#Made4Math: Volumes in Calculus

I blame @bowmanimal for all of this.  A few months ago he blogged about conceptualizing volume in calculus before formalizing.  At the time, I had just started looking over past AP Calc exams, wondering how I was going introduce volume (solids with known cross sections and solids of revolution).  Volume is a calculus topic I've not taught before, and I want my students to do more than memorize the formulas.  Because, as our own textbook puts it so beautifully: "Some students try to learn calculus as if it were simply a collection of new formulas.  This is unfortunate.  If you reduce calculus to the memorization of differentiation and integration formulas, you will miss a great deal of understanding, self-confidence, and satisfaction."[1]

Anyway, I tucked Bowman's post in my "Summer Projects" folder, and, well, it's summer now, so I thought I'd best get on with it.

Solids with Known Cross Sections

Here's what came out of solids with known cross sections:

So pretty!

The idea is a type of think-pair-share activity where students conceptualize what's going on before throwing the actual mathematical definition at them.[2]  I love this because these visuals get glued to your brain.  Now when kids see:

Find the volume of the solid whose base is the region bounded by y=x^2, y=0, and x=1 and whose cross sections are semicircles perpendicular to the x-axis.

They're less likely to throw in the towel because of all the scary words and more likely to remember that green tornado-looking thing.  I hope the conversation that goes on in their darling little heads is, "Need to add an infinite number of infinitesimally thin semicircles...no prob...I've got the tool for infinite sums, an integral, baby!  Thanks, Uncle Leibniz!"

Bowman shares some great tips for constructing these solids (be sure to read his responses in the comments section, too), but I thought I'd add some hints that helped me, if you want to make these as well:

• I could not, for the life of me, get my cross sections to stand using tabs, so I resorted to a hot glue gun, which worked marvelously.  The cross sections seem pretty sturdy (my cat even tried to snuggle with one of them, and it endured her voracious nuzzling, so I think they might just last a few years...cross your fingers).
• I splurged and got Ghostline foam board because I'm both anal and a terrible free-hand artist.  I did not trust myself to draw nice parabolas without it.  They come in packs of two at Hobby Lobby for about $3.50.
• SQUARES ARE EVIL.  Bowman mentioned they were floppy.  Indeed, they are.  I ended up only taking the squares from 0 to 0.7, instead of 0 to 1 like the other cross sections.  This anti-symmetry was deeply depressing, but I couldn't get those darn squares to stay up once they reached a certain size.  Le sigh.

After students converse about what they've seen on the poster boards, I plan to explore this applet with them as well, so they can see a 2D visualization of the 3D object we've created.

Solids of Revolution

Since I was already on this volume kick, I started to wonder how I could create a visual that would help students understand the formulas for solids of revolution (taking a graph and rotating it about a given line).  The answer?  Foam sheets and a wooden skewer:

The graph I chose was y=sin(x)+2.5 (from 0 to 2pi) because I wanted the finished product to look kind of a like a vase and I also wanted to maximize the amount of area available to me on my foam sheets (bought in a pack of 65...of which I used all but 2).  And also because I wanted to show something other than a polynomial function.

Again, I was pretty Type-A about this little project.  The foam sheets were 2 mm thick, so I let 2 mm=1 unit and used a compass and my handy dandy graphing calculator to create circles of approximately the correct radii (overboard?  Yeah...probably so...).

The skewer was the best idea of this project because not only does it serve as a visual for the axis of rotation, but it was super easy to center all the little foam circles on it because of the imprint the compass had already made:

11 down, 52 to go...

Again, instead of being scared of the weird words and sometimes weird, unhelpful figures that go along with rotation problems, I hope my students will think, "Just adding up a bunch of super thin (dx!) circles...gonna need pi*r^2 and an integral for that.  Thanks again, Uncle."[3]

And here is a fantastic Geogebra interactive worksheet we can explore as well.

I hope my students gain a great deal from these two summer projects.  I know I was really thrilled by the mathematics that was being exposed while constructing them.  For example, with known cross sections, decreasing the base by the same amount each time did not create even gaps between cross sections since the rate of change is smaller as we get closer to (differentiable) mins and maxs.  I had a similar struggle with the vase:  the closer I got to a min or max, the harder it was to get the right radius because the radii were changing so slowly.

Update
The next year I had my kids make their own solids.  See post here.


[1]  Larson and Edwards, Calculus of a Single Variable 9th ed., p. 42.
[2]  Anytime I can use the phrase, "There are no right or wrong answers here," I know it's a good activity.
[3]  In the words of Steven Strogatz, "Infinity to the rescue!"

Loves Me, Loves Me Not: Using Differential Equations to Model Love

A couple days ago I picked up Steven Strogatz’s The Joy of x from my library.  After reading the table of contents, I immediately decided to start with Chapter 20:  “Loves Me, Loves Me Not,” in which Strogatz uses differential equations to model love.

Obviously, I have to use this next year with my calc kids.

A slightly shortened version of the chapter can be found in a New York Times article here.  Please go read it if you never have!  But, I recommend having kids read it straight out of the book (or a photo copy of the chapter), which has a lovely graph to accompany the situation being modeled as well as the differential equations right in the meat of the text.

This week, I attended a 4-day workshop by MAX Teaching, to improve literacy skills across all disciplines.  On the last day, we got to put some of our new-found knowledge to the test, creating different activities for the upcoming school year.  I typed up a summary of “Loves Me, Loves Me Not,” and then, with the help of one of the MAX consultants (also a calc teacher!), we created this “Interactive Cloze”:

Loves Me, Loves Me Not

Here’s what you do with this Cloze (copied verbatim from Max Teaching with Reading and Writing:  Classroom Activities for Helping Students Learn New Subject Matter While Acquiring Literacy Skills):

1. Give to students a copy of the Interactive Cloze passage that you have created to summarize the reading and focus on key vocabulary terms.
2. Students individually guess by writing (preferably in pencil) the terms they think will best complete the passage.
3. Small group discussion to compare guesses—students may change some.
4. Silent reading to determine better responses from the text.[1]
5. Small group discussion to attempt a consensus on correct terms.
6. Large group discussion to achieve class consensus.

So, that’s that.  I’m pretty excited to try it out.  I’m also excited to expose the kids to mathematical reading beyond their textbook.  The plan is to give them this shortly after introducing differential equations.

[1] I will have copies of the actual chapter from Strogatz’s book for the kids to read.

Intro to Average Value

Last week I had an idea about how I could introduce average value in calculus next year.  When I've taught average value in the past, I felt like students just memorized a two-step procedure and several didn't see the connection to the definition of average that they've been using for years.  I know I won't be teaching this concept until...mmm...December?...but when you get excited about a lesson/idea, you just gotta follow through with it, right?

Introduction to Average Value by Rebecka Kermanshahi Peterson

What I like about this little packet:

• It starts with an application to motivate the discussion and the why should we study this?
• It recalls previous knowledge.
• It applies the fundamental 3-step process of all calculus topics: (1) Start with a non-calculus idea, (2) apply a limit, (3) arrive at the calculus concept.
• It lets students practice a FRQ from a previous exam, but forces them to search through the problems to find which one would require their new tool.
• Students discover a main idea of calculus using what they already know, each other, and the text (not me).

I recently read that four classroom characteristics important for brain-compatible learning are: (1) challenge (with support), (2) relevance, (3) novelty, and (4) a positive emotional climate.  I think this packet offers all four of these.

This is new for me.  I usually never post material I haven’t actually tried on students yet.  So, feedback, please!  Like I said, I have puh-lenty of time to revise and make this better.  I’ll probably be posting a few other things for next year that I would also love feedback on before I test them out on real, live kids.

Reflections from my first year as a HS Teacher

This was my first year teaching high school.  Before this year, I taught at the college level for three years.  [I talked about my decision to switch here.]

I came into this job knowing it would be different, but, in general, I felt pretty prepared for the job.

Ha!

I cried more the first two days on the job than I had the previous two years combined.  I felt totally out of my element.  I felt out of control.  I didn’t know what in the world I had just gotten myself into.

I had left my college teaching job for…this?  For kids who hated math?  For kids who were glued to their cell phones?  For kids who had full conversations with each other while I was trying to teach?

What. Had. I. Done?

And then I remembered why I took the job in the first place.  I remembered what one of my dear professors and mentors had asked me, “Rebecka, where will you make the biggest difference?”  

So, I (eventually) decided to leave my pity party and start focusing on why I had taken the job in the first place—the kids.  The loud, boisterous, glued-to-their-phones, disillusioned-with-math kids.

Slowly, but very surely, I started falling in love with these crazy kids.  I think it was the little, daily decisions, like these.  I think was it choosing to be thankful for my job and for the opportunity to love on kids who might not get that love elsewhere.  I think it was making small, conscious choices like speaking quietly and respectfully even when a kid lost his temper at me; like stroking a little girl’s hair whether she was doing what I wanted her to be doing or not; like keeping granola bars in my desk for kids who got hungry.  I don’t know if those little things changed my kids’ opinions of me.  But, I do know this:  it changed the way I viewed them.  Those little things weren’t for the students (even though at first I thought they were)—they were for me.  When I started serving my kids, I changed.  When I started being grateful for them, I transformed.

And now?

I love my job.

I can’t imagine going back to college teaching any time soon.  I love my kids.  I love that I get the opportunity to be around some of the coolest teenagers in the nation every single day.  I love that I have the chance to change their minds about mathematics.  I love that I work at a place that encourages academic research and collaboration in order to benefit the children of our community.  I love belonging to a district that just about everyone is proud to be a part of.  I love that I get to belong and make others feel belonged.

Was every day easy?

Hell no.

Was ANY day easy?

Mmmm…nope.

Were there days I did NOT want to go back into my classroom?

You bet.

Were there times I messed up like crazy with the kids?  Times I missed opportunities to love on them?  Times I lost my temper?  Times I wanted them to leave, just please leave?  Times I felt like a failure?

More than I can count.  Much more.

But, in the end, I feel the good outweighed the bad by a long shot.  Because, I’m a better person now than I was in August.  And I have my job to thank for that.

There’s a lot I want to work on.  If there’s one thing I learned this year it’s this:  you have to capture a kid’s heart before you can capture her mind.  I know I captured some hearts this year; but there are also hearts I’m pretty sure I didn’t capture.

I wrote letters to all my (140) students this week.  And I was disappointed by how many of them I really didn’t know all that well.  I wanted to write kind, personal notes.  And while I know my students’ personalities and their tendencies, I don’t necessarily know all my kids.  I know some of them.  But not all.  Yeah, 140 kids is a lot, but after a whole year with them, I should know more about them. 

So, that’s what I’ll be focusing more on next year.  What do my kids do at home?  Who are their friends outside my classroom?  Where do they want to travel and what do they want to see?  What are their dreams and aspirations?

If you have any bright ideas as to how you facilitate these conversations, I’m all ears.

This is long.  If you’ve made it this far, you deserve a medal.  But, this was a pretty life-changing year for me, and I wanted to reflect and document.  I never thought I’d be teaching at a public high school, let alone one with 3200 kids in grades 10-12.  I, myself, was homeschooled and specifically pursued a Master’s so I could go teach at the college level and skip the whole high school crowd.

Funny, right?

But this is where I belong.  A friend of mine recently had a baby girl.  As I watched her hold her daughter, I said, “Man, you are such a natural.  It’s like you’ve had her your whole life.”  She responded, “This is what I was made to do.  I’ve always wanted to be a mamma.”  In that moment, I knew exactly what she meant.  Because that’s how I feel about teaching.  I just never thought my teaching career would take me here.

I’m so glad it did.

Mistakes mean we're getting better, right?

We tried a modified version of Bowman’s Mistake Game in PreCalc this week to review for an upcoming game.  I split the class into six groups and gave each group a problem to work.  Then I gave them these:

Directions

·As a group, work your given problem correctly.  Then, have me check your answer.

·Once you have a correct answer, work the problem incorrectly, hiding your mistake as cleverly as possible.  Your "mistake" must be a true pitfall of the given problem (i.e., what kinds of conceptual errors would students likely make?).  Your error cannot be a simple arithmetic or algebraic mistake.

·When you're happy with your lie, put it on a whiteboard (no need to write out the original question).

·When every group is done, you will find the errors on the other whiteboards and vote on the group with the sneakiest mistake.  Winners get candy. :)

After everyone had looked through and analyzed each group's whiteboard, I brought the boards to the front and had a student from each group summarize the mistake one more time.  

They taped the original question face-up and the mistake face-down

Then, students voted on the best error.  We had previously discussed that the errors needed to be conceptual, big-picture mistakes.  Something that would tell me, “Uh, this kid doesn’t really know what’s going on here…”  Not something like forgetting to distribute a negative or simplifying incorrectly. 

Before the kids left, I had them give me one mistake they promised not to make, write it on a post-it note, and stick it to my door on their way out.

I plan on leaving these up as they enter the door tomorrow so they can be reminded of those promises right before they start the test.

_____________

Aside, and probably more important...

As usual, my first run at this activity wasn't perfect.  There's a lot that needs to be changed.  It's easy for me to get discouraged when an activity doesn't go exactly as I had planned.  But I've been thinking lately (dangerous, I know):  

(1) My class activities have to start somewhere; they can't just magically be perfect...isn't that what we tell our kids:  you have to practice and have patience if you want to become really good at something?  I guess the same goes with becoming good at making the students do the work.  Learning how to scaffold; learning how to ask engaging questions; learning when to step in and when to stay out.  This takes a lot of practice.  No matter how much preparation I put into a lesson or activity, I have to practice delivering it, too...and that can't be done without kids in the room.

(2)  My students have to be taught how to talk about math.  It's a language.  Providing places for them to talk about what they're learning is great...but I can't expect that the conversations will just magically happen.  If the conversations aren't flowing quite as well as I'd like, it's a-ok.  It probably means we're doing good stuff here, actually.  Because we're practicing something they're not particularly good at...yet.