Tuesday, August 8, 2017

WOBD Limits #3

I feel like this needs a tweak for some reason. Letter D seems a little...forced?....to me. Anyone have any feedback that might help? I will update if anyone has any great ideas! 

A- Taking limit at a real number x value(not negative infinity), limit is one sided, limit is taken at an infinite discontinuity
B- Function is continuous, does not have an infinite discontinuity
C- Limit exists 
D- Limit approaches positive infinity **This is the one I want to make better. Ideas?

Friday, August 4, 2017

WODB- Limits #2

Here's the 2nd in the Limits WODB series. I am open to ideas and feedback on any of these, so please feel free to suggest improvements! 

A- Can be evaluated with direct substitution
B- Limit exists at a point where function is undefined
C-  Limit does not equal defined value of function, not as x approaches 2, is a graph
D- Limit does not equal 5, Limit DNE

NYC Math Lab (Alternate Title: Elementary Teachers are Superheroes)

Real talk: one of the joys of teaching is summer....time to step back from the day to day grind and reflect on the year behind us, while looking to the year ahead. I make it my mission every summer to get outside my comfort zone for at least one major challenge. Know what's outside my comfort zone? 

10 year olds. 

Most of my best friends are elementary school teachers and I genuinely believe they are superheroes. I hug crying kids who didn't get into their first choice college or whose boyfriend broke up with them and I live through the day in and day out struggles with factoring, but they teach little humans how to read....how to share....how to exist in regular society. They work with the same group of kids all day (no bell to save the day when little Johnny is having a rough one and you're ready for him to go to history now kthanksbye) and they have to be masters of every subject. They are magicians of classroom management and engagement.  They still clean up bathroom accidents (and they do it with a smile). Seriously....superheroes. I've always known I have a lot to learn from them and NYC Math Lab was an amazing opportunity to do so. 

The Math Lab is the most unique professional development I've ever experienced. Each day, the teachers in attendance were able to observe instruction for a group of rising 5th graders in a "fish bowl" environment and then begin to work individually with a student as the week went on. Afternoons were spent analyzing the lesson and sharing ideas while looking ahead to what would be best for kids as the week progressed. It was a collection of people there to learn from each other- the students, the lab instructors, the participants. Everyone was working towards a common goal of students' individual conceptual understanding and sense-making and that made it feel like a true professional learning community. These weren't hypothetical kids....they were our kids. They had funny shirts and favorite books and bad days and amazing successes.  

While the content was far below the level I teach daily, I learned so much and have so many new routines I want to try in my classroom this year. The thoughtful way you must present complex ideas to young minds to help them understand is a lesson for all of us and it explained so much of why some of my students struggle to build on a shaky foundation. My biggest takeaways were:

A Math Community is Built Intentionally
I have learned more and more each year how much I believe that I teach students, not math. I love the rapport I have with my kids, but I have been trying to find new ways to build student confidence, voice, and engagement in my room. One of the key focuses of the math lab is building a mathematical community. Much of this community centered around the Triad of Responsibility, which had 3 key components:

  • Responsibility to Self
  • Responsibility to Partner
  • Responsibility to Community
This frame of reference for students to participate in the community gave all students power- not just those who understood the fastest, but also those who needed to most intervention. Praise was never given for being right; it was given for being an active participant in the community. 

One of my favorite moments came when a student, who was much younger than others in the room, felt pushed beyond what he was ready to think about that morning. I saw him getting teary as he spoke to the other instructor and I asked if he wanted to take a break in the hallway. Once we got outside, he told me how he felt overwhelmed and we talked about how proud I was of his perseverance and hard work. Just then, another instructor walked over behind me and pointed how what a huge contributor he'd be to the math community's discussions and that the community needed him and needed his ideas. He agreed to head back in and within 15 minutes was up at the board explaining his idea to the group. His exit slip that day explained how he had loved "being brave" and would be even braver tomorrow. He felt important, included, and valuable and he saw growth because of that throughout the week. 

It is the students job to make sense for themselves
As teachers, we put a lot of the responsibility for students' learning on ourselves. I think it's in our nature...we love our kids, care about their success, and are judged by the world around us by a test score. I know students need to understand conceptually to have a truly sound foundation on which they can build their knowledge and I try to achieve this in my planning, but so often in the craziness of the school year I wind up thinking "What else can I do for them?" instead of "How else can I give them an opportunity to make sense of this?" Too often it comes down to time. But I'm realizing the further I get into my career, the more I need to slow down and dedicate time to that sense-making for THEM. I can't make sense of it for them, I can only create favorable conditions for their success. What we spend time doing in class is a good indication to our kids of what we value, so I am working on learning to slow down and show kids that I truly value their understanding. 

Students were encouraged to talk to each other in a very methodical way. Instructors never said anything a student could say, often just reframing or encouraging points and debates among students. If it was clear a student has a question, the instructor would instruct the student to ask "What do you mean by that?" or "Could you explain ____ more?" Since it was up to the student to understand, it was up to the student to ask. They were learning responsibility for their own level of understanding. This is cliff notes from another great resource that we discussed, which is high on my reading list now. 

Sometimes the Best Thing You Can Say is Nothing
Before the students walked in the first day, we were specifically asked to not interact with the students on the first day. Observing in a "fish bowl" environment means you can't interrupt...you're a fly on the wall.  I challenged myself throughout the week to really listen to what the students were saying, something I know I don't always take the one on one time necessary to do well. It's hard to listen to students talk on and on at you about a misconception, but we worked the whole week from a place of "You know SOMETHING. Let's build on that." Often this meant inching backwards in the progression of understanding until you found something that you and the student could both meaningfully agree upon and using that as a building block for your conversation. I don't think in the week I was there we came to the right answer of how to implement this given the time constraints you have when covering a whole curriculum in a classroom of 25 kids and 1 teacher, especially given the confines a state test places upon the teacher. However, it made me deeply reflect on how much I assume when I talk to my students daily. They may say something that isn't quite right, but it sounds close and I let it go for the sake of time.....building a misconception that we might have been able to nip in the bud then and there. Below is the document we were given on "Nudges" and how to consult meaningfully with students one on one while not giving them too much (with some of my notes added for extra seasoning). 

Since this wasn't my curriculum, these weren't students in classroom, and there wasn't the pressure of a test, I was truly able to treat this experience as a "lab"....a scenario where I could experiment with my own practice and analyze the impact on a student's understanding. I saw a difference. 

I am taking so much from the experience into my planning for the upcoming school year and I am so grateful to have had the opportunity to reflect, observe, and learn with these other teachers and students. Elementary teaching is, much like all teaching, both an art and a science and there is so much to be learned from seeing where my kids start before they ever step foot in my room. My gaze has shifted and I know I'll be doing more elementary PD as the years progress. So to all the participants from that high school teacher from upstate that seems a little out of place in the intense 3rd/4th/5th grade conversation- thank you. Being out of place is my favorite way to grow. 

Thursday, August 3, 2017

WODB- Limits #1

Taking July off from blogging has been extremely restorative after a long school year and I'm ready to get back in action! I'm starting to create a series of WOBDs for warm ups in AP Calculus and am looking first at my limits unit. This one is just a start, but it's getting my blog moving back in a productive direction as August gets underway! MTBOSBlaugust was pretty powerful for me, so while I may not post daily, I'm trying to get back on the wagon this summer! 

I want my kids thinking about both continuity and limits in this one, so I've tried to vary the reason I see things "don't belong." These are just my thoughts, but we know the kids will always find more:  
A-  Continuous
B- Removable Discontinuity
C- Limit Does Not Exist at x=1
D- Does Not Have Y Intercept of 2 OR Limit DNE at x=0

More to come, but wanted to get this blog train back on track! 

Saturday, June 3, 2017

Integration by Parts Circuit

Let's face it....it's hard to keep kiddos engaged after the AP exam. Especially when they already took their final, know they got a "5" on it, and are starting to hit up the graduation party circuit. Luckily, I have great kids who are buying in to the whole "If I learn this now it might make my first week of college a little less stressful." (Huge shoutout to my former student rocking his engineering major at NC State whose email from the beginning of this year I've been able to show them!)

After learning Integration by Parts, I wanted to give the kids something self checking that they could work on with peers for practice, so I created this quick little circuit. I ended the lesson by having one student put a question on the board where tabular was appropriate and one student put a question where it wasn't and we talked more about why they made the decision they did. 

Feel free to use as is or modify! I got all the problems from this Kuta worksheet.  It took the kids about 15-20 minutes to complete. 

Wednesday, May 24, 2017

Standards Based Review in AP Calculus- Student Feedback

After reading through 50 thoughtful and heartfelt course evaluations from my kiddos, I have pulled out the major themes from the section on standards based review. Overall, the reactions were very positive. These were their favorite parts:

  • Liked opportunity to improve grade
  • Pushed kids to study what they didn't know, not just what felt easy
  • Getting a "4" on a topic built confidence for the exam on that topic
  • Loved having to take them "cold"- gave a real picture of current understanding without stress of low score
  • Gave accurate portrayal of what AP questions would be like
  • Enabled you to correct your mistakes and feel okay being wrong/asking questions

As expected, it wasn't all sunshine and daisies....the kids has some (mostly) constructive feedback on things they didn't like too:
  • Didn't like having to remediate to retake; just wanted to be able to retake
  • Deadlines for retakes stressed some kids out since we were doing multiple per week
  • Frustrating for students who understand concepts but make small algebraic or arithmetic errors and don't get the "4" they want so badly
  • Wanted time in class to retake since finding time outside class can be difficult
  • "Annoying" (Such helpful feedback, I know)
  • Hard to find motivation with senioritis kicking in big time

Overall, I thought it was a success and will definitely be using this strategy again in the future. I was able to see a huge amount of growth in the students that took it seriously and we were able to comb through misconceptions with a fine-toothed comb. I was able to grade extremely critically since students were striving for a perfect "4" instead of settling for a 95% or 97% and not really examining what they did wrong. I whole-heartedly agree that it was a lot condensed into a small time period and the deadlines were constrictive for some students and I'll be adjusting for some of that next year. 

The Google Form I used as a sign up was an absolute must - I could check it daily and was able to track student data through it, as shown here:

Overall, kids took 82 retakes on the 9 quick checks. Most kids said they wish they'd done more. And the growth mindset message seems to be getting through:

Can't believe we're so close to the end! 

Friday, April 28, 2017

Math Teachers at Play Blog Carnival #107

The Math Teachers at Play (MTaP) blog carnival is a monthly collection of tips, tidbits, games, and activities for students and teachers of preschool through pre-college mathematics. We welcome entries from parents, students, teachers, homeschoolers, and just plain folks. If you like to learn new things and play around with ideas, you are sure to find something of interest.
I'm so excited to be hosting this edition of the MTaP Blog Carnival at Give Me a Sine! If this is your first carnival, welcome! This is a great way to find some new bloggers you'll love and even share your ideas in the future! I am a high school teacher, so it has been particularly awesome for me to get to explore so many new middle and elementary level blogs as I prepared to host this month. 

Want more information about the MTaP Blog Carnival?  Want to host it at your blog?  Want to submit a favorite blog post?  Click here for more info

Since this is edition #107, let's ask the question on everyone's mind....What's so special about 107? 
  • 107 is a jackpot for prime number trivia! 
    • The 28th prime
    • A Chen prime (since 107+2 is also a prime!)
    • A safe prime (is of the form 2p+1 where p is also a prime)
  • The smallest positive integer requiring six syllables in English (if you include the "and")
  • The atomic number of bohrium
  • The "911" of Argentina and Cape Town
  • The number of legal acupuncture points
  • 33 states and the US Virgin Islands have a highway numbered 107

And now, on to the posts! 

Talking to Parents about Math Explorations
While this post contained only a copy of a letter sent home to parents after a recent Math Exploration event, I loved the message that mathematics continues outside the classroom walls. So often, kids conceptions of math are influenced by their parents prior knowledge and experiences. I love the intentional outreach to parents to embrace the math of every day life! 

Playing with symmetry in kindergarten
a) As a high school teacher, my heart basically melted here
b) I love all the different explorations the students used here to explore symmetry- from pattern blocks to mirrors to modeling with their bodies. The idea of symmetry is an integral part of my geometry and calculus classes daily, so building these intuitive understandings young is so promising! 

"Who Wants To Count My Windows?"

This great post from Joe Schwartz on working with constraints in a 5th grade classroom. He builds off the "Ant Hotel" problem to create a fun-filled learning experience for his kiddos! 

Practice Math Facts Using Your Voice!

Cool and definitely fun to play with, this tool allows you to practice math facts with your computer just using you voice. Did my husband wonder why I was yelling numbers to myself in another room? Probably. But I can see littler ones loving this for practice! Just make sure you allow it to access your microphone. 

Origami Math Game {Tutorial}

I think making cootie catchers is a right of passage in the elementary days. Crystal Wagner shares a tutorial for turning this into a math game for your kids! 

Middle School

My husband and I just closed on a house this week and I am so thankful to be a math teacher. I've gotten hit over the head with all the mathematics around me daily- even just buying the right amount of shelf liner for the kitchen or getting the right sized fire extinguisher. This post made me laugh out loud given all the high school math we've been doing and would be a great launching point to get kids talking about the geometry around them. You can also check out this one, on combinations, from the same blog: Three Sisters And Their PJ’s

Sort Students into Groups using Percents, Fractions and DecimalsThis fun activity will sort kids into groups AND get them thinking in the process. I can even see using this at the high school level- we all know fraction skills can always use a boost!
This activity draws on the proportional relationship that exists in linear functions and gets kids reasoning proportionally. This would provide a huge booster to the discussion of slope, too! 

High School

Denise Gaskins shared this fun Patty Paper Trisection activity (complete with Hints and Solutions: Patty Paper Trisection). This puzzle gets the participant thinking about how to trisect an angle, using simple tools. Straight edge and compass aren't going to help you here, folks. Give this one a try! 

MULTIPLE REPRESENTATIONS FOR TRIGONOMETRIC EQUATIONSSam Shah is one of my go-to's for quality and thought-provoking material in my upper level courses. This activity gets kids thinking conceptually about my all time favorite thing- the unit circle! Trig equations can be hard, but with the right conceptual understanding....BAM! You've got magic! 

To Whom it May Concern: Learn to Love the Why.This one is a keeper. Read the whole thing and listen to the voice clips. Kids can sometimes be our most honest and necessary feedback sources.

This isn't new, but it's been circulating in my mind for the past few weeks as we approach AP exam time. This is a Tuft's study that advocates practice testing as a shield from memory from stress. I've tried to integrate more practice testing into my review this year and I see a difference in what my kids are willing to take on. I don't have a fully formed opinion yet, but it definitely got me thinking. 

A Brief Ode to Blank PaperSometimes we give too much info to our kids. This piece by the amazing Tracy Zager (whose book is sitting on my nightstand and a must read) and advocates that by giving less, we cause kids to think more. 
General Mathiness

These 2 posts from Mike Lawler will get your wheels turning with unsolved problems and the bridge between "pop math" and real math:

This post explores linear congruential generators and how they could be a source of mathematical play. It delves into computer programming, modular arithmetic, and more! Definitely worth an exploration! 

Let me know if you had any other favorite posts of April and submit your posts for next month's carnival! Happy Math-ing! 

Monday, April 10, 2017

Call for Submissions: Math Teachers at Play Blog Carnival, April 2017!

I am so excited to be hosting the Math Teachers at Play (MTaP) blog carnival for April at Give Me a Sine! 
If you haven't heard of it, the blog carnival is a monthly collection of tips, tidbits, games, and activities for students and teachers of preschool through pre-college mathematics.
Have a post you loved from a colleague that really shaped your practice? Find a puzzle, logic problem, or other fun math-related post that caught your eye? Want to get a few more clicks on a post you are particularly proud of writing? Submit them all!  
The deets: Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up through first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.
To submit an entry, fill out this form: SUBMISSION FORM
I can't wait to see all the amazing posts you all have to share! 

Wednesday, March 29, 2017

Justifications Gallery Walk

As we get closer to the AP Exam, I'm working on helping my kids "play the game" in the way they can benefit themselves most- to be as mathematically precise in their language as possible. Although I've emphasized it all year, students tend to rely on their long term memory once you start review and that tends to be a little less precise than we might hope.

To get students thinking again about technical justifications for curve sketching today, I started with this warm up gallery walk. 

1)  Students did a silent, individual gallery walk to critique justifications (some close, some very off, maybe even some right!?). They left a post it under each one with their feedback on how to make it better.


2) Next, I grouped students and had them each work on one particular justification. Students read through the feedback and sorted it, using it to generate a new and improved justification.

3) Each group presented the feedback they'd seen on their problem and their new and improved justification to the class. This gave us an opportunity to talk as a class about what could be improved and to examine some common issues. 

After that, we did a released AP problem that required multiple justifications and I saw a huge difference in the precision of their language. It was a good jump back into more extensive mathematical writing. It could also be applied pretty easily to EVT, MVT, and IVT justifications, among other things! 

Saturday, March 25, 2017

What's New for AP Review 2017?

New Year, New Curriculum, New Review! 

I'm revamping my review system this year and trying out some changes with my kids. We have about 15 total 80 minute classes for review (assuming Mother Nature gets her act together and doesn't send us any more huge storms), which is a pretty healthy amount. I'm trying to integrate much more mixed review this year to get my kids thinking wholistically about the curriculum and am also using a standards-based approach to topical review to get them to key in on where they are struggling. Here is my basic outline so far:

Pre-Assessment (2012 Practice Exam)
I am very much against a prescribe approach to review, as every class I've ever taught has varied in their strengths and weaknesses. Moreover, I vary in my strengths and weaknesses in teaching and need to be realistic about where those are. I started this year by giving my kids a full exam right off the bat- something I shielded my kids from until well into our review last year. This is a reality check and a pre-assessment...a confidence builder or a wake up call. I did an analysis of common mistakes and will be going over these with my class when we look at the exam. 

Full Practice Test #dunkinandderivatives
Nothing in schooling really prepares you for the stamina you need during an AP test until you are smacked in the face with a 3 hour marathon and only a set amount of energy with which to work. To get my kids used to the demans and the pacing of the exam, I offered the secure exam during a full session on a Saturday. We made a deal that I'd get them breakfast if they showed up to do it, and so it lovingly became titled Dunkin' and Derivatives. Of the 65 calculus students in the school, we had about 30 show up. The ones who did are already asking if we can do another before the test day, so they have clearly seen a benefit to the experience. I chose a day in the near future and we'll go over it together as a group. I am hoping that this will become a tradition and we can get more kids participating as they years go on.  

Standards-Based Feedback
The problem with offering the secure test as a practice exam is that I can't hand it right back to them to study. I loved during our breaks hearing the kids conversations about what topics they noticed they were having trouble remembering, so while they worked I took the time to put together a topical feedback form for each of them. I left a spot next to each question so I could write my comments there to give general feedback on that particular topic. 

Here's a link to a word version of the file. Feel free to edit/tweak: 

Standards-Based Review Quizzes
Fom my review grade, I decided to use a Standards Based Grading model. I developed
my AP Calculus Learning Objectives last summer and have been using them throughout the year to help guide students' studying. I have consolidated all those learning targets into the big ideas that students need to know and am going to give small quick checks on each topic. I will use the 0-4 grading scale (at left) to mark each one. Students who get a 4 have demonstrated mastery. Anything below that means we have more room for growth, so after completing corrections on their quiz and demonstrating remediation in some way they will be able to re-assess to mastery. Higher or lower, their final re-assessment grade will be what is used in calculating their score. I will average all of their quick checks on the different topics to get one final average.  I stole the formula to convert this to a percentage from Mrs. Poulsen at Lake Placid High School in NY in a recent presentation she gave on SBG: Percentage=15(Average-3)+85

It may not be the most complex conversion option, but it is easy for my kids to understand:
4 translates to 100%
3 translates to 85%
2 translates to 70%
1 translates to 55%
0 translates to 40%

Here are my ground rules, thanks to lots of feedback from colleagues who are thinking of implementing SBG next year:
SBG Remediation Wall
  • Only 1 quick check will be taken in class. All others must be taken by appointment
    • I have a QR code in my room that will take kids to a Google sign up for times when I'm available. They must give me a minimum of 24 hours notice. That way I have quizzes ready to go (so NOT running around like a chicken with my head cut off)
  • Students may re-assess no more than 2 times per quick check (I have mixed feelings about this one, but it's a compromise for people who don't believe in re-assessment at all and also emphasizes learning it sooner rather than later since we have an AP exam coming shortly)
  • Students "ticket" to re-assess is proof of remediation. This can include extra worksheets from my giant Standards Based Remediation Wall (yes, I created this on my bulletin board...like a crazy person), notes from an online video, or extra practice from another resource. I am putting this on them, not me.
  • Re-assessments must be completed within one week of the original assessment.
I am giving my kids this tracking sheet to help them organize:

Topical vs. Mixed Review
Last year I spent a TON of time on topical review and I know it benefitted my students, especially since they were in a specific course that was built to help some of the students who might never take Calculus succeed. However, I think not starting mixed review with them earlier in the year meant that they struggled more when trying to distinguish what to do when. With that in mind, this year I structured my review this year so that each night has mixed review homework from our practice book. Then, students will have a brief worksheet on whatever we worked on in class that day. The first few days it will be topical review as we go back through the course highlights. Then, we will move more towards AP style application and justification questions. Lastly, I left the final few days of my review unplanned so we can do completely mixed review on what we need the most. I also won't give specific homework those days, since APs will be about to begin and they should be focussing on what they need most. 

Here's the review overview I gave my kids: 
Cram Session
This isn't new, but I figured I'd post it again in case anyone didn't grab it off my Twitter last year. Stacey Roshan created this awesome Cram Video for AP Calculus a while back and last year I created a student assignment to accompany it. I make it optional, but it's a great resource for kids that want to do it! 

We're also doing a giant review tournament and the winners get a WWE Tag Team belt (not a joke, got it in the toy section at Target), so that's keeping their attention pretty successfully. I'll write more about that later in the month. 

I'm sure I'll be tweaking and changing as I go, but I'm interested to see what kind of results this gives! If you so anything that you LOVE for review, please pass it along! 

Thursday, March 2, 2017

Area Between Curves Partner Task

We are starting my favorite unit of the year tomorrow....area and volume! 

Not only is it the culminating moment in the course- our very last unit- but it's also just plain FUN! I'm replacing half their test with a performance task, having them build 3D models of anything possible, and just spent way too much money on honeycomb party decorations on Amazon Prime. You'd think by now my husband would just expect weird and seemingly non-math related items to be shipped to our house regularly, but evidently I'm still surprising him. 
To start off tomorrow, I wanted to get the kids working in partners and visualizing what is actually going on when we find area between curves. We can do a lot with technology, but this one time where I want them to develop the idea by hand. Each group will be given a different set of functions and a different interval on which to graph them. They will then develop the formula for area between the curves with their partner. We can hang them all up and compare our methods to determine area. 

I'm worried about the "big idea" here...once they have that, we can get into the more complicated and interesting things we'll explore this unit. I'm excited to build some models of 3D solids with known cross sections, spend some time playing with honeycomb decor, and revisit my Volumes Performance Task from last year. This will be a unit where I will definitely miss being in a science classroom (I know, I was spoiled), but I'm so excited to get started. Let the fun begin! 

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!