Green Light

On our Sunday morning walk to church, the boys like to play "Green Light, Red Light."  They'll take off running and when they approach a driveway or just get too far out front, we'll yell, "Red Light!" and they stop.  We've incorporated blinkers and windshield wipers, but I digress. 

Jabin, our two-year-old, wants to be a big boy and run with his brothers.  He hears "Green Light" and takes off.  He hears, "Red Light" and keeps going. 

I'm thinking to myself, "What's wrong with this kid?" so I yell, "Stop!" and he stops. 

My wife was a kindergarten teacher before we started having children and she informed me that Jabin understands the "Light!" command but doesn't distinguish the colors. 

Light means run.  Stop means stop. 

It's that simple.

How often do I think my students can understand the nuance of something I see as being so simple?

I see 2x and think two times x and they may see 2x and think twenty-something. 

Man, what I say and what they hear may be worlds apart. 

You Want Iteration?

...I'll give you iteration.

If the best thing is for students to actually encounter a situation that provokes a question  and the next best thing is to give them an image (static or dynamic) of said situation, then is the third best thing to give them an applet that models the situation?

Dan:

I only know I shot this because I felt like I needed it, because the alternative is a problem involving savings accounts with different principals and different monthly deposits and none of my kids have savings accounts.

Yeah, I'm not a big fan of the bank account problems either.  But I like race cars.  Lots and lots of race cars.  My students like 'em too.  Red ones.  And green ones. 

Since I don't know how to make the same guy show up on a picture/video in two different places and I don't really feel like going out to the park to run (a bunch of times), I figured I'd let the computer give me some iteration. 

So...

 What Can You Do With This?

Slide 1

Slide 2

Slide 3

                   

Applet's here.

Animating GeoGebra

After having my students work on the projectile motion applets, I realized that I'd like to have them create an applet modeling linear motion during the first semester.  This will not only help nail down the ideas of linear relationships, but it will also help with the programming aspect of the more complex projectile applet they'll do in the second semester. To test it out, I created this applet this morning. 

I can copy and paste with the best of them.  That's a good thing because I know very little about HTML and almost nothing about javascript.  So, it's time to pay it forward.  Big thanks go out to Linda Fahlberg-Stojanovska for setting me up with the code.  So, if you'd like to animate a GeoGebra applet, do this:

Choose your variables

In the race car applet, I needed variables for time (t0), velocity(v1 and v2), head start (s1 and s2) and race distance (f).  (Start with "t0" for time.  We'll change it later.)

If you would like to use the "manual animation" feature on your applet, set the condition to show object to "manual."

Define Your Motion Point


I've attached the green car to Point A.  (Notice that my time variable is 't' and not 't0.')  More on that later.  You can change the size of the image by determining the location of the corners.  Be sure you define your image using the motion point (Point A in this case). 

When to End Animation
At this point, t0 should have a minimum of 0, but the maximum is whatever you arbitrarily decided upon when you started the applet.  Now we have to decide when to end the animation.  I decided to let the animation end when the Green Car (Point A) reaches the finish line (variable 'f').  All we do here is use the distance formula to determine when the race ends.  So basically, we want to know when...

(v1)(t0) + s1 = f

...which simplifies quite nicely to

t0 = (f-s1)/v1

So regardless of how we change our variables, this is when the race will end. 

Redefine t0
Change the maximum for 't0' to 1 and make the increment '0.01.'

Define 't'

Once you have defined t based on when the problem ends, go back through and redefine all points that used 't0' in terms of 't.' (See above)

Now that everything is in place, hide all of the sliders by right clicking and selecting "show object" so they don't show up in the applet. 

Export

Save the ggb and jar files

Exporting to .html will result in about 6 different files. Make sure to save them in the same folder. The two files you're most interested in are the .html and .ggb files. 

Go to the folder and right click on the .html file and open with Word Pad.

Select all of the code, delete it and replace it with this animation code.  From here, all you need to do is change the red text according to your title, slider names, variable descriptions and initial values. 

Make sure to save and then double click on the html file and you should get something that looks like this:

If there is anything I need to clear up, don't hesitate to ask.

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.

____________________________________________________________________________

^[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.

So Will It?

In my previous post, I described how we developed the Standard Form of a linear equation.  Students are already comfortable with using slope (rate of change) and the y-intercept (initial condition) to determine the equation of a line.  We are currently spending a lot of time recognizing the rule of 4 and how each representation simply tells the same story from a different perspective.  So, next I wanted to see what students would do with trying to decide whether or not a point is on a line.

Prompt:

Question:  Will the point go through the line?

Students made their guesses and we split the room into the two camps:  Yes and No. 

Cheaters were cast out!

Alright, I didn't really cast him out.  But we did have a short conversation

on how reliable the paper would be in determining whether or not the point is on the line. 

We need more information:

Not quite enough to determine slope intercept form, but enough to get a fairly good guess as to the slope of the line.  Students wrote their guesses for the equation and all groups came up with: 

y = -3/4 x + 4

Are you sure about this equation? 

Nah, we need a bit more information. 

Now we have both intercepts; enough to verify our slope and enough to write the equation in Standard Form.

But, do we have enough info to determine whether or not the point is on the line? 

So we verified two different ways: 

1. 
   
    Continuing with the pattern determined by the slope.
2. 
   
    Plugging the point into both equations to see if they are satisfied.

And finally:

 

The coolest part about this is that a colleague and I developed this lesson in about 10 minutes.  It's pretty rewarding when minds get together and focus on interesting ways to deliver instruction centered around a simple question.  Have to give credit to Dan for stoking the fire.

Why Didn't I Think of This Before?

Now that we are finished with 7th grade standards, we start to take the concepts that are considered pre-algebra and stretch them into algebra.  My students have a solid understanding of slope as rate-of-change and I have been really emphasizing multiple representations.  They can handle an equation in slope-intercept form pretty well. 

I wanted to introduce them to standard form and have usually done this by giving the equation and having them graph.  In years past, I found many students really struggled with For some reason, I decided on a different approach this year.  This year I gave students the x and y intercepts and asked them if they could figure out an equation that would fit.  I introduced this equation as:

____ x + ____ y = ______

It became a puzzle and eventually kids nailed the idea that if we have the points (2, 0) and (0, 3), we can write the equation as: 3x + 2y = 6.  And after 5 or 6 examples, I gave them the points: (e, 0) and (0, f) to which they responded with: fx + ey = fe. 

Now, given the equation, can we find the x and y intercepts? 
Not a problem. 

The game play at the beginning of the lesson really opened them up to the idea which made any actual instruction I had to do much easier. 

In the end, I say it was a win.

Triangle Ratios

The Prompt:

A car is traveling NE at a rate of 60 mph.  What is the rate of the car from the perspectives of Persons A and B?

Students pretty quickly figured that in one hour, the car had traveled 60 miles and used the Pythagorean Theorem to determine the Northern and Eastern rates.

Where we are going

                         What happens if the car is traveling more North than East? 

How We're Getting There

I made this worksheet allowing students to construct right triangles containing an angle of their choosing, checked the first construction and had them gather data.  They manipulated each triangle so they could gather three sets of lengths for similar right triangles.  (Note:  Be sure students construct the triangles by defining the angles, perpendicular lines, etc. so that the triangles don't lose their integrity as they are manipulated.)

Draw Conclusions
After data gathering, I had them find the Sine, Cosine and Tangent for their chosen angles and find a connection between those values and the ratios they had already calculated.

No joke: every kid made the SohCahToa connection--now I don't have to tell that stupid story.

Problem Solving
I've really been looking at how I encourage problem solving in my classes and hoped they'd be able to find all the information about a right triangle given one angle and one side.  I really liked how some of them dove in even though they had no idea (or at least they thought that) where to start. 

Are you sure your right? 
Yeah, pretty sure.

How could you verify.
By doing this:

Extension Tomorrow we will go full circle and answer the original question regarding a car traveling a given rate at a particular angle towards the north. 

Limited Tech Version
Project GeoGebra up front and have class work on the same triangles.  It's still quicker than drawing the darn things yourself. 

Low Tech Version
Bust out the old protractors and build yourself some triangles the old fashioned way.  This still may not be a bad idea for some students who prefer the hands-on approach, although many of my tactile kids like having the ability to interact with the shapes in GeoGebra. 

Are You Ready For Some Football?

Jason Dyer recently wrote about using systems in the context of playing football.  I couldn't leave well enough alone and so I made an applet modelling the relationship between the projectile motion of a football and the linear motion of a receiver. 

I know, the picture is a little cheesy, but it's fun and the kids love it. 

Original .ggb file or html.