#### I will be in investigating the relationship the between the t-total and the t-number. I will be using grids of different sizes to help me solve and find a formula.

The Investigation of T-Shapes Introduction I will be in investigating the relationship the between the t-total and the t-number. I will be using grids of different sizes to help me solve and find a formula. Investigation The total number of the numbers on the inside of the T-shape is called the T-total. The number at the bottom of the T-shape is the T-number. 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. 7 18 19 9 T 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 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 works for a 9 by 9 grid. The full formula for this size grid is: - X=5T-63 This time the centre column of the T-Shape is going up in

#### Modelling Motorway Toll Charging

Computer Science 111 Third Coursework Problem Motorway Toll Charging Problem The M111 is a newly opened toll motorway, with eight junctions at which the entry and exit of vehicles from the M111 is monitored. I must design and implement a program that can read the tolls data from a file and process the information to produce a table of users for that week and a summary. Form of Data Input tollsinfo file input Gravelly L123ABC J 23 15 30 Ashton ABN1123 J 11 0 56 . . . . . . . . . . . . . . . . . . . . . . . . Fratton L123ABC L 1 12 31 Form of Data Output (Results) Registration trips in wk tolls due speeding violations total due Number serious dangerous _______________________________________________________________ L123ABC 7 11.76 1 0 31.76 ***********************Summary*********************** Total revenue for week was £1129.23 Total number of different users was 24 Total number of peak journeys was 45 Total off peak journeys was 65 Total number of serious speeding was 11 Total number of

#### How I can get started. I could start by by drawing different T-shapes and using 12 different grid-sizes. I could explore as many aspects of the task as possible, explaining why and how, and develop the task into new areas.

Ryan Chimene 11V Mathematics Coursework (T-Shapes). What does the task tell me? This task tells me that I could investigate a number of different number grids and discover a lot of surprising information by so doing. It also tells me about the t-number and T-total and how I can create or identify one. What does the task ask me? This task is asking me to predict and check for facts and to work on different size grids. I have also been asked to write down questions which might occur to me and probably help me to find out more about the task. I have to link words, tables, diagrams, graphs, calculations and algebra. Lastly, I have to show how patterns have been used and to test my conclusions by providing proof of my results using mathematical models and new results from work previously done, using clear reasoning to prove my results. How I can get started. I could start by by drawing different T-shapes and using 12 different grid-sizes. I could explore as many aspects of the task as possible, explaining why and how, and develop the task into new areas. Which connection is possible? The connection between the graphs and the tables of my results. I had to find-out the type of pattern which would show-up if I used the data I placed in my tables. Is there a result to help me? Yes, the tables which I constructed by collecting a number of t-numbers and then working out

#### Maths Grid Coursework

I will now investigate different size grids. An 11 by 11 1 2 3 n=24 2 13 14 23 24 25 T=n+ (n-11) + (n-23) + (n-22) + (n-21) T=5n-77 Now test this: 24 25 26 n=47 35 36 37 T=5n - 77 46 47 48 T= (5 x 47) - 77 T=158 Check 24+25+26+36+47 =158 So this formula does work. A 6x6 grid 1 2 3 n=14 7 8 9 3 14 15 T=0 + (n-6) + (n-12) + (n-13) + (n -11) T=5n-42 Now to test it. 8 9 10 n=21 4 15 16 20 21 22 T=5n -42 T=(5x21)-42 T=105-42 T=63 Check T=8+9+10+15+21 T=63 So the formula is correct. I will now compare the formulas for the different grid sizes. 11 x 11= 5n - 77= 11x7= 77 9 x 9= 5n - 63 = 9x7= 63 6 x 6= 5n - 42 = 6 x 7= 42 5n is consistent because the T shape is always the same size. (With the same amount of numbers in side the T shape) The numbers which are underlined are all divisible by 7. This means that if you know the grid size you can multiply it by 7 and it would give you the number that you would subtract from the Tnumber. From this we can make a general formula for any size grid and any Tnumber. g=grid size The General Formula is: T=5n-7g N.B This formula only applies to a T this way up: Formulae Formula Shape of 'T' )T=5n-3 T(in 9 by 9 grid) 2)T=5n-7g` T(in any size grid) From these formulae I will investigate whether you could find the Tnumber from them instead of the Ttotal. Formula Shape Of T

#### I am going to investigate the relationship between the T-Totals and T-numbers when the T-shape is translated in different sizes of grids

T-Totals I am going to investigate the relationship between the T-Totals and T-numbers when the T-shape is translated in different sizes of grids. A good way of showing translations is by using vectors. To give you an insight of how the grids look I have used 3 different grid sizes which I will be investigating further on. In each column of the grids we see that every time 9,8 or 7 is added to the number and it follows this sequence and the numbers on a row when added contain, (9 by 9 grid) 81,(8 by 8 grid) 64 or(7 by 7 grid), 49 numbers. row The T-shape drawn on grids will look like this This is called the T-Number, I will refer this as N When adding all the number together we will get the T-Total I will refers this as T. In the next table I have calculated the T and N which gave me the following results: T 37 42 47 52 N 20 21 22 23 We can see from this information that every time the T-Number goes up one the T-Total goes up 5 if we were to skip one place than we would have to add 10 instead of 5 as you see in the above the table it shows us 37 and when skipping one place you get 47 the same thing is done to 42 and skipping a place will give you 52. Which will give me the following pattern: T + 5 N + 1 ratio 1:5 Now we got this information so we can find the formula. I have found a formula which is 5N - 63 = T I have

#### I am going to investigate T-Shapes.

GCSE Maths Coursework I am going to investigate T-Shapes drawn on a 9 by 9 square as 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 total of the numbers inside the shape is the t-total. The bottom number in the t-shape is called the t-number. I have worked out a formula to work out the t-total from the t-number. I am using N as the T-Number and T as the T-Total. N-19 N-18 N-17 N-9 N = N-19+N-18+N-17+N-9 = T=5N-63 I have now worked out a formula, which will enable me to calculate the t-total form the t-number. Firstly I must test the formula on some results I have already worked out. T-Number T-Total Manually Using Formula 20 37 37 21 42 42 22 47 47 23 52 52 24 57 57 25 62 62 26 67 67 As you can see the formula works and can be used to calculate any result. The results increase by five each time, this is why the T-Number is multiplied by five. Sixty three is then taken away to get the T-Total. Now I have worked out a formula for a 9*9 square I am going to look for similar trends in 9*9 to 14*14 which will give me 5 results. 2 3 4 5 6 7 8 9 0 1 2

#### Maths Coursework- Borders

Joanna Burton 10s 23rd June 2004 Maths Coursework- Borders QUESTION Figure below shows a dark cross-shape that has been surrounded by white squares to create a bigger cross-shape; The bigger cross-shape consists of 25 small squares in total. The next cross-shape is always made by surrounding the previous cross-shape with small squares. Part 1- Investigate to see how many squares would be needed to make any cross-shape built in this way. Part 2- Extend your investigation to 3 dimensions. Introduction - I am doing an investigation to see how many squares would be needed to make any cross-shape built up in this way. Each cross-shape is made by using the previous cross-shape and adding another layer of white squares, making all the inner squares black. The first cross-shape in the sequence is a single black square. To start my investigation I must draw the first 7 cross-shapes. This will enable me to see a pattern in the shapes so I can make a table and record how many black and white squares there are in each cross-shape I have drawn. From my table I must use the results to work out formulae for black, white and total number of squares. After this I will test the formulae on a pattern I have already drawn and on one I have not already drawn. I will be working systematically in my investigation

#### T-Total. I will be looking at T-shape drawn on different size grids. I will also be comparing and finding link and formulas between T-number and T-total.

T-Total In this investigation, I will be looking at T-shape drawn on different size grids. I will also be comparing and finding link and formulas between T-number and T-total. The different size grid will be: 10 by 10, 9 by 9, 8 by 8, 7 by 7, 6 by 6 and 5 by 5. I will use grids of different sizes to try out transformation, combinations of transformations and reflection. I will also be making tables to identify formulae. .In this experiment I will for the formula of a T, with the T-number facing southwards. My first table is the result from a 10 by 10 grid, this on page 1. Number T-total T-number 40 22 2 45 23 3 50 24 4 55 25 5 60 26 6 65 27 Here is the grid formula. n + n-10 + n-20 + n-21 + n-19 = 5n-70 and therefore formula for 10 by 10 is 5n-70. Here is 9 by 9 grids result from page 2. Number T-number T-total 20 37 2 21 42 3 22 47 4 23 52 5 24 57 Here is the grid formula n-19 + n-18 + n-17 + n-9 + n =5n-63 and therefore the formula for 9 by 9 grid is 5n-63. Here is 8 by 8 grids result from page 2. Number T-number T-total 8 34 2 9 39 3 20 44 4 21 49 5 22 54 Here is the grid formula n-17 + n-16 + n-15 + n-8 + n = 5n-56 therefore the formula is 5n-56. From the results above I can predict that the result for a 7 by 7, will be 5n-49. The formula I used to predict 7 by 7 is The gird size The formula 0

#### T-Total the investigation relies on investigating relationships between our three variables; t-number, t-total, and grid size

T-Total Introduction The T-Total investigation involves looking at the relationship between what are known as the t-number and the t-total in an n grid. The t-number can be thought of as the number at the base of the t shape, and the t-total the sum of all the numbers in the T. Therefore taking the following grid: n 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 The t-number is 31, grid size is 9 and t-total is 31+22+12+13+14=92 Therefore the investigation relies on investigating relationships between our three variables; t-number, t-total, and grid size. Further variable factors may be added along the way to increase the scope of the investigation. Variable Declaration Grid size n T-total t T-number x Individual numbers in 'T' a,b,c,d (see below) b c d a x Individual variables within the 'T' We shall first look at how we can derive t through looking at n and x. Contents Investigating t, n, and x 3 Verification 4 Alternate formula 5 Transformations 6 Vector Translations 8 Variable dimensions 9 Conclusion and Evaluation

#### T-Totals Investigation.

T-Totals Investigation: In this investigation I will be exploring the different relationships between the letter 'T' and its numbers and totals using a variety of different grid sizes and translations to effectively find any correlations between them. 2 3 1 20 This is the 'T' shape that I will be starting with. The total of the numbers inside the T-shape is: 1+2+3+11+20= 37 This is called the T-total. The number found at the bottom of the whole T-shape is called the T-number. In this case, the T-number for this T-shape is 20. Part 1: Investigate the relationship between the T-total and T-number. Starting with a 9 by 9 grid, a table for T-shapes going horizontally across the grid can be drawn: 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 T-number 20 21 22 23 24 25 26 T-total 37 42 47 52 57 62 67 +5 +5 +5 +5 +5 +5 A T-shape with the T-number 27 cannot be drawn as this would give an incomplete 'T'. As can be seen above, there is always an addition of 5. Next, I have drawn a table for the T-shapes going vertically down the same 9 by 9 grid: 2 3 4 5 6 7 8 9 0 1 2 3