# Balloons +Tangerines

I totally stole adapted Sam Shah’s beautiful and inspired Related Rates and Balloons Lesson for my 8th grade math class.  I remade the graphic organizer to be a little friendlier looking for my middle school students, and to give them a chance to practice drawing with a compass and ruler.  They particularly liked the language of the degenerate circle.  From hence forth, that’s how we’ll be referring to all points.

After wrangling about the balloons not really being round, and about the fact that circles only exist in your mind anyway (you really need to enjoy these moments if you’re going to be spending time with 13-year olds), we measured the circumference, surface area, volume, and weight of balloons inflated with one, two, three, four and five breaths, and then calculated radius and diameter.  To connect with our study of functions and graphs from last month, we plotted and graphed our results and compared the linear and not-so-linear functions.  This was a great hands-on way of comparing rates.  I saw light bulbs go on for several kids when they realized that one function was getting bigger faster than another.

Two things I didn’t foresee:

• Some of my kids are actually small enough that they couldn’t blow up the balloons.  Not a big deal, but interesting to remember.
• Many kids wanted to inflate the balloons with more than 5 breaths.  I encouraged this, but somewhere around 8-10 breaths, my dollar-store balloons began to give out, and we were all a little on edge from popping balloons by the end of class.  Next time, limit to 5 breaths, or buy better ballons!

I like using the highly technical tangerine peel method to help explain the surface area of a sphere, (Thanks to Miss Quinn for reminding me of this!) though mapmakers might have something to say about trying to flatten the sphere in this way…