SWBAT determine whether data tables show quadratic or cubic functions and find functions to match quadratic and cubic graphs and data tables.

What relationships are being shown here? Students apply their understanding of surface area and volume functions to understand different representations of these relationships.

60 minutes

Throughout my attempt to implement the Common Core Standards this year, I have been surprised by the fact that simply varying the representation I provide can really change the way students think about a problem. Once I find a problem that seems rich enough, I have been trying to think about as many different ways as I can to ask my students questions about this problem.

The Surface Area and Volume Investigation for today is based on this concept—I stick with quadratic and cubic functions that could be generated by finding the surface area or volume functions for a rectangular prism. I wanted students to be able to understand the problems both concretely (by building or visualizing a specific prism) or abstractly (by looking at the data tables or graphs out-of-context).

Rather than set up this investigation using levels, I wrote various types of problems that I thought would push people in different ways. I wasn’t sure which problems would be the most challenging, so I told students to get started and as they worked I asked them questions about what made sense and didn’t make sense.

Often I have found that when I give students different representations of the same problem, they are initially very confused, so I think it is really important during these kinds of investigations to respond to every question with a question.

For instance if a student struggles with **Part_2**:

**Student:** I don’t get how to find the rules for these data tables.

**Possible Teacher Responses***: *

- What do you know about these data tables?
- What patterns can you find? Think about vertical and horizontal patterns.
- Is there a method you can use to determine whether this is a surface area or a volume data table?
- How does determining what type of data you are dealing with help you actually fund the function rule?

After our first day working on this investigation, I realized that the first two parts of the investigation were more challenging than I thought, so I created some additional problem sets for tomorrow’s lesson.

10 minutes