Teaching Notes:
In this lesson, students look at a parabola as a squared function. Specifically, they will look at sequence of numbers (ordered pairs) that would form a parabola. In this lesson, the use of Microsoft Excel helps students to see the correllation between consecutive differences and a parabolic sequence.
The Process:
The Exploration:
How many consecutive differences did it take to reach the column of constants? Will this always be the case with Parabolic numbers? Make a conjecture and put it to the test: try different parabolic functions in column B. (We started with "x2" you may wish to try things like "5x2" instead.) You may also wish to look at other, non-parabolic functions, like "3x" or "7x3+5".
The Solution:
Parabolic functions have the nice property of reaching the column of constants after exactly two consecutive differences. In fact, the actual pattern for any polynomial of degree n is that it will reach the column of constants after exactly n consecutive differences. Some students may come close to this conjecture after doing the exploration. If so, the teacher should push them to test it and convince themselves of it.
Homework Problems:
Tell whether or not the following sequences of numbers are parabolic or not:
Bonus:
Solutions to Homework Problems: