Possibly relevant standards.
A. Algebra: representation, operations and modeling
- Linear equations and functions
- Quadratic equations and functions
- Exponential and logarithmic equations and functions
- Higher polynomial and rational equations and functions
- Functional representation and operations.
1. Using the parabolas at right, estimate the equations of the curves.
2. Discuss: how do you recognize a quadratic pattern from a table, and what does that have to do with the familiar parabola shape of the graph?
3. Use the idea of parallel and perpendicular slopes to give the coordinates for vertices of a square that has no sides parallel to an axis or to y=x. What is the area of your square?
4. Find a symbolic rule for one of the patterns below and look for connections between the symbolic and the visual. Extend the pattern one step to check your rule.
Problem 1 gives lots of nice information. Whether the student uses standard (harder) or vertex form, what the coefficients mean to them, and if they are consistent in applying that understanding amongst the different parabolas. Almost no one uses the roots, and I've yet to see someone apply regression to points.
Problem 2 pointed out that students over-identify parabolas with quadratics, as most students who struggled talked about increasing slope and symmetry as opposed to first or second differences. (Also that there was a homework that they didn't get to!)
Problem 3 was interesting in that no one started with the points. Everyone who tried it gave linear equations, established parallel and perpendicular, and struggled with how to get the sides to be equal length.
Problem 4 was the surprise. Most people chose the pattern on the left, and used visual identification of pieces to generate a direct rule. Only one student tried the pattern on the right. But instead of finding the quadratic pattern I intended, she noticed that each element was as long as the three previous elements summed. Definitely true for the picture... and got me to wondering if it would be quadratic. Nope. but definitely non-quadratic. I made a Google spreadsheet to compare, and this was unusual that the quadratic matched the sum of the first three pattern so well. I got interested in the ratios as they seemed to be converging. (Here's the spreadsheet if you want to play around for yourself.)
Finally I gave in, and looked up the sequence in the On-line Encyclopedia of Integer Sequences, where it is known as the tribonacci series. (Cute, eh?) This limit ratio is the solution of x^3 - x^2 - x - 1 = 0. I think I'll call it the Golden Ratio.
May your quizzes ever surprise you!