Author Archives: Mr. Chase

Congratulations, Ray!

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In follow up to my post from Tuesday,

Congratulations to Raynell Cooper who won big last night in the final match of the Jeopardy Teen Tournament!!

From Jeopardy.com

Here’s an excerpt from the press release on the Jeopardy website, at which you can also watch a video interview with Ray:

CULVER CITY, CALIF. (March 2, 2011) — Raynell Cooper, a senior at Richard Montgomery High School from Rockville, Md., has won the “Jeopardy!” Teen Tournament, earning the $75,000 grand prize.

Regarding his win, Cooper said, “It was unreal. I honestly didn’t think I’d make it that far.” He added, “I’m so happy that all my hard work paid off.”

One of the major highlights for Cooper was seeing Alex Trebek in person. He said, “I don’t get the chance to see celebrities often, so it was pretty amazing.” He also enjoyed meeting his fellow contestants. “We all got along wonderfully. They were excellent players and incredible people.”

Cooper, 16, hopes to attend George Washington University and will use his earnings towards tuition. He also plans to buy a car and donate to charity to help underprivileged youth in his community.

In his spare time, Cooper enjoys participating in his school’s drama club. He is also captain of the academic team and student government vice president.

(more)

Teaching domain and range incorrectly

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What’s wrong with these high-school math questions?

  1. State the domain of the function f(x)=\frac{1}{x}.
  2. Where is the function f(x)=\ln(x) undefined?
  3. State the range of the function f(x)=x^2.

As a math teacher, I’ve asked these questions before too. But I always ask them with a bit of a cringe. Do you see what’s wrong with them?

Domain

A function is only well-defined when it is defined with its domain. A function f is an association between two sets A and B that assigns only one element of B to each element of A. The set A is called the domain and the set B is sometimes called the codomain. If  f is a function mapping elements from A into B, then we often write f:A\to B. For instance, consider the function f:\mathbb{R}\to\mathbb{R} defined by

f(x)=e^x

The domain of this function is \mathbb{R}, since that’s how the function is defined. Notice I explicitly gave the domain right before defining the function rule. Technically, this must always be done when defining any function, ever.

We might ask a student, “What is the domain of f(x)=e^x?” But this is a poor question. The function rule isn’t well-defined by itself. There are many possible domains for this function, like the set of integers \mathbb{Z}, the set of positive reals \mathbb{R}^+, or rational numbers between 20 and 30. What a teacher probably means is, “What is the largest possible subset of \mathbb{R} that could be used as the domain of f(x)=e^x?” In this case, the answer is \mathbb{R}.

So I hope you see why question (1) at the beginning of this post is not a very precise question. Likewise, question (2) is not very precise either. “Where is \ln{x} undefined?” has multiple answers. The answer the teacher is looking for is (-\infty,0]. The question would be better worded, “What real values cannot be in the domain of \ln{x}?”

 

Range

What about the range? The codomain in the example f(x)=e^x is also \mathbb{R}. But the range of f is (0,\infty). The range of a function is defined to be the set of all y\in B such that there exists an x\in A with f(x)=y.

But the range depends on the choice of domain. So asking questions like (3), “State the range of the function f(x)=x^2” aren’t well defined for all the same reasons as above. The desired answer is probably [0,\infty). But the domain of f(x)=x^2 could be the integers, in which case the range is the non-negative integers. We’re not told. So in the case of (3), the more precise question would read, “State the range of the function f(x)=x^2 with x\in\mathbb{R}.”

 

Should we change our teaching?

Maybe. But maybe not. I think I’ll still ask the questions in the imprecise way I started this post. Using the more precise questions would be unnecessarily confusing for most students. But we as teachers should be aware of our slightly incorrect usage, and be ready to give a more precise and thoughtful answer to students who ask.

That being said, I think there’s room for more set theory and basic topology at the high school level. I’m a bit sad I didn’t learn the words onto, surjective, one-to-one, injective, bijective, image, and preimage until very late in my post-high-school studies. I’m not sure all students are ready for such language, but we shouldn’t ever shy away from using precise language. That’s part of what makes us mathematicians.

I ❤ precise language!

Why I hate the definition of trapezoids (again)

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Sorry, I thought I got it all out of my system in my first post about trapezoids last week :-). Allow me to rant a bit more about trapezoids. First let me remind you of the problem. Many Geometry books, our school district’s book included, state the definition of a trapezoid this way:

“A quadrilateral with one and only one pair of parallel sides.”

In case you didn’t catch the point of my first post: I think this is a poor definition and should be abolished from all Geometry curriculum everywhere. Here are some pictures I recently came across on the internet depicting the hierarchy of quadrilaterals. These picture agree with the above definition. Let me just say once more, I completely and totally disagree with these pictures, and I think you should too. That is to say, all of the following pictures are WRONG.

BAD:

 

And I could go on and on. Now here are two good ones.

GOOD:

To be fair, the first set of pictures are only partially wrong. They have good intentions. Typically, the first breakdown of quadrilaterals in those pictures is by “number of parallel sides.” The first lines that come off of the word ‘quadrilateral’ divide quadrilaterals into three categories usually:

  • No parallel sides (i.e. the kite)
  • Exactly one set of parallel sides (i.e. the trapezoid)
  • Two sets of parallel sides (i.e. the parallelogram)

So the pictures aren’t wrong, per say. They just depict different information. The problem comes when teachers ask, “Look at this diagram and tell me: Is every rectangle a trapezoid? Is every rhombus a kite?” The answer to both questions is ‘yes.’ But students instinctively answer ‘no’ when using the first set pictures, and you can see why.

The problem is a historic one. If you go back to Euclid’s Elements, Definition 22 in Book 1, you can see the origin of this problem right away (a translation from the Greek):

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

In the above definition from Euclid, here are the (not perfect) translations of each figure:

  • Euclid’s square –> Our square
  • Euclid’s oblong –> Our rectangle
  • Euclid’s rhombus –> Our rhombus
  • Euclid’s rhomboid –> Our parallelogram
  • Euclid’s trapezia –> Our…trapezia/trapezium?

The last definition is a bit confusing, since we don’t have a very well-agreed upon name for this figure. But notice that ALL of Euclid’s definitions are exclusive. A square is never an oblong. A square is never a rhombus. Each of the above quadrilaterals, according to Euclid, is mutually exclusive. These exclusive definitions persisted into the 19th century.

But sorry Euclid, no one likes your definitions anymore. I hate to say it, because everyone loves Euclid.

In his defense, he wasn’t using these names for the same purpose we do. Nothing about his language is very technical and he doesn’t say ANYTHING else substantial about these definitions. He doesn’t use them to make categorical statements about quadrilaterals or to give properties that might be inherited. The names he uses are of little consequence to the rest of his work.

Can we lay this issue to rest yet? A parallelogram is always a trapezoid. Say it with me,

A parallelogram is a trapezoid.

A parallelogram is a trapezoid.

A parallelogram is a trapezoid.

Anything you can say about a trapezoid will be true about a parallelogram (area formulas, cyclic properties, properties about the diagonals). A parallelogram is a trapezoid.

For more posts on this topic, visit here and here.