Exponentials, Differentiation, and Real-World Connections

Exponential growth and decay is one of the best units because there are so many real-life examples. There’s carbon dating – we listened to this NPR story about using radiocarbons to tell if ivory is illegal or not. This connected to one of my favorite field studies of the year, a trip to the National Geographic museum. They offer free movies on Tuesdays, and the one we saw was a film by Dereck and Beverly Joubert about lions and the impact of poaching. The filmmakers were actually there at the museum, and some of the students even got to ask them questions – not bad for a last-minute government shutdown scramble trip! It also connected to a trip I planned the the Smithsonian Museum of African Art where we viewed ivory artifacts and had another discussion about the ethics of ivory.

There’s also – tragically – the exponentially-growing incarceration rate in this nation. We spent some time talking about mass incarceration, though not as much as last year, when we watched all of The House I Live In. That was time well spent, absolutely, but possibly not as mathematically rigorous as it could have been.

I gave the students a list of some common chemicals, like ADHD medications and antidepressants, as well as caffeine. This outlined their chemical makeup just because that was fun to draw, as well as the equation for their decay inside the bloodstream. I figured this out from their half-lives, readily available online. I would have had them do it but we’re doing logarithms next, so they don’t yet have the knowledge to do so.

Sometimes students want to choose a medication not on the list that they are actually taking, which is so great! It doesn’t get much more real-world than a process taking place inside the student’s own body.

While I love this assignment, it has its challenges. It surprises me how much difficulty some students have with the exponential growth formulas. When I tell them “plug in t=1,” it’s not always immediately obvious to them how to do that. I spent a lot of the instructional time explaining exactly which keys to press on the calculators. As straightforward as this assignment seems, this class could have seriously benefited from more explicit, written-out directions, which I did nothing in the way of.

This particular class has a wide variety of mathematical skill levels, prior knowledge, ages and grade levels, learning needs, and personalities. At any point this week, students were working on any one of several different potential things. Some breezed through the chemical graph, the assessment, this other activity I wasn’t even planning on using, teaching themselves the next topic, and that assessment. Others required what some may consider outrageous levels of hand-holding through the chemical graph, and are just now finishing up the assessment for that. I’m still going to track the second assessment for the students who made it that far, but I won’t hold it against the ones who didn’t. I know those guys are more than capable of learning the second small chunk of material, but it doesn’t seem fair to keep everyone else in a holding pattern.

I can’t decide if the huge range of the class’ products is a problem to be solved with more explicit direction-writing and tighter behavior monitoring, or if it’s an inevitable result of the differences in the learners and should be taken as a sign of my flexibility and ability to differentiate. I’m sure it’s some combination of both – things like this often are.

Playing with Blocks … But with an Algorithm

Aside

In my Math 2 class we’re working on geometry topics, specifically lines and angles. To investigate the relationship between the number of sides in a shape and the sum of the interior angles, first we looked at paper triangles. We ripped the corners off to verify that added together, the angles create a straight line, aka 180 degrees. Then I asked them to figure out the measure of each individual angle in an equilateral triangle. Excitingly, the class came up with 3 different methods of doing this, that was a nice moment.

To investigate farther, I gave them each a large handful of pattern blocks. Before giving them the assignment, I let them just play around with the blocks. This was a great way to deal with students finishing the previous assignment at different times – some just got more time contentedly playing with blocks.

Most of the angles of these shapes are 60 degrees or 120 degrees, just not the white quadrilateral. I thought this was kind of boring, but I didn’t hear any of them expressing that. It seemed like it was helpfully repetitive for them, giving them a process.

Over the summer I read this professional development book about executive function, which warned that the types of students I have in my classes, students with ADHD and LD, may have a difficult time in math unless they have an algorithm. It specifically called out investigative learning and explorations like this as being a bad idea for students with executive function difficulties. This upset me because I had spent most of the summer extremely excited to implement some of these investigative approaches into my classes, convinced that it would be perfect for my learners. Having a process – in this case, comparing the 60-degree angled triangle to the angle in question – was definitely important for them, but I think having the ability to move pieces around was just as important. Having them accept as fact that the big angle in the blue pattern block was 120 degrees is not nearly as convincing as holding it up to a green triangle and seeing the straight line.

After this investigation, I absolutely understand what the book was getting at. My students do prefer to have a process and a set of steps. Unfortunately for them, not everything in the world has a set process. I don’t mind scaffolding the things that do and giving them as many algorithms as are out there, but even (especially?) students with executive function challenges need a space to explore. I do slightly regret not assigning this as a group task, because students could definitely have benefited from working together more.

My tentative plan for Monday is to write the sequence 180 360 540 720 on the board, and ask them what these represent. I want them to tell me next that the angles increase by 180 each time, and that the relationship is (n-2)*180. Some of the students aren’t quite done with the investigation yet, and I’d hate to spoil it for them. I think I need someone to come collect data on me in this class, to see if I’m spending more time with some students than others. I feel like at the end of class there’s always a surprise student who I never noticed was lost.

I have always been fascinated by languages. Being able to switch seamlessly between languages has always seemed like the coolest, most useful skill to me. Earlier this year I was thinking about the idea of counting in other languages, specifically counting in languages that are not base-10. Counting 1, 2, 3, … 8, 9, 10 and then starting again 11, 12, is something that we take for granted but several languages do this really differently. Super cool!

The task I assigned the students comes directly after some time spent on logarithms and logarithmic scale. I got the idea to do a linguistics, number-word task because they both have to do with bases. I also thought this would be a good, less-algebraic precursor to modular arithmetic.

I first gave the students a passage from The Crest of the Peacock, Non-European Roots of Mathematics that outlines four different types of number systems. This includes our base-10 system, and the illustrated body numbers I took a picture of out of the book:

As one of the students pointed out, if you wanted to say you were 17, you’d say “I’m neck years old,” which is totally how that worked.

Then they were asked to fill out a blank Pascal’s Triangle, where all the outside numbers are 1’s, and each number is the result of adding the two numbers above it:

So all the boxes on the outside are 1’s, and the box below the 1 and 2 would be 3, and so would the other one like that. What numbers come next?! ARE YOU HAVING FUN YET?!

Sidenote, I wrote a whole Pascal’s Triangle task for one of my other classes, and that was awesome too.

They compared their Pascal’s Triangle to an Ancient Chinese Pascal’s Triangle I found in order to understand Chinese numerals. There is a whole part of the book that’s about Ancient China that has the exact same graphic of the Pascal’s Triangle. I’m giving it to my one student who is way ahead and completely done to read tomorrow. The number 9 is nowhere on the Chinese triangle, but it does go all the way to 8, and 10’s on there, so I asked them to figure it out. These are the important pattern recognition skills that humans need in life! This is what the Common Core means by “look for and make use of structure”!

Then here’s the next part of the task. It’s a partially filled-in chart of number words in Ventureno, part of the now-extinct family of Native Chumashan languages spoken in present-day California. Students once again had to “look for and make use of structure”:

Can you figure it out? They had a pretty hard time with this because it’s not as strong a pattern as some of them.

I’m so excited about this task, and if there were an award for “activity you would have most wanted to do in high school,” I would totally nominate myself. I’ll be back with more dispatches soon, I’m trying to blog the most out the rest of the academic school year.

Arts and Sciences Festival

Every year at HGS we celebrate the arts and sciences with this event. It’s a great opportunity for the community to come together and admire the work of our students. Because what we’re doing here is so unique, we have a wonderfully supportive community of parents, families, former staff, board members, friends, and I suppose anyone reading my blog. What up, community member!

I also showed off some stuff that I use in class just because it looks cool and I’m proud of it. A classic example: the table of non-Euclidean Space. You can read about what the students did with these fuzzy/round objects here

As you can see, presentation is crucial. Black tablecloths make this look like it’s basically in a museum and not some weird lumpy things I crocheted over the summer, although that’s what they are. Check out that super fuzzy one, isn’t that great?

The rest of the displays were less self-centered, focusing on actual student-generated work. How generous of me, I know. Lit by lamplight were the results of the flag project:

Then I posted the trig straw graphs. Highlight of the evening: I was asked actual mathematical content questions about these!

On a loop I played stop-motion videos students created for parametric graphing. As the value of t increases, the graph is created. The one being broadcast while I took this picture was constructed out of sour skittles, and shows the graph of y=tsin(t); x=tcos(t). Above the board are the only parametric motion projects that haven’t already fallen totally apart – ping-pong balls and scotch tape are not the most durable materials.

Potentially the only strictly art project I showcased were the tessellations. In math 1 we’ve been working on transformations, so translations, rotations, and reflections. This led into an examination of symmetry, both reflection and rotational. Tessellations involve both of these topics, as well as bringing in congruence, which we’ll be encountering next.

It stresses me out to think about how little time we have left this school year. This event marks the end of the third quarter, meaning that I only have 25% of the year left. That is simply absurd. I think we should have an extra couple of weeks, but maybe a little while into the summer when I’m feeling kind of bored already. Seems reasonable.

Next week we’ll be doing triangle congruence, proof by induction, a math-words task about numeral bases, and discussing exponential growth through the example of mass incarceration. If any of it is as blog-worthy as I’m planning, I’ll be here!

Giant Flags

We’ve been working on transformations in the coordinate plane in two of my classes, and in each that work is leading us in different directions. The best part about writing my own curriculum is that I can orchestrate these mathematical story arcs with themes and threads and a plot progression.

In Math 2, we finally finished our work with probability, elections, and fairness. I say finally in spite of really enjoying the work we did. It was just a long long long unit. The first task I assigned was a flag enlargement project. Each student was given a very small flag, a scale factor, and a huge sheet of graph paper with a 1-inch grid. For some cross-curricular action, I asked them to look up the origins of their nation’s flag. It turns out that most of the time red is on a flag, it is symbolic of bloodshed. It also turns out that all of my red markers are completely dead by now.

Most of the class was able to complete this task using linear measurement and multiplication alone. My student with the Seychelles was struggling until I handed her a protractor – so much easier!

This experience with flags leads perfectly into similar figures, which leads perfectly into proofs. See what I’m saying about a story arc?

Determining if two triangles are similar isn’t the most exciting thing, yet I’m very excited about it. Remember all of those rules, SSS, AAA, and SAS? These say that even though being similar means that every single angle is the same and every single side length is proportional, with triangles you can figure this out by testing just three things instead of everything. Because of their work with the flags, they had the exact prior knowledge to understand this concept. Most of them had not measured angles (nobody except for the Seychelles) and yet their angles were the same – that’s SSS! The case of the Seychelles verifies AAA. By the time we got to SAS I was looking pretty credible, plus they had fun with some SASsy puns, which I will consider an emotional connection with the mathematics.

I gave them a fairly boring worksheet of similar triangle practice where they had to determine if two triangles are similar. Examining angles and ratios (I used the language “is the scale factor the same for every pair of corresponding sides?” to connect to dilations) and using AAA, SSS, and SAS to justify their conclusions was secretly their very first exercise in proof.

And let’s talk about proof for a minute. The way that mathematical proof is taught in school is ridiculous and everybody knows it. Lucky for me, I have the leeway to not only create a plot progression in my class, but also completely abandon traditional proof pedagogy. I’m skipping these guys up to more college-level proof. It’s not more difficult but it is more logical, more interesting, and more like the way that mathematicians actually operate. Next week we’re going to start on proof by mathematical induction, a highly algebraic proof type. Our first exercise: prove that adding any 3 consecutive numbers results in a sum divisible by 3. I love this because you can actually test it out and convince yourself that it should be true. And then the proof. I’ll let everyone know how the rest of this goes.