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  • Level: GCSE
  • Subject: Maths
  • Word count: 2961

This is an investigation to find a relationship between the T-totals and the T-number. The diagram shows a 9x9 grid, with each individual cell having one number in it starting on the

Extracts from this document...

Introduction

Lianne Haley

COURSEWORK INVESTIGATION

T-Totals

The Problem, the Plan and possible extensions

This is an investigation to find a relationship between the T-totals and the T-number.

The diagram shows a 9x9 grid, with each individual cell having one number in it starting on the top row 1-9.

The diagram shown has an upright T-shape, the total of the numbers inside the T-shape is 1+2+3+11+20 = 37, and this is called the T-total.  The number at the bottom of the T-shape is called the T-number.  The T-number, for the example T-shape given, is 20.

I need to be systematic in my approach so initially I will be investigating the relationship between the T-totals and T-numbers when the T-shape translates on the 9x9 grid, starting with the 1st row then the 2nd row etc.  This will keep it simple for me to spot any patterns.

Later I will be investigating T-shapes on different sized grids, again translating the T-shape to different positions on the grids to find a relationship between the T-totals and the T-numbers.

I can also use grids of different sizes again and try other transformations and combinations of transformations and investigate relationships between the T-totals, the T-numbers, the grid size and the transformations.

Hypothesis

1

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image00.png

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...read more.

Middle

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KEY

T-Shape Numbers

T-Numbers

T-Totals

Differences Between T-Totals

1+2+3+8

14

28

5

2+3+4+9

15

33

5

3+4+5+10

16

38

5

4+5+6+11

17

Predict: 43

Calculation shows to be correct

Predict: 5

Again the T-Totals increase by +5.

I will check again.

6x6 Grid Row3 (13-18)

image11.png

7

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KEY

T-Shape Numbers

T-Numbers

T-Totals

Differences Between T-Totals

13+14+15+20

26

88

5

14+15+16+21

27

93

5

15+16+17+22

28

98

5

16+17+18+23

29

Predict: 103

Calculation shows to be correct

Predict: 5

I have jumped to Row 3 to prove my hypothesis was correct and it was.

So if my calculations of +5 are the same for a 9x9 grid and 6x6 grid it will be the same for an 8x8 grid and a 14x14 grid.


I will tabulate my results.

8x8 Grid Row 4 (25-32)image12.png

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49image13.png

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KEY

T-Shape Numbers

T-Numbers

T-Totals

Differences Between T-Totals

25+26+27+34

42

154

5

26+27+28+35

43

159

5

27+28+29+36

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164

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28+29+30+37

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169

5

29+30+31+38

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174

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30+31+32+39

47

Predict: 179

Calculation shows to be correct

Predict: 5

14x14 Grid; Row 9 (113-126)image14.png

99

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155image15.png

image16.png

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KEY

T-Shape Numbers

T-Numbers

T-Totals

Differences Between T-Totals

113+114+115+128

142

612

5

114+115+116+129

143

617

5

115+116+117+130

144

622

5

116+117+118+131

145

627

5

117+118+119+132

146

632

5

118+119+120+133

147

637

5

119+120+121+134

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642

5

120+121+122+135

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647

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121+122+123+136

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652

5

122+123+124+137

151

657

5

123+124+125+138

152

662

5

124+125+126+139

153

Predict: 667 Calculation shows to be correct

Predict: 5

My results confirm that with any number grid the T-Totals increase by +5.


Testing the Formula

With my results recorded on the table above for the 14x14 grid I will check my original formula to make sure this is correct.

n = 144

t = 5n-63

t = (5x144)-63

t = 720-63

t = 657                                Incorrect

n = 152

t = 5n-63

t = (5x152)-63

t = 760-63

t = 697                                Incorrect

This shows me that different sized number grids require a different formula as predicted in my hypothesis.

In this case:

n = 152

t = 662 (123+124+125+138) +n

123

124

125

138

152

152-123=29, 152-124=28, 152-125=27, 152-138=14

So to work out the T-Total, this simplifies to:

t = ((n-29) + (n-28) + (n-27) + (n-14) +n)

123

124

125

138

152

n-29

n-28

n-27

n-14

n

image17.png

this simplifies to:

t = 5n-98        (98 being the total of 29+28+27+14)

The formula is dependent on the size of the grid.


On a 9x9 grid the formula is t = 5n-63                                63 is equal to 7x9

On a 14x14 grid the formula is t = 5n-98                                98 is equal to 7x14

So on a 6x6 grid would the formula be:                                t = ((5n-(7x6))

Results

I will work this out using the table previously shown:

image18.png

7

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KEY

T-Shape Numbers

T-Numbers

T-Totals

Differences Between T-Totals

13+14+15+20

26

88

5

14+15+16+21

27

93

5

15+16+17+22

28

98

5

16+17+18+23

29

103

5

n = 29

t = ((5n-(7x6))

t = 5n-42

t = (5x29)-42

t = 145-42

t = 103                        Correct

I will let g = number grid size

E.g.        an 8x8 grid would be:

g = 8

So the formula will now be:

t = 5n-7g

I have an 8x8 grid, what is the T-Total if the T-Number is 46?

n = 46

t = 5n-7g

t = (5x46)-(7x8)

t = 230-56

t = 174                        Correct

See below an excerpt taken from my results from the 8x8 grid, see page 7.

29+30+31+38

46

174

5

128

4

...read more.

Conclusion

With a 90°clockwise transformation of an upright T the T-Number is neither the lowest nor the highest, but the highest number minus the T-Number is equal to 7.

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image08.png


With a 270° clockwise transformation, the T-Number is, again, neither the highest nor the lowest number but the T-Number minus the lowest number is equal to 7.

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33image09.png

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Justifying the Formula

Question

I need to find out the T-Total for a 270° T-Shape who’s T-Number is 55 on an 11x11 grid.

Answer

n=45

t=5n+7

t= (5x45) +7

t=225+7

t=232

Check

36image10.png

45

46

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58

Conclusion

I have found that each different translation requires a formula similar to each other translation for different sized grids.

I believe I have fully justified the explanation of this occurrence by checking my findings and tabulating results.

My initial hypothesis was correct stating that each T-Shape would increase by 5 thus finding the relationship between the T-Number and T-Total was +5 for any translation on any sized grid.

Further Extensions

If I had time I could explore the relationship between different sized T-Shapes on different sized grids, for example, extended T-Shapes (5 on the top row and 3 on the bottom row etc.) and elongated T-shapes.

...read more.

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

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