One Good Thing: Complex Numbers

I had twins this summer, so I'm only teaching part time this year (which turned out to be an excellent idea, babies-wise). So what better way to transition myself back into the world of teaching than the reflection of the MathTwitterBlogosphere Blogging initiative?!

This week: One Good Thing.

This week, we introduced Complex Numbers in my Math 3B (Algebra 2 level) class. I've been loving the opportunity to dive into what numbers are all about with this. First off, I love the "get ready to have your mind blown" aspect of complex numbers. I sometimes record pieces of class from my Smartboard using ScreenFlow, either for students who are absent or for topics that students might want to review later. This Intro to Complex Numbers lesson starts with reviewing the types of numbers they have experienced thus far, with hand-wavey tie-ins to the history of human civilization & their individual histories of learning math. And then, dah, dah dum... Imaginary Numbers! Go to 12:06 to watch my fun smartboard trick about it at 13:15.


But I also love getting into why the rules of complex addition/subtraction/everything else work, and going back and forth between algebraic and graphical representations, and between complex and real number worlds. For example, the rule of multiplication is totally bizarre (and will remain unproven in our class) - it involves addition, for pete's sake.

Multiply the radii, add the angles, in case you're rusty.

Which raises the question of whether this is totally divorced from regular multiplication. So we look at it. Spoiler alert: NO. IT TOTALLY WORKS.

Do you see that pretty connection between polar representation from the graph on the complex plane?

And then there is the idea of One, which we need if we are going to talk about reciprocals of complex numbers, which we need in order to divide. What is "One" in complex numbers?" What does "One" even do? And the best thing is that they don't need to believe me that "one" should be 1+0i as opposed to 1+1i, or anything else. 

So if we are talking about "one" in terms of what one does, then we can talk about reciprocals in terms of what a reciprocal does (result in One when multiplied by the original number), rather than their middle school definition of 1/original number. And then we can divide! 

My students ask me about why we are learning complex numbers, and rightly so. And I can talk about how complex numbers are useful in higher math, and with computers (so they don't just stop when they get to the even root of a negative), and how they make many things possible (like laser eye surgery machines) because the coding requires computers to not freak out over the square root of a negative number. But why do we, right now, care about them? Because this is cool. Because math can be intrinsically interesting all by itself. We are not going to get to "real world" applications, and sure, this lays the groundwork if you want to go there later on in your life. But the purpose of this, right now, is just talking about numbers and how they work, and how they would work if we just changed the rules and had to think differently. Like, what do you mean by "one"? 

...

And of course, after doing these lessons this term, I realized that I didn't discuss Zero as being the Additive Identity, which would have really grounded our discussion about subtracting complex numbers as adding the opposite. Next time!

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!

What my Students did with Visual Patterns

Hey folks! So, the discussion in yesterday's class about the VisualPatterns.org (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 VisualPatterns.org, 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 VisualPatterns.org for the inspiration for my students!

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?

Exploring Twitter

So this week's Mission (number 2 for those keeping track at home) over at Explore the MathTwitterBlogosphere was to use my Twitter account, @LauraVHawkins, for math good. And now, I reflect!

One thing that I learned to appreciate was the power of the hashtag. Not just to be clever, it turns out. Adding #MToBS to the end of my tweets made them reach a whole new audience. Look and see my most successful tweet (by some measures):

2 Retweets and 3 Favorites! All from people who were not previously following me! So that's cool.

Tweeting a conversation is a bit too quick for me right now, though. I replied to a tweet about using the textbook less in class, and they replied back to me, and I... let it just lie there. And it seemed too lame to respond a day later, so...

Oh, but it was interesting to spend the week in a "How could I Tweet this?" state of mind. I do that sometimes with Facebook. When something super cool happens that I want to tell someone about, I start thinking about the status update I may write. Last weekend, we went to an amazing brunch at Lazy Bear underground restaurant. Not only was I dancing in my seat at the deliciousness of every course, (because there were so. many. courses), but I also took a picture of each course.

Want to see? Of course you do.

So one of my thoughts and hopes is to approach my math teaching more like other aspects of my life in terms of social media. What can I capture in words, photos or video to help me remember, celebrate or reflect on later?


Here's one attempt (inspired by 2 absent student). How do we find the solution to this problem using the rectangle model? (Where 2y^2+2y is the area of a rectangle, and 2 is the "length" so we just need to find the width.) I sadly didn't record it right the first time, though. So this video is missing the reactions of the first time around, but still cool. And that is what I will leave you with. Happy Weekend!


3.4A Problem 3 with 3D rectangle from Urban School of San Francisco on Vimeo.

Math is Happening When?

Hello world! Welcome to my first entree into the world of blogging about my experiences teaching math. (You can see my first entree into the world of blogging about other things in my life here, if you're curious.) A little bit about my teaching history, for context:

Palo Alto and Namibia, Peace Corps

I started my teaching career as a long term sub in my hometown of Palo Alto. There, the wonderful Sue Duffek was my mentor, who I clung to tenaciously, since I had no teacher training at all. After that, off to Namibia where I taught Maths and English in a small village near the Angolan border.

Look at Meme Laura! Imagine trying to get good math done on that tiny blackboard with a limited chalk supply. The classroom looks pretty nice, and it was, except for the lack of electricity or running water on campus. During the rainy season, things were rough. It's hard to learn when you're wearing those same thin uniforms, but soaking wet and cold. Also you don't have a pencil.

San Francisco - Public Charter School

After 2 years there, I came back to the US and started teaching at a wonderful charter school, here in San Francisco. I had the privilege of working with some amazing educators there, including Mark Isero (whose Classroom Kindle Project is as an inspiration for the power of technology, hard work, and social media to benefit students). And my students there were amazing as well - truly remarkable assortment of young people. LHS taught me more about teaching in my 5 years there than I could imagine learning in any other school.

San Francisco - Small Private School

Aaaand, during those 5 years I worked so hard that I almost burnt out. I met my now-husband, and decided I wanted to be able to be in my personal life a bit more. The Math Gods smiled down on me, and I jumped at the opportunity to be able to teach with the most amazing math educator I have ever known, Henri Picciotto. They were also looking for someone to step into his enormous shoes and attempt to lead the department as he entered into retirement from full-time teaching. So now here I am, in my 4th year teaching at a wonderful small private school in San Francisco (and 3rd year as the Math Department Chair). 

This year, one of my goals was to continue to get more involved in the online math ed community. So when the inspiring Dan Meyer advertised the folks at Explore MTBoS who have created 8 weeks of missing to do just that, I signed right up! And here I am, writing my first blog entry!

Wish me luck.

What about the title, though?!

Also, my title come from one of my most-used reminders to students, "Life is happening now." Copyright my friend Lindsay Stiegler, a fabulous personal trainer who does not support just sitting there when you could be running a lap, or thinking about math.