Saturday, June 11, 2011

Teachers Investigate Quadratics

In our spring grad class we've spent the week looking at assessment and instruction in the context of quadratics. The pedagogical side of class was spent talking about Skemp, making a assessment concept map on Mindmeister, a rare mindmapping tool that allows realtime collaboration...

...a questioning framework (shared in this old blogpost), a video of a teacher leading a lesson on solving quadratics (from the new to me resource of Inside Mathematics, with videos, coaching, lessons and problems), sharing a variety of articles on assessment, and watching Shawn Cornally's TEDx talk.  The assessment articles included one new to me, "Using Assessment for Effective Learning," Clare Lee, Mathematics Teaching, Jan 2001.  That led me to two books (that are available online through our library), Language for Learning Mathematics, by Lee, and Learning to Teach Mathematics in the Secondary School, in which she has several chapters.  Shawn's talk is worthy of a post of it's own... really has me thinking.

Between the classes, the teachers tried to find quadratic data of their own.  One speculated about pulse rates in fight or flight situations (concave down quadratic?) but couldn't find a good set of data. Another found a neat class of problems about water draining in a quadratic pattern.  Sadly, my remaining physics doesn't let me know why that would be. One of the students did a great little think aloud using Jing (click for the short video; I don't know how to embed someone else's screencast) with a picture of a record dirt bike jump.
 He did a quadratic fit to this using GeoGebra.  The picture does not seem to be from perpendicular to the plane of the jump, which raised the nice question: is a parabola seen from askew still a parabola?   (The other nice tip he had was to use SmartNotebook for within the screencast... interesting.)

To me, one sign that students are genuinely investigating is that they get to questions that I have to think about or don't know the answer to.

With the focus being on deepening their understanding, I asked the teachers to choose from these investigations, with the addition of the water draining.

1) Galileo Galilei (essentially that means his father was named Galileo, too; so he’s Galileo Jr.) conclusively disproved Aristotle’s idea that heavier things fall faster. He took Aristotle to task for never figuring out a good way to test it. Next he wanted to study how and if speed changed during a fall. But it was too fast for his available tech. So he devised several clever ways to slow it down. One way was to roll a ball down an inclined plane rather than dropping it. (Why would this have the same information as falling?) He did one experiment by building bumps on the ramp and spacing them until the sound of the clicking over them was periodic. That’s hard to replicate in class. But another experiment was to time a rolling ball down a plane, and then find a point where it took half or a quarter of the time. (Read more at but after you experiment.) Try that out, take some measurements and discuss what you find. How can you get a variety of data to look for patterns?

2) There’s a famous relationship between quadratics and second differences in the output, if the inputs havea constant difference. Choose a couple of different quadratic functions and experiment, generating and organizing data.
a. What is the relationship?
b. Why is the relationship?
c. How does the relationship connect to either the standard form or vertex form of a parabola?
3) We often care about the roots of quadratic functions. Is it easier to solve for the roots a function in standard or vertex form? Why? What would the quadratic formula look like for vertex form? Could you use that to find or derive the quadratic formula for standard form?

4) Not much for visual learners here. Can you devise a pictorial growth pattern that has a quadratic relationship between input and output? Well probably it’s easy!
But there must be more interesting patterns than that. Can you make a pattern with a ≠ 1? b and c ≠ 0? How would you visualize first and second differences in such a pattern? Would the function for your pattern have roots? What would they mean?

I felt choice was important, especially for experienced students who already knew a lot about quadratic functions.  They did an excellent job with their choices, looking for something new to do.  Here are some results of their investigations.  Afterwards, they came together and shared the results and had a great discourse.

(1) Not to gender stereotype, but the boys were rolling things even before they finished reading the first question. The first attempt was rolling a tennis ball on about a -1/3 slope.  They got this data at right.  Barely any difference between linear and quadratic.  They were surprised by the small size of the a value. They thought it was quite interesting that this was a situation with distance as an independent variable and time as the dependent. (I thought that was cool but hadn't noticed or expected that.) They noticed that if they didn't know to get a quadratic, they probably would have stopped at linear, because r=.985 is clearly good enough.

I asked if they could slow it down at all.  So they immediately changed the slope and switched to a hot wheels car that would run truer.

An in between occasion, before (0,0) was included.

 This interesting question came up: is averaging the data and finding the regression curve any different from finding the regression curve on all the data? (For this one it turned out the same.) One teacher included (0,0) as a point, which raised the question should the other one using all the data use one (0,0) or three? Should they add (0,0) - because if there was some experimental effect it would not be in a theoretical point like that.  How can you tell from the data that the car is speeding up? How did Galileo notice that falling wasn't uniform?

Seems to me that real investigations always raise interesting questions.  Canned investigations raise many fewer.  The big question came up: how did Galileo know that rolling was essentially like falling?  While this was the most dramatic investigation, each of the others raised great questions and connections, too.

(2) Only one student looked into differences, with some consultation.  She quickly found a pattern relating the second difference to a.  But I put up the following GeoGebra sketch.  (File here, or as a webpage.)  The teacher checked her work because of the 4.5 2nd difference.  She realized it was because of the delta, and set to work finding the relationship between the 2nd difference, a, and delta. Excellent math detective work. We discussed the new applications of difference equations because of computers, and the weird connection with derivatives.  They noticed that b and c had nothing to do with the second difference, and how that made sense if the second difference detected quadratic behavior.

(3)  This was the one I thought was the least interesting, but the teachers found several cool bits.  First of all, they did it by rewriting the vertex form into the standard form, and then used the quadratic formula to get a vertex form quadratic equation.  But then they liked it well enough that they wondered why they had never seen it before.  It was noticed that the work they did identifying the standard form with the vertex form gave an equation for the x-location of the vertex for the standard form, -b/2a.

I mentioned that I would never have thought of doing it the way they did, and they started wondering what I meant.  Another teacher saw a possibility, and solved directly from the vertex form, getting another quadratic equation that was even simpler.

(4) Two groups tried this.  They captured their work in photo, so it's probably just best to show.

 The teacher on the left was surprised that there was a 3-dimensional way to think about it.  The teachers on the right were surprised that there was a non-3-dimensional way to think about it.  The rightside teachers liked the idea of colorcoding from the leftside, and made an adjustment.

Left as an exercise for the blog-reader what the function rules are for these sequences.  The discussion revolved around trying to make the pattern understandable to a teacher that finds it very hard to visualize.  The unanswered question was: what would roots mean in this situation?

Our closing reflection was on what elements of the investigation allowed them to deepen understanding.  They brought up the closing discourse as well as collaboration, the manipulatives, the new questions, and trying to collect their own data.

Pretty good lessons to draw, and some quality mathematics.

Galileo Photo Credit: from Flickr, tonynetone


  1. A question on the last visual representations. Figuring out the functions they represent is easy enough and in the 3 dimensional I see a clear connection with the quadratic equations as high school students work with them. Each color representing a term in the quadratic. But I am not sure what the colors in the 2 dimensional represent.

  2. The red is the linear term. She did blue and green to distinguish between the 2 x^2 areas.

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