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  • Level: GCSE
  • Subject: Maths
  • Word count: 1424

In this investigation I will be looking at the relationship between the T-total and the T-number with simple drawings and try to identify a general rule.

Extracts from this document...

Introduction

Maths Coursework

Introduction

This piece of coursework is about T- totals. I will be looking a T- Shape (see below) drawn on a 9x9 number grid. The number at the bottom of the shape is called the T- number. In this investigation I will be looking at the relationship between the T-total and the T-number with simple drawings and try to identify a general rule.

image00.png

Drawings

To help me in my investigation I have produced 3 drawings, the 4th being my prediction in a grid which is 9x9.                                                                

image01.png

(T-n = T- Number     T-t = T-Total)

T-n= 21 + 12 + 1 + 2 +3 = 39 (T-t)

T-n= 41 + 32 + 22 + 23 + 24 =142 (T-t)

T-n= 61 + 52 + 42 + 43 + 44 =242 (T-t)

(Prediction)

T-n= 81 + 72 + 62 + 63 + 64 = 337 (T-t)

Table of Results    

With the T-shapes drawn in the grid above I can now put my results in a table to see if there are any patterns occurring.

image02.png

Identifying the pattern

As you can see from my table of results that the T-total is increasing by 100 every time you add 20 to the T-number.

Prediction

I predict that the next T-total will be 337 and the T-number will =80.

...read more.

Middle

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T - n = 16 + 9 + 1 + 2 + 3 = 31(T-t)

T - n = 24 + 17 + 9 + 10 + 11 = 71(T-t)

T – n = 32 + 25 + 17 + 18 + 19 = 111(T-t)

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T – n = 18+ 10 + 1 + 2 + 3 = 34    (T- n)

 T – n = 27 + 19 +  + 10 + 11 + 12 = 79   (T- n)  

T – n = 36 + 28 + 19 + 20 + 21 = 124    (T- n)

Table of Results

My results for the 7x7 and 8x8 grid will be put into a table, so that I can see if there are any patterns emerging.

7 x 7 Grid                                                                             8 x 8 Grid

T - n

T- t

18

34

27

79

36

124

T- n

T - t

16

31

24

71

32

111

General Rule

These are the nth terms for Grids 7x 7 and 8 x 8.

7 x 7                                                                                      8 x 8

n - 15

n - 14

n - 13

n - 7

n

N - 17

N - 16

N - 15

N - 8

   N

T-total = n + n – 7 + n – 15 + n – 14 + n – 13                T-total = n + n – 8 + n + 17+ n – 16 +n -15

                          =5n – 49                                                          = 5n – 56

Identifying the pattern

There is a clear pattern on the 7 x 7 grid as the T- n goes up by 8 the T- t increases by 40 every time. On the 8 x 8 grid the T-n goes up by 9 every time and the T-t increases by 45.

I have also noticed that every time the grid size gets smaller the nth term decreases by 7 but the 5n does not change.  

9 x 9 =

5n – 63

8 x 8 =

5n - 56

7 x 7 =

5n - 49

...read more.

Conclusion

Overall Rule

I am going to use all the nth terms from the 3 grids and put them in a table and try to find a rule that will work on any grid size.

n - 17

n - 16

n - 15

n - 8

   n

           7 x 7                                          8 x 8                                                    9 x 9

n – 15

n – 14

n - 13

n - 7

n

n-19

n- 18

n-17

n-9

n

9 =

n

+

n - 7

+

n - 15

+  

n - 14

+

n – 13

8 =

n

+

n - 8

+

n - 17

+

n - 16

+

n - 15

7 =

n

+

n - 9

+

n - 19

+

n - 18

+

n - 17

G =

n

+

n - g

+

n – 2g + 1

+

n - 2g

+

n – 2g –1

n = T - number

G = Grid size

General Rule

I will now solve the rule.

T-total=n + n–g + n–2g +1 + n–2g + n-2g – 1 = 5n – 7g

The rule to find any t – total in any grid size is 5n – 7g.

Testing overall rule

To prove that my rule does work I will test it on 6 x 6 and  10 x 10 grids

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T- n = 32

5n – 7g =(5 x 32) 160 – (7 x 6) 42 = 118 (T-t)

T-t = 32 + 26 + 19 + 20 + 21  = 118

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T-n = 19

5n – 7g = (5 x 19) 95 – (7 x 5) 35 =  60 ( T- t)

T – t = 19 + 14 + 8 + 9 + 10 = 60

...read more.

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