Category Archives: Math on the web

Lego Price Statistics

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Do you ever get the feeling that Lego Bricks are becoming more expensive? When we were kids, boy, it felt like they were cheaper, right? I mean, the biggest sets were $150 at most. I have a HUGE Lego collection, and it definitely seems like Legos back in my day were more affordable.

Trouble is, that’s not really true. It turns out that Lego bricks have actually gotten cheaper, by almost every measure you can think of (weight/number of pieces/licensed sets). Check out this incredibly thorough post on Lego Price statistics over time. The article is entitled, “What Happened with LEGO” by Andrew Sielen. It’s very thoughtfully done.

[ht: Gene Chase]

Half-your-age-plus-seven rule

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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

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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

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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

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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!

Pictures with equations

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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.

 

“Japanese” Multiplication

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My brother sent me a link to this video that teaches “Japanese” multiplication (thanks Tim!):

I learned about this technique in my History of Math class, and Vi Hart talked about it in a video back in 2011:

She does a nice job showing why there’s nothing particularly special about this Japanese “visual” multiplication. Here are a few reasons why it’s not better, as far as I’m concerned:

  1. It’s not faster (sometimes it is, but most of the time not). As Vi points out, counting the number of dots in a rectangle by hand is ridiculous.
  2. It’s painful when the numbers are bigger than 1, 2, or 3 and when there are more than 2 digits in the numbers (just try multiplying 976 x 8937 for example).
  3. Zeros make things difficult (use dashed lines?)
  4. Carrying is still required.
  5. It’s perhaps more error prone, since it relies on your counting all the intersections.

In the end, to multiply two numbers you still have to multiply all their digits by each other and deal with carries, no matter which method you choose. I think it’s still worth teaching various methods of multiplication to students in an effort to make the abstract more concrete.