Category Archives: Lessons

Polynomial Division: An #OpenMiddle Problem

In Precalculus, we’ve just wrapped up our work with polynomials – including long division. I was definitely not looking forward to this topic – I haven’t taught it before (or been taught it formally) and the #mtbos had me a little bit.

On our practice set, at the bottom I added this problem as an extension for students who completed the traditional practice set:


I got some interesting responses that made me pretty pleased- but I didn’t (and haven’t) check these for accuracy.

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What I didn’t want to do verify all of them….so instead I had students do it! I typed up the questions onto a worksheet and passed them out to class the next day. Their assignment: Yesterday students created these problems and claimed there was not a remainder. Choose four problems and determine if the students are correct. This was a solid move – I didn’t have to check all the work and students got more practice.


Next time I need to wrap our work back together before assessment. Most students did really well on the assessment but closure could have brought this activity to the next level. We also could have given feedback on how to improve their work (One thing you did wrong was… etc.). I will also include more time for all students to create a problem to feel more invested in the work we did on the second day of instruction.


Free-Response Question Fridays

Very few tests are well designed, but I believe the AP Calculus exam is a pretty well designed and consistent exam. One thing that amazes me is that in 6 Free response questions (FRQs) they test a majority of a course. While that is great for the test, it isn’t good for teaching. It makes it hard to use these items in class in the fall continuously, but I want students be familiar with the format and types of questions asked way before they could confidently complete an entire AP Question. I’ve watched my previous iterations of Calculus struggle with the FRQs, but never implemented a system to “train them” because I always started in the middle of the year. This year, my first full academic year of AP Calculus, I’ll be implementing FRQF’s -> Free Response Question Fridays to address that gap.

Every Friday, students will work on an actual AP FRQ (even if they aren’t computing answers to all the parts). I casually had read of Polya’s How to Solve it and his 4 step problem solving strategy on the #MTBoS, but hadn’t implemented it rigorously. I sat down this week and read it (well…I read the first two sections and skimmed the dictionary….). I planned on implementing Sarah Carter’s S.O.A.R. acronym – which is how I’ll be introducing it to kiddos in August. (P.S. I made some nifty INB notes for SOAR last year, hung a kite in my room…and never referenced it again after September. Hence the deliberate planning this time around).

Speaking of giving credit to where credit is due, in my internet search to find something like an AP Calc version of this AP Stats FRAPPY resource, I found this problem solving guide by Florida DOE “Research Based Strategies for Problem Solving in Mathematics”  (PDF). They break down all four parts of the process and have activities and prompts for each of the areas. I’ve taken some of their ideas as I’ve written my FRQF’s.

I’ve written the first month or so of my FRQF’s – focusing mostly on Polya’s first step in the process – understanding the problem. The first involves mostly just ensuring that students know what the problem is asking and understanding the situation. We’ll be breaking down this year’s ridiculous Volume problem (#5). Problem not included, because College Board be CRAZY.

Students are answering questions like these:

  1. Summarize what this problem is about in your own words.
  2. What do the variables r and h stand for?
  3. What are the units to describe this funnel?
  4. What is the height? What is the radius?
  5. What are the restrictions on the values for h?
  6. What is the shape of the cross sections of the funnel?

I’m hoping that by focusing on the thought process behind answering questions like these by May we’ll be able to let our knowledge shine instead of being bogged down by words.


P.S. Interested in looking at FRQF’s? Comment or tweet at me (@jakewinfield) and I’ll share my work. I’d love feedback too 🙂 Since I’m using AP Released Items I’m erring on the side of caution.

Deriving Laws: Sine/Cosine

After our NCTM illuminations lab for the law of sines, we worked on practicing these. We spent one day solving for side lengths, the next reviewing finding angles. Then students somehow thought these were different procedures….I’m not sure how I could have made that clearer but having this entire course better mapped out & concise is probably the main cause.

Since the Illuminations inquiry lab went well for the law of Sines, I decided to try the lab for cosines. It didn’t work so well – it involved a lot more of me walking them through finding the law of cosines instead of them being able to find it on their own. I ditched it after one period and just gave a formula. While it isn’t the proudest moment, I’m definitely proud that all my students could use it correctly. We’re wrapping up this all tomorrow – then its on to other things 😀

Law of Sines: Investigation

I’ve recently gone on an Illuminations kick. NCTM provides solid resources for my Pre-Calculus kiddos and I’ve used them twice in the two weeks I’ve been at school.

Today we did this activity on investigating the law of sines.  I’m incredibly impressed with how well it went, and would recommend it. I strongly believe in inquiry in class and this was a wonderful scaffold for me and the students.

Every student in both periods was SILENTLY engaged working on finding the solutions to the second half of the packet. Students typically struggle  when proving general cases, but this scaffolding and being facilitated as a whole class for the first 6 questions was incredibly helpful.  It felt like a huge victory that students could manipulate and solve problems like these. I haven’t seen that much productive struggle in those classes in a while which was amazing.

One student asked why I had them do the activity instead of just being told the law. I explained that just writing something down won’t help her learn it or apply it correctly. I certainly have a better understanding of where the law of sines comes from after today than before.

Tomorrow we’ll be learning and practicing using our new discovery. I’m excited to see what happens. 🙂

Water Rockets!

Last week was spring break and to celebrate the day before break we ended a unit on quadratics by launching water-bottles into the air and creating quadratic equations for them.  I couldn’t find any resources online for what I wanted us to do, so I made this day long (90 minute) project.


– 2 liter bottles, filled a third of the way with water

– Launch set up.  Its a tube, one way valve, stopper and a gadget to hold up the bottle. (Our science department had a set up I borrowed for the week).

– Bike pump to pressurize the bottles.

– iPad to video the launches

– Timers & Clipboards for groups

Our Packet:

I created a packet for this project to guide student work so I didn’t have to walk students through every part.  We worked on parts 1-4 with part 5 as extra credit.  No one was able to get to part 6 :/

To keep everyone (especially those busy-body eighth graders) engaged, everyone had a job.

1. A videographer to video tape launches with my iPad so we can examine them together in class.

2. A recorder with their packet on their clip board

3. Everyone else was a timer to share with their clip board.  I didn’t have enough timers so some students were on their phones.

Rocket time!

I added pressure to the bottles with the bike pump, students pulled the trigger and launched them into the air.  Most of the time was us having fun and preparing rockets.  🙂

Once we came back inside, students worked together to complete their packet and finding the initial velocity.  Every group ended up with different times so each answer was different.  I liked this because I know that those students were engaged with their own work instead of just getting other people’s answers.  Students needed a lot more help working on part 4 (finding the initial velocity) than I expected which was a challenge.


– A highly engaging activity to work on before break – a lot better math than a Disney movie I’ve done in the past.
– Students were engaged in applying mathematics to the real world.

– If nothing else, I had an opportunity to build relationships with my students which makes the entire day worth it.  Some students that struggle to do work were EXCITED to help out to make the activity happen.  Oh and some students with the “videographer” job were GOLD.  I now have videos that will make me smile any day 🙂

Things I’d change next time:

– All of my students had difficulty with the fourth part – finding their velocity and creating their equation.  I’d recreate that or find a way for it to be less confusing for students.  Also, no one even saw part six – the extension activities.

– Some groups weren’t clear that EVERYONE had a job.  When I made that clearer, the results were better (this is just a lesson for me about maintaining clear expectations always).

– The project could have ended with a LOT more comparison to the original predictions.  It didn’t go full circle and doing so could have enriched student’s mathematical thinking.

Final thoughts: Once again, I see potential for the next time I do this – especially right before a break as a summative work.  I’m pleased with what my students and I accomplished 🙂

p.s. Look out for a video recap – there’ll be a highlight reel coming soon (hopefully by the end of the week if I can figure out how to edit videos).

Projectile Motion: Barbie Drop

This is my second activity on projectile motion.  The first activity (on the missing airplane) is here.

Students came into class and were greeted by Barbie up close to the ceiling.  It took between 10 seconds and 5 minutes for a student to find Barbie.  One student immediately knew what we were going to do:

Mr. Winfield – Are we going to drop Barbie?

Yes, yes we are!  See, I’ve always wanted to do NCTM’s Barbie Bungee but I’ve never fit it into my scatterplot lessons.  So instead, I decided – let’s change the math and DROP her!

Why should we drop Mermaid Barbie?  Students were immediately able to tell me we could figure out how long it would take to hit the ground using the math we know.

We began with estimating her height and her drop time.  One student who is over 6 feet tall became our human ruler which was rather ingenious of them.  Then, instead of measuring, I gave them the accurate answer for Barbie’s height.  I missed out on another great opportunity to get students involved in every part of the lesson.  Students then created their formula and found how long  it would take for barbie to fall (0.733 seconds)

Now, the math isn’t real unless you drop Barbie.  We brainstormed roles we needed – timers, droppers and I wanted a videographer.  One person from each group got to drop Barbie.  Everyone else who had a phone was timing Barbie’s fall and one person was using my smart board.  I was in Google Docs making a spread sheet of times.

After a dozen drops between the two classes who were able to do the activity we had an average of 0.68 seconds and I deleted some of the bigger outliers before doing the math.  I’d say we were pretty close to Barbie’s predicted drop time.

We were able to end our discussion with possible reasons for error in our results.  Once again, I could have pushed deeper.

Next time, I’ll be better prepared.  I see this as a HUGE opportunity to push the math, understanding and application of the quadratic formula.  Next time around it’ll be better – guaranteed.

What went well:

+ So many students were ready to drop Barbie and add to our data set.

+ Some people botched their drops so we were able to decide if we should keep some of the messed up attempts and WHY.

+ Students were talking about math class in other classes – what more could you want!

What I’d Change:

– Everyone got the same answer for their prediction.

– I could have pushed more work on students – averaging, prediction, measuring.

– I forgot my iPad that day so I don’t have a video. A student recorded in both classes, but I’ve yet to get the videos :/

Barbie is still hanging out in her “Bath tub” 8.625 feet in the air.  I’ll post a picture soon.

Projectile Motion: Flight 370

Last week we had wrapped up quadratic functions, and struggled through the quadratic formula.  I decided to spend some time on formula for projectile motion to review the quadratic formula and couldn’t find great problems or activities. So I created these two activities.

1. Malaysian Airlines Missing Flight

2. Barbie Drop

I’m going to split these into two posts to share them with y’all.

The world, including myself, has been captivated by the missing plane, Malaysia Airlines Flight 370.  I see the situation full of math – just look at this infographic from the Washington Post.  Some of this math was an almost perfect fit for what we were doing and I couldn’t pass it up.  We had already done little work with projectiles so my students were familiar with the formula.  Our question: How long it would take for the plane to fall to the ground?

We began by watching the first minute of this video and predicting what may have happened to the plane:

We worked through this worksheet.  For problems 1 and 2a, I insisted that the only wrong answer was no answer, which helped boost student confidence.  We broke it down and answered question by question so everyone could work through it.  Students did the heavy lifting practicing the quadratic formula and were working with a purpose.  They figured out what else we needed to create our equation and worked away.

What worked:

+ One class worked non-stop.  They had INTENSE discussions about what could have happened.

+ My Pre-AP students were introduced to projectile motion with this problem.  Its hard not to be hooked with 230 people missing.

+We were successful using big numbers in a complex formula 🙂

+ Students still want to know where the plane is – its incredibly engaging.

What I’d change next time:

– Some classes weren’t engaged with this problem.  Those classes also the ones that were introduced to the problem the day before (when I hadn’t yet found the video).

– I simplified the problem a bit and eliminated the glide opportunity.  That’s okay, but I didn’t tell my students how or why I did that – I could have forecasted to their future science and math classes.

– The worksheet needs to be better organized and the questions from number 2 split up.

– With a little more information and prep, this could have been done with peers instead of walked through. I’m okay with how it turned out because it was a review of the day before to prep for our second activity.