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:

Quiz Quiz Trade

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! 

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. 

Recursion Notation Challenge

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!! 

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.

Inverses of Exponentials and Logs Circuit

Steal away! 

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:

Discovering the Derivative With Average Rate of Change Lab

Phase II: Emphasize Average Rate of Change

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

Average and Instantaneous Rate of Change Notes

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.