maths coursework t-shapes

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Introduction: 

In this investigation I will establish the link between the T-Total and the T-Number. The aim of my coursework is to find any pattern or link between T-Total and T-Number. By obtaining a formula, I will be able to find the T-Total if I am giving the T-Number and vice versa; without having to draw number grids. Furthermore, by obtaining various formulae, I will then be able to deduce a general formula for all grid tables.

After investigating the relationship between the two, I will then use grids of different sizes and translate the T-Shape to different positions; to see whether the pattern changes or stays the same.  

In this coursework I will be using the terms T-Total and T-Number frequently, so we need to know what they mean.

We have a grid nine by four with the numbers starting from 1 to 36. There is a shape in the grid called T-Shape. This is highlighted in the colour yellow.

This is a T-Shape drawn on a nine by four number grid.

The total of the numbers inside the T-Shape is 1+2+2+11+20=37.

This is called the T-Total.

The number at the bottom of the T-Shape is called the T-Number.

The T-Number for this T-shape is 20.

Method:

I will carry my investigation out in the following steps:

  1. Firstly, I will draw number grids of four, seven and six; plotting 3 adjacent T-Shape in each grid. My results with then be tabulated, this make it easier to compare the results and find a formula for each number grid.
  2. The formulae obtained from the grids, I will then be used to deduce a general rule which will apply to all the different grid numbers.
  3. My next step will then be to investigate further; checking whether different sized grids affects the results and the pattern we obtain. If this occurs, I will then have to find a different pattern or relation connecting T-Number, T-Shape and grid size.
  4. I will then try other transformations and combinations of translations; investigating the relationship between the T-Total, the T-Number and the translations.
  5. The last step will be to determine a formula which takes into account transformations and different grid sizes.

Below is an example of a nine number grid. In my investigation I will use the three, seven and six number grids. The below steps will be repeated for the seven, six and three number grids in the main part of the coursework.

Nine Number Grid


Table of Results

Another way in which we can work out the formula is by using algebraic terms, to replace numbers.

The T-Number is 21 we can work out what the numbers are in algebraic terms.

Ntot= N-19+ N-18+N-17+N-9+N

        = 5N - 63

This is an example of how I am going to work out the formulae for each of the number different number grids.

To further my investigation I will also be using rotations of 90 and 180 degrees clockwise. Below is an example of a 90 and 180 degree rotation in a nine number grid.

Another transformation I will use in this coursework is translation using vectors (    ). Below is an example of a (    ) translation on the yellow highlighted T-Shape.

 

The last thing I’ll do in my coursework is double transformation; below is an example of a 90 degrees clockwise rotation on the yellow highlighted T-Shape and then a (    ) translation on the red highlighted shape.

Hypothesis:

I believe that there is a relationship between the T-Number and T-Total. My hypothesis is that as the T-Number increases, so does the T-Total. This is because we know the formula for the nine number grid is 5n- 63, therefore a big the T-Number (n), will result in a big T-Total. We can prove this by putting the formula into practise:

If the T-Number is 30, the T-Total will be (5*30) - 63= 87, whereas if the T-Number is 15 the T-Total will be (5*15) – 63= 12. This proves that there is a relation between the T-Number and T-Shape; the higher the T-Number, the higher the T-Shape is going to be.

On the other hand, I think that changing grid sizes will affect the pattern, however only to certain extend. I predict that when the grid sizes change, the formula will also change slightly. For example, the formula for grid table is 5n – 63, whereas the formula for the seven number grid will be 5n – x.  This shows that the 5n stays the same; however second difference changes with respect to its size; if the number grid is six the second difference will be bigger than when it is seven by seven.

The aim of my investigation is find a link between the T-Number and the T-Total. To achieve my aim I am going to draw three number grids, with 3 adjacent T-Shapes on each. The results of each number grid will then be tabulated and a pattern or relationship will then be worked out; this way we can obtain a formula.

Seven Number Grid        

Table of Results (Seven Number Grid)

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We can test this formula by finding the T-Total, when we know the T-Number. The T-Number is 82, so the T-Total must be:

5n- 49 = x

(5*82) - 49= x

410 - 49= x

361 = x

We can also work out the T-Total by using the above grid; (67+68+69+75+82) = 361, however this tends to be time consuming, which is why we use a formula.

The next step of my investigation is to draw a six and four number grid, with 4 adjacent T-Shapes on it. ...

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