Why I hate the definition of trapezoids (part 3)

Yes it’s true. I’m writing about trapezoids again (having written passionately about them here and here previously). I’ve been taking a break from blogging, as I usually do in the summer. For us, school starts in just two weeks. So I thought I’d come out of my shell and post something…and of course I always have something to say about trapezoids :-).

Let’s start with the following easy test question. Don’t peek. See if you can answer the question without any help.

Which of the following quadrilaterals are trapezoids?

which of these is a trapezoid

Before giving the answer, let me first just remind you about my very strongly held position. I believe that instead of this typical textbook definition (the “exclusive definition” we’ll call it) that reads:

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

the definition should be made inclusive, and read:

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

So the test question above was easy, right? Quadrilaterals (A) and (C) are trapezoids, I hear you say.

Not so fast!! If you’re using the inclusive definition, then the correct answers are actually (A), (B), (C), (D), and (E). But it gets better: If you were using the the exclusive definition, then NONE of these are trapezoids. In order for (A) and (C) to be trapezoids, under the exclusive definition, you must prove that two sides are parallel AND the two remaining sides are not parallel (and you can’t assume that from the picture…especially for (C)!).

Can you see the absurdity of the exclusive definition now?

I finish by offering the following list of reasons why the inclusive definition is better (can you suggest more reasons?):

  1. All other quadrilaterals are defined in the inclusive way, so that quadrilaterals “beneath” them inherit all the properties of their “parents.” A square is a rectangle because a square meets the definition of a rectangle. Likewise, parallelograms, rectangles, rhombuses, and squares should all be special cases of a trapezoid.
  2. The area formula for a trapezoid still works, even if the legs are parallel. It’s true! The area formula A=\frac{1}{2}h(b_1+b_2) works fine for a parallelogram, rectangle, rhombus, or square.
  3. No other definitions break when you use the inclusive definition. With the exception of the definition that some texts use for an isosceles trapezoid. Those texts define an isosceles trapezoid has having both legs congruent, which would make a parallelogram an isosceles trapezoid. Instead, define an isosceles trapezoid as having base angles congruent, or equivalently, having a line of symmetry.
  4. The trapezoidal approximation method in Calculus doesn’t fail when one of the trapezoids is actually a rectangle. But under the exclusive definition, you would have to change its name to the “trapezoidal and/or rectangular approximation method,” or perhaps ban people from doing the trapezoidal method on problems like this one: Approximate \int_0^4(4x-x^2)dx using the trapezoidal method with 5 equal intervals. (Note here that the center trapezoid is actually a rectangle…God forbid!!)
  5. When proving that a quadrilateral is a trapezoid, one can stop after proving just two sides are parallel. But with the exclusive definition, in order to prove that a quadrilateral is a trapezoid, you would have to prove two sides are parallel AND the other two sides are not parallel (see the beginning of this post!).

Arithmetic/Geometric Hybrid Sequences

Here’s a question that the folks who run the NCTM facebook page posed this week:

Find the next three terms of the sequence 2, 8, 4, 10, 5, 11, 5.5, …

Feel free to work it out. I’ll give you a minute.

Done?

still need more time?

..

give up?

Okay. The answer is 11.5, 5.75, 11.75.

The pattern is interesting. Informally, we might say “add 6, divide by 2.” This is an atypical kind of sequence, in which it seems as though we have two different rules at work in the same sequence. Let’s call this an Arithmetic/Geometric Hybrid Sequence. (Does anyone have a better name for these kinds of sequences?)

But a deeper question came out in the comments: Someone asked for the explicit rule. After a little work, I came up with one. I’ll give you my explicit rule, but you’ll have to figure out where it came from yourself:

a_n=\begin{cases}6-4\left(\frac{1}{2}\right)^{\frac{n-1}{2}}, & n \text{ odd} \\ 12-4\left(\frac{1}{2}\right)^{\frac{n-2}{2}}, & n \text{ even}\end{cases}

More generally, if we have a sequence in which we add d, then multiply by r repeatedly, beginning with a_1, the explicit rule is

a_n=\begin{cases}\frac{rd}{1-r}+\left(a_1-\frac{rd}{1-r}\right)r^{\frac{n-1}{2}}, & n \text{ odd} \\ \frac{d}{1-r}+\left(a_1-\frac{rd}{1-r}\right)r^{\frac{n-2}{2}}, & n \text{ even}\end{cases}.

And if instead we multiply first and then add, we have the following similar rule.

a_n=\begin{cases}\frac{d}{1-r}+\left(a_1-d-\frac{rd}{1-r}\right)r^{\frac{n-1}{2}}, & n \text{ odd} \\ \frac{rd}{1-r}+\left(a_1-d-\frac{rd}{1-r}\right)r^{\frac{n}{2}}, & n \text{ even}\end{cases}.

And there you have it! The explicit formulas for an Arithmetic/Geometric Hybrid Sequence :-).

(Perhaps another day I’ll show my work. For now, I leave it the reader to verify these formulas.)

Women & Math

Here’s a thoughtful TED talk from Laura Overdeck of bedtime math. I’ve highlighted this website before and I think it’s such a brilliant idea for kids! Laura was the lone girl in her astrophysics undergraduate studies, so she has a great vantage point from which to give this stellar talk (pun intended! :-) ).

She has lots of great points. In particular, I liked these three simple recommendations for women–and for everyone:

  1. Don’t play the lottery.
  2. Don’t say you hate math.
  3. Don’t ask others to calculate the tip.

 

Improper integrals debate

Here’s a simple Calc 1 problem:

Evaluate  \int_{-1}^1 \frac{1}{x}dx

Before you read any of my own commentary, what do you think? Does this integral converge or diverge?

image from illuminations.nctm.org

Many textbooks would say that it diverges, and I claim this is true as well. But where’s the error in this work?

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^+}\left[\int_{-1}^{-a}\frac{1}{x}dx+\int_a^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^+}\left[\ln(a)-\ln(a)\right]=\boxed{0}

Did you catch any shady math? Here’s another equally wrong way of doing it:

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^+}\left[\int_{-1}^{-a}\frac{1}{x}dx+\int_{2a}^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^+}\left[\ln(a)-\ln(2a)\right]=\boxed{\ln{\frac{1}{2}}}

This isn’t any more shady than the last example. The change in the bottom limit of integration in the second piece of the integral from a to 2a is not a problem, since 2a approaches zero if does. So why do we get two values that disagree? (In fact, we could concoct an example that evaluates to ANY number you like.)

Okay, finally, here’s the “correct” work:

\int_{-1}^1 \frac{1}{x}dx = \lim_{a\to 0^-}\left[\int_{-1}^{a}\frac{1}{x}dx\right]+\lim_{b\to 0^+}\left[\int_b^{1}\frac{1}{x}dx\right]

= \lim_{a\to 0^-}\left[\ln|a|\right]+\lim_{b\to 0^+}\left[-\ln|b|\right]

But notice that we can’t actually resolve this last expression, since the first limit is \infty and the second is -\infty and the overall expression has the indeterminate form \infty - \infty. In our very first approach, we assumed the limit variables a and b were the same. In the second approach, we let b=2a. But one assumption isn’t necessarily better than another. So we claim the integral diverges.

All that being said, we still intuitively feel like this integral should have the value 0 rather than something else like \ln\frac{1}{2}. For goodness sake, it’s symmetric about the origin!

In fact, that intuition is formalized by Cauchy in what is called the “Cauchy Principal Value,” which for this integral, is 0. [my above example is stolen from this wikipedia article as well]

I’ve been debating about this with my math teacher colleague, Matt Davis, and I’m not sure we’ve come to a satisfying conclusion. Here’s an example we were considering:

If you were to color in under the infinite graph of y=\frac{1}{x} between -1 and 1, and then throw darts at  the graph uniformly, wouldn’t you bet on there being an equal number of darts to the left and right of the y-axis?

Don’t you feel that way too?

(Now there might be another post entirely about measure-theoretic probability!)

What do you think? Anyone want to weigh in? And what should we tell high school students?

.

**For a more in depth treatment of the problem, including a discussion of the construction of Reimann sums, visit this nice thread on physicsforums.com.

Half-your-age-plus-seven rule

Looking for a great application of systems of linear inequalities for your Algebra 1 or 2 class? Look no further than today’s GraphJam contribution:

You might just give this picture to students and ask THEM to come up with the equations of the three lines.

There’s also a nice discussion to be had here about inverse functions, or about intersecting lines. And there might also be a good discussion about the domain of reasonableness.

Here are the three functions:

f_{\text{blue}}(x)=x

f_{\text{red}}(x)=\frac{1}{2}x+7

f_{\text{black}}(x)=2x-14

This is especially interesting because I never think of the rule as putting boundaries on a person’s dating age range. Usually people talk about it in the context of “how old of a person can I date?” not “how young of a person can I date?” Or rather, if you’re asking the second question, it’s usually phrased “how young of a person can date me?” (All of these questions relate to functions and their inverses!) But in fact, the half-your-age-plus-seven rule puts a lower and and upper bound on the ages of those you can date.

As far as reasonableness, is it fair to say that my daughter who is 1 can date someone who is between the age of -12 and 7.5? I don’t think so! I’m definitely going to be chasing off those -12 year-olds, I can already tell :-).

For my daughter, the domain of reasonableness might be x\geq 18!

Friday tidbits

Happy Friday! Hope everyone has their kids registered for the AMC next week. If you haven’t already subscribed to the AMC problem-a-day from the MAA, you should! It’ll keep you sharp :-).

Here are a few nice things seen ’round the web recently:

  • The Scrambler, by Dan Meyer & co. Here, Dan challenges us to analyze a classic carnival ride, and asks us to predict where you end up at the end of the ride. And by Dan & “co”, I mean “comment” folks who have generated lots of fun solutions and applets. Dan made a great interactive version here, too.

  • And finally, this lengthy article “Reflections on mathematics and Democracy” by Lynn Arthur Steen is well worth the time [ht: Gene Chase]. He thoughtfully discusses the need for math education among the citizenship. Is “usefulness” to the democracy the highest goal of secondary math education? Do we aim to create quantitatively literate citizens? Or do we put them on the Calculus track and prepare them for college-level STEM careers? Does teaching “quantitative literacy” even count as Mathematics with a capital M? This is obviously something I’ve been thinking a lot about recently. Here are a few of my favorite excerpts:

Ten years ago I addressed the first question posed to this panel in Mathematics and Democracy—a collection of essays from a variety of professionals both inside and outside mathematics.4 (These essays are available for free downloading on the MAA website.) The chief message of this volume is that the mathematics taught in school bears little relationship to the mathematics needed for active citizenship. That mathematics we called quantitative literacy (QL) to contrast it with traditional school mathematics which, historically, is the mathematics students needed to prepare for calculus.

Mathematics and quantitative literacy are distinct but overlapping domains. Whereas mathematics’ power derives from its generality and abstraction, QL is anchored in specific contexts and real world data. An alternative framing of the challenge for this panel is to ask whether perhaps QL might be a more effective approach to high school mathematics for all.

What we forget, however, is that when NCTM initiated its standards work, most mathematics teachers did not actually believe in the goal of teaching mathematics to all high school students. Whereas now we argue about how much and what kind of mathematics to teach in high school, three decades ago debate centered on who should learn high school mathematics. At that time, the curriculum was designed to efficiently sort students into those who were capable of learning high school mathematics and those who were not. So between grades 7 and 9, somewhere between one-third and one-half of the students were placed in a course called General Math—an enervating, pointless review of arithmetic.

Another decade has passed, and our ambitions are now much higher: a common core for all, with everyone emerging from high school ready for college. In one generation, the political view of high school mathematics has progressed from something only some need (or can) learn to a core subject in which all students can and must become proficient. That’s quite a rapid change in ends, which has been matched by a major change in means. The very idea of a common curriculum enforced with common assessments was all but unthinkable back in the 1980s.

Math is not linear

This presentation has been around for a few years, but I’ve never highlighted it here on my blog. It’s a Prezi presentation by Alison Blank and it’s called Math is not linear.

It brings up an excellent point that math is often talked about as being a sequence with one class coming before another, when in fact math isn’t always like this. It is absolutely true that there’s room for prerequisites (it’s not a great idea to take Calculus 3 before Calculus 1), but much of mathematics can be approached at any time.

math courses–perhaps more like a tree then a line?

Math majors realize this when they get to college. Once you take a few basic classes like Calculus, Linear Algebra, and Statistics, you can take almost any other course you want. Occasionally there are other prerequisites (a two-course sequence in Differential Equations might require students to take them in order). But generally, you can order your college math courses in many different ways.

I’ve noticed this in grad school too. Here’s the order I’ve taken my classes so far with my comments:

  1. Differential Equations (took it in undergrad too, which helped!)
  2. Cryptography (might have been good to do abstract algebra first, but we learned the practical things we needed to know along the way)
  3. Abstract Algebra (when we talked about cryptography and elliptic curves, it was total review!)
  4. Real Analysis (took it in undergrad, but boy I understood it a lot better the second time around)
  5. Statistics (took it in undergrad, but I loved learning it from a more theoretical, Calculus-based approach)
  6. Queueing Theory (stat required, for good reason; just started this course, so I can’t tell you much more)

Did I have to take these courses in this order? Not necessarily. I’ve found that cross-references are made between courses constantly. In my queuing theory class this past week, we solved a differential equation. So there! :-)

Could we do this in high school? To some extent, yes. I think it’s still important to have prerequisites, however. And to get to Calculus, you have to take a *somewhat* linear path. (Do you agree that Calculus is the pinnacle of high school mathematics?) Along the way, though, we should indulge in lots of mathematical tangents, as the Prezi suggests.

In some ways we already allow our students a bit of latitude. We offer AP Statistics, which can can be taken almost anytime after Algebra 2. And we offer Higher Level math, which gives a nice taste of college-level math.

As Alison Blank suggests, could we teach a little topology to high school students? Certainly we could. Weren’t we all interested in the advanced math topics before we actually took the class? I remember learning all the fun and interesting results from topology way before I ever read through a topology textbook. I still haven’t taken a class in topology! (But I read a lot of Munkre’s book and did a lot of the exercises, and I think that counts for something :-).)

The point is, I think we can talk about fun and interesting math ideas way before we’re “allowed” to talk about them!

That’s certainly what my dad did with me at home. Thanks dad!

I dedicate this post to all those math teachers who went on a mathematical tangent this week. Keep up the great work!

Patient problem solving

There are lots of qualities that make someone “good” at math. Knowledge and skills are important, and so is creativity. But perhaps the most important qualities are patience and persistence.

I was inspired to write this because of this post by Alexander Bogomolny.

So many of my students shut down as soon as they see a problem, especially those students who have had a bad relationship with math over the years. Some students even give up before really understanding (or reading!) a question. This is even more true when it comes to ‘word problems.’ If you haven’t yet seen this now famous TED talk from Dan Meyer, I encourage you to watch it now (and visit his great blog!). I think he might be the first person I’ve heard use the phrase “patient problem solving” so I’ll give him the credit for that :-).

The importance of patient problem solving has broader application than just math, of course. In so many areas of life, we give up too easily when faced with a problem. We don’t realize, that if we just looked a the problem a little longer, if we came back to it a few more times, if we dove a little deeper, the problem would crack.

When I do my graduate class homework, I find great value in starting the homework problems as soon as possible. Sometimes the problems just need to percolate in my brain!

For those of us who teach, it’s important to keep ourselves fresh and engaged in mathematical problem solving on a regular basis so that we can (1) remain familiar with what real mathematicians actually do, and (2) relate to (and empathize with) our students who are being faced constantly with fresh problems.

The folks over at Math Fail recently encouraged us in that direction, saying “If you didn’t focus on mathematics today, you should at least keep the gears in motion by trying to solve this mathematical puzzle”  and then they proceeded to give us a good puzzle. Go try it now! (I solved it this morning with my wife as we were driving to church!)

I want my students to experience the immense satisfaction that comes from having solved a stubborn problem. I want them to know that if you bang your head enough against the problem, eventually it will crack! Alexander Bogomolny made this point in the the inspiring blog post I mentioned above, saying

Still, there is great satisfaction in having solved a problem – even a simple one, and extra satisfaction in being able to appreciate an elegant proof; this kind of satisfaction is multiplied manifold after you devised a solution on your own. Yes, it all may start with inspiration, but to keep the flame burning involves hard work…the more you sweat, the greater the satisfaction.

Some of my students are starting a big math paper this week, in which they choose their own topic. One of my hopes for them is that they get to experience the deep satisfaction that comes from actually doing mathematical thinking and solving hard problems. There’s also great satisfaction in coming up with good mathematical questions! And they’ll have a chance to do that too.

On a related note, when reading math textbooks, students sometimes don’t understand that reading math is very different than reading other subjects. In other subjects you might be willing to devote 5 minutes per page. But in math, a reader shouldn’t be discouraged if it takes 20 minutes or more to understand a page of text. Math is dense!

Teaching and Grace

Thanks to my dad for the hat tip. If you haven’t yet seen it, this talk (available in text & audio) is a MUST READ for all teachers. Thank you, Francis Su, for sharing your thoughts on teaching and grace!

Your accomplishments are NOT what make you a worthy human being.

I highly recommend, over and above any other teaching book I’ve read, Teaching with Love and Logic. The message of that book is similar to the message Professor Su shares. Let kids know you like them for who they are, not for how they perform in your class!