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:

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.

Student Creations

Last year, my kids blew me away with this.  This year, I was a little more prepared for what we might be able to do with projectile motion.  We spent quite a bit of time on vertical motion as part of our standard curriculum, but once we finished with our required standards, we turned our focus towards trig ratios and applying them to motion problems.  I built a few applets using GeoGebra to help my students visualize the motion and it sparked an end of year project that these kids are really proud of. 

Abel, Matt and Robert

These kids were the first to figure out how to model the projectile.  They used the rest of their time trying to dial in the effects.  We couldn't figure out how to make the backgrounds of pictures transparent, so they spent a bunch of time defining polygons to cover the white areas.  The definitions were really tricky because they had to be defined in terms of the point that was being projected otherwise the image would move but the polygon would remain static. 

Check their applet here.

David, Jett and Sartaj

This group really spent some time dialing in their applet.  In my opinion, it's probably the most aesthetically pleasing. 

Check their applet here.

Sierra, Brandon H. and Brandon J.

 

The tricky part of this applet was in defining the condition to display the "Bullseye!" text.  Since the center of the board is an ellipse (to establish a perspective) these students had to define four points to represent the vertical and horizontal extremes of the ellipse.  They then had to determine a set of inequalities which would describe when the point of the dart actually fell within the range of those four points.

Check their applet here.

Marco, Brandon M. and Lazaro

The thing I really like about this applet is how careful they were with their facts.  The fence height can change from 3' (Dodger Stadium left/right field) to 37' (Fenway Park's Green Monster).  They had to define many points in terms of other points in order to get the fence to be dynamic. 

Check their applet here.

Fareen, Alec and Breanna

This group took this project by the horns, big time.  They tackled two different motion problems in one.  They have a projectile and the bird flies in a linear path defined by an angular velocity.  They ran into a snag because their scale was so large that the applet ran incredibly slow.  So they spent some time tweaking the axes in order to end up with a really cool applet.

Hit the duck and you'll see their sense of humor--trust me.

Check their applet here.

Jodie, Abraham and Destin

This group had a HUGE vision for this project.  They wanted the pitch to come in as a projectile and then leave the batter with a greater angle and greater velocity.  The timing on this was difficult at best.  They managed to get two projectiles occuring at different times, but had to adjust the time slider to do so.  There were times that this one stumped me.  I really appreciated the challenge they took on. 

Check their applet here.

Creston, Mackay, Jared and Alex

Let's blow up a castle.  What else can you say?  This group really paid attention to detail.  Heck, they even made the clouds move.  Hit the castle and get a mushroom cloud.  What's not to like about that?!

Check their applet here.

Frankie, Alex and Dil

If you knew these guys, you'd see how appropriate a flying monkey is to their applet.  Again, with the details.  Determining the condition to show the final image took some time.  How close does the monkey have to get to the target in order for the launch to be a success?  They mulled it over and drew some strong conclusions.

Check their applet here.

My Role

I asked a lot of questions.  Direct instruction was necessary on things specific to GeoGebra like the coordinates of point B can be understood as (x(B),y(B)) but nearly all of the manipulation of the equations was done by them.  If a group got stuck on how to make the animation end, the standard line of questioning would go something like:

"What do you want the applet to look like when the animation ends?"

"In order to get that result are we more interested in the height of your projectile or the distance?"

"How can we describe the height of a projectile?" or "How can we describe the distance it's travelled?"

Once they were able determine which model they needed to use (vertical motion or linear motion), we'd set up the equation.  A lot of them looked something like:

h = -16t² + v sin(α) t + s        or           d = v cos(α) t

and they'd play with it until they solved for t.  Sometimes we'd have to think of the velocity in terms of something times t and go back to the original equations h = -16t² + vt + s or D = rt in order for them to realize that v sin(α) and v cos(α) are just rates. 

I don't think I've had more fun over a two week period in the classroom. Ever.

The best part was when the groups would finally export their applet to .html and then we'd go back to my desk where I showed them how to replace the current code with the animation code.  The looks on their faces when they saw something they had created actually do what it was supposed to do was priceless. 

Yeah, we'll prolly do something like this again next year. 

Tell 'em what you think in the comments. 

(Note: I had planned on having students write their reflections and link to their projects on our class blog. However, due to a time crunch at the end, I've posted them all here. They'll be checking this post for your feedback.)

Four!

Yeah, yeah, I know it's "fore."

But anyway...

The golf applet is up and running.  Kids got a kick out of it and are now designing their own applets.  We've done our work with developing trig ratios, solved a few problems involving right triangle trig and have previously worked with the fact that v_x and v_y are independent of each other given v₀.

I gave students a choice for their final projects.  They could try to re-create one of the projectile apps I've made or they could design their own.  They all opted to design their own.  I've got everything from monkeys flying through a castle window to slow pitch softball to tossing paper in a trash can.  It's really cool to see them make lists of what they need to do, decide which parts of the applet they want to be defined using sliders and have them manage time by deciding which parts of the problem to work on in class and which parts can wait until they get home.  (Note:  Middle schoolers try to paint the walls and hang pictures before the foundation is built)  This has been a great opportunity to talk about how to actually plan a project.  They have to work this stuff out on paper before they are even close to being ready to put anything into the computer. 

The math they are dealing with behind the scenes here is phenomenal.  I had a couple of groups solve:

h = -16t₀² + v₀sin(α)t_{0 [1]}

by plugging the variables into the quadratic formula because they wanted to find a way to make the animation stop at a certain point.

I'll post their projects once they're completed. 

If you'd like the .ggb and .html files for the applet.

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^[1] We had to use t₀ because we'll need it for animation later.  It made the equation a bit more complicated, but they worked it out.