T-Total and T-Number - page 1
Keywords: T-Total and T-Number
By Grant867 on 07/12/2008
Level: GCSE Key Stage 4 (Years 10-11)
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T-Total and T-Number
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: -
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
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.
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
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 it to here
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
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
We all ready know the answer to the one in red. To work out the one in green all we have to do is work out the difference in the t-number and in this case it is 54. We then times the 54 by 5 because it rises 5 ever time the t- number goes up. Then we + the t-total from the original t-shape and we come out with the t-total for the green t-shape. This is another way to work out the t-total.
What we need now is a formula for the relationship between the t-total and the t-number. I have found a formula which is 5t-number-63 = t-total.
The question is how did we work out this formula and what can we do with it?
The formula starts with 5* the t-number this is because there is a rise in the t-total by 5 for every t-number. We then –63 which do by working out the difference between the t-number and another number in the t-shape. This has to be done to the other 4 numbers in the t-shape. Here is an example: -
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
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 t- shape has a t-number of 32. Now to work out the difference between the t-number and the rest of the numbers in this t-shape
Working out: -
32-13=19
32-14=18
32-15=17
32-23= 9
TOTAL= 63
This will happen to all the shapes this way up. To prove this I will do another.
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
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 t-number is 70. Now to work out the difference between the t-number and the rest of the numbers in this t-shape
Working Out: -
70-51=19
70-52=18
70-53=17
70-61=9
TOTAL=63
Again
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