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.

An Idea for Professional Sports League During the COVID-19 Pandemic

  • NBA could have all the teams live in one community and play games in one arena. The teams would be separated for the most part from the rest of the community. The NBA could “rent” hotels or convention centers for whole season. The personnel would have to either not leave or be screened somehow. Or, perhaps the NBA could “rent” staff from hotels, restaurants, trainers, etc. and have them travel with the league.
  • It would be cool if fans could attend, but how could you insure they didn’t carry the virus? If cities created pop-up communities (like the idea above), the maybe the people living in these pop-up communities could attend the games.
    • One alternative: Create a segmented screen that would be in the stands. Individual fans could broadcast their reactions at home. Could they also connect and combine all the sounds made by the individual fans? If this were an episode of Black Mirror, holograms/avatars representing each fan could be sitting in the stands.
    • Fans could pay to be in virtual stands. As a benefit, maybe they could ask questions to coaches and players—and they would be the only ones to have access to this. They could receive a program from the game, maybe autographed items, etc. Basically, this would be a way for the league to make-up for lost revenue.
  • Alternative to fans: NBA and other leagues could charge a fee for streaming?
  • To benefit multiple cities, the NBA could stay for a month in one city. (Remember all the teams would be living there.)
  • Logistics may be too difficult, but the NBA could think of shortening the season and running a modified version during the summer or at least make plans to do this next year. Also, in a shortened season, it would be cool if they had a Final Four type format—do or die, one game playoffs—maybe letting everyone in.
  • Downsides
    • NBA players would be away from their families for a long time, unless their families could accompany them. (With schools being out, it might be feasible.)

NFL should be thinking about this already. The NFL has way more players, though, so this would be a challenge. One cool side effect is the economic benefits—to the city they’re in and/or the staff they bring.

Could this concept apply in other situations? For example, what about musicians? Theater? Movies? I liked the idea of a group setting up in one city/community and hiring or generating a lot of economic activity as a result.

Possible Future Challenges and Opportunities for U.S. Communities During the COVID-19 Situation

We’re living through a crazy time. I recently read an expert (current NIH director, I think) say that we may be 8-11 days behind Italy. That is, Americans will eventually experience what Italians have been. Rationing of medical resources, which the Italians have been doing, is what worries me the most. I say this to communicate that I think we’re in a dire situation.

I want you guys to know that because I’m about to discuss something that may seem insensitive or foolish. Namely, I think the pandemic has created (possibly) unique opportunities, and those who think imaginatively can use this time to be very productive. I’ll give some examples in the comments section.