Category: Algebra

Categories
Algebra Geometry

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”!

Categories
Algebra Middle School

Composite Functions and E.S.P.

An opener I have used in Algebra 1 encourages discussion of composite functions, but more importantly, allows me to show of my amazing ESP powers!

To start, you’ll need a wizard and/or magic hat.  A cape is also acceptable.  If all else fails, the cardboard crown from Burger King will suffice.

Wizard

Have students choose any number, and write it down, out of view of you…the wizard!  Encourage students to choose a number which is easy to work with.  Now, step-by-step, have students perform the following operations, keeping track of their answers as they go along, and always hiding their paper from you.

  1. Take your number, and double it.
  2. Take your result, and add 4 to it.
  3. Take your result, and divide it by 2.
  4. Take your result, and add 4 to it.
  5. Take your result, and double it.

Now, have students share with you their result.  Using your incredible predictive powers, you will be able divine their original number.  Utilize temple massaging and mysterious gestures to maximize the effect…..

If a student says “46”, their original number is 17.

If a student says “84”, their original number is 36.

If a student says “20”, their original number is 4.

I usually do this activity twice with a class.  The first time around, I suggest the students use a simple number.  In the second round, I’ll allow them to try to stump me, which usually leads to some awful mental math experiences for me.

So, what’s happening here?  In this activity, students are asked to do three different things to their results, which can be described by functions:

  • let f(x) = 2x
  • let g(x) = x + 4
  • let h(x) = x / 2

The “secret” to this ESP puzzle is revealed in this 2-minute video, used making the great FREE iPad app, Doceri:

So, as students reveal the result, your job is to subtract 12 from it, then divide by 2.  This will always yield the original number, unless the student mis-applied any steps along the way (this happens!).  Also, in the second go-round, I will always have a student whose answer is 50.98, or something worse.  Have your mental thinking cap on!

This leads to a great discussion of the “inside-out” nature of composite functions, and the ability to recycle functions.  Challenge your students to come up with their own function puzzles, and amaze the world with their math ESP!

Categories
Algebra

Experiencing Linear and Angular Velocity

Linear and angular velocity can be one of the trickier units for high school math students.  The formulas and computations themselves aren’t all that imposing, mostly just some quick multiplications and divisions, but the vocabulary and scenarios can be intimidating.

We encounter some new terms here and their symbols, like angular velocity (ω), arc length (s), linear velocity (v) and using angles (θ) measured in radians rather than degrees.  It’s a lot of new ideas to absorb, and the problems often require a student to assess and utilize multiple formulas.  An activity which I wrote a few years ago has been taken over and perfected by a colleague.  “Trig whips” allows students to experience and communicate linear and angular velocity, and provides a great excuse to get outside and move around.

TRIG WHIPS

This activity has a simple premise.  In groups of 4, students stand shoulder to shoulder.  One student shoulder acts as the center, and the group works together as a radius, walking in a circle.  Each group records:

  • Individual student distance from the center
  • Time required to complete 3 revolutions
  • Angular displacement

A worksheet for this activity allows students to keep track of their data.  Encourage students to switch positions and experience the effect of being the outside point.

My colleague reports that one class was able to complete the activity with a “record” 11 students.  Not only did the outside student need to run hard to keep up with the group, but the inside students worked together to slow the middle.

If we had a group of 20 students, how fast would the outside student need to “walk”?

UPDATE: More kids, a longer conga line, more chaos, more fun!

Human Chain from Bob Lochel on Vimeo.

Contribute your thoughts about this video on Dan Meyer’s 101qs site!