Perimeter difference = 4
Therefore:
P = 4b
P = 4 × 9
P = 36
Example 2:
Number of matchsticks on base = 11
Perimeter difference = 4
∴ P = 4b
P = 4 × 11
P = 44
Then I’m going to draw the matchstick staircase on the second graph paper to test my prediction, which contains nine matchsticks on the base.
To test my prediction and the general rule I’m going to count the number of matchsticks on the perimeter, which I have drawn on the second graph paper, which contains nine matchsticks on the base and it is thirty-six matchsticks on the perimeter. Therefore my prediction proves that using general rule to find out the number of matchsticks on the perimeter is right.
Investigation to find out the total number of matchsticks in a staircase using a different general rule.
Now I’m going to investigate the total number of matchsticks in a matchstick staircase using a different general rule and I am going to put the number of matchsticks on the base and the total number of matchsticks to find out the difference in total number of matchsticks in a table.
t = total number of matchsticks
b = number of base
When the 1st difference is not the same but the 2nd difference is, the formula will follow the quadratic pattern:
t = an2 + bn + c
But I am going to use different letters.
t = total; b = base
t = xb2 + yb + c
x is the coefficient of the 1st term and is always ½ the 2nd difference.
So in my example x is ½ of 2 = 1.
∴My formula will begin with 1b2 = b2.
To find ‘c’ I have to find the value of t, when b = 0. So I am going draw another table.
In my table when b = 0, t = 0 so c = 0.
Now I have to find the value of ‘y’.
x + y + c = the value of t when b = 1 (from the table)
x + y + 0 = 4
1 + y + 0 = 4
∴ y = 3
So my formula should be:
b2 + 3b = t
Now I am going to test my formula:
Example 1: when b = 6; t = 54
b2 + 3b = t
62 + 3(6) = 54
36 + 18 = 54
I am going to try another example to make sure whether my formula is right or not.
Example 2: when b = 2; t = 10
b2 + 3b = t
22 + 3(2) = 10
4 + 6 = 10
Therefore I would say my formula is right and using this formula I will predict that the total number of matchsticks in the diagram when the base is 9 is:
b2 + 3b = t
92 + 3(9) = t
81 + 27 = 108.
Introduction
I am going to do another investigation to find out the matchsticks on the perimeter and the total number of matchsticks in a double matchstick staircase. I have drawn four diagrams (see the graph attached).
Investigation to find out the number of matchsticks on the perimeter in a double matchstick staircase using the GENERAL RULE.
Table to show the number of rows, number of matchsticks on perimeter and the total number of matchsticks.
r = number of rows; P = number of matchsticks on perimeter; t = total number of matchsticks
And I am going to make another table to find out the perimeter difference from the number of rows and the number of matchsticks on perimeter.
When the perimeter difference is the same therefore the general will be linear pattern:
P = 6r - 2
I am going to do an example to find out the number of matchsticks on the perimeter using this general rule.
Example 1: When r = 3;
P = 6r - 2
P = (6 × 3) - 2
P = 16
Example 2: When r = 9;
P = 6r - 2
P = (6 × 9) - 2
P = 52
Then I’m going to draw the double matchstick staircase on the graph paper to test the general rule, which contains nine rows. When counting the staircase it contains 52 matchsticks on the perimeter. Therefore that double matchstick staircase proves that the general rule is right.
Investigation to find out the total number of matchsticks in a double staircase using a different general rule.
Now I’m going to investigate the total number of matchsticks in a double matchstick staircase using a different general rule and I am going to put the number of rows and the total number of matchsticks to find out the difference in total number of matchsticks in a table.
t = total number of matchsticks
r = number of rows
When the 1st difference is not the same but the 2nd difference is, the formula will follow the quadratic pattern:
t = ar2 + br + c
But I am going to use different letters.
t = total; r = rows
t = xr2 + yr + c
x is the coefficient of the 1st term and is always ½ the 2nd difference.
So in my example x is ½ of 4 = 2.
∴My formula will begin with 2r2.
To find ‘c’ I have to find the value of t, when r = 0. So I am going draw another table.
In my table when r = 0, t = -1 so c = -1.
Now I have to find the value of ‘y’.
x + y + c = the value of t when r = 1 (from the table)
x + y + -1 = 4
2 + y + -1 = 4
∴ y = 3
So my formula should be:
2r2 + 3r -1 = t
Now I am going to test my formula:
Example 1: when r = 3; t = 26
2r2 + 3r -1 = t
2(3)2 + 3(3) -1 = 26
2(9) + 9 -1 = 26
18 + 9 -1 = 26
I am going to try another example to make sure whether my formula is right or not.
Example 2: when r = 4; t = 43
2r2 + 3r -1 = t
2(4)2 + 3(4) -1 = 43
2(16) + 12 -1 = 43
Therefore I would say my formula is right and using this formula I will predict that the total number of matchsticks in the diagram when the rows are 9 is:
2r2 + 3r -1 = t
2(9)2 + 3(9) -1 = t
2(81) + 27 -1 = 188.
