Find out the relationship of the dots inside a shape of different sizes.

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DOTTY PATTERNS

AIM:         I have been set a task for my coursework to find out the relationship of the dots inside a shape of different sizes.

PLAN: I have planned to use a specific quadrilateral shape for my investigation in which lines will be 45o         (diagonal), one dot to the other; touching each others ends and being closed from all sides. I will be using         the following technique for my investigation. First of all I will commence with the shape-size being 1 cm2         increasing it every step by another 1 cm2. At the same time I will be counting the dots inside that         particular         shape. I will be using this method until I find a pattern; thereafter I will generate a suitable formula from that         pattern.

METHOD: I will be using more or less 5 diagrams and possibly the 6th one for my prediction.

DIAGRAM 1

DIAGRAM 2

DIAGRAM 3

It seems that a pattern is forming for both. Firstly for the dots and area you add 4, there after you double the number, secondly for the perimeter you just add four at each level.

        

        DIAGRAM 4

        

DIAGRAM 5

        

 

I have realised that the comment I made before for the dots and area is incorrect. What I said was that after starting with four the number will double itself on each stage i.e. 4, 8, 16… etc.

PREDICTION: I think that the shape with the area of 6 cm2 will have 61 dots and its perimeter will have 24 dots.

If we look at the table we can see that for the difference for dots and area it is starting with four and at each stage another four is added. As for perimeter and area just add four at each stage.

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ANALASYS:        

FORMULA FOR DOTS

Seeing that there are 2 differences in this table, due to this I might be using a quadratic formula.

        So;

        an2 + bn + c

        

        a= is half the number of second difference

        n= any size

        c= output when input is ONE

TEST FORMULA:

        (2n2) + (-2n) + 1

        (2 x 42) + (-2 x 4) + 1

        32 – 8 + 1

        32 – 7 = 25

                        FORMULA CORRECT

TEST PREDICTION:

        ((2n2) + (-2n)) + 1

        (2 x 62) + (-2 ...

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