.

First of all I am going to turn my T-shapes facing left.

70

 78

79

80

91

The T-total is:

70+78+79+80+91=398

The T-number is 78.

I was unable to finish off Part 3 due to running out of time. But if I were able to complete Part 3 I would have got examples of many T-shapes facing different directions up, down, left and right. I would have also put in many results tables and included lots of information of all sort of things, from examples to expressing results algebraically.  

...read more.

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