I. The objective is to maximize the number of pieces of pizza resulting from each cut. The pieces do not have to be the same size.
II. Determine the maximum number of pieces resulting from each number of cuts.
III. Now, determine the following:
The Pizza Problem
Method 1: Algebra. This is the most complex method, but involves a lot of fundamental algebra skills as well as some important critical thinking.
A. First, note that at 0 cuts, there is 1 piece(the entire original pizza). Also note that since the difference in the maximum number of pieces increases by one each time, the matching equation must be of degree 2.
B. Therefore, we can start with the general quadratic equation:
ax^2 + bx + c = 0
and plug in values to produce three specific equations:
- a(0^2) + b(0) + c = 1. Therefore, c must equal 1.
- a(1^2) + b(1) + 1= 2 and
- a(2^2) + b(2) + 1 = 4.
C. Then, solve the resulting system of equations: a + b = 1 and 4a + 2b = 3
D. Therefore, the equation is:
y = .5x^2 + .5x + 1
Note, this can also be solved using matrices.
Method 2: Geometry
Assuming the class has already discussed triangular numbers, note how the triangular numbers: 1, 3, 6, 10, 15 compare to the pizza numbers: 2, 4, 7, 11, 16.
A. Since the equation for triangular numbers is n(n+1)/2; we can obtain the pizza numbers by n(n+1)/2 + 1.
B. Thus, y= x(x+1)/2 + 1 would be the appropriate equation. Though this doesn't seem like the same formula derived by method 1; a little algebra would prove them equivalent.
Method 3: Use Quadratic Regression on the data points to obtain:
y=.5x^2 + .5x + 1
This equation(both forms) can then be used to determine y when x=10, 16, 40, ect.
The Pizza Problem
First, note that after two iterations, the difference in the differences is the same. Therefore, we know the relationship describing the pattern is of degree 2 or quadratic.
We can now derive the formula for this pattern through a variety of methods.
Method I. Algebra
We can use the general form of a quadratic(ax^2 + bx +c =0) and the early values from the pizza problem to create a system of equations and then solve this equation for a, b, and c. Hint 1: This problem becomes much easier if we realize that at 0 cuts, there was 1 piece. Hint 2: Matrices are also useful in this situation.
Method II. Geometry
The "trick" here is to compare the pizza pieces to the triangular numbers and slightly adjust the formula for those triangular numbers:
Method III. Calculator
Simply enter the data into lists and run a quadratic regression to determine the appropriate formula: