# The target of this first part of my investigation is to graphically prove, that in the drawn rectangle there are 20 squares.

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Introduction

Aim 1:

The target of this first part of my investigation is to graphically prove, that in the drawn rectangle there are 20 squares.

The design I am studying is a 3 by 4 rectangle.

I will count the total number of squares contained in this rectangle, and check whether my answer corresponds with the one recoreded above.

The method I use to calculate the total number of squares is the follwing:

In order to prove that my method used for finding out the number of squares contained in a certain design works, I will find out the number of squares contained in this design.

This design is a 4 by 6 rectangle.

I will again count the total number of squares contained in this rectangle, and graphically display my working out.

Aim 2:

I am going to extend my investigation by attempting to find a formula that enables me to calculate the total number of squares, for rectangles of any sizes.

Method:

I will begin by changing one feature of the rectangle, and see how the total number of squares contained in that rectangle varies. The feature I chose to vary is width, therefore I will analyze a series of rectangles which will have same values for width, but a different length each time.

Middle

This is the form in which my ultimate formula will be set in. I could leave this formula as my ultimate formula, but this wouldn’t enable me to reach the goal I have mentioned in my aim. In order to complete my task, I am going to investigate this formula further, by generalising it more. The coefficient m, and k vary as the length and the width of the rectangle vary, therefore before using the above formula, I would have to calculate the coefficent by looking at my table. Only when I found out the coefficient I would be able to work out k ( variable to be added to find out the total number of squares). This means my formula is not efficient enough, because it requires me to do other calculations other than the one summarized by the formula.

Aim 3:

The final aim of my investigation is to find out a relationship between the width and the length’s coefficient m, and a relationship between the width and the variable k, in a rectangle of any length or width. The formula I will produce, will enable me to find out m and k just by knowing the width. By replacing m and k in the formula above

Conclusion

k= - w ( w + 1) ( w – 1 ) / 6

for the rectangles we analyzed the value of k always seemed to result negative, that is why I put a – in front of the formula. This may not be true for every rectangle, but we could not find out due to lack of enough data.

I have reached the goal of aim 3, now I will display my ultimate formula:

T = w ( w + 1) ∕ 2 l - w ( w + 1) ( w – 1 ) / 6

When

l ≥ w

Where:

T= total number of squares

w= width of rectangle

l= length of rectangle

I found out that this formula works only if the length of the rectangle bigger or equal to the width of the rectangle. In my set of data I only had one rectangle that had these characteristics; a rectangle of length 2 and width 3.

I will now prove that my formula works for rectangles of any width or length.

- w= 4 squares

l= 3 squares

T= 20 squares

20= (4 ( 4 + 1 ) / 2 ) * 3 4 ( 4 + 1) ( 4 – 1 ) / 6

20= ( 20 / 2 ) * 3 - 60 / 6

20= 30 - 10

20= 20

- w= 5 squares

l= 7 squares

T= 85 squares

85= (5 ( 5 + 1 ) / 2) * 3 - 5 ( 5 + 1) ( 5 – 1) / 6

85= (30 / 2) * 3 – 120 / 6

85= 45 – 20

85= 25

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