Valuable Resources

To all parents/teachers/students,

I posted a brain teaser link before to the Khan Academy which has been a great resource for me. It contains a vast selection of educational videos related to almost all content areas. Most of which are narrated by Salman Khan (the site founder). These videos have proven to be a valuable resource for students outside the classroom. He is featured in a TED Talk where he talks about how he got started and offers some interesting ideas for today’s classrooms. Take a minute to explore his website and think about what he discusses during his TED Talk. Just something to think about! Let me know what you think.

http://www.khanacademy.org/

Think about this!

An interesting question was posed to me the other day that I would like to share.

You are in a boat in a fresh water lake with a rock in your hand. You throw the rock into the lake. How is the water level affected?

Seems like a simple question but it turned out to be quite a thinker!

This simulation might help you visualize what could happen.

http://phet.colorado.edu/sims/density-and-buoyancy/buoyancy_en.html

Post your thoughts please!

How can this be?

 Take a look at these triangles. They are made from the same shapes but the bottom one is missing a section. How can this be? What are some possible explanations you can come up with?

 

Maybe this animation will help:

http://www.arcadeboss.com/game-1869-16-Triangle-Puzzle.html

Logic

Logic is closely related to proofs within mathematics and is used by all of us in everyday life. It allows us to deduce and calculate a rational and desirable outcome in a given situation. Here are some important aspects of logic use in everyday life:

• It enables clear thought through a rigorous demand for truth.
• Because thought processes are clarified, the use of logic enables consideration of all available options for opinion and action, and decreases the persuasive power of popular opinion.
• Because available options have been fairly considered, the use of logic increases the likelihood that subsequent opinions and actions will adhere most closely to the truth.
• Therefore, logic's relevance to everyday life is that its use improves the likelihood of satisfactory outcomes in the day-to-day decisions each person makes.

Try this game out and see if you can complete the tables.

http://www.logic-puzzles.org/init.php

Some of these brain teasers also relate to logic. Take your pick and don’t cheat! Try and post some of your answers or questions!

http://www.khanacademy.org/math/brain-teasers

Why are Proofs Important??

The only way to verify that a claim about mathematics is correct is to prove it.  Numerical evidence or the fact that the claim is true in some physical situation is suggestive but is not verification. We all know that 1+1=2 but think about why it is true. Because someone told us and we believed them? Take a look at its proof and think about how important they are to mathematics.

P1. 1 is in N.
P2. If x is in N, then its "successor" x' is in N.
P3. There is no x such that x' = 1.
P4. If x isn't 1, then there is a y in N such that y' = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.

Then you have to define 2:
Def: 2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1' = 2 Q.E.D.

Here’s a classic proof where 2=1. Can this be true? See if you can identify the fallacy in the link below?

http://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html