**Goodie:** “A technique/question that can be applied in many places and teaches thinking.”

**Baddie: **“A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

**February 2009 Goodie of the Month** – Real Fun and Learning with Quadratics

In the second half of Algebra 1:

A typical standard is: Apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

A typical question for this is: A ball is thrown straight down with a speed of 20 [ft/s] from a height of 80 [ft]. When will it hit the ground?

A typical application of technology is:

Tell the student that the function is y(t)=-16t^{2}-20t+80. They know they can’t graph with *t* so they switch to *x*, which they graph on their graphing calculator.

They see that parabola crosses the *x*-axis. They find the intersection and write *x*=1.7 and get their points.

Now ask them "Where does the ball hit the ground?". They will point to the intersection point – totally forgetting that this is *vertical motion* and that the ball hits the ground at (0,0)!

Ask them "What is the units on your answer?". You will be lucky if they give you [seconds] and not [feet]!

So why is this a “Goodie of the Month”?

The problem isn’t the standard. Nor is it the question. Both are excellent. The problem is the technology – it is undoing the learning.

Let’s change the technology! If the animation below doesn’t work – open this link: Vertical Motion

**Here are some “good problems”.**

- Set ho = 80[ft] and vo = -20[ft/s]. Run the animation. Point with your finger to the place
**where**the ball hit the ground. Now find the place on the graph where it says**when**it hit the ground. - Set ho = 80[ft] and vo= 20[ft/s]. Run the animation. Notice that the ball goes up before it goes down. Why is this? Reset the animation and using the step forward + and step backward – buttons, stop the animation when the ball is at its highest point. Point with your finger to the place
**where**the ball is at its highest point. Now find the place on the graph where it says**when**it is at its peak. What time is this? - Set ho = 0[ft] and vo= 100[ft/s]. Find when the ball hits the ground. Do this via the animation and algebraically using the function. When is the ball at its highest point (remember – parabolas are symmetric!)? What is this highest point? Do not forget units!

**Here are some “good questions” for the function: ** *h*(*t*)=ho+vot-16t^{2}

- The function h(t) gives height in [ft]. So each member of this function must give [ft].
- ho is (initial) height. So its unit is [ft]. It is all by itself so this member is in [ft].
- vo is (initial) velocity. So its unit is [ft/s]. How does this member give [ft]?
- What do you think the unit of “16” is so that this last member gives [ft]?

- In what part of the function is gravity playing a part? In which of the above problems is the only force gravity?
- Why do you think there is a plus sign in front of vo and a minus sign in front of 16? That is, what does it mean in mathematics/physics for an object to have a positive velocity? Does gravity increase this velocity?
- Make up a problem that describes this situation: ho = 0[ft] and vo= 100[ft/s].

**Links for this interActivity (worksheets, downloads, etc.): Open Metadata**