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.

And Yet Another Reason

Many of my students have really had a tough time with solving equations with absolute value on both sides. If the test happens to be multiple choice, then they just work backwards by plugging the answer choices into the original equations. I found a way to combat that and still be able to administer the test via our network.

Short answers are allowed and the Test Player has a space for students to explain their process and submit a numeric answer. All I have to do is manually enter the number of points a student earned on each problem.




Once students have completed an exam, they receive a feedback sheet explaining what they missed and why. The rationale for each problem has been created for most of the problems in any test generator. However, since I've been creating many of the problems myself, I have to build the rationale into the problem using the algorithm definitions used in the problem itself.
So this:

Becomes this:

This becomes very convenient when it comes to online study guides as students and parents can go through the problems, see if they are correct and then, if a problem was missed, see why it was missed.

The 2 Product Property

...or more formally known as "Completing the Factoring by Using the Multiplicative Property of 2." We're going to release it under Creative Commons, so feel free to use it in your classes too. Just make sure you give credit where credit is due.

Here it is. Bask in the glory!


Beautiful isn't it? So elegant. So simple. So WRONG!

I've been teaching this stuff for a while and haven't seen this misconception before. I thanked the student for providing us with the best wrong answer I've ever seen. He looked at me kinda funny then realized I was dead serious. I loved this.

What an opportunity to discuss why we do what we do. This student had the process for solving a quadratic equation by factoring cold. But he didn't understand why we set the equation equal to zero. I don't know about your students, but mine do pretty well with the What. It's the Why that drives this class. I was a bit surprised at how long it took for a student to step up and explain the error in a way that wasn't, "if you plug in the numbers they don't work."

I do my best to make my students think, but they still try to become good little algorithm followers.

The fight goes on.

OK. I Really, Really Like It

Love may be too strong of a word. But now that the good folks in the IT department got me set up with ExamView player. Now I can administer tests via our local network. It's really a pretty simple process.

Publish existing test to LAN:

Decide who has access to the test:

Give it a name and password:

Scramble the questions. You can also set the algorithm values so each student actually gets a different test.

Decide if you want them to be able to check their answers as they go as well as the type of feedback they will receive once they have finished the test.

From there it is simply a matter of students going into a shared folder and opening up the ExamView player. They have to know where you have saved the test, but other than that, it's pretty smooth sailing.

Once students have finished the test, simply ask for the results and thar ya go!

So far, I am using only multiple choice tests. I have mixed feelings about this. However, now that I can write the questions myself and program the misconceptions into the distractors, I'm not mad at myself for doing it. I will start playing around with using numeric responses and see how it goes.

All in all, I really like the idea of being able to provide different versions of the same test depending on who is taking it and when. It also makes for a super easy re-assessment tool.

Try That With a Dry Erase Board!

...or Quadratics Unit (continued)

So once we got all the graphing out of the way, it became about solving quadratic equations. We took a lot of time developing the concept of being able to identify key points/characteristics of a parabola by examining the two forms of the equation. (ie. Graphing (vertex) Form and Standard Form)

The one thing that isn't easily identified by examining either equation is the x-intercept(s). So we started solving quadratic equations. We got out of sync really quickly, though. So I threw out a tweet asking for what you all had in regards to solving quadratics and Sam shot me this.

So I took Sam's hard work and re-mixed it to fit what my students need and came up with this (and the corresponding Notebook file). Most of Sam's problems were perfect so I just added a little to bridge the gap between what we had already done and where we were headed.

This "Quadratic solving" boot camp lasted three days and culminated with the kids deriving the quadratic formula. Realistically, I would have about four or five students who could actually derive it. The rest can follow the steps, which is what they will be expected to do on our CST's.


(the quadratic formula derivation is part of the Notebook file above)