Please Critique my Presentation of the Quadratic Formula

[Bacle2]

Hi, Everyone: I have a job interview tomorrow where I must give a 10-min presentation on the quadratic formula for an intro class , where we are assuming students know both how to factor and how to complete the square. Please comment:

Homework Statement

O.K. We are given an equation ax²+bx+c=0 , where a,b,c are real numbers, and we want to find the values of x that satisfy the equation -- if these values exist *, i.e., we want to find values of x , so that when we plug them in, in the formula, we get a 0. As an example, if we have x²+2x+1 , we sub-in , say, x=1 , and get:

1₂+2.1 +1= 4 ≠ 0 , so 1 does not work. We can see also that values like x=0 or x=1 don't work either.

We will see there is a formula that helps us find these values, and we will derive the formula. We will then see some examples of how to use it




* We're not allowing complex numbers yet.

Homework Equations

The formula we will use is: -b/2a ±[[itex]\sqrt{b²-4ac}[/itex]]/2a

The Attempt at a Solution

Let's see how we get the formula:

We start with the initial equation:

ax²+bx+c=0 , and assume a≠ 0 .

We then divide thru by a , to get:

x²+(b/a)x+c/a=0

We then go on to complete the square , by adding and subtracting (b/2a)²

to get:

x²+(b/a)x+ (b/2a)²- (b/2a)² +c/a=0 , so:

(x+(b/2a))²= (b²/4a²)-c/a=

[b²-4ac]/4a².

Now we solve for x, by first taking the square root:

x=-b/2a±[itex]\sqrt{b²-4ac}[/itex]/2a

And this does it.

Let's see some examples:

First , let's apply it to our original formula x²-2x+1:

here we have :

a=1 , b=-2 , c=1 , so we sub-in the formula to get:

x= -(-2)/2(1) ±√ [ (-2)²-4(1)(1)]/2(1)

=1± √0/2 , so we get that x is a double root .

Thanks for any comments.

 

[ArcanaNoir]

You have visuals or plan to draw on the board, right?

 

[Dick]

Well, for one thing you stated an original 'formula' (I'd call it an equation and add =0) of x^2+2x+1 and said that 1 is NOT a root. Then at the end you said "let's apply it to our original formula x^2-2x+1" and showed 1 IS a root. But you've changed the original formula! Are you just teasing them with that maneuver? It's definitely confusing.

 

[micromass]

You might want to talk about what happens if [itex]b^2-4ac[/itex] is >0, =0 or <0.

You also might want to introduce parabola's and tell them that is where it crosses the X-axis.

I would give examples of the quadratic formula first, and only then derive it in the end.

 

[Bacle2]

Thanks all for your comments.

Well, for one thing you stated an original 'formula' (I'd call it an equation and add =0) of x^2+2x+1 and said that 1 is NOT a root. Then at the end you said "let's apply it to our original formula x^2-2x+1" and showed 1 IS a root. But you've changed the original formula! Are you just teasing them?

.



No, my bad; I just wasn't careful enough; I should have written x^2-2x+1 the first time around

ArcanaNoir wrote:

"You have visuals or plan to draw on the board, right? "

Yes, I will be drawing on a board. Good point; this is still difficult, even after a few years of teaching.

Micromass Wrote:

"You might want to talk about what happens if b2−4ac is >0, =0 or <0.

You also might want to introduce parabola's and tell them that is where it crosses the X-axis.

I would give examples of the quadratic formula first, and only then derive it in the end. "

You're right about describing what happens if b2-4ac< ,> or =0 , but the problem is I only have 10 minutes for the whole presentation. Introducing the idea of parabolas may take a good chunk of time, but, given that they already know how to complete squares , this may be easier under the time constraint. I may assign the parabola approach as an extra-credit. And introducing complex numbers may take too much time.

Re giving examples at first, that is a close call: it helps keep their attention, but they may end up staring at the new formula and not paying attention to the rest. A speech prof. commented on this, and I am kind of in the fence here.

 

[Mentallic]

You're right about describing what happens if b2-4ac< ,> or =0 , but the problem is I only have 10 minutes for the whole presentation. Introducing the idea of parabolas may take a good chunk of time, but, given that they already know how to complete squares , this may be easier under the time constraint. I may assign the parabola approach as an extra-credit. And introducing complex numbers may take too much time.

I don't think he meant for you to introduce complex numbers, but instead that you should give examples of parabolas in each case. What does it mean to have [itex]\pm a, a>0[/itex] in the quadratic formula? What about [itex]\pm 0[/itex] and even [itex]\pm \sqrt{-a}[/itex]? Don't mention complex numbers, but rather show how the parabolas behave.

Re giving examples at first, that is a close call: it helps keep their attention, but they may end up staring at the new formula and not paying attention to the rest. A speech prof. commented on this, and I am kind of in the fence here.

Take a few seconds to break down the formula? I always liked that a parabola is symmetric, so you can maybe draw some parabola, then x=-b/2a is the line of symmetry and the [itex]\pm[/itex] tells us how far left and right to go to find where it cuts the x-axis. This would probably come before explaining the discriminant.

 

[vela]

Re giving examples at first, that is a close call: it helps keep their attention, but they may end up staring at the new formula and not paying attention to the rest. A speech prof. commented on this, and I am kind of in the fence here.

You need to decide on what your objectives are. What exactly do you expect the students to learn in the 10 minutes? Your choices will depend a lot on your audience. Obviously, what they already know is important. Do they already know about graphing quadratics? If not, discussion of parabolas will probably not be very fruitful, i.e. it's a topic for later. If they know about graphing, it might be a great way to motivate discussion of the discriminant. You also need to think about what level of learning the students are at. If you're addressing average high-school students, for example, you might focus on just getting them to identify what a, b, and c are and plugging it into the equation correctly. To this end, a bunch of examples in the beginning might be more useful than the derivation. If you have college students instead, you might place more responsibility for learning on the student and not have to spoonfeed them as much.

As micromass suggested, you should definitely talk about the discriminant and what happens in the different cases. You might want to use a few examples where you can factor by hand, since the students supposedly know how to do that, and show you get the same result.

 

[Bacle2]

Thanks all Again. I ended up using only what I was told students knew: factoring and completing the square. I gave an example of an expression ax^2+bx+c ; actually using x^2+x+1 (to make clear what the quadratic was for) , showing that x=1, x=0 did not work. Then I stated that one could always find the values for which the quadratic formula works , if these values existed (i.e., are real), and gave the quick proof of the quadratic in my first post.
Then I stated that if one could factor a quadratic (using the second part that students know to factor) into an expression (x-r1)(x-r2) , then r1 and r2 satisfied ax^2+bx+c , but that if the factoring was not obvious, we could use the square.

I just tried to stick to the two things I knew the students had covered, as vela suggested.

Anyway, I did get the job; they must be pretty desperate! :) . Thanks to all again.

 

[Bacle2]

In case someone is looking for a position like this -- at intro level-- I was told by the interviewer that he was looking for a presentation suited to the material that the students were said to already know, which is what vela suggested. I don't know how general this advice is, but just as data for anyone else in a similar position. So no more $0.99 pizza slice dinners.

 

[vela]

Congrats on the new job!

 

[micromass]

I'm glad you got the job!

 

[Bacle2]

Thanks again for the good wishes and the comments .