Quadratics Revisited: The Falling Object Model (part 2)

I worked on this applet for a while, but Frank offered some good pushback which got me to rethink how I wanted to go about this lesson.

Day 1+


Falling Object (Question) from David Cox on Vimeo.

Show the first part of the video and discuss what we notice. I

"9.8 meters per second per second."

I ask, "Alright, so what does that mean?"

"9.8 meters per second per second!"

"So what's the per second per second all about?"

"Hmm."

Show part 2.

Obvious question: "When is it going to hit the ground?"

Pass out the laminated picture and some Vis-a-vis markers (I knew those would come back in style).

"So what do we need to know?"

"The amount of time between each ball."

"The video was exported at 6 frames per second."

"Huh?"

"6 frames per second."

After a brief discussion, we were pretty clear that 6 fps meant that 1/6 second passed between strobes.

Let's go to work

Most kids realize that the strobes' rate of change has a rate of change and they're able to come within a few hundredths of a second from the answer.


Falling Object (Answer) from David Cox on Vimeo.

Day 2+

Project the same still and ask:

"Ok, so how fast is the ball travelling?"

"Huh?"

"How fast is the ball travelling? You guys were talking about 9.8 m/s² yesterday. What's that all about?"

"It's gravity."

"What does that mean?"

We discuss how gravity is an acceleration but we change from metric to ft ('cause that's what's gonna be on THE test) so the 9.8m becomes 32ft.

So how fast is the ball travelling after 1 second? 2 seconds? 3 seconds?

led to

What is the average rate after 1 second? 2 seconds? 3 seconds?

which led to

Then how far has the ball traveled after 1 second? 2 seconds? 3 seconds?

Once we got comfortable with 16t being a rate, it wasn't too far of a stretch to be able to say that 16t² is a distance.

Can we generalize the height with a given starting height (s)?

Day 3+

Now what if I threw the ball down at a rate of 10 ft/s?

The major preconception here was that many students wanted the initial velocity of 10 ft/s to be an acceleration so they just added it to the 16 ft/s². I tried to stay out of it as much as I could and cooler heads prevailed. We eventually decided that the height of a ball after t seconds could be modeled with:

h(t) = -16t² - vt + s

The key to unlocking this function was for each term to be able to stand alone on its own. Once everyone realized that 16t², vt and s are all distances and 16t² has to be negative since the distance has to be subtracted from s in order to get the height, we were good to go.

Now let's have some fun with rational expressions.

To LCD or Not To LCD

Is finding the LCD necessary?  I know that it makes the numbers smaller when adding fractions, but does it help conceptually?

Would teaching fraction addition like this:  $\frac{a}{b}+ \frac{c}{d} = \frac{ad+bc}{bd}$

and then reducing later make adding polynomial fractions like  $\frac{x}{x+2}+ \frac{3x}{x-3}$ a bit more intuitive?

Does teaching LCD make an already difficult concept more difficult?  Or does it help?

Here are a couple of applets that illustrate the point.  I'm wondering which one would be more helpful for 5th - 7th grade kids.

With LCD and without LCD.

If you want the original GeoGebra files:  LCD, NO LCD

I Made Me Do It

or Dave Made Dave Do It

or @calcdave and @dcox21 are pretty much the same person so this is really a post about a guy talking about himself to himself (or would that be to himself about himself?).

Whatever.

So apparently Dave was eaves dropping on one of my conversations (the nerve of some people) and started asking some interesting questions. I particularly liked the triangle image and couldn't help myself especially since Jason encouraged me via GoogleChat last night (or was it GoogleTalk?) to keep doing this GeoGebra stuff.

Oh, and Jason and I live in California (which makes us pretty much the same person too, except that he lives in the Bay Area and I live in the Central Valley so we are the same person kinda like Massachusetts and Texas are the same state in that they are both part of the contiguous United States) and he lives, like right next door to Dan who's been to Canada.

So I guess y'all have to thank Canada for this applet, eh.


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Adventures in Pedagogy: Check for Understanding

or Context Clues (part 2)

Nevan's helping me build planter boxes by holding the 2x6s in place while I screw them together when I ask:

"Nevan, are those two pieces flush?"

"Yeah, Dad."

"Do you know what 'flush' means?"

"Mmm. No."

"What do you think it means?"

"I assumed it meant 'on track' or something."

So when I'm asked, "how do you know that they know?"

That's how.

The challenge is to care enough to ask.


- Posted using BlogPress from my iPad

Adventures in Pedagogy: Percentages

Sitting here in the car with the boys as my wife runs into Target to pick something up "really quick." (I believed her. She can be quick—in Target, right?)

Anyway, I find a spot that gives us a clear view of the entrance and Dawson says:

"Hey Dad, see that white car in front of us? Why don't you park in front of it? It's closer."

"Can't. See that other car? It's like halfway in the parking space."




"Halfway? That's a bit of an exaggeration."

"Alright, then what percentage of the space is it taking up?"

"About 1/11."

"I'll give you that, but what percentage?"

"About 9 point something percent."

"Not 10%?"

"No, a little less."

Right about then, my wife returned. She was pretty quick.

And she brought me a Drumstick.

- Posted (on location) from my mobile

Adventures in Pedagogy: Problem Solving

Dawson is playing a game on his DS and I plop down next to him and ask:

"Whatcha doin'?"

"Oh man, Dad, I'm on this level that took me forever to figure out."

"Really? So what do you do when you get to a level and you don't know what to do?"

"I experiment."

"What do you mean?"

"I try stuff and keep trying until something works."

"You mean you don't just sit there, throw your hands up and say, 'I don't know what to do, so I quit?'"

"Nope."


- Posted from my iPhone

Similar Triangles and Algebra

Dr. Hung-Hsi Wu pg. 58:

The reason for the critical need of a definition of similarity is that a working knowledge of similar triangles is absolutely essential for students to achieve algebra. Without this knowledge, they would have no hope of understanding the interplay between a linear equation of two variables and its graph, which is a major topic in beginning algebra.

So, why do you think that is?


- Posted using BlogPress from my iPad

Fraction Multiplication

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.



This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com





This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


Please forgive the crowding, I had to minimize everything in order to fit them into the post.

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.    

Let 'Em Drive

The Problem:

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

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.