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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 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

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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: 43Calculation shows to be correct Predict: 5

Again the T-Totals increase by +5.

I will check again.

6x6 Grid Row3 (13-18)

 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
 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: 103Calculation 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)

 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
 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 44 164 5 28+29+30+37 45 169 5 29+30+31+38 46 174 5 30+31+32+39 47 Predict: 179Calculation shows to be correct Predict: 5

14x14 Grid; Row 9 (113-126)

 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
 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+134 148 642 5 120+121+122+135 149 647 5 121+122+123+136 150 652 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

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:

 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
 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

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.

 31 32 33 40 41 42 49 50 51

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.

 31 32 33 40 41 42 49 50 51

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.

n=45

t=5n+7

t= (5x45) +7

t=225+7

t=232

Check

 36 45 46 47 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.

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

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