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. 

Joy Ride Project

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! 

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! 

The Final Countdown: AP Week is Here!

Here's the motto I'm trying to instill in my kiddos this week:

During the middle of last year, a colleague and I were discussing the fact that our school only offered BC Calculus and we wished there was an option for students who were interested in pursuing Calculus but might not be ready for the demand of BC. Heck, my bachelor's is in pure mathematics and I did just fine having only been a product of AB Calculus. We approached our administration about this and luckily for both ourselves and our kids, they allowed us to take on the challenge of designing and implementing a full-year AB course. The course would be an "Honors Calculus" course in the fall, leading directly into an AB course in the spring.  It took some selling to get our enrollment up and I am forever grateful to the 20 kids who took the risk of a brand new Calculus course with me. The enrollment is already more than doubled for next year and that definitely wouldn't have been possible without this pilot group. 

I feel a little like the cast of one of those youth sports movies sometimes when we are in class- despite the fact that Emilio Estevez had consistently better hair than me in any number of those movies.  We are an unlikely mix.....17 boys and 3 girls (I promise our demographics are more evenly split for next year!!) with extraordinary variation in past experience with math, intrinsic motivation, and consistency in completing tasks. But they all showed up every day and tried. Cue the Mighty Duck's-esque montage where everyone learns that "Ducks Fly Together" and learn to skate without falling.  I've gotten to watch this unlikely mix of kiddos turn into a team who can function at a really high level (or more concretely, get about 5 to 9 points on the famous 2007 AB 3 problem).

Because of the extra time built into our course, I really do feel like I've given the kids all the tools I possibly can at this point in the game. I'm okay with not trying to "cram" any extra information in their brains on the last day....which is a feeling of peace as a teacher that I don't normally feel. With my unlikely group of mathematicians and the mindset that we've prepared with the most gusto we possibly could, I'm taking a very different approach on our last day in class tomorrow. Here's the plan:

1) A sincere "thank you"
Y'all.....I'm so proud of these kids. Because I teach mostly in a STEM program, this is the third straight year I've taught this particular group of 20 and they've been a huge part of my journey as an educator. Beyond all of that, they've been with me through an incredibly difficult year personally. They've accepted me on my bad days and pushed me to be better even on my best days. They've driven me a little crazy, of course. But I am so grateful for the hard work and determination I've seen from each and every one of them. They've made me laugh and learn and grow. I don't know if we ever thank our kids enough. I'm trying to remember to do it more. 

Good Luck Gift Bags (because my inner middle school teacher still hides deep below the surface)

2) A little more shop talk
Because I don't think you can emphasize "DO NOT write 'it' if you mean f'(x)!!!!" enough. I don't mind saying it for the 16,000,001st time if it means they remember. And for Heaven's sake, don't leave off the + C !

3) A real, honest talk about anxiety, nerves, and some brain chemistry
Here's why: I am someone that naturally appears incredibly collected and outgoing. I seem to thrive on being constantly interacting with others. This is a completely intentional choice I make everyday- I am much more of an introvert than anyone would suspect.  It was actually my time as an AP student that started my journey to discovering this. AP US History basically turned me into a robot; it was the highest expectations to which I'd ever been held in a subject area that wasn't my strength. Friends tried to talk some sense into me, but I assumed it was just stress and it would pass. As I got older, I started to become more cognizant of the amount of anxiety I felt on a regular basis (although I'd still fight you if you tried to tell me I was at all abnormal in what I felt). Only this year did my anxiety finally bubble over into something I needed to address and doing so has totally changed my perspective. 

I think a lot of kids feel this kind of anxiety and pressure. Our guidance counselors get crying AP students in their room regularly. We expect to see some mini-breakdowns as exams approach. And let's be real....most 16 and 17 year olds aren't equipped to deal with it. So here's what we're doing:

Activity #1:  Article Reading

I'm going to have the kids read this silently, marking a "?" next to anything they have questions about and a "!" next to anything they want to remember for the test. Then we're just going to talk and see where it goes. 

Activity #2: Journaling Activity

Since my very first standardized testing week as a 1st year teacher, this article has been in my arsenal. I love it. I love the idea that psychologically the symptoms of being "pumped" are the same as being terrified.  I try to get kids to get pumped to show what they know, not fear what they might not know. 

"In a study published last year in the journal Emotion, Dr. Beilock and four co-authors found that with students anxious about math, the more stress hormone they produced, the worse they did on a test; students with low math anxiety did better the more cortisol they produced. “The first group,” she said, “felt the rising anxiety in their bodies and reacted by thinking, ‘I’m really nervous about this test. I’m afraid I’ll fail.’ ” They choked. “The second group told themselves something like, ‘I’m really psyched up for this test! I’m ready to go!’ ” Dr. Beilock recommends consciously adopting positive self-talk. Remind yourself that damp palms and a pounding heart accompany all kinds of enjoyable experiences: riding a roller coaster, winning a sports match, talking to someone you have a crush on."

We are going to end the class by actually doing the journaling activity as an exit ticket. I am encouraging each of them to leave their fears in my classroom by writing them down and putting them into a box. With those left behind, I am going to encourage them to take only their knowledge, skill, and confidence with them into the test. My hope is after the exam we look at some of their fears and see if they were worth "carrying" with us or if in retrospect it was okay to leave them behind. 

I'm incredibly proud of them and am hoping to leave them feeling excited to show all they've learned. Any I will definitely be reflecting with them on how to make the course better. 

Good luck to everyone's AP classes these next 2 weeks! Cheers to all your work to prepare an incredible group of kids for an incredible challenge! 

Learning to Trust the Core: My Weekend with the #CoreAdvocates

I feel very fortunate to have spent the weekend with some amazing math educators from all over my state, digging into the Common Core State Standards. Even though the standards are in their sixth year and we've certainly seen a shift in many classrooms towards a new way of teaching, I think we've all seen that the kinks definitely aren't worked out and there are lots of people who are still reluctant. In all honesty, who could blame them? When you've been doing something for a long time in a particular way (not to mention being a student who learned a particular way), it's difficult to feel like you're being asked to reinvent the wheel. When a huge part of your evaluation is based on a multiple choice test, hard choices sometimes get made. You hope you're doing what's best for kids and try to trust your own professional judgement. 

[Brief aside: I feel so lucky to have taken the path I did to education. My graduate advisor, Dr. Jean Schmittau, was someone who was truly ahead of her time. She advocated for conceptual, student-centered teaching for decades before the words "Common Core" were ever utter together and truly pushed her students to think deeply about mathematics and pedagogy. Let's put it this way....one of our assignments was to make a concept map that showed the concept of multiplication and it's relation to the body of mathematics as a whole. I had to go to Kinko's to get mine printed because it was roughly 5 feet wide. We spent almost a month of 3 hour classes trying to answer the question "What is multiplication?" and often we were given another week to think about it because a room full of people with pure mathematics degrees didn't answer her well enough.  She and the wonderful members of my cohort shaped me into a teacher who is always seeking to improve and challenge myself and my students. I was reminded of her in so many ways this weekend. ]

We opened the weekend with this big question:

Why are we here? 

Pretty existential, right? We were challenged to examine that question from 3 different perspectives and I couldn't think of a better way to share all that's swirling around my brain right now.

Why are we HERE? 
There is something so valuable for a teacher's soul when they are able to congregate with like-minded people in a neutral site. You are able to immerse yourself in a world that centers around what you're studying and have discussions that might be difficult in a place you are too "comfortable." It may seem small, but I think it's a huge part of what makes conferences so powerful.

Why are WE here? 
 A topic that came up over and over again this weekend was the idea that teachers need to "own the standards." It's a picture that I think holds a lot of power in education. If we truly "own" what we are teaching and examine it as practitioners, we will be cognizant of how these standards connect with the conceptual structures our students have created in their schooling. A huge part of this is creating teacher leaders and I was happy to be in a room full of them. I honestly believe that these people are the key to any real change- teacher leaders need to buy in to it and model that these changes can be effective. Ideas that teacher leaders can demonstrate are good for kids (and good for teachers) are the ones that spread like wild fire. 

One aspect that we discussed was the idea of thinking critically about our own instruction and that of our colleagues. My biggest concern about doing this as a department is always that people will feel they are begin "evaluated" if the proper level of trust isn't established beforehand. Teaching is so deeply personal and it takes trust in your peers to allow them into your room to reflect on both your teaching and their own. But what if you do? The trust, collaborative spirit, and sharing of ideas that could happen is almost unfathomable and contrary to the isolationism of teaching in the past. It's using the 21st century skills we ask our students to have every day.

One of the things I'm taking away from the weekend that I felt most passionate about was a really powerful tool to help start these conversations among mathematics coaches, administrators, and peers. 

Achieve the Core Instructional Coaching Tool

This tool provides feedback on what I would consider the 3 most important aspects of any lesson: alignment to standards, appropriate selection of instructional strategies, and student opportunities to engage with the mathematics. Digestible language, jumping off points for conversations, and no wording that rings of "judgement" of the teacher....love. We discussed using it as a goal setting tool in our PLC's and as a tool to start to generate trust among a department. It's something I'm carrying with me as I continue to pursue teacher leadership in mathematics.

WHY are we here? 

I consider myself someone who is pretty well versed in the Common Core, as my master's

centered around conceptual teaching and I completed it right before the Common Core was implemented. We used the standards as our textbook. I've got to be on the savvy-iest end of things, right?
 
WRONG! But let's be honest, I didn't know what I didn't know. 

http://www.360results.com.au/wp-content/
uploads/2013/06/tumblr_mh6eljSrzR1rx06nvo1_500.jpg

The picture here rings so true to me....we examined the standards as a whole, not just our own grade level. And more importantly, we examined them from a perspective of 2 of the 3 major shifts- Focus and Coherence.  So often as teachers we are focused on the "checklist" of topics we need to teach, the day by day "calendar" we've created, and not looking at what's happening outside our own course. For this weekend, I hit "pause" on my high school teacher mentality and tried to examine the standards from a mathematics educator perspective. 

Here's my biggest takeaway from this weekend:

We need to learn to trust the standards. 

The key word I've started to imagine with the regards to the standards is this: design.  These were written, revised, and critiqued by people who have spent their careers immersed in mathematics education. These were designed intentionally to create a cohesive understanding of mathematics that balances conceptual understanding, procedural fluency, and applications.  They are saying this: "Trust me. There is a method to my madness." These were my "ah-ha" moments: 

•  Situation: A 3rd grade teacher who is introducing multiplication teaches how to "carry" because it makes it so much easier for students to multiply big numbers!
• Explanation: This teacher is trying to do right by their students. They are trying to give them a strategy that will always work. However, in doing so, they're neglecting the coherence that is built in the standards to really develop students' conceptual understanding of multiplication so they can apply it later. Fluency with the standard algorithm for multiplication isn't expected for 2 more years (5.NBT) and if we introduce a "trick" like this too early it can actually be a detriment to the understanding our students are building. 
• Situation: A teacher is so used to teaching surface area in 7th grade that they continue to teach it. It's in the textbook, right?
• Explanation: The core of the major work in elementary school is focused on developing students' understanding of number and operation. Geometry can be a tool to do that, but it is regularly listed as a supporting standard. This doesn't mean it should be ignored; it just means it can be used as a lens through which you can refer back to the major topics. It isn't until 8th grade that a geometric topic is considered part of the "major work" and with good reason- the lower grades are trying to go deeper into content. As we advance in the grades, we can have a wider focus. Surface area formulas are not specifically memorized in any grade because it lacks the conceptual basis that the Common Core demands. Just because it's in our textbook doesn't mean it needs to be our focus.
• Situation: High school teachers have always heard that there is supposed to be "less content" in the Common Core, but our courses seem to be even more jam packed.
•  Explanation: The high school standards were not written to decrease the amount students learn each year. This was never their intention. By focusing on fluency in the younger grades, high school is able to become a time where concepts are connected. Mathematics is a story and high school is the time when we weave together all of the plot points.

These standards were not written arbitrarily. They have a plan and every teacher has a role to fill to make sure students are getting the most from the instructional shifts. We need to see ourselves as part a whole; we are a K-12 team that is created to support the success of each individual student. Synergy, synergy, synergy.

This is by no means comprehensive of all the things I'm leaving this weekend having learned, but it's certainly where my mind is focused right now. I feel so inspired to keep learning and leading and I'm grateful I was able to be explore so much this weekend!

Tasks vs. Projects in Calculus

So it's Friday afternoon and in an effort to really knock one out of the park (and avoid the pile of ungraded tests staring back at me), I have some serious thoughts going on about my next Calculus PBL, which I am trying to design with a fellow AP Calculus teacher at my school. I am lucky enough to teach in a STEM program that received a huge federal grant and, as a result, has access to lots of things others don't. Too often I'm thinking of ideas without enough time to actually go through the enormous process that is ordering anything through aforementioned federal grant. So here goes nothing:

Disclaimer: This is a brainstorm....brain dump....random collection of ideas that I hope other amazing AP Calc teachers might be able to expand upon? I offer no solutions in this post...just questions.

We are getting 2 new 3D printers next week and have a training on using them (YES!) and there is obviously a direct relationship to all the applications of volume we discuss in AP Calc. I've done some searching....which always just leads to more questions and ideas. I started from watching these 2 videos:

Solids of Revolution

NCTM Performance Task

This possibility seems like more of a logistical issue than a conceptual one. I know how to make this a demanding task for kids and have seen amazing applications- from the "Goblet Design" project from my wonderful coworker or the vase volume performance task I had the opportunity to play around with at NCTM Nashville (shout out to Brett Doudican from Coordinated Achievement on this one). It would be fun to 3D print these to scale and then use displacement to check the accuracy of our volume calculations. There are so many ways to give kids constraints and have them develop something amazing. It's task-based and, more importantly, not a project for the sake of doing a project. 

Cross Sectional Volumes

Logistically, I actually think this is easier. In about 15 minutes of playing around I was able to create 2 designs that would model different cross sections. My kids have experience with 3D printing...this wouldn't be mind blowing for them.  My struggle is having them print 3D models seems like the thing I hate most....a project for the sake of doing a project. Might it be a fun way to spend some time after the AP exam? Sure. But if I want to invest any time in this, I want it to be something that's worthwhile mathematically as well. I want something that can be low-floor, high-ceiling since our school ranges in ability level from Honors Calculus to BC Calculus (to a kid who is taking an MIT Quantum Physics MOOC for fun....seriously). Can you tell that I have lofty goals here?

So here's where I need my MTBoS friends...

What sort of tasks and applications do you associate with this topic?

I want to do more than just create a model, so how can I challenge my kids with creating a model within given constraints and do it meaningfully? My coworker and I have been trying to think of how to get kids to apply this concept in the real world and really use the model to serve a purpose. 

Please feel free to share, comment, and question!! We promise to tell you where this road leads us and share any resources along the way!  

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. 

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!)

One of Those Days...

I've been watching way too much Keeping Up With The Kardashians in an effort to numb my brain during end of quarter grading, so here it is, the peak and the pit of today.

Pit: The server goes down. In the middle of 5th period. For 2 periods. When I have tablet and Smart Board activities planned for all! Coworkers lost all files. Thank you, Dropbox, for saving my life. 

Peak: My Geometry class bringing a staples "easy" button to class and it exclaiming "That was easy!" every time we successfully finish a tough problem. Special right triangles ain't never been so much fun. 

Notice...the peak is the kids. The pit is all the other nonsense. 

A Geometric Valentine's Day

We're just digging into geometric mean in Geometry and the inevitable question of "Why are there 2 types of means?" always arises. 

Here's a timely explanation of one way math & geometric mean are used in real life....just in time for Valentine's Day!

Happy Valentine's Day to all of you!

Update: Discussed this with my class last period...they immediately wanted to know if I used eHarmony or Match.com (despite the fact that they know I'm in a long term relationship) and which was better. They decided they should probably test both to figure out which one worked better.  I guess I'm glad they're trying to test their conjectures? Scientific thinking at its finest. 

Mathematical Bakery

Working with generalized proofs today, we ran into a problem that said (a+b)(a+b)...we were using the distance formula. Oh the crazy answers I heard from my Geometry students!! Sure, we haven't done a lot of multiplying of binomials this year, but still....I was a bit frightened. I explained to them that at the end of this year we're going to do a pretty hefty Algebra review to make sure they're in shape to take Algebra 2 next year. I told them "factoring is cake compared to what you'll see there!" The following conversation ensued:

Student 1: "So....you're saying Algebra 2 is like a scone?"

Me: "What??"

Student 1:" Well if factoring is cake, Algebra 2 is a scone....it's a little harder."

Student 2: "Pre-Calculus must be like a stale scone!"

Student 3 (from across the room): "And Calculus is like a really hard loaf of bread!"

Student 1: "Good thing we have studying...that's our coffee! We can dip our scones into it and it will soften them and make them taste better."

Student 3: "I don't think I'm going to study in Calculus."

Me: "Why not?"

Student 3: "Because....old bread dipped in coffee? Gross."

Groups of kids like that remind me why I love my job, especially on the days when you're ready to hide under your desk.

A Very Fractal Christmas, Part II

Here enjoying my last (very snowy) night in the northeast before heading back to warmer weather and lesson planning tomorrow, but I had to share how well our tree came out!!

All in all, it was about 3 feet tall and the kids were so excited they offered to come in before and after school to finish it. A great end to a long first semester!

May everyone have a happy new year and enjoy the fresh start January brings with the kiddos! 

A Very Fractal Christmas

Today is our last real day of classes before the break and, as would be expected, most classes have a test or quiz. Tomorrow we have the parties and the annual Christmas dance and then we'll have 10 glorious days off! One of my classes somehow got a day ahead  and took their test yesterday. I was at a loss of what to do with them until I found this great idea: build a 3-D fractal Christmas tree! I plan on introducing the idea with this video and seeing where it goes from there!

I'm looking forward to seeing how it goes, especially since this is one of my more challenging groups behaviorally. The fact that they get to do this when no other class does might just be privilege enough to motivate them!...or let's hope that, anyway!

Will update more this afternoon with pictures!

Higher Level Confidence

Despite 4 weeks of practice and warnings over and over that I would never just give them a question like "find the slope between these 2 points," my Algebra class was still flabbergasted that I had the nerve to give them a world problem on a test. And let's just say....I gave them pretty much all word problems.  As you may have guessed, it was our first real, honest to God, Common Core aligned test. For everyone's testing pleasure- no questions below comprehension level here! And as of today, I have 26 new enemies.

The tests are graded and the students did wonderfully, but ohhhhhhhhh baby are they mad. I had 4 angry emails right after...all from the same parent....whose student got a 93% on the test. It's not that my kids don't know the material, but any challenging question shakes their confidence. They have had their hands held and been given knowledge level tests and quizzes for so long....I don't even think they know that they're capable of applying their knowledge!

Does anyone have some tips for helping to build students' confidence with higher level thinking? I always thought the challenge would be helping them be capable of it, but we've done so much practice that they've demonstrated to me that they can handle it. I know as we practice more, confidence will build. But, really, how do you break the "this should just be regurgitation" mindset? 

Giving back the tests tomorrow will be fun, though. That should help in the confidence department! 

Proofs by David Bowie

When I was in college, I felt like proofs had personality. I have no doubt this came from years of pouring over them- from intrigued amusement to hopeless despair (I'm talking to you, hyperbolic planes). They joined the list of mathematical characters friends and I would create to numb some of the mathematical stress: the Jacobian who liked going on African safaries, the data my professor loved to put in the freezer, the triangles evil cousin Cryangle. And proofs? These were just some:

• The cute little ones which were made you giggle....toddlers, anyone?
• The elegant ones which so effortlessly seemed to make such a profound impact. These were definitely always wearing elbow length ladies gloves and pearls.
• The beastly ones that took up a lot of space without really saying anything. Any Planet Fitness members out there? These made my inner lunk alarm go off. I usually walked away going..."Well that's just dumb!"
• The ones with a trick up their sleeves. These always struck me as the salesmen of the group.
• The wise old grandfathers....proofs by induction. Figuring out what we definitely know is true. Building on that experience. These always seemed so reasonable to me. 
• The ones that are just really, really, really annoying. 

Then there was my favorite type...the smart, sassy, sarcastic ones....indirect proofs. The let you think something was true until BAM...they proved you wrong. They had attitude. They were the Real Housewives of my mathematical world. 

My students have not spent the endless hours pouring over proofs that I have and they also haven't needed such a mental break from the stress of a 500 level math class that they would begin to understand these characters. However, I wanted to do my favorite proofs some justice and let the kids do some exploring.

I set up 2 stations:

1) A set of sudoku puzzles 

2) This video from the 1980's movie The Labyrinth

1, 2, 3....

DEBATE! CONJECTURE! CONTRADICTION! 

All of this, of course, after the debate and conjecture about Jim Henson and David Bowie. Middle schoolers of 2012 cannot fathom the 1980's.

 

Great first day of the lesson. Wrapping it up tomorrow with discussing how we gave hypothetical conjectures, then tested them to try to see if there was a problem with them. 

Word of the Week: Centroids

Each week I have my students for a Math Lab. This was supposed to be an extra enrichment or intervention period, but it's turned into organized chaos. I have 20-30 students in 3 different courses all vying for help and attention. It's that class that I dread leaving for a sub....chaos. 

My Geometry students are at the top of the food chain and it breaks my heart to give these critical thinkers "drill" practice when they could be exploring. Today was one of those days when you're not sure if an activity it will bomb or fly. And this one? It flew!

We have been studying centers of triangles (orthocenters, incenters, centroids, and circumcenter). This has had 2 highlights for the kids: a foldable that they used to keep them all straight and giggles when one student kept slipping and saying "circumcision" during an oral presentation. They are always cooperative and brilliant, but I wanted to give them some reference into why this stuff might matter. Good thing I spent 3 semesters of college teaching physics! Centroids.....centers of gravity....suspending random objects around the classroom? Sounds like it's worth a try. 

I used this video on center of gravity to introduce the topic. One their interest was piqued, this video related the information to our chapter. I had each student design a triangle out of cardboard and find its centroid, then suspend it. Then they had to find some weird shape other than a triangle and try to find its center of gravity. Can you find a centroid for that? How is it alike and different? All questions to be examined! 

Anytime you get to hear "whoa!" and "that's so cool" is a good day in my book. I had to shoo students out of the room when the bell rang and we spent the first minute of class later that day balancing our triangles on smaller and smaller points around the room. One student got it balance on the head of a pin....he took it home because he just HAD to show his parents. Here's a great activity that I wish I had thought of before today....could have been integrated nicely. 

The only problem? The 20 Algebra students I had in the class found it nearly impossible to focus on their linear equations project with all excitement! They wanted to build their own centroids and were even starting to use the vocabulary. I love when, at the end of class, I catch students who aren't even in the course watching videos I've posted for other classes (also a problem with Vi Hart's videos!). As I was walking out to my car I found 3 triangles, centroids labelled, that had been made by students outside my class. Love when math leaves the 4 walls of my room! 

Any other ideas for integrating more physics into high school math? Obviously with calculus, but with lower level? Share share share!

Success: Redefined

There are lots of different ways I define success for my students. Yes, I am happy when my they "succeed" on a quiz and get a good grade. This success comes easily to some and only by labor and pain for others. No matter how small the victory, I try to celebrate it with that student. Then there are the times when the student who struggles through math every day makes the game-winning goal at the soccer game. Those moments are just as proud for me and often more enlightening. Getting to see someone at their best makes you re-examine your biases about them- something every teacher needs to do sometimes. Flash to tonight...

I played basketball in middle school. No one said I played it well, but play it (and love it) I did. I am an avid college basketball fan.....including an NCAA tournament trip and quite nearly 1.000 college basketball games throughout my life. So after a wonderful weekend, I spent my Sunday evening at school watching some serious hoops action. The JV and Varsity teams were having their first scrimmages and were also assisting with drills and practice for a local Special Olympic team. 

Tonight defined for me a third type of success....the type where I see my students succeed as people. These 12, 13, and 14 year old students tirelessly walked through drills and talked with people who are very different from them. My students were not afraid or "too cool"; they were inviting, gracious, and supportive. After drills and games, the Special Olympic team scrimmaged for the crowd. My students cheered on every success and supported every failure. They offered hugs and congratulations to all the players. 

One of those moments that reminds me that I teach kids, not math. Really amazing kids. Just something to be thankful for during this Thanksgiving week. 

Dimensional Analysis

While working extensively on rates of change, my class ran into the speed bump known as dimensional analysis. For some of my students, it came easily....they have a natural intuition about these types of relationships that makes it much easier for them. For others, it was agony. I was the dentist and they were having their mathematical teeth pulled. No one cares that there are 5,280 feet per mile when google can just tell you the answer, Ms F! Even Bill Nye's new "Solving for X" video was annoying to them....and if BILL NYE can't make it work, who can??

The turning point for all the kiddos came when I posed the following questions to them:

How many difenwarps is 2.931 yipyaps if there are 436.9 difenwarps per yipyap?

After a heated argument about the pronunciation of "difenwarp" (which, mind you, included them informing me that my pronunciation was wrong despite the fact that I invented the word), even the fast kids started to realize that their intuition wasn't always going to serve them in unfamiliar territory. 

I decided to introduce KWL charts to them as a means of gathering information. I always liked the idea of them, but for other subjects....kids want to know about their bodies or wars. Kids are less likely to want to know about combining like terms. Here's a sample of the one I used with them:

We included what we KNOW...the amount we start out with, what we WANT to know....the units we are aiming at, and anything and everything we've learned at that might help. After a few practice problems, I let them loose. Here were the results!

Ohhhh happy day! Seems like most of the resources I could find were high school chemistry labs. I'm thinking next year maybe we bake...they have to convert, they get a delicious baked good as a reward!  Any other good ideas for dimensional analysis?