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.

America’s never been first

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The following is an insightful article, “Debunking Education Myths: America’s Never Been Number One in Math” by Liz Dwyer. It gives a bit of quantitative ammo to support the feeling I’ve always had: America has never had any “good ol’ days” when it was #1 in math. Here’s a taste of her article:

Has America really fallen behind the rest of the world in academic achievement? According to a new report from the nonprofit Brookings Institution, all the doom-and-gloom commentary suggesting that we’ve fallen from the top spot simply isn’t true. And, even more surprising, America’s results are actually on the rise.

National panic ensued last December when data from the Program for International Student Assessment tests revealed our less than stellar international math results. Even worse, high schoolers from our competitor du jour, China, scored the top spot. But the report’s author, Tom Loveless, writes that, “The United States never led the world. It was never number one and has never been close to number one on international math tests. Or on science tests, for that matter.”

Back in 1964, American 13-year-olds took the First International Math Study and ended up ranking in 11th place. Considering that only 12 nations participated, including Australia, Finland, and Japan, our next-to-last performance was pretty abysmal. Other international tests American students have taken over the years have also never showed that we were in the top spot. It’s a myth that we’ve fallen from our glory days.

(more)

Hat tip to Alexander at CTK.

 

Helping students who are stuck

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How do you help a student who is stuck on a problem? A student who can’t seem to find the mistakes s/he makes? I really like this recent idea from Maria Droujkova, shared by Denise of Lets Play Math. Original source here.

When a kid is feel bad about being stuck with a problem, or just very anxious, I sometimes ask to make as many mistakes as he can, and as outrageous as he can. Laughter happens (which is valuable by itself, and not only for the mood – deep breathing brings oxygen to the brain). Then the kid starts making mistakes. In the process, features of the problem become much clearer, and in many cases a way to a solution presents itself.

Maria Droujkova

When will I get my school-issued iPhone?

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I’d love to streamline the attendance/homework checking/gradebook procedures. It always seems pointless to me to have to write down homework grades and attendance, then reenter it on the computer. Some of today’s teachers are already using smart-phone applications for such tasks.

 

One slick app from gradepad.com

From an article on NEA.org by Tim Walker,

It was only a few years ago that cell phones were being banished from classrooms. As far as school districts were concerned, these devices’ reputation as tools for student distraction, mischief, and even harassment easily outweighed any possible benefits in the learning process.

Banning them was—and, in many districts, still is—the easy call to make, but as cell phones have become more sophisticated, powerful, and even more entrenched in students’ daily lives, a growing number of schools have decided to open the door to what are, essentially, mobile computers.

“Educators can’t afford to be behind the 8-ball anymore,” says Mike Pennington, who teaches world history at Chardon Middle School in Chardon, Ohio, and blogs about classroom technology at Teachers for Tomorrow, a website he co-founded with colleague Garth Holman. “Schools need to embrace mobile technology and mobile learning. Students live in this world. These devices belong in the classroom.”

According to some estimates, smart phones, and to a lesser extent tablets like the iPad, will be in the hands of every student in the United States within five years. And as more schools embrace mobile learning, the number of education apps—mobile applications that run on your smart phone—are skyrocketing.

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The article goes on to mention a handful of apps that have classroom potential, including the one above, GradePad. I also liked the looks of Attendance. And one of the commenters mentions that similar apps are available for Android users as well (here). This is all very cool, in my opinion.

In fact, I have a dream…

I can imagine a time in the not-to-distant future when I walk around the room at the beginning of the period checking homework and taking attendance from a mobile device. I’d be able to see the seating chart, do random name calling, see student photos, and control my computer. If students were issued similar devices, I could have them post their work on the board, using their mobile device as a slate to operate the front board. And students would use their devices as calculators and text books as well, perhaps. All my grades, attendance data, student data, and seating charts would be synced with the network and with our online grade reporting system.

We have Promethean (“smart”) boards in the front of our classrooms, and that’s been nice. But I think having mobile devices in the classroom would be far more advantageous, revolutionizing the way we teach more than smart boards ever did.

Most of what I’ve said is already technically possible–the hardware already exists. One hurdle will be cost, of course. But the cost could be significantly offset if there was no need to purchase hard-cover textbooks (very expensive) or smart boards (also very expensive). Another hurdle will be getting networks, software, and network administrators to cooperate. For instance, our district uses multiple vendors and some of our key data systems aren’t linked, like they should be. Allowing mobile devices to connect to the school network and the internet, providing district-approved  & purchased software, and syncing the whole system with existing data systems would be a sizable task.

Three cheers for the future! 🙂

Physics of Angry Birds

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A nice projectile motion application, suitable for almost any level of high school math.

Here’s an excerpt of the Wired.com article by Rhett Allain:

You know the game, I know you know. Angry Birds. I have an attraction to games like this. You can play for just a little bit at a time (like that) and each time you shoot, you could get a slightly different result. Oh, you don’t know Angry Birds? Well, the basic idea is that you launch these birds (which are apparently angry) with a sling shot. The goal is to knock over some pigs. Seriously, that is the game.

But what about the physics? Do the birds have a constant vertical acceleration? Do they have constant horizontal velocity? Let’s find out, shall we? Oh, why would I do this? Why can’t I just play the dumb game and move on. That is not how I roll. I will analyze this, and you can’t stop me.

 

[Hat tip: Tony Sanders]


 

Why I hate the definition of trapezoids

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I should have made this post a long time ago, because it’s a bone of contention I’ve always had with trapezoids. Or…not with trapezoids–I like trapezoids–but a bone of contention I have with the definition of trapezoids. In my humble opinion, it’s a major problem with Geometry as it’s currently taught. Here’s the usual definition of a trapezoid (taken from our school’s Geometry text book, by Holt Rinehart and Winston):

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

I’ve emphasized the words “one and only one,” which is what I want to comment about in this post. (Here’s another source and another source and another source that say it that way, too.) Sometimes it’s also said, “a quadrilateral with exactly one pair of parallel sides.”

I’ve prepared a simple GeoGebra applet and posted it here. It allows you to play with the trapezoid, moving its vertices and edges. As you drag it around, at all times, one pair of sides will be parallel. But wait, it’s not always a trapezoid, is it? According to the Geometry book, there’s one moment, as you’re dragging it around, that it stops being a trapezoid and for that one second is exclusively a parallelogram. Here’s the moment I’m talking about:

Is this still a trapezoid?

That’s right, using the Geometry textbook’s definition of a trapezoid, if both pairs of opposite sides of the quadrilateral happen to be parallel,  it’s not a trapezoid anymore. At this point, the mathematical reader should be crying, “Foul! How did we ever let this happen? This definition of a trapezoid is so inelegant!!” And I couldn’t agree more.

We don’t do this with the definition of any other quadrilateral. Why do it with a trapezoid? If I were to make another little applet that lets you drag around a rectangle, would we say “it’s not a rectangle” at the moment you make the four sides equal? No! That would be absurd.

The definition of a trapezoid, in my opinion (and thankfully in the opinion of some others) should read:

“A quadrilateral with at least one pair of parallel sides.”

And the hierarchical diagram should look like this one, I found online (taken from a mathematically enlightened author):

from mathisfun.com

Here’s a nice paragraph from the wikipedia entry on trapezoid:

There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some authors[2] define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors[3] define a trapezoid as a quadrilateral with at least one pair of parallel sides, making the parallelogram a special type of trapezoid (along with the rhombus, the rectangle and the square). The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral be ill-defined.

This site and this site also get it right. So there’s hope for the Geometry community and for teachers everywhere. But please, let’s work hard to eradicate the “exclusive” definition from ALL the textbooks. It’s hideous.

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