# SL2RL-Math247

## 27 March 2009

### Goodie of the Month – Tangents to Quadratics

Filed under: Uncategorized — admin @ 2:17 pm

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.”

March 2009 Goodie of the MonthTangents with Quadratics

Well now that we have saved all that time not factoring and completing the square and considering complex roots, we might want to study tangents to quadratics.

Now you might ask why? Answer: it reinforces so many topics from Algebra 1 and ties them all together.

How to study tangents to quadratics?

Tell the student that the slope of the tangent line to the quadratic function y(x)=ax²+bx+c is always s(x)=2ax+b.

Now ask them "Given y(x)=x²+x-3, find and graph the equation of the tangent line to this quadratic at x=-2?

1. First they must graph the quadratic by finding the y-intercept at c=-3, roots at -2.3 and 1.3 and vertex at (-0.5,-3.25).

2. Then, they must find the point on the quadratic by substituting y(-2) and check that this point (-2,-1) is on the quadratic.

3. Then, they must find the equation of the slope at this point: s(x)=2x+1.

4. Then they must find the slope of the tangent by substituting s(-2) to get m=-3.

5. Then they must find the equation of the tangent line using the point-slope formula: y= s(-2) (x+2)+y(-1).

6. Finally, they must graph this line y=-3x-7 and see that it is indeed tangent to the quadratic at the point.

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

Just look at all the skills it reinforces from Algebra 1! And everything is visible and checkable!

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## 8 March 2009

### Baddie of the Month – Teaching complex numbers with the quadratic formula

Filed under: 8-12,algebra,applets,education,K-12,math — Tags: , , , — admin @ 12:14 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.”

March 2009 Baddie of the MonthTeaching complex numbers with the quadratic formula.

Who was the dingbat who first decided to work complex numbers when teaching the quadratic formula?

And why? Because you can get complex conjugate numbers from the quadratic formula?

A car has brakes. Do we teach hydraulics to someone learning to drive a car?

Storyline: We are teaching quadratics. Everything we discuss is totally real. (In fact, we usually "fix" our problems to be with integers, but that is subject of previous rants.)

We are factoring, finding intercepts, drawing graphs in the Cartesian plane – all real numbers.

In the middle of this, we start teaching a totally different subject – namely complex numbers.

And then we go back to real numbers and real applications of quadratic functions.

Let’s face facts.

(a) Complex numbers have NO relation to quadratic functions or their applications that we will work on.

(b) Complex numbers have NO visual representation on the graph of a quadratic function*.

Ergo – complex numbers do NOT help us understand quadratics.

Conclusion: Do NOT mention complex numbers at all when teaching the quadratic formula.

Simply state that when D<0 (negative discriminant) the quadratic function has no roots and therefore does not cross the x-axis. Don’t mention "real roots". Don’t go anywhere else with this discussion at all.

STICK TO THE SUBJECT MATTER AT HAND – Quadratic functions and their applications.

—————–

This is the material in a typical textbook in the chapters for quadratic functions. (This is actually a reasonable toc. Some left me gasping for breath.)

Let’s go. We only have 2 months. No problem – we will fix everything to be integers.

1. Here is a quadratic function y= x². Let’s make a table of points. (Hmm – I’ve only ever graphed a line. I know I can graph a line using any 2 points. What the heck are all these points?)
2. The graph of a quadratic function is a parabola. (Hmm – So a parabola is the graph of a quadratic function?)
3. Here is the general expression of a quadratic y=ax²+bx+c, where x and y are variables and a, b and c are constants. (Hmm – They all look like letters to me.) Also, m²-2km+k² is a quadratic. (Huh?)
4. Let’s factor ax²+bx+c. Here are a bunch of rules. (Dang, I can’t factor this quadratic x²-3x-2. Teacher said it was because I copied wrong x²-3x+2. Do these rules work?)
Here are some more rules. (Hmm – Why do we want to factor? What are those numbers anyway?)
5. Let’s complete the square. Here is the plan. The point (h,k) is the vertex. (Hmm – Why is it the vertex? Why is there a minus in front of h and a plus in front of k?)
6. Let’s graph quadratics by completing the square and transforming the graph of y=x². (Say what? You go left when? First upside down? And then stretch?)
7. Here is the quadratic formula. We prove it using completing the square. (Wow, look at all those letters and equations. Now – square root. Plus and minus sign. Never seen that operation before – cool.)
8. Using quadratic formula, let’s find the roots of a quadratic. (Roots? Is there a function here? Is there an equation here? Linear functions have roots?) The roots are the factors. (Hmm. Roots look easy to find. Couldn’t we just factor that way and skip all that factoring stuff?)
9. Let’s graph a quadratic by completing the square. Now use the quadratic formula to find the roots. The roots are the x-intercepts of the function. (Hmm. Roots look easy to find. Aren’t parabolas symmetric? Why can’t I just find the vertex by going halfway between the roots and skip all that completing the square and transforming the function?)

Okay – I am sure they got all that. Let’s pause and do a totally different subject.

1. D is called the discriminant. D can be positive, zero or negative. If D is negative, the quadratic function doesn’t have real roots. (Real roots? Are there fake roots?)
2. If D is negative, the quadratic function has complex roots, which are complex numbers. You remember: z=x+iy. Complex roots come in pairs called complex conjugates.
*Wait – we can make this worse. Let’s graph complex numbers and their conjugates in the plane. (No kidding – my son did this in the middle of learning to graph quadratics.)
(Whoa – I thought we were talking about quadratic functions and graphing parabolas. What do I get with complex conjugates? Where do I put these on the graph? Whaaaaaaaaaat?)

Now back to quadratics. Back to the reals – are we totally confused yet?

1. Now let’s look at applications of quadratic functions.

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