Great NCTM problem

Yesterday I presented this problem from NCTM’s facebook page:

Solve for all real values of x:

\frac{(x^2-13x+40)(x^2-13x+42)}{\sqrt{x^2-12x+35}}

We’ve had an active discussion about this problem on their facebook page, and you should go check it out and join the conversation yourself. Go ahead and try it if you haven’t already.

Don’t read below until you’ve tried it for yourself.

Okay, here’s the work. Factor everything.

\frac{(x-8)(x-5)(x-7)(x-6)}{\sqrt{(x-5)(x-7)}}=0

Multiply both sides by the denominator.

(x-8)(x-5)(x-7)(x-6)=0

Use the zero-product property to find x=5,6,7,8. Now check for extraneous solutions and find that x=5 and x=7 give you \frac{0}{0}\neq 0 and x=6 gives x=\frac{0}{\sqrt{-1}}=\frac{0}{i}=0. This last statement DOES actually hold for x=6 but we exclude it because it’s not in the domain of the original expression.The original expression has domain (-\infty,5)\cup(7,\infty). We could have started by identifying this, and right away we would know not to give any solutions outside this domain. The only solution is x=8.

Does this seem problematic? How can we exclude x=6 as a solution when it (a) satisfies the equation and (b) is a real solution? This is why we had such a lively discussion.

But this equation could be replaced with a simpler equation. Here’s one that raises the same issue:

Solve for all real values of x:

\frac{x+5}{\sqrt{x}}=0

Same question: Is x=-5 a solution? Again, notice that it DOES satisfy the equation and it IS a real solution. So why would we exclude it?

Of course a line is drawn in the sand and many people fall on one side and many fall on the other. It’s my impression that high-school math curriculum/textbooks would exclude x=-5 as a solution.

Here’s the big question: What does it mean to “solve for all real values of x“? Let’s consider the above equation within some other contexts:

Solve over \mathbb{Z}:

\frac{x+5}{\sqrt{x}}=0

Is x=-5 a solution? No, I think we must reject it. If we try to check it, we must evaluate \frac{0}{\sqrt{5}} but this expression is undefined because \sqrt{5}\notin\mathbb{Z}. Here’s another one:

Solve over \mathbb{Z}_5:

\frac{x+5}{\sqrt{x}}=0

Is x=-5 a solution? No. Now when we try to check the solution we get \frac{0}{\sqrt{5}}=\frac{0}{\sqrt{0}}=\frac{0}{0} which is undefined.

The point is that, if we go back to the same question and ask about the solutions of \frac{x+5}{\sqrt{x}}=0 over the reals, and we check the solution x=-5, we must evaluate \frac{0}{\sqrt{-5}} which is undefined in the reals.[1]

So in the original NCTM question, we must exclude x=6 for the same reason. When you test this value, you get \frac{0}{i} on the left side which YOU may think is 0. But this is news to the real numbers. The reals have no idea what \frac{0}{i} evaluates to. It may as well be \frac{0}{\text{moose}}.

There’s a lot more to say here, so perhaps I’ll return to this topic another time. Special thanks to all the other folks on facebook who contributed to the discussion, especially my dad who helped me sort some of this out. Feel free to comment below, even if it means bringing a contrary viewpoint to the table.

________________________

[1] This last bit of work, where we fix the equation and change the domain of interest touches on the mathematical concept of algebraic varieties, which I claim to know *nothing* about. If someone comes across this post who can help us out, I’d be grateful! ūüôā

Summer Odds and Ends

I promise I’ll start blogging again. But as followers of this blog might know, I like to take the summer off–both from teaching¬†and blogging. I never take a break from math, though. Here are some fun things I’ve seen recently. Consider it my own little math carnival :-).

I love this comic, especially as I start my stat grad class this semester @ JHU. After this class, I’ll be half-way done with my masters. It’s a long road!¬†[ht: Tim Chase]

Speaking of statistics, my brother also sent me this great list of lottery probabilities. Could be very useful in the classroom.

These math dice. Honestly I don’t know what I’d do with them, but you have to admit they’re awesome.¬†[ht: Tim Chase]

These two articles about Khan academy and the other about edX¬†I found very interesting. File all of them under ‘flipping the classroom.’ I’m still working up the strength to do a LITTLE flipping with my classroom. My dad forwarded these links to me. He has special interest in all things related to MIT (like Khan, and like edX) since it’s his alma mater.

I’ll be teaching BC Calculus for the first time this semester and we’re using a new book, so I read that this summer. Not much to say, except that I did actually enjoy reading it.

I also started a fabulous book, Fearless Symmetry by Avner Ash and Robert Gross. I have a bookmark in it half way through. But I already recommend it highly to anyone who has already had some college math courses. I just took a graduate course in Abstract Algebra recently and it has been a great way to tie the ‘big ideas’ in math together with what I just learned. The content is very deep but the tone is conversational and non-threatening. (My dad, who bought me the book, warns me that it gets painfully deep toward the end, however. That’s to be expected though, since the authors attempt to explain Wiles’ proof of Fermat’s Last Theorem!)

I had this paper on a juggling zeta function (!) sent to me by the author, Dr.¬†Dominic Klyve (Central Washington University). I read it, and I pretended to understand all of it. I love the intersection of math and juggling, and I’m always on the look out for new developments in the field.

And most recently, I’ve been having a very active conversation with my math friends about the following problem posted to NCTM’s facebook page:

Feel free to go over to their facebook page and join the conversation. It’s still happening right now. There’s a lot to say about this problem, so I may devote more time to this problem later (and problems like it). At the very least, you should try doing the problem yourself!

I also highly recommend this post from Bon at Math Four on why math course prerequisites are over-rated. It goes along with something we all know: learning math isn’t as ‘linear’ an experience as we make it sometimes seem in our American classrooms.

And of course, if you haven’t yet checked out the 90th Carnival of Mathematics¬†posted over at Walking Randomly (love the name!), you must do so. As usual, it’s a thorough summary of recent quality posts from the math blogging community.

Okay, that’s all for now. Thanks for letting me take a little random walk!

87th Carnival of Mathematics

The 87th Carnival of Mathematics has arrived!!¬†Here’s a simple computation for you:

What is the sum of the squares of the first four prime numbers?

That’s right, it’s

Good job. Now, onto the carnival. This is my first carnival, so hopefully I’ll do all these posts justice. We had lots of great submissions, so I encourage you to read through this with a fine-toothed comb. Enjoy!

Rants

Here’s a post (rant) from Andrew Taylor regarding the coverage from the BBC and the Guardian on the¬†Supermoon¬†that occurred in March 2011. NASA reports the moon as being 14% larger and 30% brighter, but Andrew disagrees. Go check out the post, and join the conversation.

Have you ever heard someone abuse the phrase “exponentially better”?¬†I know I have. One incorrect usage occurs when someone makes the claim that something is “exponentially better” based on only two data points. Rebecka Peterson has some words for you¬†here, if you’re the kind of person who says this!


Physics and Science-flavored

Frederick Koh submitted¬†Problem 19: Mechanics of Two Separate Particles Projected Vertically From Different Heights¬†to the carnival. It’s a fun projectile motion question which would be appropriate for a Precalculus classroom (or Calculus). I like the problem, and I think my students would like it too.

John D. Cook highlights a question you’ve probably heard before:¬†Should you walk or run in the rain?¬†An active discussion is going on in the comments section. It’s been discussed in many other places too, including twice on Mythbusters. (I feel like I read an article in an MAA or NCTM magazine on this topic once, as well. Anyone remember that?)

Murray Bourne submitted¬†this awesome post about modeling fish stocks. Murray says his post is an “attempt to make mathematical modeling a bit less scary than in most textbooks.” I think he achieves his goal in this thorough development of a mathematical model for sustainable fisheries (see the graph above for one of his later examples of a stable solution under lots of interesting constraints).¬†If I taught differential equations, I would¬†¬†absolutely¬†use his examples.

Last week I highlighted this new physics blog, but I wanted to point you there again: Go check out Five Minute Physics! A few more videos have been posted, and also a link to this great video about the physics of a dropping Slinky (see above).

Statistics, Probability, & Combinatorics

Mr. Gregg analyzes European football using the Poisson distribution in his post,¬†The Table Never Lies. I liked how much real world data he brought to the discussion. And I also liked that he admitted when his model worked and when it didn’t–he lets you in on his own mathematical thought process. As you read this post, you too will find yourself thinking out loud with Mr. Gregg.

Card Colm has written¬†this excellent post¬†that will help you wrap your mind around the number of arrangements of cards in a deck. It’s a simple high school-level topic, but he really puts it into perspective:

the number of possible ways to order or permute just the hearts is 13!=6,227,020,800. That’s about what the world population was in 2002. So back then if somebody could have made a list of all possible ways to arrange those 13 cards in a row, there would have been enough people on the planet for everyone to get one such permutation.

I think it’s good to remind ourselves that whenever we shuffle the deck, we can be almost certain that our arrangement has never been created before (since ¬†52!\approx 8\times 10^{67} ¬†arrangements are possible). Wow!

Alex is looking for “random” numbers by simply asking people. Go contribute your own “random” number¬†here.¬†Can’t wait to see the results!

Quick! Think of an example of a real-world bimodal distribution! Maybe you have a ready example if you teach stat, but here’s a really nice example from Michael Lugo: Book prices. Before you read his post, you should make a guess as to why the book prices he looked at are bimodal (see histogram above).

Philosophy and History of Math

Mike Thayer just attended the NCTM conference in Philadelphia and brings us a thoughtful reaction in his post,¬†The Learning of Mathematics in the 21st Century. Mike wrote this post because he had been left with “an ambivalent feeling” after the conference. He wants to “engage others in mathematics education in discussions about ways to improve what we do outside of the frameworks that are being imposed on us by those outside of our field.” As a secondary educator, I agree with Mike completely and really enjoyed his post. Mike isn’t satisfied with where education is going. In his post, he writes, “We are leaping ahead into the unknown with new educational models, and we never took the time to get the old ones right.”

Edmund Harriss asks¬†Have we ever lost mathematics?¬†He gives a nice recap of foundational crises throughout the history of mathematics, and wonders, ultimately, if we’ve actually¬†lost¬†any mathematics. There’s also a short discussion in the comments section which I recommend to you.

Peter Woit reflects on¬†25 Years of Topological Quantum Field Theory. Maybe if you have degree in math and physics you might appreciate this post. It went over my head a bit, I’m afraid!

Book Reviews

In¬†this post, Matt reviews a 2012 book release,¬†Who’s #1, by¬†Amy N. Langville and Carl D. Meyer. The book discusses the ranking systems used by popular websites like Amazon or Netflix. His review is thorough and balanced–Matt has good things to say about the book, but also delivers a bit of criticism for their treatment of Arrow’s Impossibility Theorem. Thanks for this contribution, Matt! [edit: Thanks MATT!]

Shecky R reviews of David Berlinski’s 2011 book,¬†One, Two Three…Absolutely Elementary mathematics¬†in his¬†Brief Berlinski Book Blurb. I’m not sure his review is an *endorsement*. It sounds like a book that only a small eclectic crowd will enjoy.

Uncategorized…

Peter Rowlett submitted this post about linear programming and provides a link to an interactive problems solving environment.

Peter Rowlett also weighs in on the recent news about a German high school boy who has (reportedly) solved an open problem. Many news sources have picked up on this, and I’ve only followed the news from a distance. So I was grateful for Peter’s comments–he questions the validity of the news in his recent post “Has schoolboy genius solved problems that baffled mathematicians for centuries?” His comments in another recent post are perhaps even more important though–Peter encourages us to think of ways we can remind our students that lots of open problems still exist, and “Mathematics is an evolving, alive subject to which you could contribute.”

Jess Hawke IS *Heptagrin Girl*

Here’s a fun-loving post about Heptagrins, and all the crazy craft projects you can do with them. Don’t know what a Heptagrin is? Neither did I. But go check out Jess Hawke’s post¬†and she’ll tell you all about them!

Any Lewis Carroll lovers out there?¬†Julia Collins submitted a post entitled “A Night in Wonderland”¬†about a Lewis Carroll-themed night at the National Museum of Scotland. She writes, “Other people might be interested in the ideas we had and also hearing about what a snark is and why it’s still important.” When you check out this post, you’ll not only learn about snarks but also about creating projective planes with your sewing machine. Cool!

Mike Croucher over at Walking Randomly gives a shout out to the free software Octave, which is a MATLAB replacement.¬†Check out his post,¬†here.¬†MATLAB is ridiculously expensive, and so the world needs an alternative like Octave. He provides links to the Kickstarter campaign–and Mike has backed the project himself. I too believe in Octave. I’ve used it a few times for my grad work and I’ve been very grateful for a free alternative to MATLAB.

The End 

Okay, that’s it for the 87th Carnival of Mathematics. Hope you enjoyed all the posts! Sorry it took me a couple days to post it–there was a lot to digest :-).

If you missed the previous carnival (#86), you can find it here. The next carnival (#88) will be hosted by Christian at checkmyworking.com. For a complete listing of all the carnivals, and more information & FAQ about the carnivals, follow this link.

Cheers!

Rationalization Rant

Every high school math student has been taught how to rationalize the denominator. We tell students not to give an answer like

\frac{1}{\sqrt{2}}

because it isn’t fully “simplified.” Rather, they should report it as

\frac{\sqrt{2}}{2}.

This is fair, even though the second answer isn’t much¬†simpler than the first. What does it really mean to simplify an expression? It’s a pretty nebulous instruction.

We also don’t consider

\frac{12}{1+\sqrt{5}}

to be rationalized because of the square root in the denominator, so we multiply by the conjugate to obtain

2-2\sqrt{5}.

In this particular example, multiplying by the conjugate was really fruitful and the resulting expression does indeed seem much more desirable than the original expression.

But here’s where it gets a little ridiculous. Our Algebra 2 book also calls for students to rationalize the denominator when (1) a higher root is present and (2) roots containing variables are present. Let me show you an example of each situation, and explain why this is going a little too far.

Rationalizing higher roots

First, when a higher root is present like

\sqrt[5]{\frac{15}{2}},

the book would have students multiply the top and bottom of the fraction inside the radical by 2^4 so as to make a perfect fifth root in the denominator. The final answer would be

\frac{\sqrt[5]{240}}{2}.

Simpler? You decide.

This becomes especially problematic when we encounter sums involving higher roots. It’s certainly possible, using various tricks, to rationalize the denominator in expressions like this:

\frac{1}{2-\sqrt[3]{5}}.

But is that really desirable? The result here is

\frac{1}{2-\sqrt[3]{5}}\cdot\frac{4+2\sqrt[3]{5}+\sqrt[3]{25}}{4+2\sqrt[3]{5}+\sqrt[3]{25}}=\frac{4+2\sqrt[3]{5}+\sqrt[3]{25}}{3},

which is, arguably, more complex than the original expression. Can anyone think of a good reason to do this, except just for fun?

Rationalizing variable expressions

Now, let’s think about variable expressions. Here is a problem, directly from our Algebra 2 book (note the directions as well):

Write the expression in simplest form. Assume all variables are positive.

\sqrt[3]{\frac{x}{y^7}}

The method that leads to the “correct” solution is to multiply the fraction under the radical by \frac{y^2}{y^2}, and to finally write

\frac{\sqrt[3]{xy^2}}{y^3}.

This is problematic for two reasons. (1) This isn’t really simpler than the original expression and (2) this expression isn’t even guaranteed to have a denominator that’s rational! (Suppose y=\sqrt{2} or even y=\pi.) Once again I ask, can anyone think of a good reason to do this, except just for fun??

So how far do we take this?

Is it reasonable to ask someone to rationalize this denominator?

\frac{1}{2\sqrt{2}-\sqrt{2}\sqrt[3]{5}+2\sqrt{5}-5^{5/6}}

You can rationalize the denominator, but I’ll leave that as an exercise for the reader. So how far do we take this? I had to craft the above expression very carefully so that it works out well, but in general, most expressions have denominators that can’t be rationalized (and I do mean “most expressions” in the technical, mathematical way–there are are an uncountable number of denominators of the unrationalizable type). All that being said, I think this would make a great t-shirt:

And I rest my case.

Good discussions in the math blog world

Here are two blog posts I saw a few weeks ago. I’ve been following the comments with great interest, and the conversations have been fruitful. You should go check them out and join the conversation!

  • Critical Thinking @ dy/dan — Once again, Dan gives deserved criticism to a contrived textbook problem. Hilarious problem, and fun discussion in the comments.
  • Disagreement on operator precedence for 2^3^4 @ Walking Randomly — The title says it all, but it’s the first time I had ever thought about how 2^3^4 or expressions with carets should be evaluated. Note that it’s clear how 2^{3^4} should be evaluated. We’re just unclear on how 2^3^4 should be evaluated.