What the 'L' - L shape investigation.

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Alexander Ford 11L

WHAT THE 'L'

Mathematics GCSE

Coursework

L Shape Investigation

The diagram below shows a standard L shape, which is drawn on a 9 by 9 number grid.

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81

The total of the numbers inside the L-Shape is 1 + 10 + 19 + 20 + 21 = 71.

This is called the L-Sum.

The number in the bottom left hand corner of the L-Shape is called the L-Number.

The L-Number for this L-Shape is 19.

This investigation is in three parts.

PART 1 Is to investigate the relationship between the L-Sum and the L-Number.

PART 2. Using grids of different sizes is to translate the L-Shape to different positions. Investigate the relationship between the L-Sum, the L-Numbers and the grid size.

PART 3. Investigate what happens if you use L-Shapes of different sizes. Try other transformations and combinations of transformations. Investigate relationships between the L-Sum, the L-Number, the grid size and the transformations.

I will use the following key to illustrate the algebraic terms that I will use during my investigation:

KEY

L

L-Number

C

Number of cells in L-Shape

G

Grid Size

?

The Sum Of

X

Number Of Cells Above The L-Number

Y

Number Of Cells Across From The L-Number

To start my investigation I will use the 9 by 9 grid section above I calculated the sum of the first seven L-Shapes; shown below:

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I calculated up to seven three by three L-Shapes and found their L-Sum (as this is the last L-Shape that will fit across a nine by nine grid). Numbers eight and nine and all other numbers in those columns will not be able to produce an L-Shape as the across axis arm will not allow an entire L-Shape. The next L-Shape number in the grid would be 28.

I will display my results from these calculations into a table format as follows:

Number In Sequence

2

3

4

5

6

7

L-Sum

71

76

81

86

91

96

01

I will now use the difference method to establish and determine if a sequence pattern exists between them. Therefore, the next step is to calculate the differences between each L-Shape.

Number In Sequence

2

3

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5

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7

L-Sum

71

76

81

86

91

96

01

Difference

5 5 5 5 5 5

As the first part of my investigation is to find the relationship between the L-Sum and the L-Number I am going to times the difference of the L-Sum with the L-Number. Thus giving me 5L.The results show that the 1st difference is constant and therefore, the formula must consist of 5L.

My results shown in table format are as follows:

Number in sequence

5L

Difference

Sum of L-Shape

(L-Sum)

95

-24

71

2

00

-24

76

3

05

-24

81

4

10

-24

86

5

15

-24

91

6

20

-24

96

7

25

-24

01

From the table of results I can see that the difference between 5L and the L-Sum is a constant -24. Therefore, we have 5L -24 present in the formula. I will use this formula to prove its correctness and to look further for additional differences.

Number In Sequence

Formula

Formula Equation

Results

L-Sum

5L -24

(5 x 19) - 24

71

71

2

5L -24

(5 x 20) - 24

76

76

3

5L -24
Join now!

(5 x 21) - 24

81

81

4

5L -24

(5 x 22) - 24

86

86

5

5L -24

(5 x 23) - 24

91

91

6

5L -24

(5 x 24) - 24

96

96

7

5L -24

(5 x 25) - 24

01

01

From these answers we can determine that the formula 5L is correct in a 5 cell standard L-Shape in a 9 by 9 grid. Therefore, the algebraic formula to ...

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