Saturday, October 6, 2012

September Essential Questions (Final)

Ask yourself if you can answer all of these questions.  They were the gems that students asked during the first month of school that I wrote on my low-tech note card in my dress shirt pocket.  

Parents, ask your students if they understand and can explain. The ones in bold are for Pre-Calc only.  

My Best,

Mr. Morrissey


  1. This comment has been removed by the author.

  2. Hmm, some feedback:

    1. I assume you mean x-intercept for a line, because there's no general method to find exact solutions to f(x) = 0 for any arbitrary function (!) So I'd reword it to "Given the function f(x) = ax + b for some fixed constants a and b, where a is not 0, show that there is a unique x-intercept and explain how to find it."

    2. Assuming there's no cube root button ;) (then the easiest way would be to raise the number to the 0.33333333... or 1/3 power)

    3. I'm not sure what the point of this is. If it's to demonstrate that (-a)/b = a/(-b) = -(a/b), then I guess it's okay. The intent is not very clear at the moment.

    4 & 5. The question "Is 1 a prime number?" makes the question "What is the smallest prime number?" unnecessary.

    6. The question is imprecise because it doesn't mention that you are talking about integers -- if you were talking about real numbers, then the answer would be different. Between 1 and 5, inclusive, there are 5 integers, infinitely many rational numbers (but countably many), and infinitely many real numbers (but uncountably many). You should attend to precision and say instead, "Explain why if a,b are integers and a < b, then the number of integers between a and b, inclusive, is precisely b - a + 1."

    7. I guess that is fine.

    8. I guess that is fine.

    9. This really isn't a math question... It's a semantic issue. We can say "root of a function" and "zero of a function" and it means exactly the same thing (in my opinion). What they both mean is encapsulated by the following definition: "A root (or zero) of a function f is a number r in the domain of f such that f(r) = 0."

    10. This is somewhat vague. What kind of answer are you expecting?

    11. I'm lost on this one! Do you mean whether you can divide P(x) by (x-a) then by (x-b), or by (x-b) then by (x-a), and get the same result? (And do you see how much clearer this revised question is? ;) )

    12. Common denominator of an equation isn't well-defined. Besides, in the equation x/a + y/b = 6, you just have to think of it as the following equivalent equation: (1/a)x + (1/b)y = 6; note that 1/a and 1/b are just constants and you can treat them as such. You don't have to multiply by any common denominator!

    13. I would suggest the following question instead: "Why does one sometimes get extraneous solutions to an equation when following a process of solving it? For what types of equations can we be guaranteed not to get extraneous solutions?"

  3. Yongyi,

    These were all questions by students.

  4. Ah, that explains it. I guess you can't expect students' questions to be perfectly formed! I was thinking of essential questions as questions written by the teacher to be thought about over the course of the unit.