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Archive for March, 2013

a/k/a What does a regular 3.5-gon look like?

Functions come in discrete and continuous families, which are something like the Montagues and the Capulets. Very little in common. Sometimes angry with each other.

Continuous functions tell you something about the real numbers. A function that converts from Fahrenheit to Celsius is continuous because it'll tell you Celsius for any value of Fahrenheit, including decimals, rationals, irrationals, any real number.

Discrete functions, meanwhile, only tell you something about sets of numbers you can count — the whole numbers, for one example. The function that tells you what your tax credit is for the number of kids you have is discrete because it won't give you a credit for your fractional 2.34 children.

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Another discrete function is the one that takes the number of sides of a regular polygon and tells you the measure of one of its inner angles. A regular triangle has three sides and its inner angle is 60 degrees. A regular quadrilateral has four sides and its inner angle is 90 degrees. A regular pentagon has five sides and its inner angle is 108 degrees.

That's a recipe for a regular pentagon right there. Draw a 108 degree angle between two segments with the same length.

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Then draw another 108 degree angle on the last segment.

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And another, and another, until the segments reconnect and you have a regular polygon with five sides.

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We can write a table:

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We can graph those values:

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We can also write an equation:

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That equation perfectly describes the discrete values in that graph. But the equation is stupid. It doesn't know it's only supposed to describe those discrete values. We can put in other values and, like a sucker, it'll give us a number, even though it isn't supposed to and even though that number won't make any sense.

Like n = 3.5. A regular polygon with 3.5 sides? No such thing. But if we throw n = 3.5 into that function, it gives us the number 77.1 degrees.

Maybe that's just gibberish, the result of pushing this function machine beyond its warranty. But maybe it isn't.

What if we tried to draw a regular 3.5-gon in the same way we did the regular 5-gon up there?

We'd lay down a 77.1 degree angle.

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Then another on top of that one.

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Then another. And another. And another. And another. And one more. And we're back where we started.

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Blam. The regular 3.5-gon exists!

So different representations of functions (the table, the graph, the polygons, the equations) show and reveal different features of the function. Sometimes they reveal dirty, interesting secrets. The domain of the function — the part that says, "I only work with discrete numbers." — is like a product warranty. But warranties were meant to be voided. Push your way past the warranty, hack away, find something interesting, and show it off.

BTW. One of you enterprising programmers should create the animation that runs through continuous values of n and shows the regular polygon with that many sides. That'd blow my mind. I can only do the discrete values.

BTW. Malcolm Swan demonstrated the 3.5-gon on the back of some scratch paper in the middle of a design session here in Nottingham. That kind of throwaway moment (often before tea, of course) has been a lot of fun these last two months.

BTW. But where is the 3.5 in that shape? Maybe you see how the number 3.5 turned into the number 77.1 and how the number 77.1 turned into that star shape. But where is the 3.5 in the star? I'll hint at it in the comments but I'll encourage you to think this through. (It may be helpful to see 3.5 as the rational number 7/2.)

2013 Mar 28. I love you guys. I fall asleep for a few hours and wake to find out it's Christmas. Some interesting visualizations of rational regular polygons from Marc Garneau, Eric Berger, Stuart Price, and Josh Giesbrecht.

2013 Mar 29. More applets. One from Andrew Alexander and the other from Khan Academy.

Featured Comments

Daniel:

So, we can now draw a p/q-gon for any natural p,q. Does it allow us to run continuously through polygons? what about irrational number (so many of them between any two rationals…). To make it really continuous we need to have polygons for them as well. According to the above construction, I don’t thing that is possible because an irrational polygon will never meet its starting point (if it will, it will contradict its irrationality).

Matt:

The visual math discovered is cool, but what I’m really amazed by is the fact that the online math ecosystem allowed people to quickly create interactive visual demos in at least five different free, visual environments! Scratch, Desmos, Geogebra, Logo, and Sage. (JavaScript might be a sixth, though Andrew’s nice JavaScript demo wasn’t created within a visual tool, and KA’s JavaScript demo was pre-existing.)

Isn’t that just awesome?

I updated 101questions today to include a single major new feature: a lesson editor.

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Creating webpages like this soaks up too much of my time. I have to upload files in three different places. Changing a single word in the lesson means firing up an FTP client. Changing anything about an image takes ten minutes at least. None of this is creative work.

So I put together the task editor I want to use. You can add supporting materials — photos, videos, questions, teacher notes, student notes, links, and more. You can re-order them quickly, all from the browser. More fun is that other users can download them quickly. Click the "Download" button and Internet pixies will zip all the resources up and send the file to your computer.

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I've been using it for a couple of weeks and I'd like you to use it also.

I've added other features some of you have asked for:

Better tagging.

You can add tags like "pizza" or "basketball" or "money." You can type a few key mathematical terms into the Common Core search bar and it will locate standards for you. Of course, all of this will make the search engine much smarter.

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A smarter search engine.

People e-mail now and then telling me in kind terms how awful this spreadsheet is. I'm in total agreement. Unless you're fluent in Common Core shorthand, it's impossible to find tomorrow's topic today. So now you can head to my page on 101questions, click Search, and then click "Search this user." Type in what you're looking for. Click "Has lesson" to narrow down my material to everything that's been a little more developed. Click the grade boxes to tighten the results down even more.

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Try it out. Add some tags to your old material. Leave me some comments here. I'll need as much useful criticism as you can offer. Let's make this great together.

Bill Jerome, in an excellent post aimed at people who perceive an obvious connection between learning analytics and Netflix recommendations:

The more a user stays engaged with [Netflix and Amazon], the more profit they generate. The comparisons to those kinds of analytics pretty much end there. Unfortunately for those looking for the easy path, our outcomes are complex and the inputs aren’t actually that obvious either.

Then later:

Now what happens if we tell a student they aren’t achieving learning outcomes when in fact we are wrong about that? The potential for demotivating the student comes at a high cost. This could happen with errors in reporting the other way, as well. If learning analytics inform a student they are succeeding but in fact they are not prepared for their next exam or job, the disservice is just as bad. Getting learning analytics wrong on the learning dimension is a recipe for disaster and must be done carefully and with understanding.

As far as I'm concerned, between this post and Michael Feldstein's earlier "A Taxonomy of Adaptive Analytic Strategies", the e-Literate blog has cornered the market on nuance and insight in the learning analytics discussion.

BTW. Probably related: What We Can Learn About Learning From Khan Academy’s Source Code.

Great Classroom Action

Brian Miller posts a smorgasbord of applied proportions activities, each of which poses students as crime scene investigators. He also gives purpose to the skills he wants to teach by fitting them inside a larger, more enticing question:

The basic premise is as follows: I show the Bone Collector clip first. I tell them we need to figure out the shoe size of the killer because we need to make sure that the killer is not in the room. Then I concede that I realize they are not trained investigators. Thus I tell them that over the next couple days we will be doing some investigator training, to get them ready to take on this case.

In some recently consulting, I was asked how to make to make trig identities less of a slog. I didn't hesitate to send along Section 2 of Sam Shah's worksheet. Section 2 really needs its own post. Briefly: watch how he sets students up for a major cognitive conflict as they all graph (seemingly) different trig functions only to compare and find out they're exactly the same. "Okay, are you all graphed? On the count of three there will be the big reveal. One … TWO … REVEAL! Whoa. Really? REALLY? Yes, really." (I don't think I've seen anybody else pull off Sam's worksheet persona, which is some kind of cross between "reality TV host" and "Vegas lounge act.") The point of the worksheet isn't practice. It isn't instruction. It collects student hypotheses about a discrepant event which they'll be working on "as we go through the next few days." If math class were a movie, Sam's worksheet would be the trailer.

Julie Reulbach encourages us to "save the math for last" in a very nice modeling activity. But questions like "what do we need here?" are modeling. They are math. Is it more accurate to say, "Save the computation for last?"

Never before had “test points” seemed so obvious to them. Test points were not just random points, they meant something. They told a story. The next day, when we finally got to the actually inequality lesson, foldable, and then homework, the students really understood the need for a test point. They also easily understood the horrific workbook word problem.

Bruce Ferrington asks students "What is my area?":

My chief interest here is to look at how the kids are going to approach this question. I want to see if they can come up with any short-cuts that will speed up the calculations, so they don't have to cover their entire body in 1cm grid paper and count out each square. Look what they did!

2013 Mar 22. Sam Shah wrote up his thinking behind the worksheet.

Featured Comment

Julie Reulbach:

Maybe, save the “procedures” for last? You know, the actual lesson with the steps and the summary of all of the rules they discovered but didn’t realize it.

Tom Hoffman's perspective on Rhode Island's summative graduation exam is worth your time:

Another question I thought was typical showed two spinners that would give you random numbers from 1 to four. It wanted to know the probability that the sum of the two would be a prime number. I drew a complete blank, until I realized I could easily write out all 16 combinations and just circle the ones that resulted in a prime number. That more clearly took mathematical reasoning, problem solving and content knowledge.

I like the question, and I like the direction it should push math curriculum. But I'm also aware that if even if kids have been taught probability, if they haven't been taught it in a way that encourages flexible and resourceful problem solving — rather than pulling numbers out of stereotypical word problems and following procedures — they will be completely screwed.

I'm glad parents, policymakers, and stakeholders are taking these exams (or shortened versions of them) and reflecting on their results. But again we should be careful not to write expansive prescriptions for what we teach kids based on the test results of grownups. The proposition, "A middle-aged bureaucrat hasn't used algebraic expressions in three decades and turned out fine therefore we shouldn't teach algebraic expressions to fourteen year-olds" has yet to be nailed down for me.

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