Strategy: Math Notes
Start with the Dunk Tank Problem from 2008 MCAS:
Give students a Math Notes organizer (template including space for: facts, questions, diagram, possible steps to try, solution) and have them fill it out (working independently first, then collaborating after completing as much as they can on their own). This question is a good one to use as it includes hidden questions (ex: you need to know the volume of the bucket before you can determine how many buckets it will take to empty the tank). There is a diagram of the tank, but students need to come up with their own sketch of the bucket. The solution requires use of multiple formulas but also asks students to reason with the results, not merely calculate by plug and chug. Since there are so many parts to this problem having an organization system is important. Most students can complete the individual parts of this problem but have trouble figuring out what those parts are. The Math Notes structure allows students to see the pieces of the puzzle spread out in front of them, which helps many students determine how to assemble everything into a solution.
Strategy: Integrated Math Engagement
By PreCalculus, students have studied many functions and their graphs, ranging from linear to logarithmic. While the shapes of the graphs, the forms of the equations and the applications vary widely, all graphs are shifted and stretched in the same way. Start class by asking students to plot the parent graph for all the types of functions they know (linear, absolute value, quadratic, cubic, exponential, logarithmic). Then ask students to graph a wide variety of shifts and stretches of these functions. They may start with tables but will quickly want to find shortcuts. Without answering questions, continuously refer students to the parent graphs. Eventually, bring students together as a class to make observations. Finally, write “y=a*f(b*x+c)+d” on the board and have students use the observations they made to describe the effect of each constant (a, b, c, d) on the graph of y=f(x).
Too often students study functions in isolation and memorize a different formula for each one. In reality, no matter what the function, multiplying stretches and adding shifts. Once students recognize this connection they will be well prepared to graph any function. In order to discover this connection, students need to see a variety of functions at once, in an integrated activity.