# SL2RL-Math247

## 22 February 2009

### Goodie of the Month – Fun and Learning with Quadratics

Filed under: applets,education,ICT,math — Tags: , , , , , — admin @ 3:23 am

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 MonthReal 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)=-16t2-20t+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

 m ft 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)

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-16t2

• 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].

## 25 January 2009

### Goodie of the Month – A Good Question for Algebra 1

Filed under: algebra,applets,education,ICT,math — Tags: , , , , — admin @ 2:01 am

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.”

January 2009 Goodie of the MonthA Good Question for Algebra 1

Two ships are sailing in the fog and are being monitored by tracing equipment. As they come into the observer’s rectangular radar screen, one ship, the Rusty Tube, is at a point 900 mm to the right of the bottom left corner of the radar screen along the lower edge. The other ship, the Bucket of Bolts, is located at a point 100 mm above the lower left corner of that screen. One minute later, both ships’ positions have changed. The Rusty Tube has moved to a position on the screen 3 mm left and 2 mm above its previous position on the radar screen. Meanwhile, the Bucket of Bolts has moved to a position 4 mm right and 1 mm above its previous location on that screen.
Assume that both ships continue to move at a constant speed on their respective linear courses. Using graphs and equations, find out if the two ship will collide.

Why do I like this question?

• Students can understand it and it is fun.
• They can graph it on paper or using a graphing program.
• It involves finding the equation of a line through 2 points (twice) – good reinforcement.

Why do I think it is a “good question”?

1. The graph looks like every 2×2 system of linear equations they have solved in Algebra 1.
• It looks like the boats collide at the intersection point (see below).
• It seems like all they need to do is solve the system and be done.
• … until you say “Where is time on the graph?”.
2. The student can build a animated simulator that “shows time” – easily!
3. The kids can make the boats collide – what fun!.
• They can move the starting points until they get the boats to collide.
• They can also adapt the simulator so that they can change the slopes and get the boats to collide. Directions here.
• My thanks to David Cox for seeing this!
4. You can get all kinds of mathematics out of them.
• You can get them to calculate when each of the boats reaches the intersection point in the original question.
• You can get them to check the math on their “colliding simulator” to see if the boats really do collide, where and when.
• You can ask them about a 3D graph and what this would look like when the boats don’t collide and when they do.

To animate, click on the play button at bottom left of graph.

To animate manually, right-click on slider and deselect “Animation on”. Then, click and drag the point on the slider.

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)

Source: I found this question asked on answers.yahoo.com. My webpage for this question is: mathcasts.org/mtwiki/Gq/BoatCollide

## 12 September 2008

### Race Car Activity – Exploring Slope and Intercepts in the Real World

Filed under: applets,education,ICT,math — Tags: — admin @ 10:23 pm

Click and drag the slider points to adjust the cars speeds and positions. Then use the animation buttons.
(The animation buttons may no longer work because there are multiple animated pages on this blog. If they don’t, please go to the webpage – they will work there.)

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)

Here is the webpage: http://mathcasts.org/mtwiki/Activity/CarRace
My thanks to Jon Ingram for showing me how to do this!   See how! (after September 15)