Thursday, January 10, 2013

How I Sextupled the Time It Takes To Teach End Behavior

In Algebra 2 . . . I used these cards:

Hi Students! Here's a deck of cards. Each one has a polynomial function and its graph.

The degree of each polynomial is determined by its highest exponent. Talk with your group and agree on the degree of each one.

The leading coefficient is the number in front of that term with the highest exponent. Talk with your group and agree on the leading coefficient for each card.

Now we are going to focus on the ends of the graphs. Look at the left side. Does it point up or down? What about the right side? What you are looking at is called the "end behavior". Group the cards according to end behavior. Show me when you're done.

Next I want you to look for similarities in each group. What do the degrees have in common? What about the leading coefficients?

In 5 minutes or so, all of the groups had identified what the groups had in common according to even/odd degree and positive/negative leading coefficient.

We did some practice once they had this figured out.

Oh, and then I played some music and showed them some polynomials one at a time and they showed end behavior with their arms. And there was dancing.

This used to take me 4 minutes. I would write four rules on the board and they would copy them down.

Today it took 24.

But it was fun. And later I saw somebody end-behavior dancing in the hallway.

This approach is better than just telling them, right?


  1. I will be teaching this soon. Such a great idea Amy! I can't wait to use this.

  2. I have no idea what you're talking about. Yet...I can't stop reading this blog.

  3. I think the question of which way is better ties to a belief about how kids learn... Is talking at them the best way for them to learn new things and create connections in their brain, or is them actively thinking about what causes what in functions better for them. Alternatively, what exactly are they learning when you tell them to copy down a chart versus playing with function cards.

    For what it's worth, yes - I believe that this approach is better than just telling them.

  4. Thanks for the comment, Irrational cube! I have been a pretty traditional teacher in the past, but I am really making an effort to get my students to use intuition and deeper thought. It is a process for both of us, but we are getting there. Occasionally (like the day of this lesson) I pause and ask myself if it is worth the extra time. I still think the answer is yes, but sometimes I need to reflect and re-assure myself that I am moving in the right direction.

  5. Thumbs up all the way! Awesome! Good job!

  6. It like anything in teaching or life. Its multiple perspectives on the same thing that ultimately lead to understanding. We had an old computer game when I was a kid that mandated you examine the question from every angle before you determined your answer.

    Certainly the statistics I am working with for my job demonstrate exactly that. Not to mention you made it a positive learning experience for your students

  7. Hi Amy, Dan from Desmos here. We'll be releasing a card sort authoring tool in the near future. We'd like to release it preloaded with some useful card sorts and we thought the card sort you posted here fit the bill. Would you let us consider re-creating it in our software and releasing it when we launch?

    Some conditions:

    We won't sell your card sort.
    We will release it for free use and modification by teachers.
    We'll credit you.

    Let me know what you think!


    1. Hi Dan, I would be honored to have you and Desmos use this card sort. I'm so happy you asked! I look forward to the rolling out of the new tool!


    2. Awesome. Thanks, Amy. We'll keep you posted.