In this investigation we have to find a relationship between the numbers in the T and the T-Total number.
Matthew Worthington 10M Maths GCSE Coursework T-Total investigating monitoring cover sheet Introduction In this investigation we have to find a relationship between the numbers in the T and the T-Total number. We are using a nine by nine grid and we will move the T is a systematic way to see if wee is able to find a relationship between the numbers across the grid. 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 The total of the numbers inside the T are: + 2 + 3 + 11 + 20 = 37 The number at the bottom of the T is called the T number: 2 3 4 0 1 2 3 9 20 21 22 28 29 30 31 To make sure you get a different T number each time, you need to move the T shape around the number grid. 2 3 4 2 21 4 5 6 4 23 7 8 9 7 26 T- Number T - Total 20 37 21 42 22 47 23 52 24 57 25 62 The pattern in this table is for the T- Number it is plus one every time and for
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
I am going to investigate how changing the number of tiles at the centre of a pattern, will affect the number of border tiles I
Contents Page 1 ~ Introduction Page 2 ~ Patterns for 2 centre tiles Page 5 ~ Patterns for 3 centre tiles Page 8 ~ Patterns for 4 centre tiles Page 11 ~ Patterns for 5 centre tiles Page 14 ~ Summary for patterns with a single row of centre tiles Page 16 ~ Patterns for 4 centre tiles Page 19 ~ Patterns for 6 centre tiles Page 22 ~ Patterns for 8 centre tiles Page 25 ~ Patterns for 10 centre tiles Page 28 ~ Summary for pattern with a double row of centre tiles Page 30 ~ Summary for single and double rows of tiles Page 31 ~ Patterns for 6 centre tiles Page 34 ~ Patterns for 9 centre tiles Page 37 ~ Patterns for 12 centre tiles Page 40 ~ Patterns for 15 centre tiles Page 43 ~ Summary for patterns with a triple row of centre tiles Page 44 ~ Conclusion Borders Coursework Introduction For my experiment I am going to investigate how changing the number of tiles at the centre of a pattern, will affect the number of border tiles I will need. I will do this to find patterns and a formula, to link back to each set of patterns. Each formula will be tested by using a larger border, but with the same number of centre tiles, this will ensure my formula is correct. I will then try to find a general formula, that will enable me to predict the border for any size centre tiles. I will also do the same for the total tiles in the pattern. Key N ~ Pattern B ~ Outer border
T-Totals Coursework.
T-Totals Coursework Maths UF 2003 By Jack Palmer Contents . INTRODUCING T's (Including T-Numbers & T- Totals) 2. MY FIRST EQUATION (Concerning 9 x 9 grids) 3. ROTATING T's (Upside down and back to front) 4. GRID SIZE AND THE T-TOTAL (i) Original T's (ii) Upside Down T's 5. STRETCHING T'S 6. VECTORS Maths Coursework T-Totals! ) This is a T-shape! It allows us to gather information into algebraic formulas to explain the relationships between numbers. This is the T-Number. It is the central part of our research. If you add up all the numbers in the T, you will find the T-Total! For the T above, the T-Total will be 1 + 2 + 3 + 9 + 16 = 31. 2) Using algebra, we can work out a formula for this T. On a 9x9 grid a T would look like this: From this we can see that if: T number = n = a 2 = b 3 = c 1 = d 20 = n a = n-19 From this we can see that the T-Total b = n-18 will equal: c = n-17 d = n-9 1 + 2 + 3 + 11 + 20 = 37 e = n Using the algebraic formula for each of the numbers we can see that: T-Total = (n-19) + (n-18) + (n-17) + (n-9) + (n) = 5n-63 We can see that if we apply this formula to a 9x9 grid we can find the T-total, and we can prove this by testing it on 1
Investigating the links between the T-number and the T-total on a size 9 grid
Investigating the links between the T-number and the T-total on a size 9 grid Richard Smith 5? 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 I will be using the number at the bottom of the T as the T-number, so in these two examples the T-numbers are 20 and 51. This will be the layout of each T: - T2 T3 T4 T1 T-number Take the differences between the T-number and T1, 2, 3 and 4. n-19 n-18 n-17 n-9 n 2 3 1 20 These differences are the same throughout the grid (size 9). Examples n-19 n-18 n-17 n-9 n 32 33 34 42 51 n-19 n-18 n-17 n-9 n 6 7 8 26 35 If you take all the differences, which add up to be -63 and take that from 5 (the amount of numbers in one T) multiplied by the T-number, it gives you the T-total. Here is the formula: - 5n-63=T-total I will now test this formula using some of the T shapes above. 6 7 8 26 35 2 3 1 20 32 33 34 42 51 5n-63 = T-total 5(20)-63 = T-total 37 = T-total Also: + 2 + 3 + 11 +20 = 37 5n-63 = T-total 5(51)-63 = T-total 92 = T-total Also: 32 + 33 + 34 + 42 + 51 = 192 5n-63 = T-total 5(35)-63 = T-total 12 = T-total Also: 6 + 17 + 18 + 26 + 35 = 112 As I have proved, the formula is correct for the T shape in a grid size of 9.
T-Total investigating monitoring
Adam Taylor Maths GCSE Coursework T-Total investigating monitoring cover sheet Introduction In this investigation we have to find a relationship between the numbers in the T and the T-Total number. We are using a nine by nine grid and we will move the T is a systematic way to see if wee is able to find a relationship between the numbers across the grid. 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 The total of the numbers inside the T are: + 2 + 3 + 11 + 20 = 37 The number at the bottom of the T is called the T number: 2 3 4 0 1 2 3 9 20 21 22 28 29 30 31 To make sure you get a different T number each time, you need to move the T shape around the number grid. 2 3 4 2 21 4 5 6 4 23 7 8 9 7 26 T- Number T - Total 20 37 21 42 22 47 23 52 24 57 25 62 The pattern in this table is for the T- Number it is plus one every time and for the T-Total
The Relationship between the T-number and T-total
Mathematics coursework by Nanjie Lu The Relationship between the T-number and T-total Question One * Investigate the relationship between the T-total and the T-number. First of all I want use letter n for T-number, use letter T for T-total, use letter g for size of number-grid. This question already told me that the size of number-grid is nine. So I am copied a size nine number-grid below: In this table I chose a group of numbers to compare them and try to find out a relationship between one number and the total amount of this group, the numbers are below: 2 3 0 1 2 I chose the number ten for basic number, I am going to use letter X for this basic number. Also I am going to use letter Y for the total amount of this group. So the result is showing below: X-9 X-8 X-7 X X+1 X+2 Next step is work out the how much Y is (the total amount of this group). Add them all together equals: So the relationship between the Y and X is: Y = 6X - 21. I am going to check the answer above to see if it is right: * X = 10 * Y = 1 + 2 + 3 + 10 + 11 + 12 = 39 * 39 = 6 × 10 - 21 Now I can say this formula can be used for these numbers in the group above. Now I try to use same way to find out the relationship between the T-total and T-number is a size 9 by 9 number-grid. I chose three T-shapes from the table above, they are: 2 3 1 20 6 7 8 6 25 59 60 61 69
Relationship between T-total and T-number I am going to investigate T-totals and T-numbers on a 9x9 grid.
Relationship between T-total and T-number I am going to investigate T-totals and T-numbers on a 9x9 grid. The T-total is when a T-shape is placed on a grid and the numbers in the T-shape are added up to give a total. My aim is to find a relationship between T-totals and T-numbers. I am going to translate the T-shape 5 times on a 9x9 grid and then do the same on various sized grids. T total = 1 + 2 + 3 + 11 + 20 = 37 [All numbers added together] T number = 20 [number at the bottom of the T shape] I am going to translate the T shape across the 9x9 grid. T total = 1 + 2 + 3 + 11 + 20 = 37 T number = 20 T total = 2 + 3 + 4 + 12 + 21 = 42 T number = 21 T total = 3 + 4 + 5 + 13 + 22 = 47 T number = 22 I have translated the T shape across the 9x9 grid 3 times. So far I have noticed that when the T number increases by one, the T total increases by 5. I predict that if I move the T shape across once more I will get a T total of 52. I am going to do 2 more T shapes to confirm my prediction. T total = 4 + 5 + 6 + 14 + 23 = 52 T number = 23 T total = 5 + 6 + 7 + 15 + 24 = 57 T number = 24 I have tested my prediction and have found that my prediction was right. Now I am going to see what the relationship is between the T total and T number when I translate the shame down the grid. T total = 1 + 2 + 3 + 11 + 20 = 37 T number = 20 T total = 10 + 11 + 12 + 20 + 29 = 82
T-Total. I rotated the T round 180 and took another four readings and worked out that the Equation for this position was N=5N+63
Mathematics G.C.S.E. T-Total To start off with the task in hand, I set about the first question, by using the original grid of 9 squares using the first t-shape pictured. After this I took 4 readings on the original shape and worked out an algebraic relationship between the T-total (TT) and the T-number (N). After these readings I worked out: N=5N-63 After this I rotated the T round 180? and took another four readings and worked out that the Equation for this position was N=5N+63 To prove this formula, I worked out every number in the T in terms of N and came up with: N-19 N-18 N-17 N-9 N This helps you to work out T numbers when you know only N After this, to try to get some work done I tried to work out some proof for the 'other' T shapes rotations these were: N N+9 N+17 N+18 N+19 This also proves the theory I made earlier that the opposite angles are the opposite, i.e. plus and minus. Now I will try to work out proof for other rotations N-7 N N+1 N+2 N+11 N-11 N-2 N-1 N N+7 These conclude my theory that opposite rotations are opposite signs After these readings, I worked out equations for the T at 90?and 270? and arrived at the equations: 90?: N=5N+7 270?: 5N-7 After these initial set of results I worked out that the opposite T's were always positive and negative: 0?:Negative 80?: Positive 90?:Positive 270?:Negative This find, now