Bring an end to the rationalization madness

In less than a month, we’ll be hosting the one and only James Tanton at our school. We’re so excited! I’m especially excited because he’s totally going to help me rally the troops in this fight:

He posted this a few years ago, but I only stumbled on it recently. I’ve been looking for Tanton videos to use in our classes so we can get all psyched up about his visit! Needless to say, I was loving this video :-).

For more on why I’m not such a big fan of ‘rationalizing the denominator’ see this post.

Composing power functions

I presented the following example in my Precalculus classes this past week and it bothered students:

Let f(x)=4x^2 and g(x)=x^{3/2}. Compute f(g(x)) and g(f(x)) and state the domain of each.

 

As usual, I’ll give you a second to think about it yourself.

..

 

 

..

 

Done yet?

.

 

Here are the answers:

f(g(x))=4(x^{3/2})^2=4x^3, x\geq 0

g(f(x))=(4x^2)^{3/2}=8|x|^3, x\in\mathbf{R}

The reason that the first one was unsettling, I think, is because of the restricted domain (despite the fact that the simplified form of the answer seems not to imply any restrictions).

The reason the second one was unsettling is because they had forgotten that \sqrt{x^2}=|x|. It seems to be a point lost on many Algebra 2 students.

 

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.)

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!

Pictures with equations

Check out this awesome blog post by Richard Clark on the Alpha Blog.

 

Follow the link to see lots of great pictures made with equations. These pictures are so complicated it makes you wonder, is there any picture we can’t make with equations? My first answer is NO.

Think about vector-based graphics. Vector graphics, for those who’ve never heard the term, are pictures/graphics that are stored as a set of instructions for redrawing the picture rather than as a large array of pixels. You’ve used vector graphics if you have ever used clip-art or used the drawing tools in Microsoft Office, or if you’ve ever used Adobe Illustrator, or Inkscape. The advantages of vector graphics include very small files and infinite loss-less resizeability. How can vector graphics achieve this? Well, like I said, vector graphics are stored as rules not pixels. And by rules, we could just as easily say equations.

So the answer is certainly YES we can make any picture using equations. I think the harder question is can we make any picture using ONE equation? Or one set of parametric equations? Or one implicit equation?

What constraints do we want to impose? Do fractals/iterative/recursive rules count?

I am curious to find out how the creators of these picture-equations came up with them. It seems infeasible to do this by trial and error, given the massive size of these equations.

Oh, and if you haven’t yet seen the Batman Curve, you better go check that out too.

 

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! :-)

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!