I gradually produce formulaes, to eventually get to a general one which gives you the total number of squares, knowing just one feature of the rectangle ( width or length).
I will explain the method in which I managed to come up with these different formulaes. Since I am varying the width, the formulaes will be representing a relationship between the length and the total number of squares, in any chosen rectangle. In order to make my explanations and working outs clearer, I will use an example. My example regards the relationship between the length and the total number of squares of a any rectangle of width= 5 squares. I have analyzed 3 rectangles with same width, but different lengths:
The first step to work out a formula was to calculate the gap between the total number of squares, in the rectangles with different lengths. In this case we notice that the gap is always of 15 squares, and we could describe the sequence in which the total number of squares varies, as a linear sequence ( would result as a straight line if represented graphically). I multiplied 15 to the values of: 4, 5 and 6, and I noticed that in order to obtain the respective total number of squares, I always had to subtract the same value: in this case 20. I used this method for evey width present in my table, and I found out that this formula worked.
T = mL + k
Where:
T= total number of squares in the rectangle
L= length of the rectangle
m= difference between the total no. of squares for the different lengths
k= variable which can be either negative or positive, which is added to the first part of the formula in order to give the total no. of squares.
W= 1 square
T= L
e.g 2 = 2
W= 2 squares
T= 3L - 1
e.g 5 = (2*3)-1
W= 3 squares
T= 6L - 4
e.g 20 = (4*6) - 4
W= 4 squares
T= 10L – 10
e.g 30 = (4*10) - 10
W= 5 squares
T= 15L – 20
e.g 85 = (7*15) – 20
W= 6 squares
T= 21L – 35
e.g 217= (12*21) – 35
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 with the formulaes I am investigating, I will have produced my ultimate formula that gives me the total number of squares.
Method:
I will record the single formulaes in a chart, this will facilitate me in finding a formula. I will begin by investigating the relationship between the width and the length’s coefficient m. I will then investigate the relationship between the width and the variable k. I will conclude my task by testing if my final formula works for every recatngle O have analyzed.
I will now draw a table displaying the width, and the respective formulaes broken down in two parts:
The purpose of this table is to analyze separately each part of the formula, to identify a pattern within each part. I will begin by analyzing the first part of the formula. We notice that the values of the coeficcients of the length, increase following a specific trend. I found out that if I subtracted each value to the subsequent one, the value of the result increased by one each time.
3 – 2 = 2
6 – 3 = 3
10 – 6 = 4
15 – 10 = 5
21 – 15 = 6
From the pattern I have identified, I may be able to produce a formula which expresses this sequence. The numbers present in this sequence are triangle numbers; they are called so because they can be represented graphically in a triangular shape. It is precisely this latter statement which gave me a clue to find the formula.
I will use an example, to clarify my explanations.
( I underlined the part of the formula relevant to my point)
If I imagine a rectangle which width is equal to the square of the width considered (4 squares )and its length is equal to the width plus 1, by using the formula w ( w + 1) I would be able to find out the area of this rectangle. If I divide the result by 2, I would obtain the area of a triangle which is exactly half of the rectangle previously considered: w ( w + 1) ∕ 2. Since the coeficcients of the lengths present in the first part of the formulaes are triangle numbers, this formula might relate the width and the coefficient of the length.
m= w ( w + 1 ) / 2
w ( w + 1) ∕ 2 } 4 ( 4 + 1) ∕ 2=
= 4 * 5 ∕ 2
= 10
I will apply this formula to different examples in order to see if it works for rectangles of any width.
5 ( 5 +1 ) / 2 =
= 5*6 / 2
= 15
6 ( 6 + 1) / 2 =
= 6*7 / 2
= 21
This is the first part of the formula I produced. This formula enables you to find out the length’s coefficient m, without having to calculate it by looking at the table . In other words, if the there is set of rectangles, which we know the width, this formula enables us to evaluate the difference beween the total number of squares of each rectangle.
I will now concentrate on the second part of the formula. By looking at it I can also identify a pattern. As the width increases, the numbers representing the variable k are tetrahedral numbers. These numbers can be represented graphically in a pyramidyal shape.
I have analyzed my table and I managed to produce a formula representing the relationship between the width and the variable k. the method in which I approached this formula, was different as for the previous one. I could not find a geometrical explanation due to the complication of the pyramydial shape. My approach was purely mathematical, based on the ideal of formula manipulation. With the following formula I would be able to find the variable k, just by knowing the value of the width, for rectangles of any length or width:
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.
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
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
