Wednesday, October 23, 2013

Mission #3: What Else is Happening Out There?

So it's week 3 of the Explore the MathTwitterBlogosphere extravaganza. Our task was to look through some of the really cool sites they had curated and find one that inspired us. Waiting for my computer to back up before I shut it down and headed home, I started looking through the list. 101 questions is cool and fun, although I haven't yet found anything I can apply. Estimation 180 is a fantastic idea, but I've been feeling crunched for time in class (the wandering mathematical discussions that I indulge because they're awesome, while cringing at the time that passes!) and don't think I have space for it.

Then. Oh my goodness, you guys. THIS. Visual Patterns. 3 figures. Different kinds of patterns. Some of the diagrams aren't even sequential! Some are 3.D.

This was the first one. I can't even tell what's happening! How many squares were being added?! I had to sit back down, grab a piece of scratch paper, and figure it out.

And then I did.

And then... I CHECKED IT. ON. THE. WEBSITE. See how they asked me how many squares were in step 43? Bam. No waiting until the next class to see if they got the right answer. It's right there.

So then I had to open my computer back up and change Friday's homework. B period Math 1 (9th graders, Algebra-level), you are about to figure out Pattern #109. Granted, I'm going to also make them document their thinking on paper. Because I'm more interested in their process than the answer for the 43rd figure. Does that take away from the coolness of doing mathematical thinking on an outside website? Not sure that it does, or that I care too much.

I'm excited because this is right in line with what we've been doing - finding rules and patterns. But it's presented in a different way than they're used to, and as I always tell my students, "you know I love when there's more than one way." They've pretty much mastered linear patterns at the point (we're halfway through our first term), but they need practice and periodic reminders, which this is perfect for.

Plus there's a nice variety of kinds of number patterns, from my initial investigation. So I'm going to lead them in slowly (Friday's Pattern #109 is linear and it gives them the first 3 figures). Freshmen can respond in interesting ways to change. "Wait, I was supposed to click on the link to open it?" So let's make the math super familiar at first.

Then maybe Pattern 82. Still linear and give them the first figures in order, but with legos. So that's fun. Then Pattern 111. Still linear, but the change in x, if you will, is no longer 1. Then maybe Pattern 103, a smooth quadratic. Pattern 102, a more tricky quadratic (in the category of "well, it's not linear, so try quadratic. Close, can we make it work?"). Then hit them with Pattern 108 - quadratic, but way above their level to figure out algebraically, so they have to use the shape. Back to a more reasonable quadratic with Pattern 101. Go to Pattern 95 - a straightforward cubic, but in THREE DIMENSIONS? Throw in a Pattern 107, which looks nonlinear perhaps (it's made of squares after all) but isn't. Pattern 29 which is in context so therefore trickier to apply strategies that may have become rote. Have them pick one - who can find the most interesting pattern that we haven't done yet?

Okay, now they're probably tired of the site. But spread out over several weeks, some in class and some at home, and the discussions we're having are about how you found the pattern. What was your strategy? Did you draw the next one? Make a table? How did you decide what kind of pattern it was? I'm betting we develop an even stronger toolbox of pattern-seeking and rule-finding strategies. And that is what I'm all about.

A non-Math 1 pattern

Just for fun, look at this one.
I found the table pattern (adding cubes). And that leaves us with a constant fourth difference. So this must be a 4th degree function. I could find it in a huge system of equations. Is there a better way?


  1. Hi Laura! OMG, whose site is this?? So awesome. The person who started it must be awesome too. She's my new cool hero, for sure. (Okay. Enough before I really throw up on myself :)

    I'm super thrilled that you're using the site with students. That's why it's there!! It really is a labor of love, and of course MANY teachers have contributed to the site, including Chris Robinson who helps me with all the coding.

    Pattern #100... I never (very rarely) use table of values and find the difference until it's constant. I just look at it. I'm seeing that step 43 has 43 sets of cubes, the biggest (at the bottom) is a 43x43x43, then 42x42x42, .... all the way up to 1x1x1. So, it's like sum of consecutive integers, in this case sum of consecutive cubes. Since we're dealing with cubes, shouldn't it be 3rd degree?

    Spread the love, Laura. Thank you so much for sharing!

    1. Hi Fawn! Thanks for the comment and all the love! And thank you even MORE for the awesome site!!

      And I definitely agree that that's how you could calculate the value of step 43 for Pattern 100. But for finding the rule (because that's a lot of multiplying and adding unless you're using a computer program for it), it's definitely 4th degree even though each step is made up of cubes. (Which is part of what I like about it - it's tricky that way!) Now I'm going to have to sit down and figure it out :)

      My tweet is blowing up on twitter (by my standards, anyways) so hopefully you'll get a lot more visitors to your site, which means a lot more kids will get to benefit from the hard work you did. Thank you!!

  2. Nice post, I'd like to know what your students thought of it?
    Pattern 100 is indeed 4th degree and it happens to be the squares of the triangular numbers (according to the online encyclopedia of integer sequences). I'm wondering how to make sense of that visually....

    1. Thanks for your comment! How crazy that Pattern 100 is the squares of the triangular numbers! I didn't realize it until you pointed it out. Makes me wish I knew more about them to understand why that would be! I'll update with my students' responses on Monday!

    2. This may help:

  3. I'm totally geeked out on this site as well. it's completely fascinating to me how much math there is in one pattern: tables, graphs, simplifying expressions, expanding brackets, equivalent expressions, quadratic, linear,'s amazing! THanks for the post.