Thursday, November 29, 2018

Did I Finally Figure Out How to Teach Recursion?

Recursion has always been a weird topic for me- one where I try all sorts of different things to try to get kids to tie the relatively basic conceptual idea to the funky notation that is sometimes associated with it. I've taught it at the middle school level (next=now blah blah blah), in Algebra 1, in Algebra 2, and in Pre-Calculus and I inevitably always have some kids who struggle make the transition from the idea to the notation and requirements. So, yet again, I tried tweaking this year. Here's what I did. 

I started by putting this slide on the board: 
Students had 30 seconds in groups to tell me the exact sequence about which I was thinking....this obviously caused some unrest. Some groups were prepared with random guesses, some got indignant that they didn't have enough information, some argued among themselves about which side to take. I let groups guess and every time I said "WRONG!" until someone finally pointed out that this was rigged. 

I let them know that I'd allow the class to ask 2 questions to try to get it right. They needed to prioritize those questions in their groups and then we continued the discussion. The 2 questions they asked:

1) What's the pattern?

We discussed- was this strong enough to give me the EXACT sequence I was thinking about? Nope! 

2) What's the first term?

That's it! We need to know where to start and how to proceed from there! Those 2 magical ingredients of a recursive formula were just generated by the class- score. 

Next, I challenged groups to complete this table together. They had great discussions, challenged each other, and the wait time to try to get them to ask questions at the end was excruciating because they really felt confident about it. 

Welp, guess they don't need me now. They just discovered how to write these without me ever teaching it. I'll be in the Math Office drinking my latte with my feet up on my desk. 

After some wrap up notes, we played a round of Quiz Quiz Trade- one of my favorite Kagan Strategies. Below are my rules:

And here's the actual document:

These were small tweaks to how I normally teach recursion, but they made a huge difference for my kiddos this year. The extra work we did with notation previously made a big difference as well (see my last blog post). I loved seeing the kids excited about a topic that has sometimes caused anguish for others in the past! 

Friday, November 16, 2018

Building Understanding of Sequence Notation

Sequences and series are never a particularly hard topic for my Algebra 2 kids- conceptually, at least. They get it. They've been looking for patterns since they were tiny little humans and it's a fun puzzle for many of them to do so in class. What they struggle with consistently is the notation we demand of them at this level of mathematics. Some mastered it in Algebra 1, but many just show up and pray they do it right this time or memorize what their teacher told them to do instead of trying to battle the sense-making involved to understand and be able to apply this knowledge. This activity is one I use to combat that and it builds to have students generate their own equation for recursive sequences. 

Phase I:
We start by doing a sorting activity, where students are given the cards at right and asked to order them. This is an easy enough task- definitely low floor for some of my weaker students. This is a springboard for discussing the "why" and asking questions like:
  • Which comes directly after n? How do you know?
  • Which comes directly before n? How do you know?
  • If you have n-3, which would come 2 after that? 
You can really extend it as far as you want to go here. 

Phase II:
From there, we look at the actual expressions we use for terms. 

 Using what we previously did, ordering these isn't usually much of a challenge. From there, students work in their small groups to extend:

Phase III:
Here is where the wrap up discussion as a class occurs and we begin to test our understanding of the notation. In small groups, we first just examine the differences between position in the sequence and actual value of the term:

Then, we start translating from words into notation. 

I've always heard it said that students are more receptive to an idea if they think it came from themselves or another student....this activity has been a huge help in clarifying misconceptions with the help of other students, not just a teacher re-explaining it the same way for the 400th time. 

If you have anything else that you love for teaching recursion, sequences, or series, please share!! 

Monday, November 5, 2018

Limit Definition of the Derivative Carousel Activity

I've been working on building up the limit definition of the derivative conceptually with my non-AP Calc class for the last week (see my recent post on what I've been doing so far ) and today we started putting it all together! 

To into this activity, I used an awesome warm up from Math Teacher Mambo (Thanks, Shireen, for sharing on the AP Calc Facebook group). As soon as she shares it publicly, I'll link it here. It's so, so good!

Future Home of Warm Up Activity Link

After completing the warm up, here's how the carousel worked:

1) Put students in to groups and have them start at a blank poster or piece of VNPS

2) Each group completes one step at their poster, then rotates. They will then check the next group's work, correct it, and then add the next step to the poster. Here were the directions:

  • Draw a blank axis- I told them just first quadrant- and a function of their choice
  • Sketch a secant line
  • Label x, x+h, and h
  • Label f(x) and f(x+h)
  • Write an expression for slope of the secant line
  • Transform expression into slope of the tangent line

Here were some of our results:

Awesome conversations ensued, including whether it would make sense for f(x) to equal f(x+h) and what that would mean for the secant line (Helllllloooooo, Rolle's Theorem!). Kids were explaining to each other, critiquing each other's work, and doing a lot of sense making among themselves! 

We'll see how this translates to retention beyond today on their next quiz, but I loved seeing the progress that they're making. I think many could explain it better than a few of my AP kids right now- a good "challenge accepted" moment for me to amp this conceptual understanding up more in AP, too. 

Inverses of Exponential and Logarithmic Functions Circuit

Quick post to share a new circuit I just finished.

 My kids needed some more practice and I wanted them to begin reflecting on everyone we'd done in our logs unit before we get into more difficult applications and review for our Quarter 1 exam....this took care of 2 birds with one stone!

This is just the basics and doesn't use any laws of logs, so it could be amped up if you wanted to do so.

Steal away! 

Thursday, November 1, 2018

Discovering the Derivative!

I knew when I took on non-AP Calculus this year that I was going to have to up my game. Last time I taught a non-AP Calculus course I was a first year Calc teacher and I was truly staying only a few days ahead of the kids. I didn't have the perspective of deeper conceptual understanding that added years of teaching a course can bring. I was on a mission this year to build strong conceptual understanding for these students so as they moved on to Calculus in their college career, they would have a strong enough background to face any challenge.

The whole concept behind the limit definition of the derivative can really elude kids. They get that slope is rate of change and they get the mechanics of the derivative rules, but they can be very unsure as to why we're taking a limit and what that "h" means anyway. I decided to tackle this by building up the concept, slowly and intentionally.

Phase I: Have students experiment using what they know about average rate of change and see how they relate it to instantaneous rate of change (on their own, with no intervention from me)
Phase II: Make sure they understand average rate of change and the structure behind the notation
Phase III: Move students slowly from their understanding of average rate of change to instantaneous, moving notationally from middle school to college level

I was really happy with how it went and truly impressed with the ingenuity of some of my students. While there are things I will definitely tweak for next year, I felt like my kids walked out with a firm grasp on what the hot mess of notation known as the limit definition of the derivative means. We did no evaluating, no calculating. We just worked on concept and structure.

Phase I: Engineering Design Challenge

There are profound benefits to having a work wife who is a physics teacher and that is being able to raid her lab room for supplies when you're feeling inspired in math class. I just wrapped up a graduate course about utilizing engineering and engineering practices in the STEM classroom and this phase was very much inspired by that. It also helped serve as my final exam project, so should out to my students for writing at least half my paper for me.

I told the students that their objective was to find the instantaneous velocity of a marble on a track using only items found in the classroom. No downloading a radar gun app on their phones. No running to the physics lab to grab a photogate.

Groups took all different approaches, some of which suprised me! I knew many would time the whole length of the track and soon realize that the smaller their interval, the more accurate their approximation should be. Some tried to control for the initial velocity by creating ramps, using the acceleration due to gravity to help them calculate. They really got creative with their reasoning, worked well together, and gave each other great feedback. 

I used a BeeSpi to determine the actual velocity of the marble and we found a percent error to determine how accurate groups were in their measured value. And basically, the coolest thing happened. Since we used the engineering design process- which uses iteration of trial, redesign, feedback, etc- the students continually tweaked their process. And each and every group independently determined that you wanted an interval as tight to the actual BeeSpi as possible. We wanted the size of this interval to approach 0. Without that, we'll never have the most accurate prediction. Couldn't have lobbed it in for me any better the next day to build the idea of the limit definition.

If you're interested in the activity and actual documentation of my kid's work, check out the whole document here:

Phase II: Emphasize Average Rate of Change

Here's the document I used for notes and partner tasks: 

I wanted kids to understand not only how to calculate slope and it's meaning (duh, they've been doing that forever), but I wanted them to really examine the structure. Here's what the filled in notes looked like:

I also wrote a matching activity here where students would match the structure to the interval and function to the solution, but I wrote it at 11 pm and when I opened it the next morning I realized it made no sense so we will be revising that for next year :)

Phase III: Moving from Average to Instantaneous

These notes and activities are found in the notes above. 

We talked about how the slopes of these secant lines (average rates of change) approached the slope of tangent line (instantaneous rate of change) and the groups chatted about how this all related to what we did the day before. Smaller interval, more accurate prediction. 

Partners then worked through this exploration (adapted from Calculus God Mr. Korpi ), showing the impact on the slope of the secant line as the size of the interval approaches 0. Students were able to articulate this easily and knew the slope was approaching 1. (They also had a good debate over what the word astute meant, so add that to the list of things they learned today). The very last question took some prompting and we never got to a formula, but the idea you see written here is a good summary of the conversation we had. They made sense of this themselves, I just wrote it down.

Then, we went to the big momma. I made her a few years ago and I love her. She builds from the middle school understanding of slope to the limit definition of the derivative. 

The kids walked out today and could explain this in their own words, could write the limit definition for a given function, and could identify the function and x coordinate from a "disguised derivative" given to them. All in all, I'd call that a big win. 

Monday, October 1, 2018

Google Forms Pre-Assessment

My non-AP Calculus class is about to start our limits unit and I wanted a better picture of what they remember from last year to inform how I lay out the unit. A little background: my kids are coming from a variety of places including Pre-Calc Honors, Pre-Calc Advanced Topics (an advanced functions class- the "non-honors" version for our school), crash course community college Pre-Calc over the summer, and some even from Algebra II. I wanted to try something new to get feedback from kids that would also mean I didn't have to wait until I saw them next, so I threw together a Google Form using my answer key. 

For each question, I posted my full solution to the problem, then asked students to tell me if they got it correct or not, how confident they felt on the skill, and then an extra space for comments. Comments were optional, but kids have been sharing some interesting feedback on there. Here are some screenshots of the form:

It's been most interesting to see the correlation (or lack there of) between kids getting a question correct and their confidence. It's helpful to see that kids got something correct, but still didn't feel confident in it....a nice insight outside of just what percent of kids "knew it." 

I shared a link on the handout and told kids they could only receive credit if they filled out the form, so there's incentive to actually do it. I'm using it to plan out where students will need support and where we can move faster than I'd expected. 

Definitely a trial run, but so far so good! 

Tuesday, September 25, 2018

Unit Circle & Exact Value of Trig Functions Review Activities

The Unit Circle is my jam. Last year, I even went to a tattoo parlor and asked them to slap one on my forearm (to which they replied that I'd need to do it much bigger if I wanted the detail I that saga continues). When I taught Pre-Calc, it was truly my favorite week of the entire year. But now that I'm teaching all things Calculus, there's less time for the beauty and elegance and just a short window for rapid fire review. My AP students are assumed to know it backwards and forwards. My school level Calculus students needed a bit more work with it based on my pre-assessments, so I did a few new things to practice with them! 

Unit Circle Hula Hoop Puzzle

I gave each group a hula hoop and had them align it with the floor tiles to form a set of axis. From there, each group was given cards with all radian measures and all coordinates of the unit circle. 

You could go even further with this and include degrees, but I wanted to make sure the students were starting to think in radians from very early on in the year. I figured I would give them 2 minutes to try to label everything, but this turned itself into a 15 to 20 minute activity with an amazing debrief. 

As I walked around the room, I noticed some great strategies. I took note of them for next year and I want to create some kind of guiding questions displayed to lead our conversation. Here's where the convo wound up going today:

  • Where did your group start? 
    • Students started with quadrantal angles (which was a word they couldn't tell me before this convo, so glad it came up), then divided from there
    • They were able to check themselves by thinking about the order the radians should go around the circle. I saw one group specifically calling each other out for putting ⅚π in the 3rd quadrant because it clearly had to be less than π. The idea made sense and I saw a few kiddos who'd clearly just tried to memorize their way through this in Pre-Calc have the logic behind it click. This took a huge working knowledge of fractions which is somehow still a struggle for kids who've successfully made it to Calculus. 
  • What did you do when you got stuck?
    • For my group, this seemed to be on the coordinates. The angles took some discussion, but they were able to reason through it together. The coordinates were a different ballgame.
    • Students were at least able to sort the coordinate into quadrants. Many were able to think about the reflections that take place to make angles with the same reference angles have related coordinates. All of these ideas were integral to where we'd go after this- reviewing the circle and how to use it. 

Exact Trig Values Speed Dating

After a brief review and a few practice problems, we were ready to practice! Instead of just a worksheet or a Kahoot and trying to get my first block to wake up already, I decided to make them start wandering. No pre-planning required...this one was easy to wing! 

1) Get a whiteboard & marker
2) Find a random partner
3) Answer the random exact value question Mrs. G puts on the board with your partner
4) Boards up!!
5) Class Discussion (if necessary)
6) Find a new partner and repeat

This not only gave me the chance to get kids working and talking, but I liked that I could go over each question and check in with kids I saw struggling. 

I have a whole library of other activities I've done with Pre-Calc classes in the past, but I really liked these for a group that only has 1 day to review all of this!