Monthly Archives: May 2012

Letting Go in Algebra Class

A problem from the May, 2012 issue of Mathematics Teacher by Jennifer Kaplan and Samuel Otten outlines a Max/Min problem from calculus which presents a scenario accessible to Algebra 1 and Pre-Algebra students.  In the problem, a dog chases a ball thrown from the beach into the ocean, as shown in the picture.

Beach problem

The challenge is to minimize the amount of time it will take the dog to travel from “me” to the “ball”, if he can run 4 meters per second on the beach, and travel 1 meter per second in the water.

In the past few months, I have been looking for interesting problems to try with different levels of classes, and have made observations about how students approach non-routine problems.  Today, I worked with an 8th grade Honors Algebra 1 class.  It was a nice day, so we gave kids the chance to move outside, work in groups, and take 40 minutes to work on their ideas.  While some groups preferred to work with pencil and paper, others took immediately to the chalk we provided to begin sketching their ideas.

Algebra Chalk

The teams have until next week to develop a solution to present to class.  Many groups after the allotted time seemed to have a process for working towards a solution.  But while I was happy with the persistence the class showed in working through the problem, only one group considered using a variable during their discussion, and that one group only considered x as a distance the dog needs to travel on the beach, and did not pursue it further.  The general procedure so far has been to collect data, make a table, or narrow down by guessing and checking.  So, here are my questions:

  • Should I worry that so few students can apply the algebra they have learned?
  • How can I coax groups to utilize some algebra, without being overly helpful?
  • Should math teachers always feel compelled to demonstrate the “right way”?  Is a non-algebraic solution less valid than an algebraic one?

The traditional math teacher in me can’t wait to jump in and walk students through “my” way to solve it.  But the facilitator in me was thrilled and impressed by the rich discussions taking place today.  It’s hard to let go of old ways of doing things.

Next week, students will produce 60-second videos where they will present their solutions.  Looking forward to the variety of arguments we will certainly see.  Also hoping to work on this problem with our non-honors and academic students.

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?

Instantly Grade Google Docs with Flubaroo

Back in February, at the annual ASCD conference, I saw a presentation by Google, where they demonstrated a number of tools your can use with their documents.  One of those tools was Flubaroo, which I just now had my first opportunity to test drive.  If you are comfortable with making a Google Doc quiz, then Flubaroo is simple script you can use to provide student feedback.

To test it out, I created a short quiz, and shared it with colleagues.  The video below steps you through how I used Flubaroo to grade it:

Think about some of the ways we can use this:

  • Have students submit homework or quick quiz responses.
  • Have a few computers with a Google Doc quiz serve as an activity in a learning center
  • Allow students use phones or laptops to respond to short prompts.
  • Provide daily basic skills assessments, instantly graded

I’d love to hear how others are using this tool.

Why Do Kids Need to Factor?

One of the joys of my job as a math coach is having conversations with colleagues about the hows and whys of math class.  Often, the best conversations come from unplanned meetings, just chewing the fat about what is happening in classrooms.  Earlier this month, I visited a second-year high school teacher with a quick question that turned into a deeper conversation after I noticed her posted objective: “factor polynomials where a > 1”.  The objective was for our academic-level algebra class.  I asked how things were going with the class, and the teacher expressed the usual frustration of teaching kids the process of factoring, which led to her asking what I would do differently…what did I have in my “bag of tricks” to teach factoring?

I had to think hard about my answer.  Like most math teachers, I had taught Algebra I many times, and had gone through the process struggles students have with traditional factoring.  I’ve never really subscribed to a “trick” for factoring:  I have some colleagues who use a chart method, while others attempt grouping.  But it occurred to me that the entire premise here was flawed…just how important is factoring to teach and learn?  My second-year teacher friend thought for a minute, then gave the answer many teachers would give:

They’ll need factoring in Algebra II and other classes they take down the road.

Sigh…

Is that really the best answer we, as math teachers, can give for learning factoring?  It’s just a cog in the polynomial machine: add – subtract – FOIL – factor – simplify rationals – graph.  How exciting!

In a previous chat with a different teacher, I suggested that we upset the entire process.  Why do we save fun, neat stuff like projectile motion for “later”?  Dive right in, look at some non-linear graphs, and develop new ideas about quadratic functions, symmetry, intercepts, and vertices right away.  If you have never been to the PhET (interactive science simulations from the University of Colorado), go there now.  Check out the projectile motion applet:

phetprojectiles

Have fun tossing Buicks across the sky.

Think about the number of kids who never get to experience these great math models, because we beat them with a stick with pages of FOIL and factoring worksheets, before they ever get to see a parabola.

So, what of factoring?  Let’s say I wanted to re-write the following function in factored form:

Is it wrong to have students graph the function and look for intercepts?

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If we have an intercept at x = 2, then we know that (x-2) must be a factor.  From there, we can piece our way to the second factor of (3x-7).  Is it “wrong” to teach students to look at polynomials this way?  I suspect many would call it blasphemous, but somehow I know that my kids know more about the inter-connectedness of functions, intercepts, and polynomials than theirs.

I want my students to utilize and switch between multiple representations of all families of functions: equation, table, graph, context.    Unfortunately, most textbook ensure that these topics will continue to be taught in a linear fashion.

Let’s take that old objective of “factor polynomials” and change it to “develop and understand quadratic models for natural phenomena”

And if the reason we teach anything in math is because “they’ll need it for their next class”, then we are doing math wrong.

Census at School

At last month’s NCTM conference, I came upon the American Statistical Association booth in the exhibit hall, and enjoyed discussing their resources for K-12 students.   I had utilized a number of ASA’s many resources in the past, and encourage you to check out Chance Magazine and the ASA’s Project and Poster Contests for K-12 students.  But Census At School was new to me, and I hope to encourage teachers in my district to utilize the wealth of data here.

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On the site, students can contribute data about themselves including:

  • Demographics: including gender, location and age
  • Measurements: height, foot length, finger length
  • Interests: sports, activities, online behaviors
  • Favorites: foods, drinks, school subjects
  • Issues: the site has an interesting slider where students communicate their opinion on a number of social issues.

I have done many class data collection activities like these for years, but what excites me most about Census at School is the “Random Sampler” feature.  With it, you can download a random sample of participating students, selecting what states, grades, and genders to include in the sample.  The data can then be downloaded as an Excel document, and allows for myriad comparisons.  To test drive this feature, I asked for a random sample of 50 high school students from Pennsylvania (my home state) and California.

How do attitudes on recycling differ, between kids from PA and CA?

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How do number of texts sent and receive differ in this sample of students?

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Looking for something active and productive to do with your kids in those last few school days? Set up some data collection stations, and contribute your class’ data to Census at School!

Desmos Online Graphing Calculator

Recently, I have been noodling around with the Desmos Online Graphing Calculator.  I have used Texas Instruments products exclusively and extensively in my classes for years, but am always on the lookout for tools that are easy to use, functional, and (most importantly) cheap!  The Desmos calculator aligns quite nicely with my personal motto, “If It’s For Free, Then It’s For me!”

The calculator has an interface which is intuitive, and it’s easy to dive right in and start graphing:

desmos1

Inequalities can also be graphed easily and nicely:

desmos2

Trig functions are formatted in readable form as you type them, and you can choose to have the x-axis “count by pi”, which is a pretty cool feature:

desmos3

Points can be traced.  Check out how the minimum here is communicated with appropriate symbols:

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Yep, we can do piece-wise functions too:

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What appeals to me about this calculator is that it is web-based, fires up quickly, and is ready to use.  This site should be on everyone’s links for students, shared on Edmodo, or whatever resources page you use.  While I love my TI software, it often takes too long to load, and you need to be a real Nspire user to navigate around.  The site is also usable with Ipads:

Horsham-20120502-00199, Uploaded by Photobucket Mobile for BlackBerry

But keep in mind that this is a stripped down calculator.  It graphs stuff, and that’s about it.  You’ll still need your TI’s to perform calculations, analyze data, or do more in-depth analysis, like intercepts or integrals.

Sometimes, less bells and whistles are better.

EDIT: check out the comments for news from Eli, founder of Desmos, who gives some more great information about this tool!