Engaging students: Finding the volume and surface area of spheres

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Austin DeLoach. His topic, from Geometry: finding the volume and surface area of spheres..

What interesting word problems using this topic can your students do now?

I found an interesting word problem that has to do with finding the size and density of Pluto using satellite images and data at https://spacemath.gsfc.nasa.gov/Geometry/6Page143.pdf that would be a good way for students to practice finding the volume of a sphere among other things. This problem could not be used at the very beginning of the section, but it is definitely interesting and could be very engaging for some students. There are multiple parts to the problem, but the third part has students calculate the volume of Pluto using the scale of measurement that they discovered in an earlier part. Students would then use their calculated volume to determine the density of the planet and compare it to other common things by using the given mass of the planet. Not only is this practice for the students to be able to calculate volume of spheres, but it helps them by showing further applications and how their calculated volume can be used to make more scientific discoveries. Problems like this are very good for students to see so that they can recognize real-world application for what they are learning in school, even if it is simplified for the sake of the class.

How could you as a teacher create an activity or project that involves your topic?

One idea that I think is interesting and engaging for students is taking an orange, measuring the diameter, then seeing how many circles of the same diameter the removed orange peel can fit in. There is a short demonstration video at https://www.youtube.com/watch?v=FB-acn7d0zU to see what I mean. This is a good activity because it is very hands-on for students to be actively engaged, and it also helps students recognize that mathematical formulas are not just thrown together, but there is reasoning behind all of them. This will also help the students remember the formula for the surface area of a sphere, as they will be able to think back to this activity and remember the time that they discovered the formula on their own. There is potential to be messy with this activity, but because it is such a memorable activity and will genuinely engage the students and let their curiosity about mathematics come to life, it is worth it if you can set aside the time for clean-up afterwards.

How has this topic appeared in pop culture?

A big place for volume and surface area of spheres to come up in pop culture is in sports. One recent situation can be seen at http://www.espn.com/espn/wire/_/section/ncw/id/18605942 where the Charleston women’s basketball team had to forfeit two victories because their basketballs for those games were not regulation size. The team accidentally used NCAA men’s basketballs (which have a circumference of 29.5-30 inches) instead of the standard women’s basketballs (circumference of 28.5-29 inches). Because the balls were not regulation size, the victories did not count. Students could use the given circumferences to find the surface area and volumes of each ball and see how significant the difference is, then discuss with their peers what the significance of different sized basketballs is. Although this is not an advanced practice idea, it is still a way for students to compute volume and surface area, as well as discover the significance of each of those properties in a way that could interest them, as many students are interested in sports and do not often think of math as playing a significant role in them. Computing the volume and surface area of the basketballs would also help them recognize the relationship between those and circumference.

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