Wednesday, January 25, 2017

Joy Ride: An Introduction to Riemann Sums

This year, I tweaked the order in which I teach integration a lot. Since we begin to talk about differentials when we talk about linear approximation in the applications of differentiation unit, it seemed natural to me to flow directly into differential equations and indefinite integration. Instead of leading with the area problem, we worked on general antiderivatives and differential equations. Since my students are now familiar with moving between functions and their antiderivatives, I am now starting to move into area applications to introduce definite integrals. They are starting to ask about "going backwards" from acceleration and velocity naturally, instead of having to force the area problem somewhere that it seems unnatural. This will also give me time to re-visit u-substitution since I got to teach it without having to worry about changing bounds in our indefinite integration unit. 

For the last 3 years, I've taught in an entirely project based program. The switch back to regular ed has been an interesting one for me, as I can see the ways my pedagogy has changed but don't always have the time, resources, or freedom to implement these changes. I knew when I was given the opportunity to have a 2 hour block for a project instead of giving my AP Calculus kids a midterm that I wanted to jump on the opportunity. I adapted this Gorilla Jump activity from MAA (which is an amazing activity if you've never seen it!) into a project that would require more data collection. It was a real challenge for the kids, but they seemed to walk away with the big idea and were asking the right questions (even if they didn't have all the answers yet). 

Students worked in small groups using this Driving Simulator (made on Scratch from MIT) to generate velocity data over even intervals. 

They were free to decide what intervals to use and needed to be mindful of units as most were measuring in seconds while velocity was in miles per hour. This simulator also has weather and varies the time of day, so I had my kids react to these variables so it would affect their velocity. Some drove responsibly at 68 mph on the highway. Some just accelerated as fast as possible the whole time.  I know who to watch out for in the school parking lot now. 

Using that data, students generated estimates for total distance travelled using the lowest velocity on the interval, the highest velocity on the interval, and the average velocity on the interval. They were asked to represent these estimates graphically, which naturally leads to a rough version of a Riemann Sum. In addition, I asked them to challenge themselves to see if they could write an equation (using sigma notation if they were feeling extra fancy) to represent how to generally find the total distance travelled. This was frustrating for the kids, for sure. I used it mostly as a pre-assessment to see what they remember from the previous year on sigma notation. (Verdict: we've got some serious work to do there). They were able to discuss upper and lower bound and postulated that collecting data over smaller intervals would lead to more accurate results. All in all, it led to extremely positive conversations and I think they'll have a very solid foundation when we attach formal names and notation to these ideas next class. 

A few notes of things that jumped out at me, having implemented it once: 

  • Since units were in mph and sec, there was converting needed. The kids didn't struggle with this, but it caused the outputs to be extremely small decimals for some groups. Just something to keep in mind.
  • A lot of kids discussed ways you could have changed your driving to make your estimate more accurate instead of changing your data collecting methods. While a valuable conversation, some kids got lost on a tangent here and struggled to finish in the allotted time. 
  • Since there was no requirement that their velocity function be monotonic, using the lowest or highest velocity on the interval did not lead directly to a right or left hand Riemann sum. I know that's something that won't be a hard jump for my kids next class, but again it's something to keep in mind. 
  • The kids got really cranky having to make 3 of the same graphs by hand. If you have a way to photocopy them quickly, you'd have a lot fewer cranky 17 year olds on your hands. We didn't at the time (I wasn't in my classroom and we had limited time). 
  • The car has an odometer. It would've been interesting to copy down the actual mileage of the trip and see how far off we were, especially since we're talking about error. Missed opportunity, Gironda. 
  • There's got to be a cooler final product for this. Again, I had a limited time from in which to do this with my kids, so posters were concise and manageable. I know this could get pushed to be a lot better. 
Let me know what else you might do to make this better. I liked it enough and it was worth doing, but would like to improve it for next year! I'll update with finished products from the kids when I get them! 

Tuesday, January 24, 2017

Blended Learning: Lessons Learned

One of my biggest regret after moving districts this year was that I never sat down and put all the lessons I'd learned as a blended learning teacher together in one place. Luckily for me, at the end of last year I got to sit down with the people at Opportunity Culture, the group that had been instrumental in helping my district move towards blended learning. They put together this vignette summarizing my experiences and my best advice to new blended learning teachers. 

Pioneering Blended-Learning Teachers Reach More Students Vignette Series
A vignette written about my experiences with blended learning in Pre-Calculus. If you want my lessons learned (often the hard way), read this! 

I'm so thankful to have had this experience and to have had it documented! 

Hey! I said that! 

Thursday, January 19, 2017

Choosing a Method of Integration

One of the biggest difficulties my students face when we work through our integration unit is distinguishing between the different techniques we've learned. When told which approach to take, they can nail almost any integral. When given a mixed bag of problems to sort through on their own, the tides start to turn. I tried to spiral in a bit more mixed practice than I did last year, but it still didn't feel like quite a enough this year.

I designed this to have kids work on in phases-
1) Individually evaluate which method you would use (gut instinct, what do you think?)
2) Swap papers with a partner and say whether you agree or disagree and be ready to argue why
3) Work with your partner to try to decide on who is right. Check yourself by evaluating the integral
4) Generate a list of what features helped you identify which method to use!

I wish I'd left more time to do it in class. I will definitely budget more for it next year. 

Any other favorite activities for helping kids with this?

Thursday, January 5, 2017

New Year, New Blog [Re-Blog]

Re-blogging this from Julie Reulbach on the ExploreMTBoS site! I'm excited to participate and encourage anyone who has thought about blogging to jump in for 2017! It will totally change your teaching! 

Welcome to the Explore the MTBoS 2017 Blogging Initiative!

With the start of a new year, there is no better time to start a new blog!  For those of you who have blogs, it is also the perfect time to get inspired to write again!
Please join us to participate in this years blogging initiative!  To join, all you need to do is write just one post a week for the next four weeks.  To make it easier for you, we will post a new prompt every Sunday!  Once you have blogged, please fill out the form below.  Each week, your blogs will be posted on this site for all to enjoy!
This Week’s Theme:  My Favorites
This week, the blogging theme will be “My Favorites”, where you can post about one (or many) of your favorite things!  Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well.  It does not have to be a lesson, but can be anything in teaching that you love!  It can also be something that you have blogged or tweeted about before.  Some ideas of favorites that have been shared are:
  • A lesson (or part of one) that went great
  • A game your students love to play
  • A fun and/or effective way to practice facts
  • A website or app you love to use in class
  • An organizational trick or tip that has been life changing
  • A product that you use in your classroom that you can’t live without!
Blog Newbies!
If you are brand new to blogging, you can read Starting A Blog from the 2015 initiative.  This post will give you specific instructions on how to start a blog.
Hot Tip!  Don’t stress about your blog name!
The hardest part about blogging is often coming up with a title.  Do not let this detail derail you!  A great suggestion is to make your blog address your name.  Then, you can title your blog later – or change the title anytime you want!  To see what this looks like, check out Sam Shah’s blog.  His web address is, but the site name is “Continuous Everywhere But Differentiable Nowhere“.  No one cares about your blog name, they just want to read interesting, inspiring, and helpful posts!
Hashtag it!  #MTBoS #MtbosBlogsplosion 
Don’t forget to tweet out your blog link and add hashtags so other teachers in the MTBoS community can easily find your post!  If you are not tweeting yet, you should be!  There is an amazing community of math educators there just waiting to inspire and support you!  Check out How To Start a Twitter Account to get started!  Also, if you are brand new to Twitter or just want to get more out of it, there are more Twitter tips on Julie Reulbach’s blogpost, Tweet, Connect, Repeat.
This year, we are joining up with the #mtbosblogsplosion.  Special thanks to  Carl Oliver@carloliwitter, for jump starting blogging for many people in our community!
Hashtags to add to your tweets:  #MTBoS #MtbosBlogsplosion
Also, if you have a wordpress blog, please re-blog this post to get the word out!
Deadline: Press submit by the end of the day Saturday, January 7, 2017.
Yes, this is a quick turn around this week – but we don’t want you to put it off or delay!  Once you are finished with your blog post, fill out this form and your blog post will be featured on this site [meaning the MTBoS site this is reblogged from] next week!

Wednesday, January 4, 2017

Indefinite Integration with U Substitution

Happy 2017! 

So much changed in 2016, but one of the things I'm most happy about is that my blogging habits changed! I've always been an #MTBoS lurker, but finally getting more involved in blogging has helped shape me so much as a teacher and I'm looking forward to even more sharing and collaboration this year. Let's just say I'm on an upward trend...
I was happy to see my kiddos on the first day back from break, but watching them attempt u-substitution felt like watching this: 

To be expected, I know. We started u subs 2 classes before break and on the class before break I had a whopping 6 students show up (we went right up until 2:30 pm on December 23rd....most of my kids were off skiing or sleeping until well after my class was over). Knowing that we would be struggling to get back into the groove today, I wanted to come up with an activity that would include checkpoints along the way to encourage students to analyze their work throughout the process, not just at the very end. Cue this awesome activity from Tatia Totorica on TeachersPayTeachers on U Substitution that is FREE! Unfortunately, it used definite integrals and we aren't quite there yet. However, I loved the idea and decided to adapt to make it my own. 

U Substitution 4 Color Activity (Indefinite Integrals)

At it's core, this is a matching activity. Print it out on different color papers and have kids work in partners to find the pieces that correspond with each other. They also work to fill in the columns on their worksheet, which requires them to show their work in the appropriate columns. I liked that the weren't easily matched...they required some thinking and analysis. By the end of class, the students were cruising through these with their partners. 

Happy Integrating!