Welcome! Here is my contribution to Math Twitter Blog-o-Sphere Mission #1: What is one of your favorite open-ended/rich problems? How do you use it in your classroom?
I was looking for a way to open a discussion of functions with one of my tutoring students last summer and found a lesson on the Internet called The Shapes of Algebra, which (at least in one version) included a great Ice Breaker activity. (See references below.) This activity asks students to consider five functions – represented in five different ways (verbal, tabular, graphical, function notation, and as an application problem) – and determine which representations belong together. One example is shown below.
What I Did
After printing the file, cutting apart the different representations and mixing them up, I gave my student the pile of cards with only the briefest of explanations: Which ones go together? It was wonderful to watch how engaged he was as he first figured out what he was looking at, and then went to work puzzling out which cards described common situations. His initial groupings contained several errors, but he corrected most of them on his own as he worked through the set of cards. To finish the activity, he taped each set of representations to a piece of paper labeled with the name of the function, creating a personal reference for the rest of our lessons.
In addition to creating a introductory experience with different types of functions, this activity generated lots of great conversations about interpreting graphs, identifying slopes and rates of change, as well as the meaning of function notation. Each of the five functions are completely different (linear, quadratic, reciprocal, absolute value, and exponential) and nicely set the stage for the rest of my planned function lessons.
How I Would Use It
In a classroom, I would definitely use this lesson as a group activity. I would create “kits” of the cards so that (ideally) each group would have a complete set for a different function along with lots of cards for other functions so they had opportunities to compare and contrast different representations. I think this activity would spark a great discussion and I would follow it with The Shapes of Algebra explorations using the Function Explorations document I created. The document both structures the activity and provides a nice way to collect students’ observations about the way the functions change. Desmos is a great tool for students to use for graphing the functions, but a graphing calculator (or even pencil-and-paper!) will also work.
What would you do differently?
There are several links for The Shapes of Algebra handout (including one for an online version), but I can’t seem to find the link where I found the ice breaker activity. You can use my link, or Google (including the quotation marks) “Mary owes her mother $7” and you should find the file.