Wednesday, April 9, 2008

Math lesson of the week: two approaches to the Quadratic Formula

1. McDougal, Littell's Algebra 1, (a traditional algebra text), p. 732 

Investigating the Quadratic Formula

Consider a general quadratic equation

ax2 + bx + c = 0      where  a ≠ 0

Perform the following steps.  Then describe how your result is related to the quadratic formula.

1. Subtract c from each side of the equation ax2 + bx + c = 0
2. Divide each side by a.
3. Add the square of half the coefficient of x to each side.
4. Write the left side as a perfect square.
5. Use a common denominator to express the right side as a single fraction.
6. Find the square root of each side.  Include ± on the right side.
7. Solve for x by subtracting the same term from each side.
8. Use a common denominator to express the right side as a single fraction.


2. The University of Chicago School Mathematics Project (UCSMP) Algebra: Integrated Mathematics, (a sequel to Everyday Math), p. 574-5 

Quadratic Formula
If  ax2 + bx + c = 0 and a ≠ 0, then
The Quadratic Formula is one of the most famous formulas in all of mathematics.  You should memorize it today. [Italics not mine].

Applying the Quadratic Formula

Example 1
Solve -x2 + 2x + 27 = 0 [reference to an earlier word problem]


Recall that-x2= -1x2. So rewrite the equation as
-1x2 + 2x + 27 = 0
Apply the Quadratic Formula with a = -1, b = 2, and 2 = 27.

[A demonstration of this, followed by similar problems for students.]


In short:

Derivation vs. memorization and plug-in.

In comparison with traditional approaches, UCSMP's Reform Mathematical "discovery" approach gives students much less practice factoring and finding perfect squares and common denominators.  

But unless you've mastered these things, you probably won't be able derive the Quadratic Formula. 

To use Reform Math's lingo, instead of "discovering" it, "authorities" must "spoon feed" it to you--a tactic that Reform Math's supporters purport to abhor.


Lsquared said...

That's an interesting example. I'll have to remember that.

Now, if you're really going to get into the comparisons, you should realize that UCSMP was an early version of reform, and a more truly reform text would be Core-Plus (you could also look at Comap, but I think that's somewhat less wide spread) or Integrated Mathematics (but it doesn't have an index, so it would be hard to find the quadratic formula without reading the whole book

lefty said...

Thanks for the suggestion.

I've thus far searched through Integrated Math 2 (McDougal, Littell) and Integrated Math (Amsco) and it's the same story: the Quadratic Formula is introduced out of nowhere, with no derivation or discussion of where it comes from.

Still need to get my hands on Core-Plus and Comap.

Catherine Johnson said...

What a find!


"memorize this" AND no practice

I'd love to see how Connected Math handles the quadratic equation

Lsquared said...

Connected math (CMP) is middle school, so I expect they don't get as far as the quadratic formula.

I actually like some of the stuff in CMP--what I don't like is the stuff that they leave out. (My pet peeve is that I went looking for a good example of how reform math books handle division of fractions, and looked through the whole CMP series, only to discover that they don't handle it--they just skip it entirely. Not a satisfactory answer to how to handle a difficult topic.

Lsquared said...

Please consider thes appended to my last two remarks. It appears that I can't close a set of parentheses today.

lefty said...

No division of fractions anywhere in CMP. That's astounding. Is this true of the entire CMP middle school curriculum?

Lsquared said...

I haven't looked hard at the second edition, but the first edition had no division of fractions in the entire curriculum 6-8 (I was looking for it specifically, and I had the whole set of books on the shelf in front of me. I looked in all of the books that looked like they might possibly have it, and there was nothing).