Math Corner: Solving for One Variable

From the another thread:

Do you guys know if anyone has written down rules for solving equations with variables? Here’s what I’m getting at: When you solve an equation with a variable, the sequence of actions you will take to isolate a variable. It seems like the first action you choose can be crucial, too. For example, look at this equation:

7(2e−1)−11=6+6e

What should you do first? I saw this at Khan Academy, and they used the distributive property first–getting “rid of” the 7 outside of the parenthesis. That does seem like the best move, but I didn’t realize that initially. I think I might have tried to add 11 to both sides first. And then my instinct might have been to divide by 7 on both sides, but I think that would have been a mistake. In working on several of these type of problems prior to this one, one of the important early moves involved “getting rid of” the number being multiplied by an operation with a parenthesis. For example,

200 = 20(4c+2)

I guess you could use the distributive property first, too.

In any event, I’m trying to find a list of rules or principles that will guide a student on the sequence of operations they should choose. What should be the first step, and then the second, third, etc. I did a quick search for something like this, but didn’t see anything.

Populist Left Vs. Socialist Left

Here’s a thread on the differences between the populist left and socialist left. Off the top of my head, here’s a summary of each. The populist left opposes high concentrations of wealth, while the social populists want to create a utopian society where…I guess production of goods, services and wealth are shared by everyone. (Sorry if these are bad definitions.) How do you guys feel about these categories? Where do your politics line up, including if they fall outside of these two categories?

The Stupid Coin Flip Problem

I’m really not that interested in talking about this again, but I read a passage (about an entirely different topic) that made me think about this. Here’s the passage:

You can flip a coin 100 times and have it come up tails 60 times. It’s an unlikely result, but it happens. About 3 percent of the time, you’ll get 60 or more tails.

You can flip it again the next day, and it can come up tails 60 times again. This is possible, and no doubt has been done before. (Again. Don’t do this. It is so boring! Read more football instead. That’s a good use of your limited time on our doomed planet.)

If you flip the coin 100 times for 100 days, though, you’re not going to keep getting tails 60 percent of the time. Regression, the most powerful arm of probability, will tug the percentage back toward 50 percent. It just will. It doesn’t have a choice, and neither does the coin.

I’ll go over what stands out in the comment below.

Solve the Brewster’s Millions Challenge

Montgomery Brewster could receive a lot of money if he could overcome one challenge–namely, he had to spend $30 million dollars in one month–basically $1 million a day. In the challenge, I believe only a smart percentage could be given away. Additionally, if he bought or did anything of value, the value would be added to the remaining balance. This was from a movie, that most of you know, and maybe the amount would be different now. In any event, I like thinking about ways I would overcome this challenge. How would you solve this challenge?