Sunday, October 24, 2010

Motivating Synthetic Division and Associated Theorems and Rules

So this is what I use to help students practice the various theorems and rules typically associated with synthetic division after we've covered the rules in class. The set-up is "Mr. H found the following work shown by a student on scratch paper during a test. The student did not write the original question, but somehow Mr. H knew what the student was trying to do. Mr. H also noticed that the student could have saved a lot of time during the test had the student used the theorems and rules learned in class. What was the student trying to do? How could the student accomplish the same thing but faster and why? Comment on each step."



The theorems and rules I'm asking students to apply are:
  • Intermediate Value Theorem
  • Remainder Theorem
  • Factor Theorem
  • The Rational Zero Test
  • Descarte's Rule of Signs
  • Upper Bound and Lower Bound Rules

We use Pre-Calculus with Limits: A Graphing Approach by Larson, Hostetler, and Edwards.

Some comments that I'm looking for:
  1. p/q is work for the Rational Zero Test. It is used to find all the possible rational zeros.
  2. f(x)=2x3+5x2-37x-60 has 1 variation in sign (sign change in the coefficients) so there's 1 positive real zero by Descarte's Rule of Signs. (Work is not shown)
  3. f(-x)=-2x3+5x2+37x-60 has 2 variations in sign (sign change in the coefficients) so there's 2 or 0 negative real zeros by Descarte's Rule of Signs. (Work is not shown)
  4. 5 is an upper bound because the last row is either positive or zeros and 5 is positive. This means no real zeros above 5.
  5. 6 should be skipped since 5 is an upper bound. However, this may still be useful if we're graphing the function. The remainder of 330 means f(6)=330. So we have as point (6,330).
  6. 3 has a remainder of -72. Since f(3)=-72 and f(5)=180, and since f(x) is a polynomial (continuous everywhere), there must be a zero in the interval [3,5] by Intermediate Value Theorem. 4 is a good number to try.
  7. 2 is unnecessary since we know there is only 1 positive zero and it is between 3 and 5.
  8. 4 happens to be a zero. The quotient is 2x2+13x+15. Student should attempt to factor this quadratic to save time.
  9. 1 is unnecessary since there is only 1 positive real zero by Descarte's Rule of Signs and it's been found that 4 is that zero.
  10. -6 is a lower bound since the last row is alternately positive and negative and -6 is negative. This means no real zeroes below -6. The original f(x) should not have been used here. The quotient of 2x2+13x+15 should have been used instead, if factoring is not immediately obvious.
  11. -10 should not have been tried if all we are doing is factoring. We already know -6 is a lower bound.
  12. -5 is a zero.
  13. 4 is used here to find out the last factor. We found 4 was a zero from an earlier step.
  14. The polynomial f(x)=2x3+5x2-37x-60 was factored to f(x)=(x-4)(x+5)(2x+3)

It seems to me that, for many students, time is better spent practicing how to consistently and accurately divide using synthetic division. Here's some questions to math teachers:
  1. Do you teach these theorems and rules?
  2. If you don't cover them, why not?
  3. If you do, how do you cover them (how much time/depth)?
  4. How do you teach/motivate learning these theorems and rules?

I'd like to hear your thoughts.

Monday, October 11, 2010

Which "Average" is Used?

Found this comic called "The Dunning-Kruger Effect" (Spiked Math) which I shared on Google Reader. I think it can be a piece for discussion in Stats class. I like that I also learned about The Dunning-Kruger Effect (via Wikipedia).

Questions for a Stats class:
  • Which "average" is the comic most likely to be referring to (mean/median)?
  • Is the other one also possible?
  • Is it possible that half of the professors are above the median?
  • Is it possible that more than half of the professors are above the median?
  • Is it possible that less than half of the professors are above the median?
  • Is it possible that half of the professors are above the mean?
  • Is it possible that more than half of the professors are above the mean?
  • Is it possible that less than half of the professors are above the mean?
This reminded of a quote by George Carlin:
Think about this; think about how stupid the average person is, and then realize that half of 'em are stupider than that.


Questions for a Stats class:
  • Which "average" is George Carlin most likely referring to (mean/median)?
  • Is the other one also possible?
  • Is it possible that half of the people are stupider than the "average person" (median)?
  • Is it possible that more than half of the people are stupider than the "average person" (median)?
  • Is it possible that less than half of the people are stupider than the "average person" (median)?
  • Is it possible that half of the people are stupider than the "average person" (mean)?
  • Is it possible that more than half of the people are stupider than the "average person" (mean)?
  • Is it possible that less than half of the people are stupider than the "average person" (mean)?
Video is probably not appropriate in a high school setting, but the quote above can still be used.

Sunday, October 10, 2010

10/10/10 Ten/Ten/Ten Shi/Shi/Shi

For many ethnic Chinese, October 10th is a significant day. But this post won't be about that nor will it be about the math of 10/10/10. This post is about the Chinese language, specifically about Classical Chinese. You can probably guess from the title that shi means ten. It is the Chinese Mandarin pronunciation of ten.

Before we continue, a quick disclaimer. I am not a linguist. An undergraduate linguistics class ages ago hardly qualifies me to explain or convey accurately the linguistics of the Chinese Language. With that said, when I heard of 10/10/10. It reminded of a story my dad read to me when I was little. It's a story that made no sense when my dad told it to me, but which made perfect sense when I read it on paper.

(CC Attribution 2.0 Generic by Ell Brown)

It was famous story written in Classical Chinese by a linguist. It's called "Story of How Mr. Shi ate a Lion."

《施氏食獅史》

石室詩士施氏,嗜獅,誓食十獅。
氏時時適市視獅。
十時,適十獅適市。
是時,適施氏適市。
氏視是十獅,恃矢勢,使是十獅逝世。
氏拾是十獅屍,適石室。
石室濕,氏使侍拭石室。
石室拭,氏始試食是十獅。
食時,始識是十獅,實十石獅屍。
試釋是事。

First time my dad told it to me, he started laughing and coughing while reading the story from the newspaper. At first it seemed like he was stuttering. So I asked to see the story myself while he tried to catch his breath. As soon as I saw it I started laughing too.

Without fail, all Chinese people who read this story out loud in Mandarin laughed. The story is not particularly funny. My translation follows below:
"Story of How Mr. Shi Ate a Lion"

In a stone chamber there was a Mr. Shi, who loved lions, and who vowed to eat ten of them.
This man often visited the market to look for lions.
At 10 o'clock one day, there arrived 10 lions to the market.
Coincidentally, Mr. Shi also arrived at the market.
When Mr. Shi saw the 10 lions, he used his arrows and killed the 10 lions.
Mr. Shi brought the ten lion corpses back to his stone chamber.
The chamber was wet, so he had his servants wipe it dry.
With the stone chamber wiped dry, he tried to eat the ten lions.
As he began, he suddenly realized that the 10 lions were actually ten stone lion corpses.
Try to explain this story.
There's a translation over at wikipedia of this Lion-Eating Poet in the Stone Den which is roughly the same. The Chinese characters are still the same. Although the grammatical structure is a little different from modern spoken Chinese, the story is accessible to anyone who has read a little Chinese history or who watched Chinese historical dramas.

So why is it funny to us? This is what it sounds like when you read it out loud.

Lion-Eating_Poet_in_the_Stone_Den_Shī_shì_shí_shī_shǐ_
(CC Attribution-ShareAlike 3.0 Unported from Wikipedia)

Here's a video of someone reading it. She kinda loses it in the end.


According to my dad, people have used this ever since as an argument against Romanization of Classical Chinese. The large number of homophones in Chinese makes it incomprehensible. In fact, to anyone who has never heard of the story, the above video seems like footage of a girl practicing her pronunciation and intonation. The original Chinese characters preserve the meaning and will be obvious to any reader of the written story. Were it romanized as "Shī Shì shí shī shǐ" it would challenge even the most imaginative Chinese mind to decode it. In practical terms, no one would ever write like this. The author did it only to make a point. Lucky for me, I get to tell his story about a certain Mr. Shi who's a fan of lions.

Friday, October 8, 2010

A Comic Guide to Limits Part 1: When Limits Exist

Some may find this useful. There were some insightful comments in the old blog regarding the model used here. They dealt primarily with the conflict between the concept of "arbitrarily closer" and the granular nature of the model. The model used here clarified the difference between finding the limit of f(x) as x approached c and f(c) for students, and is intended as a supplement. I found this helped students get a practical understanding on how to evaluate limits.

To be honest, I haven't spent much on this since. I stopped working on it to reflect a bit on whether something like this is helpful or harmful. As time passed, I just tabled it. There's a few more parts to this comic series. In the other parts I examine situations when limits do not exist and continuity of a function. If anyone is interested, let me know and I might resume work on those and post them on here.

In any event. Here it is. Comments welcome.