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









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].

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

8 February 2009

Baddie of the Month – Teaching “completing the square” for quadratics

Filed under: education,math — Tags: , , , , — admin @ 4:13 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 Baddie of the MonthTeaching “completing the square” for quadratics

Yes, she is stuck on quadratics. But quadratics are so useful and can be fun and we keep teaching techniques that don’t give much value for the time and effort.

Why? These are reasons I have found online or been given "for" studying “completing the square”:

(1) one can then prove the quadratic formula , (2) one can find the vertex of the quadratic,
(3) one can graph quadratics using graph transforms and (4) one can solve integrals of the form dx/(x^2+bx+c)

My response:

  1. Students at this level cannot understand a mathematical proof. All they see is a manipulation of symbols/letters/numbers. Ask any non-math student. Ask any non-math adult. They didn’t get it. Period. So we teach them a technique in order for them to see a proof they don’t understand. (I am happy if they get accurate results from the quadratic formula with any numbers for a, b and c and can relate them to the graph of the quadratic and/or the answers to the question that was posed and judge the reasonableness of all.)
  2. It is much easier and useful to find the vertex of a quadratic by first realizing that every parabola is symmmetric and thus the vertex must be the value of the function at the half-way point between the roots*. Finding the vertex this way requires them to relate solutions from the quadratic formula to roots/zeros/x-intercepts, reinforces learning about midpoints, relates the vertex to the quadratic formula – more reinforcement  and it teaches them to find function values.  Win-win-win-win. Using "completing the square" teaches them to manipulate numbers.
  3. Does anyone actually graph a function using a graph transform – ever? After trying to teach graph transforms for over 20 years – I have decided that the best I can hope for is "a vague understanding" that "x^2+4" is "x^2 up 4" and 3sinx is "3 times taller than sinx" and cos2x is "2 times faster than cosx". Trying to combine all these is a total waste of time.
  4. Solve integrals – good grief. Maybe we should teach them partial fractions in algebra 1 too.

How to find the vertex: The x-value of the vertex is x= -b/2a. Substitute this value into the quadratic to get the y-value.

(BTW: nowhere did I find the semi-plausible defense that “completing the square” is also used to find the center of a circle and identify conic sections in the 2nd half of Algebra2. By then, I agree that completing the square is not an unreasonable technique to teach.

*roots found using the quadratic formula I am sure :) .

Related topics:

Past: January 2009 Baddie of the Month: Hand-factoring a quadratic with a≠1.

Future: Baddie – Teaching complex numbers in the same 2 month span as you teach graphing of quadratics.

Future: Goodie – Good questions for Quadratic Equations/Functions

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