Mathematically Correct vs. Mathematically Helpful: Polar & Rectangular Forms

There are some topics, especially in Pre-Calculus, where I feel like I need to help kids realize the difference between making mathematically correct moves and making mathematically helpful moves. Trigonometric identities is one of these. Converting between polar and rectangular equations is another. If you present it the right way, kids appreciate the elegance of a mathematical "trick"- something one of my beloved mathematics professors would call a "cute" move. These moves the ones that are more than just correct- they are actually helpful in getting you towards your final goal in a problem. 

I have always found that in these kinds of situations learners respond in one of two ways:
either with excitement and curiosity or resentment and bitterness. Students who have come to my class from a previous course that emphasized rote memorization and procedures sometimes feel like they're "doing everything right" but still can't come up with the right answer- a huge source of frustration. Students who have been asked to reason on their own, look for patterns, and solve puzzles in previous courses often find these types of problems invigorating and exciting; they live for the "thrill" of discovering the right move to make.

One way I've tried to combat this and get everyone more on the "I can solve it!" page is to try to make these problems into puzzles whenever possible. I love the way this Time Magazine article discussed the human desire to problem solve:

Human brains have an extreme form of consciousness: they’re ravenous for new innovative solutions to problems in the world, ravenous for optimizing our lives, for building pyramids of knowledge. 

I have found that the more I try to celebrate the "cute" and elegant mathematical moves and help kids see the repeated reasoning that goes into these problems, the less resentful they become. 

Here is a puzzle activity for converting polar and rectangular equations. Directions I give the students: 

I have partners sort them into groups and try to put the "steps" in order. They then need to create a "booklet"  (fold a piece of paper in half so there are 4 sections- 2 on the front, 2 on the back) with one problem on each "page", glue the steps down in order, and explain the reasoning between each step in their own words. I love the conversations I hear between partners, especially on a problem where they "just know" what the next line should be because it seems correct, but they need to think about how to justify it. This really helped my kids dig into the "why" of these sorts of problems. 

Polar and Rectangular Conversion Sort

Coherence & Canoe Trips: Common Language for Understanding Vectors & Polar Coordinates

First, let me begin with this:

Okay, now that we've gotten that out of the way...

One challenge that I find with teaching Pre-Calc is that it's the culmination of all the content the students have learned up to this point. Very few of the topics are completely new to the students, so often there is a "we already know this" attitude when we intro a topic. However, the course requires more than just the knowledge of each specific skill; it demands an integrated understanding of the concepts. Disconnected thinking just isn't going to cut it here....coherence among pedagogical approaches needs to be a priority if we want kids to have an integrated conceptual framework in their minds. Kate Nowak mentioned it in her talk at NCTM Nashville- you don't want to piece together a string of "short stories" in your class; you want to reveal to students the "whole novel." 

My goal years ago was to develop some sort of common experience to use with my kids to talk about the concepts of vectors and polar coordinates since they are so closely related but often taught separately. I wrote and have been tweaking these explorations for the past few semesters.

Canoe Trip Part I- Finding Vector Components


Canoe Trip Part II- Converting Between Rectangular and Polar Coordinates

Photo from http://smsta.msta.org/wp-content/uploads/2014/09/460507657.jpg

Both of these are introductory level activities and I've found that some kids latch on to the "rule" that they develop quite easily, while others cling to the "drawing a triangle will always work" approach. Both methods are 100% valid to me, so I don't make much a distinction at this point in the concept (although the "rule" method is much faster!). When I see those right triangles being drawn out, I get the "teacher stock photo" feeling. My heart feels like this picture- a student choosing to use the concept instead of just blindly memorize the formula! I have a "moment", for sure. 

I like the way these work right now, but I'd like to make them more open ended in the future. Maybe give different groups different scenarios and have them critique each other. Maybe ask them to create their own scenario to model a situation. Unfortunately, these topics always wind up getting smushed at the end of the course and we don't get to have the "play time" I'd like. Anyone have any tweaks or any ways they love to teach these topics? I've love anything to improve or enhance these! Comment away! 

Thanksgiving

We have an amazing art teacher at my school who dedicates the day before Thanksgiving Break to gratitude each year. She has each student write notes, some anonymous and some signed, thanking someone who has influenced them at school. This year, one of her students chose to write this to ALL teachers instead of just picking one she sent it out to all of us. It was my email I received all day:

I'll be the first to admit that I love a good pity party. I think about not having to choose between paying off my student loans and using my 20's to travel the world. I think about having evenings and weekends to relax....maybe even go out! I think about being evaluated on my own merit alone, not on the amount that I'm able to overcome the host of outside influences on a  group of teenagers. It all sounds pretty appealing, especially when you're in a job that even said teenagers can see is under-appreciated.

But then I take a second and I think about those teenagers- the fun that I have with them on 8 out of 10 days, getting to witness them understand something for the first time, earning their trust enough to have tough conversations about what's going on at home or with their friends, the list goes on and on. I can't imagine a career where I don't get to share dynamic moments like that with a group of people who are excited about all that the future holds. We talk about "kids these days," but I have been lucky enough to know great kids who are going to make the world better.

So here's my yearly list of things for which I am grateful, work edition:

• The opportunity to teach in a program that supports innovation, most notably in it's allowance for failure in the interest of growth
• That I have spent my entire career surrounded by the most incredible colleagues
• The people who have shaped my teaching, whether it was in person or through online collaboration (I'm looking at you, MTBOS). The whole "standing on the shoulders of giants" thing is so true- I am truly a better teacher for having worked with each and every one of them. 
• The trust of those around me which has opened new doors to me as a teacher
• That I married an educator who not only understands my insane work schedule, but pushes me to think differently about my own practice and my kiddos
• Caffeine....in nearly any form.
• The truly amazing kids who challenge me every single day to bring my best and forgive me when I don't

Time to put on my stretchy pants and prep for my 3 favorite things: family, stuffing, and football. Happy Thanksgiving! 

Prunes and Cowboy Boots: Tales from NCTM Nashville

I've been home from Nashville for 2 days and trying to somehow simultaneously process all the awesomeness I just experienced and entertain family that is in town for the holidays....not an easy task so far. After packing up the in-law's van this morning and eating about 15 candy canes (it's the holidays, isn't it?), I'm hoping a little writing will do the trick. 

Nashville or Bust! 
This year has been my toughest yet teaching, without a doubt. I have been facing that typical year 4/5 burnout- taking on a huge number of responsibilities while trying to juggle being a newlywed and have a personal life. The balance between doing what's good for my students every minute of every day and doing what's good for my own life and sanity....it's been a tough one to try to strike. In addition to all of that, I've had a heck of a fall personally. I sliced the top part of my finger off about 4 days before the school year started trying to be an iron chef and not use a hand guard on my mandolin slicer. We've been battling puppy cancer with our (way too spoiled) rescue mutt since September- and seem to be on the other side of it right now, thankfully.  We've had other personal trials we never saw coming and all of this has such a huge impact on life in the classroom. You don't get an "off" day in teaching and when you have one, it just makes you feel guilty that you're letting the kids down. I needed some time away from the classroom to get some perspective back, to reflect, and to immersed in a group of people who understand all too well the challenges I'm facing. I'd been looking forward to NCTM Regionals as that opportunity and it was that, for sure. 

Side Note: If you've felt this way, I highly recommend this blog post by Vicki Davis. We definitely all walk wounded in teaching....gotta stay in the game. 

Prunes? Why Would You Put That in a Blog Title?
Andrew Stadel (Divisible by 3) told a wonderful story in the opening session about how advice is taken best when it is timely. Or more precisely, how another exhausted parent at a playground solved his toddler's constipation issues by telling him how they'd gone through the same issue and the answer was PRUNES! If someone had told them this advice at their baby shower or months before, it wouldn't have carried or the same weight or perhaps might have been forgotten completely.  I think about it like this: In high school, I read the book The Unbearable Lightness of Being. Or should I say I skimmed "it"....."it" being the Sparknotes. It didn't resonate with me; I didn't have the life experience to be able to connect with it. In my senior year of college, I was assigned it again and was dreading it. I was also right in the middle of a breakup of the cliche big college relationship....cue the timely part. I didn't even know if I think the book was all that "good", but it was the right book at the right time for me and it helped shape my experience at that point in my life. It was my "prunes" when I was holding the constipated toddler in the middle of the playground. 

I felt that familiar feeling of resonance in Nashville....a recognition that this was the right advice at the right time for me. I am feeling re-energized not only by the presentations I heard (most of which were great), but by the people I met. I was ready to hear these things and meet these people and I'm so thankful that I had the opportunity. 

What Did I Walk Away from Nashville With?

• Formative Assessment I got to attend the pre-conference workshop on formative assessment and left with tons of research-based ideas on how to involve my kids more in peer and self assessment and also with a new lens on the amount of grading I'm doing. I'm going to start incorporating way more self-graded or ungraded assignments which I can leave feedback on without muddling it with a grade at the top. I teach honors kids. If it's an A, they file it away. If it's not an A, they hide it in hopes no one else sees it. Either way, I spend hours grading and wind up finding papers on the floor or in the recycling bin.....frustrating to say the least. Get rid of the grade and they need to fish for the comments and the feedback more. So simple, but filled with so much power.
• Modeling I love modeling tasks...it's my bread and butter as a STEM teacher. While It might not be a new idea to me, I definitely left with a host of new ideas for every course (including Calculus! Yay!). A huge thank you to Brian Shay (@mrbrianshay), Brett Doudican (Coordinated Achievement), and Robert Kaplinsky (RobertKaplinsky.com) for inspiring modeling sessions. 
• Blended & Flipped Learning I've been doing blended learning for two years in my reach-model classroom, piloting it for our district. I've proven that the reach model works for the right kiddo- one who is self-motivated and takes the time to do their work outside of class. I learned that I really am towards the forefront of thinking on much of this stuff (which is nice validation), but I had the chance to really push my thinking further with great colleagues. One session in particular that I attended examined how people are using flipped models, examining the difference between videos that are purely instructive vs. videos that still drive student thinking using the ideas from Principles to Action. Yes, the videos replace the lecture. But in particular, we discussed using the video as a driving factor the conversation the next day. The video does not need to answer questions- it needs to generate them. So small, but so powerful. Thank you to Jeremy Strayer from Middle Tennessee State University for having such a productive conversation in his session. I also attended the packed session on flipping AP Calculus with Joel Evans and left feeling like my dive into flipped AP Calc (which I promised my class we'd start after Thanksgiving) won't be as scary as I thought. 
• Networking I finally got to meet a lot of the people whom I've been following for a long time and who have inspired a lot of my lessons....got to have my little math teacher celebrity moments. Getting to chat with so many of MTBoS colleagues who have supported me without ever meeting me was amazing.  And my final math teacher celebrity moment- finding out Kate Nowak (Function of Time) and I had the same graduate advisor. My graduate experience shaped so much of who I am as a teacher....no wonder I find inspiration from someone who worked closely with the same people I did. 
• FREE STUFF!! I won a copy of Principles to Action and have already highlighted the crap out of it. I ate about 37 tootsie rolls and 113 mints from vendor booths. I signed up for copious amounts of free resources from NASA. I got a brain stress ball from Gizmos when I told them I was part of a grant. I got a student sized whiteboard (because I don't already have 30 of these in my classroom....). And I heard lots of great country music in our time outside the conference. 

My Favorite "Overheard at NCTM Nashville" Moments

Pretty sure this was Robert Kaplinsky right before his session started to a participant talking about fear of things not working

"If everything always works in your classroom, you're not innovating enough" 

An awesome lens for viewing flipped instruction, when people were asking questions too specific about "videos"

"It's not about the videos, it's about the pedagogy." 

God Bless Puckett's Nashville

"Mac and cheese is the vegetable of the day"

A participant asked Kate Nowak how to do a particular problem she'd offered to us as a challenge. Her response was the perfect teacher answer. 

"It's pretty satisfying and magical when you figure it out, so I don't want to steal that from you by telling you."

So to NCTM, all the conference attendees, my coworkers who tolerated my crankiness on the traffic-filled drive home, and the city of Nashville, I say: Thank you, thank you, thank you!!!  

What Did I Miss?

This may be it. Every teacher's favorite question. 

Absences are a chronic issue with particular students it has a serious effect on the classroom. Math is so cumulative....everything we do builds on what we've done the day before. I had a student return back to Pre-calculus today after a week out and in that time, here's what we've covered:

• Right Triangle Trig Review
• Special Right Triangles
• Angles in Standard Position
• Coterminal Angles
• Reference Angles
• Radians and Degress
• Deriving the Coordinates of the Unit Circle from Special Right Triangles
• Finding Exact Values of Sine/Cosine/Tangent
• Finding Exact Values of the Reciprocal Functions

Any hope of completing the scavenger hunt we started with as a warm up was futile. She was just too far behind. 

Over the summer, I tested out a theory that I could create a sync-able folder that I could share with students so they could access all of my documents from every class and I wouldn't have to think about uploading them at the end of each day. God bless any of you that do this, but I do not have the presence of mind after a day of chaos to save all changes and upload them to my website (especially for 3 different preps). 

Here's the solution I have tried this semester:

Creating a "What Did I Miss?" Folder using Notability and Google Drive

I teach with Notability, so this wasn't a big leap. However, I am sure there are other apps that have an "auto-backup" feature. My documents automatically sync to my Google Drive and I share the folder via a link so anyone can view it. This has solved a lot of problems for absent students, but it's had some benefits I never considered. Many students benefit from being able to go back and re-read the notes (including students who need filled in notes as part of the IEP) that I've seen it as a huge benefit for everyone. And talk about a tremendous resource at finals time, especially since we don't use textbooks at my school. Tech is here to make our lives easier, right? 

When am I EVER going to use this?!?!

I don't think I'll get through a semester without showing this now that I've seen it! 

Happy Friday! 

Blended Learning & Design Thinking

Being a STEM teacher makes you think about things you've never considered...a lot. I've spent more time considering rations you'd need to live on Mars and the location of the focus of our parabolic planetarium at school than I'd like to admit and generally these mental aerobics have made me a stronger thinker and teacher. This has been a direct result of my time spent in STEM and life was not always so in my previous job. One thing that was usually about as far from my mathematical mind was the life of people like my friend Jon, who is a graphic designer. I've always loved looking at his latest projects for clients and he would always explain the client's reaction to it. He even designed the layout of the cornhole boards I made for my wedding, offering me 3 different options and allowing me to give him feedback on what I liked best before he sent me the final design. Seems like a different world than mine own (or at least it did at the time). 

Cue dramatic music and study the picture here:

STEM Design Cycle

This is the design cycle by which we live and die in my STEM program. It is a process that as adults we move through almost without thinking- figure out the problem, come up with a idea, see how it works, get some feedback and reflect, and then modify to make better. It wasn't until I met a very wise art-teacher-turned-STEM-guru that I started to see the world of design and my own world start to converge. This whole process is the key to everything I do as a teacher, as a mathematician, as a team member, and in my own personal relationships. This is how we grow and improve. 

Recently I attended a PD that asked us to view our blended courses from a design perspective, something I wish I'd considered more before this whole thing started 2 years ago. When I think about the creation and implementation of the course, I realize that I'm immersed in this process almost constantly. The class is by no means perfect and I regularly try to tweak and get feedback on things. Feedback can be a scary thing when you're sticking your neck out and trying something new....the more I teach blended, the more confident I feel in asking for help in it and the more critical I find myself being of my own design. We discussed and considered research from the Schlechty Center and it's some powerful stuff. I am by no means an expert in this, but here are some highlights that jumped out at me when viewing through the blended learning lens:

Design vs. Planning

All too often we choose to embrace structure and order over chaos and divergent thinking. It's easier. It's comforting. And frankly, it's how we were taught. Because design is trying to meet the needs of it's "client" (the student, in our case) and not trying fit activities into an organized algorithm, it is messy. It involves experimentation, student choice, and being willing to admit something didn't work at all.  As a teacher, it's often frustrating when students come to you without the background knowledge they need and rather than planning for the content I'm trying to teach I should be designing for the needs of my students where they stand in order to build them up to where I need them to be. Blended is the perfect opportunity to do this since it offers so many options for personalization. However, that type of personalization takes some seriously intentional work.  

Design Qualities for Creating Engaging Work

To read these, check out this document from the Schlechty Center: 

Schlechty Center on Engagement

There's so much substance in these 10 principles. They address the big issues I wrestle with daily in blended learning: motivation, failure, cooperation, feedback, choice. 

Some more interesting reading on the subject:

Working on the Work

Focus Within Realm of Teacher Control

I am learning to remember that I have no power over many of the choices the students make, especially in a model that allows for students to complete their work in an unsupervised environment. It's a large amount of control with which I've had to part. What I came away from the PD thinking was that I need to look at the factors I can control and use design thinking to improve those. Less self-blame, more self-reflection. 

One and a half semesters into teaching using this model and yesterday I still had a "real talk" with my students about why work doesn't get turned in consistently and what I can do to help them with that.  In a short conversation, my students and I discovered that the LMS we are using doesn't notify them of a certain type of assignment and we made plans to change that for the future. Such a productive conversation, but it involves being vulnerable and admitting to others that something isn't working well- that's scary. The kids are definitely seeing that I'm human and willing to improve and learn from them. I'm incredibly grateful for everything I've learned so far and the honest and constructive feedback of almost 100 kids who've been willing to fail forward with me. 

Sharing is Caring- Technology Edition

Technology is something I consider myself relatively fearless with in the classroom, as I am definitely a digital native who knows there's few things a quick control-z or restart won't fix. I attended a week long institute this summer on integrating technology into math instruction and I've also been teaching in a one to one STEM magnet program for 3 years and piloting blended learning for the past 2 in Pre-Calculus. This picture has been big for me....definitely taken somewhere from the Twitter world! I find it to be a good lens when I am trying to integrate tech for the sake a tech- a way to check myself before I wreck myself, if you will. 

I love the quote from Goos, Galbraith, Renshaw and Geiger (2003) about using technology as a partner in education, saying that it should be “used creatively to increase the power students exercise over their learning; for example, by providing access to new kinds of tasks or new ways of approaching existing tasks." We have so much power to do that these days and it's something I'm still trying hard to learn myself. It should be transforming the tasks and challenges we give our students. 

So tomorrow is our annual PD day in my district and there are few things I love more than sitting in a room of math teachers, nerding out about our content. It is also the first time I've been asked to plan and implement my own PD for other secondary math teachers about integrating technology into secondary math and designing it has been one of my favorite tasks all year.  Trying to narrow down exactly what to share has been challenging, but here's what I've come up with: 

• Desmos Polygraph Activity (because if you haven't seen it before, be ready to fall in love)
• Discussion about cognitive demand of tasks and metaphors for technology-mediated learning
• An activity (admittedly adapted from an amazing activity we did at the institute) where students use dynamic software to find the characteristics of quadrilaterals and create a hierarchical classification system of them
• Some quick "tips and tricks" I've learned from teaching blended and STEM
• An hour of self-guided "challenges" that I've written to allow people to move at their own pace through using Desmos, Geogebra, and Google Docs in different ways

Here are all of my files for the PD if anyone is interested in checking them out!

Presentation Materials

Now, if I can somehow find the time to simultaneously be in 3 places at once, tomorrow will be a big success! 

MTBoS and Becoming "Connected"

Consider this my love letter to all the connected educators out there: 

When I started at my current school, I was given the amazing opportunity to co-plan with some seriously tremendous teachers. I was a new, excited teacher and they were curriculum writers, teacher leaders, strong collaborators, and people who valued my ideas and contributions. I was part of a true PLC....where we trusted each other, co-planned, and pushed each other to be better. I had constant support and feedback and it made me so much stronger as a teacher and a leader. I was a part of a family (and they continue to be my family today). 

Many of those teachers have moved on and I find myself as a PLC singleton, without other teachers to work with and co-plan each day. I am being asked to be that leader for the new teachers around me and while I am thrilled about the opportunity, it is also hard to lose the support system you'd worked so hard to establish. One of my goals this year was to connect on a bigger level- not just with the people with whom it was convenient to connect, but with the people with whom it was beneficial to connect. I wanted teachers who would challenge and inspire me, even if it was from 500 miles away. Enter the #MTBoS. 

I have found that (and so much more) in just a month and for that I am so grateful. I have a challenging schedule this semester and am working crazy hours (even by teacher's standards). The thing that is making it easier and more worthwhile is the amount of passion I have found in my peers online. I am still working on getting more connected and breaking down those walls to better myself, but I am so appreciate of everything I've learned so far. I am excited about everything I am still learning. I can't wait to see how this drives my teaching and leadership. 

Just a few of the things I've learned:

• Those "creative" things I'm doing....other people are doing them too. Some doing them much better. Not to self- see what people are doing better and model it!

• There are things that I am starting to hone in on as areas of focus for me as an educator. Seeing the types of people and conversations to which  I am drawn  has clarified that for me and helped me see some of my future goals more attainably.  

• Inspired by all the awesomeness around me, I decided to pursue my "Google Certified Educator" credentials. I thought I was using Google well! I knew most of the information in the reviews! Then I took my test (which was more challenging that I'd thought...it was a really good, tasked-based assessment) and have already used about 5 new strategies today that I thought of during the test. Embedding a youtube video to use as a timer since timers aren't already in Slides? YES!!! Why hadn't I already thought of that?! 

• I'm not the only one who is trying to balance it all. Almost my entire department is young and child-free. I now see all these people with their own kids and families  and are still incredible mathematics educators and I am so inspired. With the right mix of passion and hard work, you can have some sort of life and not sacrifice the quality of your teaching. 

So this is my thank you. Keep sharing your joys and frustrations, because they are making this world a smaller and friendlier place.  I can't wait to see you all at NCTM in Nashville! 

Post-it Queen

I am definitely the Post It queen of my department. My desk is constantly littered with a minimum of 12 different pads of brightly colored notes. I have a pad that I leave in each group's "supply basket" for daily group work (because you never know when you will NEED them!). I got heart-shaped post-it's from my high school best friend as an engagement gift to help keep me organized as I planned the wedding. You get the picture.

To throw gas on the flame, I was gifted over 15,000 Post-its as part of the Reddit Teacher Gift Exchange (THANK YOU!) 2 years ago. They have become a regular part of my class games, activities, and assessments. 

My favorite way to use them right now is together with the (free!) Post-it Plus app. I post questions around the room that the students need answer in pairs or groups and ask each set of students to post their answers, one per note. 

Today's was a pre-assessment on laws of exponents as we movie into our logarithms unit in Pre-Calculus.

I posted 6 problems, one on each of the major laws. I could instantly look at the Post-its and get a general idea of where the strengths and weaknesses in prior knowledge exist. Kids all got it right, we can briefly discuss and move on.

Our biggest problems seemed to be rational exponents, so from there we can delve further into the type of misconceptions the students have. 

Here is where the app kicks in! You use the app to take a picture of the post its (Pro-tip: Ask the kids to use pen or marker- pencil doesn't photograph very well), as seen here.

From there, the app creates a "board" of the notes so you can scroll through them and assess student misconceptions. It's anonymous so kids are more willing to be wrong, which I love. 

From there, I've had kids do group responses to common errors, identify patterns, and do "find my mistake" activities. I'm sure there is so much more I could do, just haven't tried yet.

It's a fun formative assessment that the kids instantly think is "cool" because of a simple app.

The "board" of answers through which we can scroll!

Happy Hump Day!

Introducing the Derivative (The Italian Way)

I'm from a big, loud, Italian family and all our biggest celebrations always involved pasta. Imagine my excitement that I can share one of the most exciting days of Calculus with my students and bring pasta along for the ride! 

Inspired by Math Teacher Mambo's post about relating graphs of f and f', I couldn't wait to test this out with my students. I was amazed- a concept with which they were likely to struggle was suddenly accessible and they were having great conversations with their partners. 

I really tried this as an experiment and spent less time that I wish I had actually manipulating the pasta, so I may re-write this to give them more practice with the "approaching" idea. 

Side Note: One of the reasons I love teaching so much with pasta is my now infamous "pasta speech." I did an activity with pasta my first year of teach about something (triangle inequalities maybe?) and haven't had to buy a new box of pasta yet. In that time, I've moved classrooms and moved schools and, quite frankly, not been concerned about the food safety of said pasta. I always give a very impassioned speech- a pasta-biography of sorts. It's humble beginnings on a grocery store shelf....it's long summer in a hot storage unit while I slowly moved trunk-fulls of stuff to my new city in my '92 Subaru....it's residency on the bottom shelf of my bookcase, close to the not so cleanly environment that is a high school classroom floor. All of this to lead to the moral of the story- don't you DARE put that pasta in your mouth! I even had a student buy me a box of pasta for Christmas and leave it on my desk with a note that says "Thought you could use an upgrade." Never gets old. 


The Lesson:

To get kids thinking about average vs. instantaneous velocity, I started with this question....

As a group, answer the following question & justify on your whiteboards numerically, graphically, algebraically, or verbally:

Alex and Erika both live next door to each other, 13 miles from school. Alex leaves home at 6:30 am and arrives at 6:45 am. Erika leaves home at 6:50 am and arrives at 7:00 am. Who passes the gas station halfway to school in a shorter amount of time?

This led to a pretty lively debate and helped me pre-assess their knowledge of physics a bit. Eventually, one student convinced the rest of the class that since we don't know their instantaneous velocities that we didn't have enough information to answer this question- which I love, because it leads them to questioning how we get that information. 

The next activity was inspired from a University of Washington activity, but I modified it heavily so my kids could really build the knowledge themselves (without quite so much jargon). 

Introduction to Derivatives Activity

We did get to some practice with the limit definition and I'm going to use this Derivatives Match Up as a warm up activity in class on Monday to give them some more partner practice. Derivasaurus Rex is very excited about all these rates of change (more on him later)!

Once in a While...

...you get a message like this from a kid. 

Taking risks in the classroom is exhilarating, but sometimes you're not sure if you're helping anyone (especially yourself) by taking on so much extra work. Teaching in daunting and exhausting and some days you just feel ready to break. 

Thankful for this student and the others who make a point to remind me why I do this (even if sometimes it's harder for me to see the reminders). Such a happy note to end my week. 

Rational Functions Challenges!

I always find my Pre-Calculus class the most challenging to teach at this time of year. Yes, even more so than prepping for AP Calculus for the first time and teaching a blended class with 50 students and about 10,000 things to grade every night. 

Why, you ask?

We run on block scheduling and there are few things which I think are more of a challenge for mathematics teachers than the gaps in background knowledge that happen from skipping semesters (or YEARS) of math in high school. By the time the kiddos get to me in 11th or 12 grade, all those "must know" skills have been forgotten. My goal is to try to remind them of those skills without having to reteach the concept completely. I know it's in there....we just need to bring it back to the surface!

We are currently studying rational functions, a topic which is so vital to their study of limits and later calculus. The whole idea of discontinuities....so important! 

This year for some of the review topics, I'm using some classroom flipping (with videos I made with Educreations). Kids who remember the material can speed through these and demonstrate mastery while kids who need the extra remediation have the opportunity to get it. My kids had 2 "flips" last night- a brief video on holes vs. asymptotes and one on horizontal asymptotes. Having the preview of these topics allows me to group the kids strategically and have them really practice in class. 

Desmos PolyGraph Activity 

Think "Guess Who?" for functions (the one pictured above is quadratics)! I loved that it let me move around the room and assess the comfort level with mathematical vocabulary.

We had a big discussion about what questions would be considered appropriate for pre-calculus level students. Here's what we decided...

From the teacher perspective, I could follow each conversation and make sure they were using academic vocabulary. I was able to circulate around the room and listen and ask probing questions to kids who needed it. It was competitive, but fun! Every single one of my students said on their feedback form for the activity they'd want to do it again.

What the kiddos are saying....

"It made me really think about the characteristics of the graph/function."

"It was fun and helped strengthen my math terminology" 

"I like the fact that you can ask questions to better understand the topic and also get asked the questions. It's easier to understand and more fun with being partners with some random person :)"

"It was a fun, creative way to show what we know with out the stress of a quiz, or test."

Once I felt confident that they had mastered the vocabulary and were ready for more collaborative practice,  the students had to complete a graphing mini-project on my giant laminated graph paper! The first 2 questions were a mini-review from rational expressions and equations and the 3rd was the actual mini-project.

Now to grade them all....

(Is the domain and range bothering anyone else?? Have some more work to do with that group, still!)