Needed some fresh inspiration for the intro to the volume/surface area unit in 7th grade. We start with rectangular prisms, which were beat to death the year before.

I needed to:

- review past concepts without just reteaching
- emphasize the fact that now that we can solve equations, our use of these formulas takes on a new look

I was doing some planning last weekend, mostly letting ideas percolate around the question ‘when do we need to manipulate dimensions of cuboids‘ and I suddenly thought of the mat that clerks at Post Office use. And one of our parents is a manger at the post office across the street. One email later I was in position of an official USPS SmartMat. You know, the one that looks like this:

(am I the only one who was always intrigued by this thing?)

The basic gist of it: place a package on the ‘red’ arrow. The horizontal and vertical axis measure length and width. The curved line which the opposite corner of the box covers is the height at which the entire box is more than 1 cubic foot. So what we have are equal-volumed measurements!

## What we did:

We started off by using 12 hands and rulers to build a cubic foot. Bigger than you think.

Then we explored these three questions.

- I put random stickers on.
*Predict*which will have the most surface area – then divide and calculate. - Let’s pick one height line – where on the line is the best surface area.
- I want a box of surface area 880 square inches – find any point which will yield this (hoping for a pattern that we could use to predict) (this is the key one for getting at my goal of finding unknown dimensions by solving an equation)

## Things That Worked

- I used prediction a few times – I’ve gotten a lot better at doing that this year – buys engagement, and gets the synapsis firing before calculation (ie, ‘I think the endpoints because they have the biggest single measurement’)
- I had a box on hand near a cubic foot. Great for prediction and demonstration of the use of the mat. Also having a box greased the wheels for a discussion why we would want to minimize surface area.

## Things That Didn’t

- The curves aren’t position so that the ideal surface area comes from the center or anything. This is okay, it just means I probably won’t be able to really get them thinking about why the optimum points are so.
- I need to break down the worksheet for the kids a little. Most where alright with it, but not everybody.

Thanks for the mat Tim!