Category Archives: Geometry

Building a Better Snowman

In a recent tour of my midde school, I observed a 6th grade teacher working with a class to use compasses.  Their goal: to make a snowman with segments of different, but given radii, 3cm, 2cm and 1cm.  Eventually, this will lead to students having their first experience with circumference, and that sneaky number pi.  But why tie students to fixed radii values?  And just what are the “perfect” measurements for a snowman?  Here are some lesson ideas for letting students explore their own snowmen, using technology, then taking it a step further by considering how much snow our snowmen will need.

Snowmen may come in all different sizes, with different accessories.  But it is pretty well established that the traditional snowman is made of 3 body segments.   Snowmen with anything other than 3 segments are blasphemous.

snowmen

WHAT ARE THE PERFECT SNOWMAN DIMENSIONS?

Clicking the image to the right will take you to an interactive graph I made using the Desmos calculator.  You can manipulate the diameter of each snowman segment by pulling the sliders provided, but the height of the snowman is fixed at 6 feet.  Like your snowman with equal segments?  Knock yourself out.  Are you more of a bottom-heavy snowman connoisseur?  There’s room on the bus for you , too.

Once you are convinved that you have built the world’s best snowman, please share your slider settings here.  I’d like to feature them in a later blog post.

INVESTIGATING YOUR SNOWMAN

For a classroom discussion, have students print out their snowman images (Desmos has a snazzy print feature), and compare their snowmen in groups.  Whose snowman would need the most snow to build?  Whose would need the least?  Or, if all of the snowmen are 6 feet tall, then will they use the same amount of snow?

This is a great time to talk about volume, and introduce the formula for the volume of a sphere.  And, since each student has their own product, have them find the volume of each snowman segment, then add them to get their required snow total.

HOW MUCH SNOW DO WE NEED?

If it snows 5″ overnight, will we have enough snow on the ground to build our snowman?  For this next stage, we can have students compare the volume of snow needed for their snowman with the volume of snow on the ground.  For example, if your yard is 10 feet long and 10 feet wide, will 5 inches of ground snow be enough to build our snowman?

BUT, we pack snow while we build snowmen.  How much less is the snow volume in a snowman vs its volume when it is on the ground?  2 times less volume?  5 times less?  10 times less?  I really have no idea.  Next time I have a few inches of snow on the ground, it will be experiment time.  With a class, perhaps debate the correct number and use it for calculations.

calculator

To help with calculations, and checking student work, I have created this handy Snowman Calculator on Excel.  You can input your a and b values from the Desmos document, along with the dimensions of a yard or rectangular area.  The volume of your snowman, along with needed snowfalls, are then given.

Think warm!  And then we will start working on sandcastles.

Doing the Translation Dance

Last month, I wrote about my talk on Encouraging Perseverance in Math Class, given at the Fall, 2012 ATMOPAV conference.  But earlier that same day, I had the opportunity to hear Scott Steketee‘s thoughts on functions: “Function Dances: Using Transformations to Make Variables Vary and Functions Behave”.

Steketee

I have found that the approach many teachers take to functions is one of notation only.  That by simply introducing the f(x) and g(x) symbols, and “covering” domain and range, algebraic functions will be understood.  Scott’s presentation provided ideas for introducing the concept of  function, without all of the scary symbols, through dynamic Sketchpad files.  The group worked through a number of progressively intricate functional relationships on iPads.  In this first example, students can grab points and look for relationships.  Some points will not move when dragged, as they are “dependent” upon other points’ movements.    Also, the dependencies vary, from simple linear relationships, to a few which require dilations or reflections.

iPad1

Later, we were introduced to the Sketchpad “Translations Dances”.  As one point (below, the point on the green outline) travels about its “domain”, we are challenged to trace the “range” of the translated point p.  These start off innocently enough, but become more diabolical as the translations begin to include reflections and rotations.

iPad2

These were addictive and appropriate uses for the iPad, and I was able to easily load the files into iPad’s Sketch Explorer through my DropBox account.

The second half of Scott’s talk was more kinesthetic, social, and potentially embarrassing, as the group split into partnerships to choreograph dances based on transformations.  My partner acted as the independent variable, and I (the dependent variable) followed her actions, using lines in the floor to act as  axis of reflection.  This would be a fun way to expose kids to functional ideas, but I made sure that no photographic evidence of my dancing ability exists!

What I appreciated most about Scott’s sketches and dances is that they allow teachers to develop functional ideas without having to wade through all of the complex language.  Through play and exploration, students can summarize their observations, and begin to characterize the relationships.  As students begin to understand the relationships between variables, we then can discuss the need to have special notation to express them.  Finally, dilations and reflections, which are often over-looked in our curriculum, become the stars of the show through fun (and addicting) Sketchpad games.  My screen grabs here certainly don’t do Scott’s files justice, so download them, play around, and enjoy the dances!

What is on Your High School Math “Mount Rushmore”?

Today was math club day.

Math club is a strange creature, where the content is always dependent on the interests and audience of kids who show up.  Some days, we will work through interesting past contest problems.  Other days, we will discuss “not taught in class topics” like Mersenne Primes or Cryptography.  One day last year, we watched and enjoyed the “Pi vs e Debate“, which is fun viewing, but will produce odd looks from those who aren’t in on the joke.

Today, I challenged students to think of the “3 most important math equations” they had encountered in their high school career.  This opener caused some discussion of what should be considered “important” as opposed to memorable.  A few students shared their responses on the board:

Top 3

This led to my presenting a  video I came across recently, “10 Mathematical Equations that Changed the World”.  The ranking is by no means scientific, and a few of the equations are well-above high school math.

This also produced discussion of the blurred line between physics and math, especially as students begin to take more challenging courses.  Ask your students tomorrow “What are the 3 most important math ideas you have learned in your life”?  Can our students summarize and prioritize their math experiences and reflect upon their learning?

As a teacher, which concepts belong on your high school math “Mount Rushmore”?  Shrink down the high school math experience to the 4 most central ideas.  Here’s mine:

  • Quadratic Formula.  Is this one a gimme?  One of the first times our students consider a general case and the gatekeeper to algebra 2.
  • A proof of the Pythagorean Theorem.  I’m going back and forth on this one, thinking perhaps law of  sines / cosines would be better.  Pythagorean Theorem alone seems too middle-school, but being able to develop and defend a proof then moves it to a summary of geometry.
  • The Central Limit Theorem.  The backbone of confidence intervals and hypothesis tests.  If you don’t teach stats, this might be out of your wheelhouse.  If you do, I think you are with me.
  • The fundamental theorem of calculus.  Ties together all of our algebraic and abstract-thinking skills in a nice tight package before we send the kids off to college.

Would love to hear your “Mount Rushmore”!