kp on the MathCounts course and SAT math

kp writes:

I took the AOPS Advanced MathCounts class this summer (I'm a coach and wanted to see what it would be like for my students and what new things I could learn from it.) I appreciated that the class taught the shortcuts but also focused on how the shortcuts worked and how you could adapt them when the problem was given a new twist. (For example, to find the number of factors a number has, first find its prime factorization, then add 1 to each of the exponents, then find the product of these numbers. They explained why this made sense, then assigned different variations of problems on this topic.)

I'm no expert on the SAT (I'm a middle school teacher), but there does seem to be a large overlap between hard middle school math and what is on the SAT. We sometimes use SAT practice problems in our MathCounts practices. Our district's merit scholars often participated in MathCounts in middle school. Perhaps that is because the type of kid who stays after school to do math is the type of kid who is also successful on the SAT, or perhaps it is because MathCounts helps to prepare them for the SAT.

The MathCounts course sounds like a blast.

Teaching the shortcuts is a great idea -- it's the shortcuts that help you see what's actually going on, I think.

I remember years ago reading an article -- it may have been a study -- about smart-works-hard type students versus the 'naturals.' The smart-works-hard types went on wild goose chases trying to solve problems, while the naturals produced short, elegant proofs and solutions. I laughed, reading that, having been on many a wild goose chase myself.

help desk - probability

from Art of Problem Solving Introduction to Counting and Probability by David Patrick, p. 128:

8.2.4 A penny, nickel, and dime are simultaneously flipped. What is the probability that heads are showing on at least 6¢ worth of coins?

I can do this by brute force, but I don't see the math.

a chapter title I wasn't hoping to see

Chapter 6: Some Harder Counting Problems

Introduction to Counting and Probability by David Patrick

help desk - how many vertical angles

Here's a problem from The Art of Problem Solving Counting Introduction to Counting and Probability by David Patrick:

How many pairs of vertical angles are formed by five distinct lines that have a common point of intersection?

p. 64

Jo in Oklahoma on exercises vs problems

re: SAT problems, Jo in OKC said...

I asked my daughter today. She took the SAT this fall and got a score in the range you mention.

She said the questions are all routine exercises.

She would agree the AMC questions and AIME questions are problems.

One of her favorite areas of math is counting. :-) I remember covering permutations and combinations in high school math. However, what I learned was just a small fraction of what's covered in Art of Problem Solving's Intro to Counting and Probability course or book.

as smart as a 5th grader

source:
Competition Math: for Middle School (Volume 1)
p. 48

More evidence that "repeating the same lesson over and over" works as well as Howard Gardner's Chinese teachers said it does. I knew how to solve this problem within a few seconds of looking at it, and the reason I knew how to solve this problem within a few seconds of looking at it is that I used the difference of two squares to rationalize denominators many, many times while working my way through Saxon Math.

Art of Problem Solving

I've been only dimly aware of the Art of Problem Solving web site for lo these many years, so I'm glad kcab steered me over to them web site today.

Here's an article on AoPS from 2007 (pdf file).

question

Are there resources you like for teaching oneself beginning probability?

Someone (either lgm or lsquared, I think) recommended the Arlington Algebra Project, which looks like it's probably terrific for my purposes - though it lacks an answer key, which is not good.

I'll also be using this site (thank you!)

Anything else?

I wonder if I should take the AP statistics course at ALEKS.


update

kcab recommends The Art of Problem Solving, which turns out to have a course that sounds like exactly what I need: Introduction to Counting and Probability

Thank you!