Comments on: WCYDWT: Polynomials

By: Math Hospital | mrmillermath

Math Hospital | mrmillermath — Tue, 03 Jul 2012 20:48:00 +0000

[…] – failure is helpful and not shameful. Â I took that theme fromÂ a Dan Meyer post you can find here.Â This is where I remind students that getting things wrong is a great opportunity to […]

By: Touzel Hansuvadha

Touzel Hansuvadha — Sun, 27 Feb 2011 23:36:13 +0000

Marcia, is it accurate to say that a polynomial is the product of two lines? I feel like a pedant, but isn’t there a slight difference? I mean, ax+b is not a line, but y=ax+b is. And multiplying y=(ax+b) by y=(cx+d) gives y^2=(ax+b)(cx+d)–is that a polynomial?

I feel like such a dick for asking this line of questions…

By: Kathy Clark Couey

Kathy Clark Couey — Mon, 25 Oct 2010 02:57:36 +0000

I just figured out why I love reading your blog. Post after post your ‘math is important message so teach it well” message inspires me . . . even though I teach science.

By: Marcia Weinhold

Marcia Weinhold — Mon, 25 Oct 2010 00:11:43 +0000

There ARE pictures for polynomials. They are called graphs. I started study of polynomials with factors. We built polynomials out of linear functions (do you know what happens when you multiply two lines together?), and played with making the polynomial do what we wanted by placing new zeroes in the polynomial (new linear factors), or making one of the factors have negative slope. We predicted what the new graph should look like by looking at where all the factors were negative or positive, and what sign the product would have (remember sign charts?).
Since Geometer’s Sketchpad started graphing functions, we have used that software to explore because you can make different graphs different colors, and the graphs are a little prettier than a graphing calculator. Adding sliders to the factors, or putting the zeroes as points on the x-axis, adds another dimension to the exploration. To get a “full” picture, you have to be able to add quadratic factors with no real zeroes, e.g. x^2 +2. For more info, see “Designer Functions” Mathematics Teacher, Aug. 2008, pp. 28 to 32.

For some applets to work with the area model (lots of fun), see http://www.fi.uu.nl/wisweb/en/ , choose applets, then geometric algebra 2D. There are four ways to play with these, and you do have to play because the site has no directions.

Thanks for your thoughts, Dan!

By: Elizabeth S

Elizabeth S — Sun, 24 Oct 2010 22:43:11 +0000

I love the discussions like this. Let’s keep taking the gloves off, putting the smocks on the students, and handing them finger paints galore.

By: Sue VanHattum

Sue VanHattum — Sat, 23 Oct 2010 15:55:21 +0000

Loving this! Thanks for the lovely polynomial picture site, Nathan.

And Dan, you wrote this right as I entered into the polynomial section of my college course (beginning algebra). I’m delighted.

I would like to head towards the inverted class model eventually. Get some of my favorite topics to lecture about on youtube, and tell them to watch before or after class (depending on whether playing with it first is important).

By: Javier

Javier — Fri, 22 Oct 2010 05:12:20 +0000

No joke. This is what we worked with today. I have a group of seventh graders taking Algebra and they have a bit of that fearless streak in them. We talked a little about binomial multiplication and then told them I was going to show them how to break the resulting trinomials back into the binomial factors.

But first, I wanted them to give it a shot. After the frustration and grumbling started one of the kids says, confidently, “I got it.” She wasn’t asking me, she was telling me. The rest of the class, just in hearing that one of their classmates was able to discover some mathematical secret, dives right back in. A few more students got the first trinomial and one went up and explained it to the rest of the class. After a few cycles, the whole class was taking the trinomials to task and I hadn’t done anything more than hint and nudge. It was magical. The best part is that they owned it by the end of the period. It was all theirs.

By: Scott

Scott — Thu, 21 Oct 2010 20:27:09 +0000

I have been trying to implement problems of this type (the messy ones) over the last few years in college algebra and math for general studies classes. Since we do not meet as often as a high school class, I often struggled with how to “get everything covered” and still do some of the more interesting constructivist/discovery learning type problems. I have been utilizing the concept of the inverted class to accomplish this. Meaning, aspects of the class that were traditionally done in class (e.g. a lecture) are now done outside of (and before) class; things that were normally done outside of class (e.g. the homework problems) are now (mostly) done inside of class. I use Camtasia to create the video lecture where students get the first exposure to the material, thus freeing class time for everyone to “get their hands dirty.”

I am still fine tuning the videos and the in-class activities. And, with common finals, I do still feel obliged to do some drill-skill type problems. So far, I am relatively happy with the results.

Thanks Dan for an incredible resource. I have been reading through some of your “archived” postings. You and your readers are really getting people to think about what it means to “teach” math well.

Scott

By: AsteriskCGY

AsteriskCGY — Thu, 21 Oct 2010 20:26:36 +0000

(2x+3)(x+2)
Wow, that took me 4 tries to get.

By: Daniel Schaben

Daniel Schaben — Thu, 21 Oct 2010 14:07:08 +0000

I am learning that math needs to be messy for kids. Think finger painting with smocks messy. Guess and check messy.

Example:

Problem: Given a standard 8 Â½ inch x 11 inch piece of paper, determine a function which gives the volume of a box (without a lid) made by cutting squares from each of the corners and folding up the sides. Let x be the length of a side of the square, and write the volume as a function of x. What would the dimension of x be to maximize the volume of the box?
I use to show the students the logic behind it by working through the dimensions on a roughly drown picture at the board. Recently. I have noticed that students just didn’t get it after showing them how to do the problem many times. So I have given up on that approach and given them a piece of paper, scissors, and told them each to create their own box. Now I have a pile of paper boxes in the back of my room (messy) Some look like whiskey flasks, some look like short Kleenex boxes. We are right now in the measuring and finding volume stage (collecting the data). We will compile that data and see if we can discover the polynomial function. V(x) = (11-2x)(8.5-2x)x. In creating the boxes we had a great discussion about domain. One student tried to cut a 5 in square out of a corner.