Thursday, November 7, 2013

Twitter PreCalculus Math Chat

So today I experienced my first Twitter Chat, thanks to Mission #5 of Explore the Math Twitter Blog o'Sphere. Verdict: Delightful!

I was aiming for the #alg1chat, but then I got carried away with Sunday night football. I knew the intriguing #alg2chat would conflict with my Monday night workout every week. And somehow I just got distracted and missed #geomchat Wednesday night. So I was very excited when my computer reminded me about #precalcchat today, Thursday, only a little after the 6:30 (Pacific) start time! 

I started just fine with my answer to the first question. I did freak out a little when Taoufik Nadji, one of the facilitators, then asked me to elaborate on how I thought about key ideas & skills, access vs. challenge, etc while planning a lesson. In 140 characters?! But I took it to 2 tweets and was able to at least give some of the flavor of what I meant. 

I felt like I was being most useful when sharing some of my best practices, like teaching two units at once in alternating class periods (with lagging homework) or doing a preview of necessary Algebra skills before diving into PreCalculus level material. I also liked when I could share a link to a more developed idea, like to Henri Picciotto's blog post on lagging homework

I was worried it would go too fast to keep track of things, but it was okay. I was even able to chop vegetables and get soup started for dinner! (But I definitely couldn't also pay attention to the football game my husband was watching. I have no idea how my Fantasy Football team is doing right now.) However I did somehow still manage to miss things. I'm not quite sure why, because I was making sure to scroll up and down my TweetDeck column. But I just scanned back down and saw a couple things I didn't read when they were originally sent out. 

Here are my two take-aways from the evening for my teaching (what this is all about, really): 
  • Periodically check that I'm regularly explicitly teaching all of the Mathematical Practices in my classes. As a private school teacher for 4 years now, I get to not have to think about state standards unless I want to. But that doesn't mean I don't want to be aware of them, especially the ones that I also value, like the Mathematical Practices. My feeling is that we do a good job of teaching them throughout all 4 years, because they reflect the values of our department. But it probably is worth checking and attending to more frequently.
  • Preview Algebra Skills. My students (like many students) tend not to emerge from 9th grade with all algebraic skills mastered and never needing revision. It makes my lessons go better when I take a minute to look at what algebra skills they'll need for the activity of the day, and spend 5 minutes having students review them before diving in. Plus it supports them learning/relearning algebraic skills they didn't quite master the first time.
All in all, worth doing!

Saturday, November 2, 2013

What my Students did with Visual Patterns

Hey folks! So, the discussion in yesterday's class about the (which I learned about through Mission 3 of MTBoS) assignment for homework was so cool I have to share it! Some context first: These are 9th graders in Math 1 (an Algebra-level class). Most of them have seen Algebraic content before, but for various reasons didn't develop a very strong procedural or conceptual understanding of the material. So one of our goals is to teach core Algebra concepts in new and different ways to give them another opportunity to learn the ideas without perpetuating inequitable patterns around who "gets math" and who doesn't.

They've done a lot of work around linear table patterns, the idea of rate of change, graphing, and function diagrams. Side note: I learned about function diagrams from the amazing Henri Picciotto, who is actually presenting on them next month at the CMC North conference at Asilomar, if you are interested in developing this additional way to represent functions. I highly recommend it!

So then I learned about, and I gave them Pattern 109 for homework last week. They liked it, saying that at first it was strange to only get the images, but then they figured it out and it was easy (y=3x). Success at dealing with the online homework, but no rich math discussion. So then I gave them Pattern 82 and Pattern 111. Pattern 82 they handled pretty well (y=2x +1), again no discussion.

Then we got to Pattern 111.

MUCH DISCUSSION. You see, they are not given Figures 1, 2 and 3 (which makes it easy to figure out how much the pattern is increasing by every time). Instead, they have to use these examples to figure it out. We had a lot of different ways:

Gabi's Method:
She made a table including the missing figures, and then looked at the "jump" in the table from 6 blocks to 12, then 12 to 18. It was immediately obvious to her that there were 2 more blocks for every new figure, so she filled in the table in purple. It took us a little side discussion to figure out how you could find the "2" if it didn't jump out at you (in red). Then she said that meant the equation must start with "2n" and then she tried 2*2 and 2*3 and realized she needed to add 4 each time to get to the correct number of blocks. Hence, her rule!

Parker took a different approach:

See Parker also many a table, where the change in x was 1. She didn't see that the y-value went up 2 when x went up by 1, so she did guess and check and found that it went up by 1. She, along with the rest of the class, assumed that it was a linear pattern - I'm glad next week they'll have to deal with some quadratics.

She used the y-intercept, or the "zero method" to first find the "+ 4" at the end of the rule. Revealingly, she previously learned "y = mx + b" but didn't realize that the "+2" on her table actually was the m in the general equation. She had to substitute in the values from the first figure to solve for m. So I stepped in and asked the class how else she could have found it. Others pointed out the connection to the rate of change she found on the table.

Finally, Lucas delighted me by his embrace of function diagrams!

Interestingly, he only needed the given values in his table. Part of function diagram is the "magnification factor." We ask "what did we do the the 3 to get 6?" A.k.a. "∆x times what = ∆y?" Sometimes we even abbreviate magnification factor as m. It's very exciting when they realize that's the SAME m as they met in middle school in y = mx + b. "Oh, that's why it's the letter m!"

That gave him "2n + ?" and he went to the 4th figure's value to figure what he needed to add to 2•4 to get 12. Then he verbally checked in with 7 and 18.

Okay, thanks for reading! I was really excited to have so many different approaches to discuss, and appreciative of for the inspiration for my students!