Math = Love: Mindblowing Facts About Derivatives and Spherical Geometry

Saturday, September 12, 2015

Mindblowing Facts About Derivatives and Spherical Geometry

So, I mentioned in my previous post that I recently had my first experience with spherical geometry at math teachers' circle.

The session was called Lunes, Moons, & Balloons.  You know when a session starts off with balloons and sharpies being passed out that it's going to be a good one!

We began with a pretty simple worksheet to jog our memory of circle and sphere formulas.

I was coasting along pretty well until I hit question 4.  I don't teach geometry.  I've never taught geometry.  I don't remember the last time I needed to calculate the surface area of a sphere.  What in the world is the formula?

Now, I knew the formula for the volume of a sphere.  That's ingrained in my brain.  Probably from being on the academic team in high school.  The table I was setting at had an interesting mix of people.  1 elementary teacher.  3 middle school teachers.  1 high school teacher.  2 college professors.  One of the college professors remarked that this was so nice because it was the easiest math she had dealt with all day.  The elementary school teacher remarked that this was the most difficult math she had dealt with all day.

I mentioned to my table that I couldn't figure out the formula for surface area of a sphere.  One of the middle school teachers asked me if I knew the formula for volume of a sphere.  Of course!  Then, she asked me why I didn't just take the derivative of the volume formula.

Oh. My. Goodness.  The surface area of a sphere is the derivative of the volume of the sphere.  How did I not know this?

Another person at my table immediately recognized another derivative relationship.  Circumference of a circle is the derivative of the area of the same circle.  #mindblown  I excitedly messaged my fiance to ask if he knew about this relationship.  He did...  

This is just a reminder to me of how awesome and beautiful math is.  I was a math major in college, but I still have so much more math to learn and come to appreciate.

Next, we moved on to the hands-on aspect of the meeting.  We drew lunes on our balloon.  And, then we ended up drawing a triangle.  A triangle with angles that add up to more than 180 degrees or pi radians.  Again, mind blown.

Actually, most of my table was solving the problems in their heads.  But, I decided I needed to draw it out to see it.  

I'm glad I did because the problem became more and more complicated and soon everyone was crowded around my balloon.  They were thanking me for illustrating the problem because it helped them to see how to solve it.  It was just a reminder that visuals are important.  So often, my students don't think they are because they can do simple problems without them.  Maybe this is a sign I need to do more extended problems that eventually need a visual to help solve them???

So, the point of this post?  Math is cool.  And, I'm still learning.  I hope I never stop learning.

What's the most recent thing you have learned about math?


  1. I love the relationships between volume-surface area and area-circumference! It's just x^3 and x^2 and x (volume, area, linear length)

  2. You need to get yourself a Lenart Sphere as a present. Once you're sick of that, play with hyperbolic geometry, where the sum of the interior angles of a triangle are always less than 180 degrees. I always show these two cases to my middle school babies, then leave the models out for the rest of the year. They go along with my motto "there is ALWAYS more math".

    1. I don't even know what this is. Must research!

  3. I only recently started really appreciating spherical geometry. It is so fun! When I taught it to my topology/geometry class, I did this activity with grapefruits and hair rubber bands:

    Personally, one of the math things I learned most recently was from this comic: I had a similar reaction to the person in the last panel. (The fact is that the sum of reciprocals of integers that don't have a 9 in them converges.)

  4. Can you please share the resources from this workshop, lunes, moons and balloons? I do teach geometry and just the little you mentioned sounds awesome!

  5. Here's a recent epiphany I had in class:
    Consider y=mx+b, where m>0. m is the slope of the line and since m is a constant, we see the slope is not changing, therefore the y-values are getting bigger, but the rate at which they are increasing is not changing.
    Now consider y=x^2 and "pretend" it's an equation of a line (duh, we know it's not, but just go with it for a minute). If y=x^2 is a line, then what's the "slope" of that line? y=x^2 is the same as y=xx, so it stands to reason that x is the slope of the "line." Notice that for this function, when the y-values are increasing, so is the rate at which those values are increasing because m=x.
    You can do the same for y=x^3, only now, m=x^2 which makes sense because the y-values are increasing even faster than with y=x^2.
    This occurred to me when I was teaching an introductory lesson in AP Calculus and this idea was unfolding in my brain as I was teaching it and it was fantastic.
    You're right - math is stinkin' cool!

    1. Wow! I never made that connection before! Thanks for sharing!