Tag: quadratics

Categories
Algebra High School

Applications of Quadratic Functions

My class just completed its unit on quadratic functions, where we looked at all of the old favorites: completing-the-square, quadratic formula, -b/2a.  We also looked at “sideways” parabolas (those of the form x=…), and the formal definition of a parabola, including the focus and directrix.

But beyond the important algebraic processes to be mastered, I wanted students to appreciate the many vital applications of parabolas.  Towards the end of the unit, I posted the following task on Edmodo:

Research an application of parabolas, and explain how the properties of parabolas make them an effective shape for your chosen application, including specific vocabulary. Find a picture of your parabola in action, and use an application like Geogebra to find its equation. Turn in electronically as a single slide or page which could be posted.

Many of the questions students had about the assignment dealt with my intentionally non-specific instructions: Do I have to use the focus? What if I can’t find anything about the directrix? How do you use geogebra? How much do I need to write?  This was one assignment where I needed to “play dumb”; I wanted to students to think about what was essential, and craft explanations carefully.

HOW THE CLASS DEVELOPED THE RUBRIC:

On the day the assignment was due, I printed out all student papers.  I told students that the assignment would be worth 20 points, but I wanted their advice on how I should grade it.  What should I look for?  What evidence of understanding should I see?  After a rough start, where we argued the benefits of “creativity”, the class settled on a nice list of 4 ideas:

  • Proper use of vocab: while not all vocabulary words are required, those that are mentioned should be used properly.
  • Understanding of the application: pretty self-explanatory
  • Equation: is it correct? does it model the situation?
  • Structure / Flow: this was the compromise for creativity.  A good paper should have a logical structure which a reader should be able to follow.

The next step was to assign points to each category.  I told students that the assignment was worth 20 points.  With their group, I told students to come up with a method for allocating the 20 points.  After a minute or two, all groups had contributed their point-allocation ideas, which were recorded on the board:

 points

I then took a pseudo-average from the group results, and a 7-7-3-3 point structure was agreed upon.

NEXT STOP – SPEED-DATING

Armed with this rubric, I placed students in groups of 3, and used a version of Kate Nowak’s great speed-dating method to have students peer-assess their work.  With the clock set for 4 minutes, students shared their parabola discoveries, and discussed ideas for improving their paper.  After 4 minutes, the students (labeled A, B and C) moved to a new group.

  • A’s moved one group to their left
  • B’s moved one group to their right
  • C’s stayed at their current group

So, in 12 minutes, students had a chance to have their work evaluated by 6 peers, and see how their work stacked up to the class standard.  At the end of the “speed dates” I gave students 2 more days to revise their papers, based on their peer reflections, and this revised paper would be what was graded.

The class found some great applications of parabolas, and the chance to reflect and revise not only made the papers so much sharper, but also allowed students to share each other’s applications.  Some examples were:

Bridge Design:

Bridge

Satellite dishes:

Satellite

And solar cookers:

Cookers

And bringing in the solar cooker application allowed me to share this video about a home-made solar cooker.  Cool stuff for kids to see!

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Categories
Algebra Statistics

What’s the Probability That Quadratic Will Factor?

A comment from my post last week about the need for factoring led me to re-visit a question I have posed to classes before, but never allowed to move beyond the “gee, that’s interesting” stage.

Given a polynomial in standard form, with random non-zero* integer parameters a, b and c, what is the probability that the polynomial will factor?

I’ve pursued this question with classes before by writing a polynomial on the board, with blanks or boxes in the a-b-c positions.  Sometimes, I would take “random” shout-outs from the class to fill in the boxes.  With another class, the randint function on a TI calculator was used to generate our abc’s.  The point was to demonstrate that a large majority of quadratics are not factorable, and that despite the nice, rigged, problems we encounter in textbooks, we should spend far more time considering what to do with the messy ones.  But I’d never put pencil to paper and thought about the theoretical probability.

After my post on factoring last week, Jim Doherty mentioned a speaker he had encountered find an experimental probability that a quadratic would factor, and cited 7%.  That number seemed reasonable to me, but perhaps a bit on the high side.  I set up an Excel document to generate three non-zero integers (more on this later), and rigged a system to check for perfect-square discriminants.  I recorded experimental results, in groups of 1000 trials, and kept a running total.

Excel document

Quadratic Graph

After 25,000 trials, I found that 7.26% of the quadratics would factor.

*While this endeavor started off innocently and quickly enough, I had to start over after I realized my Excel document allowed for zeroes.  It took a little logical Excel rigging to exclude them.

So, there must be a theoretical probability out there someplace?  Anyone know how to do it?