• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Investigation – T-Shapes

Extracts from this document...

Introduction

                        Kyla Middleditch

Investigation – T-Shapes

The task set is to investigate the way the totals within a 2D ‘T-shape’ can be calculated on a grid of various sizes in various positions.

1 . Investigate the relationship between the T-Total and the T-Number.

Diagram 1.

 The area defined by cyan shading on Diagram 1 is the “T-Shape”.image00.png

I will call the total of the numbers inside the T-Shape the T-Total.

I will also consistently use one number square of the T-Shape through out my investigation to relate my formulas to. I will call this number in the shape the T-Number, this number is indicated by the bold blue font on the T-Shape in Diagram 1.

For the purpose of consistency and methodical working all diagrams in Part 1 will be displayed on

    9x9 grids.

For the T-Shape shown in Diagram 1 the total is: 1 + 2 + 3 + 11 + 20 = 37

I continued to investigate the totals of the following T-Shapes, I created each new T-Shape by moving the shape one square to the right each time across the top row of the grid.

image01.png

...read more.

Middle

From my table of results (Diagram 7) I can predict that a T-Shape in the following location (Diagram 8) will have the following T-Total:

62 + 5 = 67.

However, I would like to investigate producing a consistent algebraic formula that could be used to find the T-Total anywhere on a 9x9 grid.

                                                Diagram 9.

image04.png

I will return to my predicted answer example, Diagram 8 and use the formula to work out its T-Total.

T-number of Diagram 8 = 26

For the purpose of developing algebraic formulae, the T-Number will be expressed as n.

T-total = (n) + (n – 9) + (n– 18) + (n – 19) + (n – 17)

We can simplify this to 5n – 63.

We get –63 because it is the total of the differences between the T-Number and the number in each of the other four squares. I can test the formula by using the numbers given in Diagram 8: (I subtracted each number in the T-shape from the T-Number except the actual T-Number square)

(26) + (26 – 9) + (26– 18) + (26 – 19) + (6 – 17)

This can be simplified as: 130 – 63 which equals a T-Total of 67.

To check the formula I can work out the T-Total by hand:

7 + 8 + 9 + +17 + 26 = 67.

...read more.

Conclusion

Diagram 11.

r = new number which must be investigated.image06.png

From Diagram 10 I can do the following calculations relating them to the framework of Diagram 11:

(8 – 5 = 3)  = N-r

(8 – 2 = 6 = 2 x 3) = N-2r

(8 – 6 + 1 = 3) = N-2r+1

(8 – 6 – 1 = 1) = N-2r-1

These calculations show that r=3.

So far the 3x3 formula looks like this:

(n) + (n-3) + (n-6) + (n-7) + (n-5) = T-Total                  Simplified as: 5n – 21.

Diagram 12.

image07.png

My previous formulas, for the 3x3 grid and the 9x9 grid, do not work on this 4x4 grid, they give the following answers:

5n-21 = 29

5n-63 = -13

        I know these formulas give me the wrong answer on this grid because I worked out the T-total by hand and it is: 22.

        I will use Diagram 11 and its explanation to try and work out a formula for the 4x4 grid.

N = (10)image06.png

N-r = (10 – r = 6)

N-2r = (10 – 2r = 2)

N-2r-1 = (10 – 2r - 1= 1)

N-2r+1 = (10 – 2r + 1 = 3)

10 – 6 = 4            10 = 1 = 11

10 – 8 = 2

10 – 1 = 9

        From looking at the equations above, the only number which would satisfy each one, is 4.

...read more.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE T-Total essays

  1. Number Stairs Investigation

    By drawing all my formulas together I can see a pattern occurring. The first set of numbers are in a CUBIC SEQUENCE; 6, 10, 15 etc. 3-step 4-step 5-step 8 x 8 6x + 36 10x + 90 15x + 220 9 x 9 6x + 40 10x + 100

  2. T-total Investigation

    8 by 8 I can now put expressions inside the T to check if my formula is correct. T- 17 T-16 T- 15 T-8 T The centre column is going up in 8's because of the grid size. With my T set out like this I can now check that

  1. Maths GCSE Coursework – T-Total

    If we note the same form of table we have used before, we can find the "magic number", the above graph shows a vertical translation of the T-Shape by +3, were v=46, t =210 which translates to, v=16, t=60. Middle number (v)

  2. Objectives Investigate the relationship between ...

    n+n+1+n+2+n-8+n+12 =n+n+n+n+n+1+2+12-8 (gather like terms) ='5n+7' Therefore my algebraic formula for finding the T-total of any 90� rotated t-shape is '5n+7' I will now test this formula to see if it works, I will test it using the T37 T-shape Substituting the T-number in place of 'n' I get: 5x37 + 7 = 192 The formula works!

  1. T shapes. I then looked at more of these T-Shapes from the grid in ...

    N - 17 N - 16 N - 15 N - 8 N 10 * 10 Grid 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

  2. T-shapes. In this project we have found out many ways in which to ...

    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

  1. T totals. In this investigation I aim to find out relationships between grid sizes ...

    - 56 t = 180 - 56 t = 124 Which is the same answer as before proving this formula works. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 On a 4x4 grid we can try the same method of generalization to

  2. The object of this coursework is to find the relationship between the total value ...

    The total value of this T-Shape is 40 (1 + 2 + 3 + 12 +22 = 40) and the N number is 22. To see if there is a sequence between the T-Shape, I will move the shape down one row.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work