SL2RL-Math247

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

10 January 2009

Baddie of the Month – Factoring a Quadratic with a≠1

Filed under: applets,education,math,sloodle — Tags: , , , , , — admin @ 5:13 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.”

I am going to try to blog a baddie and a goodie per month. We shall see and of course – this is my opinion.


January 2009 Baddie of the MonthFactoring a quadratic with a≠1 “by hand”.

Okay, I can mostly understand learning to factor “by-hand”:   x²+3x+2  or   x²+x-2.  

Once you understand the principles and get the technique (my scheme), factoring a quadratic by hand with a=1 is faster than using the quadratic formula.

But, I absolutely and totally do not understand the reasoning behind other factoring-by-hand techniques!

Why not: Factoring techniques

  • serve no useful purpose – once factored, with a≠1 you must still solve the individual factors.
  • don’t always work – MOST quadratics even with a=1 and real roots CANNOT be factored by hand.
  • are hard to learn, there are many “special cases”, they take alot of time to teach, …

What to do and why: Use the quadratic formula for all your factoring needs.

  • We are going to teach them the quadratic formula anyway.
  • It always works – either we get real roots and can factor or we get non-real roots and know we cannot factor.
  • By using the quadratic formula all of the time, the connection between quadratics, roots, x-intercepts, graphs of quadratics becomes clear.
  • Repetition of a single technique is much more likely to stay in their heads.

Conclusion: Don’t teach factoring by hand except when a=1. Use the quadratic formula.

Here’s how: ax²+bx+c=a(x-x1)(x-x2) where x1=(-b+D))/2ax2=(-b-D))/2a,  D=√(b²-4ac)

(Here “by hand” means looking for the factors without a formula like when you say “The factors of 2 are 1 and 2 and oh yes, they add to 3 (first expression) or the factors of 2 are 1 and 2 and oh yes, they subtract to 1 (second expression)”.)


Related topics to come in future blogs

Please, OMG please don’t teach completing the square – an even worse waste of time than hand factoring.

Please, please don’t teach complex numbers in the same 2 month span as you teach graphing of quadratics.

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