On Teaching By Learning

One of the things that I have learned over the years is to let go of any preconceptions I have about how a problem should be solved.  I have methods I prefer, but my students need to develop their own.  Never has this been more obvious than it was today.

We are beginning to play around with non-linear functions and so I gave my class the following problem:

You're going  to build a garden and need to build a fence around it.  If you have 120' of fencing, how would you set it up in order to have the biggest garden?

...or something like that.

I didn't specify rectangle although I figured most kids would default to that.  I didn't mention the barn in the back of the property that could be used as one of the sides.  I just kept the prompt as simple as possible so I could see where they took it.

Many assumed a square right off the bat. And one group felt pretty ambitious and looked up some shapes in their school planner only to settle on the typical decagon.  (Good luck calculating the area of that one, fellas.)

So, I'm walking around and seeing groups all proud of themselves by defining x as one side and y as the other and they're arguing about whether 2x + 2y = 120 or x + y = 60 is the better equation.  I ask a few questions like, "so how will that equation help you determine the largest garden" to which they reply, "we'll find the maximum."

"Mmm-kay," I say as I mutter to myself, "I'll be back..."

Then I walk up to a group of three boys who usually push the envelope when it comes to creative problem solving and I see the equation x + y = 60.  I think to myself, "oh, no not you too."

Then I look a little closer and I see this other thing they're working on.

xy > 900


I double take.

"Wait, what?"

Before any of them even address my incredulous look, one of them says, "let's go put it in GeoGebra."

"So how'd you come up with that?"

"Well we figured that x + y = 60 tells us how much fencing we have.  And since a square would give us 900 ft², we want to know if there is an area out there that's greater than 900."

"Uh, yeah.  That's, uh, yeah, that's exactly how I'd do it."

Not really.  This is what I really said.

These kids used a system.  Not in a million years would I have ever considered using a system to solve this problem and three 13 year-olds set me straight.

Man, I love this job.

Something Different

This year, I decided to take a much more hands-off approach when it came to student projects. There were some homeruns, but there were too many swings-and-misses. Some students opted not to even step to the plate. I suppose that's what happens when students are offered more autonomy. But, I didn't do enough to prepare them to make decisions in such an open ended environment. I think I was too hands-off.

For the final project, I gave my 8th graders seven choices; one of which was to determine the angle that would maximize the distance traveled by a projectile.

What they knew:

• Linear motion model.
• Vertical motion model.

What they didn't know:

• Vertical and horizontal motion do not affect one another.
• How vertical and horizontal motion work together to determine the path of a projectile.
• Trig ratios

Last year I had students do an investigation on trig ratios prior to working with projectile motion. But due to a shortened school year and the fact that all of my students will be taking geometry next year, I had to cut something.

It took a few short conversations for the group to get the fact that horizontal and vertical work together to determine the path and that they needed to use the vertical motion model to determine how long the ball would be in the air. From there, they could figure out how far it would go.

But there was one problem: they didn't know how fast the ball was travelling which made it impossible to determine the vertical and horizontal components.

The Process

Q: How fast is the ball travelling when it is hit?
A: I didn't specify, did I?

This led to a nice conversation on how we need to eliminate as many variables as we can.


Solution: Pick a velocity and work with it. They chose 100 ft/sec.

Q: So how fast is the ball travelling vertically and how fast is it travelling horizontally?
A: That depends.
Q: On what?

So we took turns pushing Joey around the room from behind and the side simultaneously. Each time one person pushed harder than the other.


Conclusion: If the person from the back pushes harder, Joey goes forward more. If the person from the side pushes harder, Joey moves to his left more.

Then we talked about how the velocities can be modeled using vectors and we can use what we know about triangles. Since the forces are perpendicular, we have a right triangle.

Q: If all we know is the hypotenuse of the right triangle, how do we find the other lengths?
A: Is that really all you know?


Solution: They settled on using a 45-45-90 since that is the only way they could figure out the other two sides.

Q: But what do we do for other angles?
A: Yeah, that's kinda tough, huh? Why don't you use a protractor to draw the angle you want, build the triangle you want and measure.
Q: Can we use GeoGebra?
A: Or that.

They used an applet with a fixed hypotenuse of 100 and gathered data on the other two sides.

Q: Is there an easier way?
A: Yeah. It's called sine and cosine. See how these ratios don't change as long as the angle remains constant? (it took a little longer than that, but you get the point)

They were off and running.

Conclusions

• 45 degrees maximizes distance.
• Complementary angles yield the same distance.
• Oh, and this:

I think you physics folks would say something like this:

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.

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.




Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

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?



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


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.



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


If this proves to be useful, I'll dial it all in and post the still, video and applets for download.

The Monkey Hunter

Yesterday, Dan threw out this tweet:

So, naturally, I had to try this:


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)




If I were using this in class, I'd probably animate it so students had to choose values for velocity, start height and distance and hit "GO."

Download original .ggb file.

WCYDWT: Projectiles

The camera angle messes with the perspective a bit, but I still find these images interesting. Does it mess with the perspective so much that it ruins the problem or does it just lead to another discussion on how things aren't always as they seem. I'm thinking that I'll show the video, but make digital and hard copies of the photos available to students.

Question #1


Projectile Question (YG) from David Cox on Vimeo.

Question #2

Projectile Question (PG) from David Cox on Vimeo.

Answer #1

Projectile Answer (YG) from David Cox on Vimeo.


Answer #2


Projectile Answer (PG) from David Cox on Vimeo.

Next up:
Create something that helps kids see vertical and horizontal motion independence.

Which One's Best?

Two questions:

1.  Is the question clear?
2.  Which image asks it best?

#1

#2

#3

WCYDWT: Space Shuttle Discovery

I've tipped my hand here a bit because simply showing a shuttle launch would be way too open ended. I have the original lesson plan as designed by NASA and I'm pretty sure how I'd go about presenting this lesson, but I'm curious as to how y'all'd do it.


STS 121 Launch (Initial/Final values) from David Cox on Vimeo.
 

Bottom's Up!

A few of you have asked for more explanation on using Diamond Problems and/or Bottom's Up for factoring quadratics. I figured showing you would be easier than writing about it. So here you go:


If you have any other questions, let me know and I'll be glad to see what I can do to help out.

Learning

Joe works too fast.

Joe doesn't show his work.

Joe ends with the beginning in mind.

Today Joe took a test on being able to identify the graphs of quadratic and cubic equations. He bombed it.

Joe came back to my desk while the rest of his class was finishing their tests and had some good questions. He said he didn't know how to set up an input-output table so I showed him. I plugged in one x and found y. I plugged in a second x and asked Joe what the y-value would be. Joe did the rest himself.

Joe and I graphed his results. I asked Joe to graph a simple parabola with coeffieient of 1 on the same set of axes.

Joe said, "Hey Mr. Cox, if the number in front is bigger than 1, then it grows faster."

"And what if the number is a fraction?"

"Then it grows slower?"

"Are you asking me or telling me?"

"It grows slower."

"What if the number is negative?"

We turned the paper upside down.

Joe gets it.

His grade is different than he thinks it is.

Update (can I update a post I haven't published yet?)

Joe just informed me that a cubic with a positive coeffiecient will end up in the 1st quadrant and one with a negative coeffient will end up in the 4th. Yeah, better get on that grade change.

Vertical Motion

...or Quadratics Unit Completed

One of my favorite things to do with quadratics is work with the vertical motion model. The way I introduce this is to show something like this video:




or this one:



Note: I have downloaded the video files for offline use as YouTube is blocked in my district. I have also edited these videos so students do not know what the starting heights are.

A pretty good discussion usually ensues about how a falling object is accelerating as it approaches the ground. I'll ask students what information would they find interesting. Responses usually include but are not limited to:

1. How high was he when he jumped?
2. How fast was he going when he hit the water?
3. How long was he in the air?

The easiest to answer is "How long was he in the air?" simply because we can set a stop watch to the problem and figure it out.

From there we can use the vertical motion model to find out how crazy one must be to earn a world record high dive.

Twist #1: Add initial velocity

I find the toughest thing about teaching vertical motion is the fact that my students have a tough time recognizing the resulting parabola is not the actual path of the ball, but the graph of height vs. time.

To view applet online click here.

I have found this applet to be an invaluable resource in teaching vertical motion.

Twist #2: Add horizontal velocity

I have been thinking for a while about adding a problem which includes a horizontal velocity to this unit, but really had no way to do it. Last year I threw this post of Dan's in front of my kids and they came up with this. The tough part about working with the horizontal velocity component is helping students realize that vertical and horizontal forces don't have an effect on each other. The best sources I have found in helping explain this concept are this (h/t Rhett Allain) and this.

However, recently I enlisted the help of Dr. Linda Fahlberg-Stojanovska to create a GeoGebra applet that would model a problem that involves vertical motion with horizontal velocity.

To view actual applet click here.

I really like this applet because of the iterative feature. I love that I now know how to get rid of the sliders so as to eliminate the cheat factor.

Student Response:
They LOVE the applets. When they can actually see that their calculations work out, they get pretty excited. I've had some students become interested in making their own which may turn into a pretty cool project.

For the kids who still have trouble understanding the independence between the horizontal and vertical components of this problem, I tried to appeal to their ELA sensibilities regarding point of view.

Tell the story from the vertical point of view.

Object is dropped from 's' starting height and falls for 't' seconds before hitting the ground.

Equation used:

Tell the story from the horizontal point of view.

Object travels at 'r' miles per hour for 't' seconds before hitting the ground.

Equation used: D = rt

Download Vertical Motion applet 1 and Worksheet
Download Vertical Motion applet 2 Worksheet included

Vertical Motion v. Time

Some kids have a tough time recognizing that tha parabola created when graphing the height vs. time of a falling object isn't actually the path the ball takes. This seemed to help clear it up with some students today.

Quadratics Unit

My introduction to quadratics has evolved over the years. Recently I settled on having my students do as much graphing as possible and allowing them to make the connections they need to make in regards to the equation of a quadratic and it's resulting graph. I have found that allowing my students to roll their sleeves up and start graphing allows them to become more comfortable with quadratics. This pays dividends when it comes time to solve quadratic equations by factoring, completing the square or by using the quadratic formula.

The order of things goes something like this:

Intro to Quadratics
This worksheet usually comes after we have exhausted ourselves on linears and introduces the idea that there are other relationships and, no, they don't all form a line. We do this by measuring various circular objects and calculating the area and circumference. We then graph circumference vs. radius and area vs. radius.

Graph the Magic Number
This worksheet is a little less organic, but still results in a quadratic relationship.

Stretch Factor
What happens when we graph change the "a" value? I give students a series of "Silent Board Games" (borrowed from CPM), where students are given incomplete input/output tables and are required to find the patterns to complete them. Once they are completed, they graph the parabolas on the given coordinate plane. The idea is for them to make the connections between the input/output table, the equation and the graph. Usually students are pretty quick at recognizing they can describe what the graph would look like just by observing the equation.

Graph Given the Vertex
In this worksheet, I give students random points on the plane and have them graph a parabola with a stretch factor of 1 or -1. By this time I expect them to have a grasp of the relationship between the points on a parabola and it's vertex. This allows them to see that if they can graph one, they can graph them all.

Vertical Shift/Horizontal Shift
Both of these worksheets are in the same format as the stretch factor worksheet. I have students complete input output tables and graph. By this time, they are looking for relationships between the equation and the visual interpretation of the graph.

GeoGebra Investigation
This is pretty intensive and I probably need to break it up into smaller more manageable labs. I would like to throw that out there for discussion. The main ideas that I wanted my students to explore are:

• parabolic symmetry
• line of symmetry is the average of the x intercepts
• x value of the vertex and the l.o.s. are the same
• relationship between a, b and the l.o.s.
• similarities and differences between quadratic functions and equations
• solve quadratic equations by graphing
• determining the number of solutions by using the discriminant

Find the vertex
This last worksheet has students take different quadratics and use the equation to find the vertex and line of symmetry. From there they will graph using only the vertex and the stretch factor. I then allow them to use GeoGebra to check their answers.

Worksheets (if you're interested)