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• Level: GCSE
• Subject: Maths
• Word count: 2012

# Investigate the relationship between the T-total and the T-number

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Introduction

## GCSE Maths Investigation – T shapes

Part 1 – Investigate the relationship between the T-total and the T-number

When the “T-total” equals 37, the T-number is 20.  I will investigate the other T-totals to see if there is a relationship.

 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

 T-total T-number 1 + 2 + 3 + 11 +20 = 37 20 2 + 3 + 4 + 12 + 21 = 42 21 3 + 4 + 5 + 13 + 22 = 47 22 4 + 5 + 6 + 14 + 23 = 52 23 5 + 6 + 7 + 15 + 24 = 57 24 6 + 7 + 8 + 16 + 25 = 62 25 7 + 8 + 9 + 17 + 26 = 67 26

The difference between the T-totals is:

37             42                     47                    52               57                62                   67

+5                 +5              +5             +5                 +5                 +5

I will now see if there is any relationship between the T-total and the T-number.  In order to see this I will subtract the T-number from the T-total.

T-number – T-total = Difference

42 – 21 = 21

47 – 22 = 25

52 – 23 = 29

57 – 24 = 33

62 – 25 = 37

I will put this information into a formula.

Here; n = the T number

T = T-total

So if n = 20

20 + (20 – 19) + (20 – 18) + (20 – 17) + (20 – 9) = T

1                2                 3                11

Therefore:

n + (n – 19) + (n – 18) + (n – 17) + (n – 9) = T

To simplify:       5n – 63 = T

### T = 5n - 63

I will use one of the T-shapes on the 9 by 9 grid to see if this formula works.

Eg1:        20 x 5 – 63 = 37

This is correct

Middle

94

95

96

97

98

99

100

 T-total T-number 1 + 2 + 3 + 12 + 22 = 40 22 2 + 3 + 4 + 13 + 23 = 45 23 3 + 4 + 5 + 14 + 24 = 50 24 4 + 5 + 6 + 15 + 25 = 55 25 5 + 6 + 7 + 16 + 26 = 60 26 6 + 7 + 8 + 17 + 27 = 65 27 7 + 8 + 9 + 18 + 28 = 70 28

The difference between the T-totals is:

40             45                     50                    55               60                65                   70

+5                 +5              +5             +5                 +5                 +5

I am going to see if the formula that I calculated earlier works for a 10 by 10 grid.

Therefore:  T = n + (n-21) + (n – 20) + (n – 19) + (n – 10)

## To simplify: T = 5n – 70

This formula is similar except the number that is being subtracted.  However I will see if this works with a T-shape in the 10 by 10 grid.

T = 5 x 26 – 70

T = 60

This is true because 5 + 6 + 7 + 16 +26 = 60

The equation is correct.

However, in order to find the relationship between the grid size and the T-total and T-number it must be included in the equation.

So, when n = T-number

T = T-total

W = Grid size

## This simplifies to: 5n – 7w

If the grid decreases to 6 by 6, the T-numbers will be closer together.  I noticed that on the previous 2 grids, the numbers being subtracted can be divided by the grid size and both equal –7.

I think that a 6 by 6 grid should be:

T = 5n – 42        (-7 x 6 = 42)

 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-total T-number 1 + 2 + 3 + 8 + 14 = 28 14 2 + 3 + 4 + 9 + 15 = 33 15 3 + 4 + 5 + 10 + 16 = 38 16 4 + 5 + 6 + 11 + 17 = 43 17

Part 3:  Use grids of different sizes again.  Try other transformations and combinations of transformations.  Investigate relationships between the T-total, the T-numbers, the grid size and the transformations.

REFLECTION

Horizontal reflection

 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

38   ,    43                      39  ,  44                    40  ,  45

+5                                +5                             + 5

Conclusion

210 +  -20 - -90 – 63 = T

T = 210 – 20 + 90 – 63

T = 217

This is true because the T-total is 217 for this T-shape.  I deliberately used a translation with negative numbers to check that this did not alter the equation.

COMBINING REFLECTION AND TRANSLATION

n = T-number

W = Grid size

Z = across translation

U – Up or down translation

N = new T-number

a = horizontal distance from reflection line

b = vertical distance from the reflection line

T = T-total

HORIZONTAL REFLECTION TO TRANSLATION:

5n – 7w + 10a + 5 + 5N + 5z – 5wu – 7w = T

The new T-number needed to be included in this equation otherwise it would not work, because the translation would occur from the original T-number.

This formula works because when tested on a 9 by 9 grid with the T-number being 21 and a being 2, the new t-number is 26 and when translated by 2 and -2 the T-total equals 147.  This agrees with the equation.

TRANSLATION TO VERTICLE REFLECTION:

5n + 5z – 5wu – 7w + 5N + 10bw + 12w = T

The new T-number needed to be included in this equation otherwise it would not work, because the reflection would occur from the original T-number.

This formula works because when tested on a 9 by 9 grid with the T-number being 76 and the translation being +2 by +6, the new T-number is 24.  When “b” is 1, the T-total equals 318.  This agrees with the equation.

T-shape investigation.doc                5/4/2007  11:53 AM

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