Don't worry....this is not a test.... The recent thread dealing with why a neg times a neg is a positive reminded me of the following. (BTW, I really think the thread was aimed at --how do you explain to beginners that a neg times a neg is a pos -- not really the
technical (abstract algebra type) reason WHY.
This is a set of 10 questions for you to ponder. I think they go to the heart of how well prepared a middle or high school math teacher is to go beyond simply giving the procedures and algorithms in the text. They are the type of questions which arise all the time from students who perhaps donUt know they arenUt supposed to ask them -- after all, THIS wasnUt covered in the lesson..... :-) Enjoy -- and if you feel brave, why not post your responses? Or have your kids give them a shot.
(1) Jaunty John is complaining that he can never keep straight the
difference in meaning between these three expressions: a/0 0/a
0/0 (where a is a non-zero number). Can you help him out?
(2) Sunny Sandi says she is having trouble with the idea of "slope",
in particular, as it applies to horizontal and vertical lines. Give her a good explanation which will set her mind at ease on this point.
(3) When the graphs of y = 2x^2 + 3x + 1 and 3x + 5y = 9 are
drawn they are seen to intersect. How could Marvelous Mike determine
the exact coordinates of the point or points of intersection?
We are interested in HOW MANY (real) solutions it has. How might we determine this?
(5) Give a mathematical explanation of the "rule": Minus times minus is plus. Please: no films running forward or back or people removing debts.
(6) Krafty Katie has been considering the equation y = 2x^2 + 3.
She claims that the y-intercept is 3 and the slope is 2. Comment on her observations.
(7) Solve the equation 3(2x + 7) = 2x - 4(3 - x)
(8) In studying radicals in general and square roots in particular, a student mentions that he saw in a book somewhere the phrase
"rationalize the numerator". He wonders what it means and asks you to
help him understand it. What do you tell him?
(9) When studying graphing, Brilliant Bob realizes that there is an
easy way to tell if two linear equations represent parallel lines. All he needs to do is to compare their slopes, and if they are equal, the graphs are indeed parallel. He wonders if there is a similarly easy way to
decide if two graphs are perpendicular. Can you help him out?
(10) a) Is it possible to add up enough terms of the series 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ....... so that the total will eventually exceed 10?
b) Is it possible to add up enough terms of the series 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ..... so that the total will