A while back I snitched part of a post to the AP Calculus EDG from John F Mahoney.
He wrote, "Friends - Here are some topics I've used in pre-cal and calculus for 'debates' with students..." and then he lists five items, all of which are good, some of which are potentially great because you can talk about them over and over.
I wanted to amplify his remarks a little, so here is his list intersparsed with my comments, questions, concerns, and .... well, other stuff. Anyway, pick your favorite topic, and tell me what you think with a comment.
1) Should rectangles be considered to be trapezoids? Argument for: The trapezoid rule is based on trapezoids. If the function has a piece that is constant, wouldn't the trapezoids under that piece be rectangles?"
I wrote recently about the choice of inclusive and exclusive defintions in geometry, and particularly whether a trapezoid should be defined as "at least one pair of parallel side" vrs. "exactly one pair of parallel sides." I tried to engage another teacher who answered the discussion by looking in the glossary of his book... exclusive definition, end of discussion... teaching is like that for some folks. So are squares a subset of rectangles which are a subset of parallelograms which are a subset of trapezoids, which are... (ok, I'll stop now)..what is your answer, and why?
2) What is 0^0? Since 3^0 etc = 1, it should be 1. But since 0^3 = 0, it should be 0
Remember when you answer.. you are talking to High school Calculus and pre-calculus students.
3) Consider a circle of radius r. Draw a chord at random, what is the probability that its length is > r?
Yikes, this one can get real deep real fast...
4) What if you could take logs of negative numbers? How should they be defined?
This one is good for discussion because the technology is such that some of them have played around with their computer/calculator/W_alpha and actually have an inkling.
5) Circular functions are based on the unit circle. How should we define similar functions based on the unit square - or the unit diamond? I have never tried this one, but it seems like a great project: lots of things to consider.. like what is the "unit"?, and what is the period... and what about a "unit triangle"?
A unit circle has a formula x^2 + y^2 =1... what if we used a unit hyperbola, x^2-y^2=1, How could you make definitions for sin and cosine. What would it look like
I would add a one more that I ask, and students sometimes surprise me with, and I think have a lot to do with how they think to solve problems:
6) A cube is usually what people say when you ask them what is the thing like a square but in three-space... I still ask, but they usually give that answer, or can't explain much about why they gave the one they did. But then ask them what is the three-space equivalent of a triangle. I usually ask this with the instructions to write down the name, or draw it if you can't name it... that gives everyone that long thinking time to pick their own uninfluenced by a snap answer from a classmate. I get several different responses. Of course the follow-up is why do you think that is the best one. You can send me your choice and a justification if you would.
What would you add to the list above? What is your favorite discussion opener to make kids think? Share, Please.
4 comments:
Should rectangles be considered trapezoids?
I love this. Honestly, I don't care. But there are consequences to either answer. We need a decision. We need to live with that decision.
In a discussion about the consequences I might reveal the history, and what led American textbooks to choose away from how the terms were historically used.
But we are essentially discussing a choice of convention....
By the way, is a line parallel to itself? Think carefully. Most American high school geometry books refuse to make a decision on this one. They treat the answer in some places as yes, in others as no... Do you know why?
The square question is a great one, too.
I want to play with that one.
Jonathan
Jonathan,
re:" parallel to itself" I'll take a WAG
Euclid's definition 23 says (from Heath's translation) that parallel lines are in the same plane but "do not meet" , so purists might exclude them. But this is based on that shakey fifth postulate about the angle between lines, and he defined the angle between lines to exclude when they lie on the same line (definition of plane angle)... so over 2300 years people have been playing with the definition and trying to adapt it to projective geometry and so we come up with about four ways to define them...
1) Parallel lines do not meet
2) Parallel lines meet only at infinity
3) Parallel lines have a common direction (in high school this would be slope in two-space and direction vector in three-space)
4) Parallel lines have a constant distance between them..
So 1 and 2 would say not self-parallel; 3 and 4 would say self-parallel
Just one more note about the trapezoid, rectangle idea.. I think there is good reason to expose kids to the idea that we might want to adopt both conventions and use (and define) the one we need(want) when we need(want) it.
For example, I have (almost) never introduced the usual formula for the area of a trapezoid A= (1/2)(b+t)h that I don't point out that this works for all parallelogram types AND for triangles (t=0) and thus we may think of triangles, in at least one way, as a degenerate trapezoid...
Re parallel: there's also an obligatory handwave over "equivalence relation"
Re trapezoids: neat with the all parallelograms thing, esp the extension to triangles. I'm grabbing that and using it, thanks. I'll combine it with my verbal twist "height x average of the bases"
I like discussing conventions with classes. Decisions were made, we live with them.
The big deal with trapezoids (US convention) is that we can have a neat Euler diagram: big circle for quadrilaterals, smaller circles for kite, trapezoid, and parallelogram - none of them intersecting - in the parallelogram circle two intersecting circles for rectangle and rhombus.
I really think we've become enslaved to that picture. Not that it's a worse choice, just I'd like to have thought more before making it.
The main theorem(s) of trapezoids is for isosceles trapezoids:
In a trapezoid ABCD with BC || AD
IF AB is congruent to CD
THEN angle A congruent to angle D
and vice versa.
This clearly rules out generic parallelgrams as trapezoids but not rectangles.
I say YES
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