I've been spending some time lately thinking about how to help elementary school teachers teach math conceptually even if they don't consider themselves strong in math. I keep coming back to interactive applets (read: GeoGebra) because it's kind of a one-stop shop. Yesterday, I tried a few out with a 4th grade class and they responded very favorably. So, here are the latest two I've done on fraction multiplication.
Click on "File" and "Save" if you'd like a copy of either applet.
Please forgive the crowding, I had to minimize everything in order to fit them into the post.
Thursday, March 3, 2011
Wednesday, March 2, 2011
Another First
Usually kids raise their hands to show what they know.
Today, M.C. politely asked me to get out of the way, walked up to the document camera, put her work under it and said, "I need your help, guys. Really. I want your feedback. I don't know how to do this."
The class responded. No one ridiculed her. She asked them questions. They asked her questions.
All I could do was sit down and watch.
Today, I caught a glimpse of what school should be.
Tuesday, March 1, 2011
Let 'Em Drive
The Problem:
Me: How many different ways could we do this?
Class: Quadratic Formula, completing the square and factoring.
Student 1: Mr. Cox, why does y = 0? Isn't this a function where y can equal a bunch of things?
Student 2: Yeah, but we want y = 0 because that's when it crosses the x-axis.
Student 1: Ok, so we are just focusing on one possible value of y.
Student 3: So then, if we let y = 0, then we are finding the x values that make y = 0, right? But what if we want to know when y = 1? Can we do that too?
Student 4: I guess we should just let y = 1, but we'd have to subtract it from both sides so we get:
0 = x2 + 10x + 15. So now we are finding the x-intercept again.
Me: Is this an x-intercept? What does y equal?
Student 4: Oh, no, y = 1.
Student 5: Mr. Cox, can we see what this looks like in GeoGebra or something?
We graphed it and then moved a point around on the parabola to verify our results and looked at how the points where y =1 maintained the same symmetry with the x-intercepts. Then, a kid pipes up:
Me: Hmm. Maybe. I'll tell you what...you guys give me a function that has x intercepts at -4, 0 and 3. You have 15 minutes. Go!
About 5 minutes later, J.V. walks up with this:
And I ask him, "How did you come up with this?"
J.V.: Remember the other day when we had to come up with parabolas with x-intercepts? I figured I could just work backwards like we did then.
I'm amazed at how often these kids will take the lesson into places I wouldn't have thought to go. It's just a matter of letting go a little. And the more I look for open ended opportunities, the return is exponential.
Find the x-intercepts of y = x2 + 10x + 16.
Me: How many different ways could we do this?
Class: Quadratic Formula, completing the square and factoring.
Student 1: Mr. Cox, why does y = 0? Isn't this a function where y can equal a bunch of things?
Student 2: Yeah, but we want y = 0 because that's when it crosses the x-axis.
Student 1: Ok, so we are just focusing on one possible value of y.
Student 3: So then, if we let y = 0, then we are finding the x values that make y = 0, right? But what if we want to know when y = 1? Can we do that too?
Student 4: I guess we should just let y = 1, but we'd have to subtract it from both sides so we get:
0 = x2 + 10x + 15. So now we are finding the x-intercept again.
Me: Is this an x-intercept? What does y equal?
Student 4: Oh, no, y = 1.
Student 5: Mr. Cox, can we see what this looks like in GeoGebra or something?
We graphed it and then moved a point around on the parabola to verify our results and looked at how the points where y =1 maintained the same symmetry with the x-intercepts. Then, a kid pipes up:
Are there graphs that have more than two x-intercepts?
Me: Hmm. Maybe. I'll tell you what...you guys give me a function that has x intercepts at -4, 0 and 3. You have 15 minutes. Go!
About 5 minutes later, J.V. walks up with this:
y = (x + 4)(x - 0)(x - 3)
y = x3 + x2 - 12x
J.V.: Remember the other day when we had to come up with parabolas with x-intercepts? I figured I could just work backwards like we did then.
I'm amazed at how often these kids will take the lesson into places I wouldn't have thought to go. It's just a matter of letting go a little. And the more I look for open ended opportunities, the return is exponential.
Saturday, February 26, 2011
Monday, February 14, 2011
Thursday, February 10, 2011
I'm Holding Somebody Responsible For This!
Step 1: Choose values for 'a' and 'c'.
Step 2: Move the 'b' slider from -10 to 10.
Step 3: Select all the points step 2 puts in the spreadsheet. Right click and select "Create list of points."
Step 4: Click on the "Conic through 5 points" tool.
Step 5: Click on 5 of your points generated in step 3.
Step 6: Right click on the resulting function in the algebra window and change standard form
Step 7: Compare this function to the Starting Function
Step 8: Explain to me why I never knew this before.
Step 2: Move the 'b' slider from -10 to 10.
Step 3: Select all the points step 2 puts in the spreadsheet. Right click and select "Create list of points."
Step 4: Click on the "Conic through 5 points" tool.
Step 5: Click on 5 of your points generated in step 3.
Step 6: Right click on the resulting function in the algebra window and change standard form
Step 7: Compare this function to the Starting Function
Step 8: Explain to me why I never knew this before.
Wednesday, February 2, 2011
Quadratics Revisited: The Falling Object Model
I keep hoping that I can use this to help kids derive the falling object model. I'm getting close. I exported the video at 6fps so 1/6 second elapses between strobes. I'd like some feedback on this before I roll it out to my students. How intuitive is it to use both applets together?
Once you have plotted your points, use the FitPoly function. Simply enter "fitpoly[A,B,C,D,E,F,G,H,I,2]" to plot a quadratic function.
If this proves to be useful, I'll dial it all in and post the still, video and applets for download.
Once you have plotted your points, use the FitPoly function. Simply enter "fitpoly[A,B,C,D,E,F,G,H,I,2]" to plot a quadratic function.
If this proves to be useful, I'll dial it all in and post the still, video and applets for download.
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