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7

Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work. So please don't accept this answer. Anyway, the educational answer is to see that student is using the fact that $1^2 = (-1)^2$ in the critical middle step. Show them the corresponding fact for cubes, because it's crazy, and might expand ...


4

When I asked the What are the Laws of Rational Exponents? question on SE Mathematics, I was largely thinking about this context; teaching at the level of high school or early (remedial) college math. While it wasn't the top-voted, my answer there represents my best thinking about the status of this issue in classes at that level. As I wrote: Regarding the ...


3

Should we impose that $(a^m)^n=a^{mn}$ only when $a \gt 0$? Maybe you should tell your student that he/she have discovered by himself/herself the proof that the rule $(a^m)^n=a^{mn}$ cannot be true with negative bases and rational exponents, and this is a great achievement. Try to explain that he/she proved that If the said rule is true for negative ...


2

I once tutored someone and simple linear equations was part of the course. One example I came up with related to the student's own life was saving up to buy a smartphone. The problem can be made into different variations, with a basic version such as: I want to buy a phone which costs $400. With a weekly income of a, how many weeks do I need to save to ...


1

I wouldn't necessarily call them "trap" questions, but I wouldn't be surprised if a vocabulary quiz on middle/high school math could be difficult or low scoring. Ask students to define or to give examples for "degenerate triangle", "extraneous solution", "rational number", "principle square root", "inverse function", "quotient", "numerator", etc. Even ...


1

I'll suggest that the best verbalization is the last one in the question: $a$ equals $b$, which is equal to $c$ Here's why I think so: It's the closest to a literal symbol-by-symbol reading of the chained equation, while still being a grammatically-correct English sentence. Note that this matches the top-voted (but not accepted) answer to the analogous ...


1

Most problems I see (in classes after the first calculus courses) are basic misunderstandings in what is allowed to do with equations (and even worse with inequalities). There are also misconceptions on the meaning of strings of equations/inequalities, i.e., $p^2 < p^2 + u \le p^2 + p < p^2 + 2 p + 1 = (p + 1)^2$ (here $1 \le u \le p$). There are ...


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