#### The Relationship between the T-Number and the T-Total

The Relationship between the T-Number and the T-Total The T-Total is the number that is derived from the sum of all numbers within a 'T' shape placed on a numbered grid. The T-Number is the number at the bottom of the 'T' : I have been asked to find the relationship between the T-number and The T-Total and devise a formula to derive the T-Total from any given T-Number of any translation of the 'T' shape. The grid below is 9*9 meaning that any number on the grid is nine less than the one below. This means that the number above the T-Number will always be 9 less than it, it also means that the number two squares above the T-Number is 18 less and the two either side are 17 less and 19 less than the T-Number. This means that: T-Number = TN T-Total = TN + TN - 9 + TN -18 + TN - 17 + TN -19 So T-Total = 5TN - 63 So to test the formula: Eg. 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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 + 2 + 3 + 11 + 20 = 37 And: 5*20 = 100 - 63 = 37 Eg. 2 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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

#### have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence

Page 1 Introduction I have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence. In this investigation I will be aiming to find a formula which could be used to find out the number of squares needed to build the pattern at any sequential position. Firstly I will break the problem down into simple steps to begin with and go into more detail to explain my solutions such as the nth term. In order to find this I would need to work of the formula: Term 1 Term 2 Term 3 Term 4 B=1 B=5 B=13 B=25 W=4 W=8 W=12 W=16 Pattern Dark squares White squares Number of squares 4 + 4 = 5 2 5 8 5 + 8 = 13 3 3 2 3 + 12 = 25 Hypothesis I predict that there will be a difference between each cross-section shape. Therefore there will be a quadratic sequence which can tell me how many squares would be needed in any given shape. Page2 Prediction + 3 + 1 = 5 + 3 + 5 +3 + 1 = 13 + 3 + 5 + 7 + 5 + 3 + 1 = 25 + 3 + 5 + 7 + 9 + 7 + 5 + 3 + 1 = 41 If you notice in my prediction there is a pattern of

#### 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.

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) 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. 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. My prediction is already included on the 9x9 grid it is the Green T-shape. Testing my prediction My prediction was correct as you can see on the grid the green T-shape =337. T-n= 81 + 72 + 62 + 63 + 64 = 337 (T-t) General Rule The general

#### T-Totals Maths

T-Totals Coursework My aim is to investigate the relationship between the T-Number and the T-shape on a varying size of grid. 2 3 4 5 To the left is a basic T-shape. In this investigation, the number in bold which is "5" is the T-Number. The sum of the all the numbers in the T-shape is the T-Total. For Example: 1+2+3+4+5 = 10. Therefore the T-Total for this T-Shape is 10. Using this information I can now begin my investgation. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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 I am going to start my investiation on a 9 by 9 grid. This shows all the numbers from 1 to 81. Firstly in my investigation, I am going to find a formula that relates my T-Number to my T-Total, firstly with a 9x9 grid and then onto a grid of any size. To work out my formula I have drawn two T-Shapes on my grid. The first t-shape in green has a T-Number of 20. The other T-Shape highlighted in pink has been translated one number to the right giving it a T-Number of 21. I then worked out the T-Total for both shapes. Green T-Shape: 1+2+3+11+20 = 37 Pink T-Shape: 2+3+4+12+21 = 42 T-Number T-Total 20 37 +5 21 42

#### T Shapes Investigation

Maths Coursework T Shapes Investigation Definition Abbreviation: On a grid we place a T Shape it can be any size. T Shape T Total is the sum of all the numbers inside the T Shape Tt. T Number is the number at the base of the T. T No or Tn. Part 1 Investigate the relationship between the Tt and the T No. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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 The Orange T Shape is a 3 by 2 and the green is a 5 by 3, both of these are on a 9 by 9 grid. T Shape T No Tt Orange 3 by 2 20 37 Other 3 by 2 34 107 Other 3 by 2 56 217 At this point I thought if I put a T Shape into the Nth term: For a 3 by 2 2 3 1 20 Goes to n-19 n-18 n-17 n-9 n = 5n-63 Therefore to make it a formula it becomes Tt=5n-63 Now I began to explore how to find certain numbers such as the 5 and the minus 63. The only numbers I can work with are the 3 and the 2 from the t shape, the 9 from the grid. So the obvious way to get the 5 is from the t shape. So the 3 we call the T width or W. And the 2 is the t height or H Now we have Tt = (H+W)n-63 Next I tested this out with a 3 by 2 t shape, with a T No of 20 Tt =

#### Investigating Octagon Loops

Investigating Octagon Loops Introduction I am primarily investigating Octagon Loops. In the investigation I plan to find out the relationship between the number of tiles used in the loop and the total number of free edges. Also, I will find the relationship between the number of tiles and the inside free edges and the number of tiles and the outside free edges. What I Will Do I plan to do this by first using octagon shapes and positioning them into loops, I will then put the results into a table so that it's easier to analyse. I'll start by making a loop with the least amount of shapes and then build up the number of shapes to give me a good range of values. Then I will analyse the results and hopefully discover some rules. What I did First of all I moved the octagon shapes around to find out the minimum amount it would take to make a loop. After discovering this (4) I made a table like the one in my results section and made more loops to get a more varied range of results which would make it easier for me to make a rule. I then set about trying to find out rules for the relationship between the 4 main aspects of the loops (number of tiles, inside free edges, outside free edges and total free edges). The diagrams page shows what shapes I made with the octagons to gather my results. Results My results for the octagon loops are shown below: No. Of Octagons Inside

#### In this assignment I am going to try to find the relation between the t-total and the t-number and then will express this in an algebraic form. I have been asked in the question to find the relationship between the t-total ad the t-number in a nine by

Abdul Khan W3 16/06/01 T-SHAPES Introduction This assignment is called 'T-Shapes'. In this assignment I am going to try to find the relation between the t-total and the t-number and then will express this in an algebraic form. I have been asked in the question to find the relationship between the t-total ad the t-number in a 'nine by nine' grid; I will d this by creating a table for the t-total and t-number. Hence I will try to discover the common difference and then fid the formula connecting the t-number to the t-total. 2 3 4 5 6 7 8 9 0 11 12 13 14 15 16 17 18 9 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 75 76 78 79 80 81 'T-TOTAL' Add all the numbers up including the t-number. 50 51 52 60 'T-NUMBER' The number at the bottom of the 'T' 69 algebraically Classified an 'n' I am investigating the relationship between the t-total and the t-number. 37 20 42 21 47 22 52 23 57 24 62 25 67 26 72 27 77 28 82 29 Pattern: I have noticed a pattern in my results that each time the t-number increases by 1 the t-total increases by 5. I know that; As 5 is the most common difference the must be a '5n' in the formula. 22 23 24 32 41 Un

#### T - ToTaLz WkD CwK- ChEk it OuT !!!!

Muslimz 4 lyf! GCSE Maths Coursework Tasks ) Investigate the relationship between the T-total and the T-number 2) Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T- number and the grid size. 3) Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations. Plan For my GCSE Maths coursework I am going to look and analyse at a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the T-shape. This is highlighted in the colour red. This is shown below: - The total number of the numbers on the inside of the T-shape is called the T-total. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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 The total of the numbers inside the T-shape is 1+2+3+11+20=37 This is called the T-total. The number at the bottom of the T-shape is called the T-number.

#### T-Totals. We have a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape.

T-Totals PART 1 We have a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape. This is highlighted in the colour red. This is shown below: - 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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 The number 20 at the bottom of the t-shape will be called the t-number. All the numbers highlighted will be called the t-total. In this section there is an investigation between the t-total and the t-number. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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 For this t-shape the T-number is 20 and the T-total is37 For this t-shape the T-number is 21 And the T-total is 42. As you can see from this information is that every time the t-number goes up one the t-total goes up five. Therefore the ratio between the t-number and the t-total is 1:5. This helps us because when we want to translate a t-shape to another position. Say we move

#### T- Total. You will notice that the centre column of the T-Shape is going up in 9s because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this size grid.

T- Total 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 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 Look at the T-shape drawn on the 9 by 9 number grid. The total of the numbers inside the T-shape is 1+2+3+11+20=37 This is called the T-total. The number at the bottom of the T-shape is called theT-number. The T-number for this T-shape is 20. If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this. T-17 T-18 T-19 T-9 T You will notice that the centre column of the T-Shape is going up in 9's because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this size grid. This is done in the working below:- T-total = T-19+T-18+T-17+T-9+T = 5T-63 Now to test this formula to see if it works For T-Total I will use the letter X For the T-Number I will use the letter T So X = 5T-63 T = 20 X = 5x20-63 = 100-63 = 37 Now I will do two more to check to see if it will work anywhere on the grid. T=43 X=5x43-63 24+25+26+34+43=152 X=215-63 X=152 T=49 X=5x49-63 30+31+32+40+49=182 X=245-63 X=182 I have tested the formula for three different T-numbers and the formula