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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.

## Drawings

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

(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)

## 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.

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.

Middle

31

32

33

34

35

36

37

38

39

40

41

42

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44

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49

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)

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

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

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

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

T- n = 32

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

T-t = 32 + 26 + 19 + 20 + 21  = 118

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

T-n = 19

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

T – t = 19 + 14 + 8 + 9 + 10 = 60

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

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