This is a blog for my second semester pre-calculus 30s students. Have a blast!

Friday, May 05, 2006

Way to go!!!!

Way to go Vince!!!!! Fantastic double scribe post. Hopefully everyone can take a look at this and match your efforts in following posts. Keep up the great work!!!!

Thursday, May 04, 2006

Today's post

MAY 4, 2006

COUNTER EXAMPLES

Today's lesson is COUNTER EXAMPLES. It is about proving the CONJECTURE (conjecture: statement made by examples) wrong by providing true statements, examples, diagrams, equations. This lesson is somehow easy because all you do is prove that the conjecture is wrong. However, such conjectures may lead to sudden confusion because sometimes it gives just a small possibility of proving it wrong. You REALLY have to think about it too.
Here are some examples:
1) The sum of 2 prime numbers is always even. (CONJECTURE)
examples that verify this conjecture:
3 + 5 = 8 even
7 + 9 = 16 even
11 + 13 = 24 even
examples that contradict this conjecture:
7 + 2 = 9 odd
11 + 2 = 13 odd
13 + 2 = 15 odd
2) The expression 1 / x will always be less than 1 for any value of x.
examples that verify the conjecture:
x = 2
1 / 2 = 0.5
x = 4
1 / 4 = 0.25
examples that contradict the conjecture:
x = 1
1 / 1 = 1
x = 0.5
1 / 0.5 = 2
That's today's lesson. The homework is Exercise 48.
MAY 2, 2006

VENN DIAGRAMS

LESSON LAST TUESDAY:

EXAMPLES

A upside down U B = m, u

( B \ A ) ' = o, p, s, m, u, r


A \ B = o, p, s


( A U B ) ' = r

Wednesday, May 03, 2006

Scribe responsibility

Just a quick note folks. I purposely haven't said anything in class about our scribe responsibilities over the past couple of weeks because there shouldn't be a need to. This is an enriched university entrance level course where standards and expectations are high. Everyone fully realizes that each day that there is a new lesson someone needs to be the scribe for that day. Even if it helps 1 or 2 people that night who might have been confused during the lesson then it's definitely served its purpose. Remember you are all responisble young adults and if the routine of scribing continues like it did for most of the course so far, everyone would only have to be the scribe every 21 classes (almost an actual month). Not a lot to ask for something which might make the world of difference to a few of your classmates, or possibly even people around the world taking the course. I've also noticed that it's the same group of 4 or 5 people who are always the "fill-in" scribes. The honus should be on everyone to put in their small part. Have a good day / night folks!!!

Monday, May 01, 2006

Logic and Proof

Logic and Proof: Inductive/Deductive Reasoning

If the same result occurs over again, we conclude it will always occur. This is called inductive reasoning.


ex. multiply and odd number by an even number
9 x 8 = 72
11 x 12 = 132

7 x 10 = 10

  • From this, we can conclude that the products are all even. <-- We made a statement that is called a conjecture. A conjecture is something like an unproved theorem. Basically, we made a conjecture that says that all product of odd/even numbers are even.
If two points are placed on the circumference of a circle and joined by a chord, you'll get two regions.

*note: Having beyond 5 points made it difficult to count the regions, so I only did the first three examples.

*Conjecture: The number of regions follow the pattern 21, 22, 23,... 2n
  • The 6-sided polygon proves our conjecture wrong. Using our pattern, we should get 32 regions, but the 6-sided polygon only has 31 regions.
*In deductive reasoning, we make a statement that we are absolutely certain is true based on TRUE STATEMENTS.

ex. Add 2 odd numbers.
1 + 3 = 4
5 + 7 = 12
9 + 11 = 20
  • Sum is always even?
Odd numbers:
2x + 1, 2y + 1
  • 2x + 1 + 2y + 1
  • 2x + 2y + 2
  • 2(x + y + 1)
* 2 is a factor, therefore sum is always even.

ex

That's it for today's lesson. Homework is the Inductive and Deductive Reasoning worksheet and Exercise 45.