# Shifting Populations

My grade 8s studied the populations of two groups of snails, one in a tank with no predators, and one in a tank with some fish (…who apparently find snail eggs to be tasty).  My colleague from the science department, Heather Charalambous was kind enough to host this study, and to use science class time to support some of the conceptual thinking around how and why the snail populations changed (…and to count the snails!).  Kids used spreadsheets to create linear and exponential models, compared their two models, and made predictions about what would happen to the snail populations.  We checked their predictions against the actual number of snails at benchmark dates, and examined discrepancies between their predictions and the actual outcomes.  Materials are below.

For grade 10, I collaborated with two excellent colleagues, Julie Jonsson, and Rachel Iannacone to create another permutation of this project, looking at the question: What events have the most significant impacts on the populations of cities?  In grade 10, students study U.S. History, so we tweaked this to fit into their coursework.  Students were asked to choose one American City and  to examine their city’s population from 1850-1940.  As with the 8th graders, they created linear and exponential models to help them to analyze and make “predictions,” about what they thought would happen between 1940 and 1960.  They then compared their predictions to historical data, and made arguments about the reasons for any differences.  Students who were ready created some polynomial models as well – although these models potentially fit the data better, they are complex, and challenging to defend the contextual choices.  The culmination of the work asked students to look at their city’s population changes through 2014, and to make a future prediction for what they think will happen to the population over the next 35 years.

Improvements from last year…

• Collaborating with teachers from science and social studies really helped to make this work deeper for students.  Especially with the grade 10 project, students were forced to look beyond the math to examine why populations shifted.
• The math felt like it was in service of the compelling questions, and I think that students really felt like their math skills helped them to quantify and analyze an interesting problem.
• The grade 10 project guidelines and rubric were carefully honed down for clarity and depth, and were designed as a precursor to and preparation for the I.B. Extended Essay, which most of our students will complete in grades 11-12.  This improvement was largely due to the collaborative efforts of my two partners, Julie and Rachel, who were just awesome to work with.  We were able to really hash out our different opinions and priorities, without anyone feeling threatened or marginalized, and to keep working until the project met all of our standards — this was one of the best professional collaborations I have experienced.

Better last year…

• We did not have a culminating event for either of these projects this year.  Last year, we organized a “population summit,” where students presented their findings to a panel of “experts.”  Having to present their work publicly in this way really made students up their game.  This year, we did put up their work on the math wall, but somehow it wasn’t quite the same as public presentation.  Although presenting takes time, I really want to build this into the project if we can in the future.
• Although the students did get some choice in their cities, there were a few who did not get cities that they were that interested in.  This made for less engagement, and I want to figure out how to really make them feel like they have some control next year (…even if it is just the illusion of choice).

The Materials…

SHIFTING POPULATIONS FINAL ASSIGNMENT + RUBRIC

SHIFTING Essential Questions

SNAIL Project Guidelines

Let me know if you use or adapt this work for your classes, or if you have ideas for how to improve or deepen this work, and please send me a note or find me on twitter if you’d like to see some student exemplars.  Happy to share.

# Algebra 2 Concept-Based Map (Draft)

I’m in beautiful Pomos, Cyprus, having finished my second year at an international school in Nicosia.  Pomos is an inspiring place to work and to plan for next school year, and I am anxious to share the work I am doing with you for your thoughts and feedback.

This post reflects a current draft of next year’s work for my Grade 10 Algebra 2 class (Algebra 1 to come soon). I want to begin by gratefully acknowledging some of the most important sources for materials and inspiration for me.  My online MTBoS community is wonderfully generous, I have some top-notch local colleagues, and it is a truly great time to be a collaborative math teacher.

• Although I veer in several key places, my starting point for this map was the work of David Wees (@davidwees) and the New Visions for Public Schools’ Algebra 2 curriculum.
• The kernel of inspiration for the work was inspired by Glen Wadell’s (@gwaddellnvhs) big picture thinking, in particular THIS POST, which has been churning around in my head since last June when I first read it.  The way that he begins the year communicates a clarity, which connects his whole course together in a way that I wanted to emulate.
• Thanks to Pam Wilson (@pamjwilson) as well for sharing her linear equations unit, which was a big help for me.
• Henri Picciotto’s post on “Forward Thinking”, which helped me to focus on always keeping the big questions and concepts of the course at the front of my mind when planning.
• My colleague from the English department, Laramie Shockey, and her help in understanding Lynn Erickson’s Concept-Based Curriculum Model (which is a required strategy at my school, but which I had not found useful until Laramie’s mentoring).  This process was really clarifying and useful for me. I was doing many, if not most of these things already, but this is a concise way of framing it, which helped me to pull the pieces together.

By its nature, this map remains a work in process and is a living document. To keep this as relevant and lasting as I could, I worked to pare this down to the most important concepts for the course – but any curriculum map has very limited meaning until a group of students actually arrives. The process of creating this map really helped me to gain clarity about each part of the course, and what I want students to learn.  It helped me to know what to remove from the course, and what to prioritize.  This has to work differently for each school, and yours will naturally have to be different from mine, but here is some of what guided my choices:

• Sequencing – whenever possible, begin with informal before formalizing both in the small (day by day) and the big picture.
• Include multiple exposures to ideas – for example, identify linear functions visually in Unit 1, formalize and practice skills with linear functions during Unit 2.  Compare linear functions to exponentials in Unit 3, and model with linear functions in unit 7.
• My map is based on Virginia State Standards (My school’s standards of choice) with the addition of the CCSS Math Practices, but in addition, my curriculum for grade 8 and grade 10 is geared to prepare students for the IB math program in grades 11 and 12.  In addition, I teach some very specific skills to support the grade 10 science curriculum.
• I teach equations first, and then functions. I find that students can work with functions more fluently once they are comfortable with the algebra. Although this is different from the New Visions work, I have had success with this sequence, and it seems to work with the populations I teach.
• We include right angle trig and a study of vectors during A2 to support grade 10 students who take physics during the second half of the year.
• Generally I pared down the language in this document.  While I like specific academic language, this version was developed with kid-language in mind. I want kids to actually be able to say the things that are written as generalizations when a visitor comes in to ask them what they are learning and why.
• In addition to the sequence of topics, I included a “Unit X,” which emphasizes the importance of problem-solving and cultivating the habits of mind of a mathematician in grade 10 math.

I found that the concept-based model helped me to focus on what I wanted students to know and to do, but I haven’t yet made the whole of the model useful for me.  It’s quite possible that I don’t fully understand the concept-based system, but I don’t understand the importance of the one word conceptual lens or one word for Macro/Micro Concepts, so I’ve left these out of my maps.  My school asks that teachers use a specific model (Atlas Rubicon – Yuk, \$%#&@, and Blech!!) for our internal sharing, so I do have to include them in that version. Feel free to send an email if you’d like to see these as well, but for me they weren’t that useful. Please let me know if you understand these better than I do and can lend some insight.

My process:

This process has to be personal and specific to your situation, but here were my steps.

1. Name the concept (unit)
2. Sketch/draft generalization for the unit.  Brainstorm Essential or Guiding Questions
3. List important topics, facts, procedures
4. Write the related generalization for each topic, fact or procedure
5. Revisit unit generalization based on what happened during steps 3-4
6. Translate topics/facts into “Critical Content” (What students should know) and “Key Skills” (What students can do)
7. Design formative and summative assessments
8. Cycle through 1-7 until they (mostly) match each other and I am (mostly) satisfied with them
9. Correlate with my standards to see if I’ve missed anything
10. Cycle back through 2-9
12. Organize the “Possible Learning Experiences” – this is the most fun for me – I love to source, modify and/or create and choreograph the experience for my students.  This document does not yet include this part, but I will publish it here soon.

Steps 1-7 are cyclical for me, and I think you could start anywhere as long as you cycle through these until they all match – this was one of the real moments of clarity for me. I would write a unit generalization, and then realize that it didn’t match the facts/topics. I had to decide which one I had to change, which forced me to make a clear decision about what I wanted to prioritize. I wanted to connect my guiding questions with my essential understandings. If one was in there without a clear reference to the other, I tried to visit them until there was a match, or I felt that there was a reason to include one without the other.

I would love your feedback on this map.  Does the sequence make sense? Am I missing anything critical?  Is my language kid-friendly enough? Academic enough?  Do you do things in another order that works better for you?  Thanks in advance for your thoughts!  Here are the maps in Keynote and .pdf format.

Algebra 2 Conceptual Course Map DRAFT 3.key

Algebra 2 Conceptual Course Map DRAFT 3

# Math + Art + Desmos… Connections.

“I love math and art, and I’m glad that I was introduced to Desmos, a way to use both subjects at the same time.” – Marianna, Grade 10

Drawing with graphs has been a powerful way to motivate students’ interest in understanding how equations relate to functions, and how manipulations of equations lead to transformations from a parent function.  I jumped on to Fawn Nguyen’s Des-Man project as soon as I saw the idea, and have done some incarnation of this work each year.  Each time I’ve guided students through this process, it’s gotten better and deeper, both through the development of my own approach, and from improved tools like the Desmos Des-Man interface (…which I’ve heard is currently “in the shop” undergoing some improvements) and more recently tweaks to this idea like the “Winking Boy” challenge, created by Chris Shore (@MathProjects), and posted on the Desmos Activity Builder by Andrew Stadel.

They took the responsibility of recreating their chosen image seriously, and honestly, their work exceeded my expectations. There were regular exclamations of satisfaction echoing around the room as we worked on this. They persevered. They definitely attended to precision. They argued with each other about the best equations to use. They reflected about how to make the best use of Desmos. They practiced the habits of mind of successful mathematicians.

When we shared the in-progress work for some peer feedback, kids were actually applauding each other when their work came up on the screen.  Not because I reminded them to be a supportive audience, but spontaneously.  Seriously.  And when they saw the staff creative picks at Desmos, they asked me whether they might be able to submit their work.  The whole class was taking pride in creative math work.

I asked that students reflect in writing on their learning during and at the end of the project.  I haven’t asked for students to do enough writing in math so far this year, so when they seemed to be really struggling with this, I made a fill-in-the-blank “reflection assistant” to scaffold their thinking and writing, and to give them some ideas about what to include in their written analysis.

A few highlights from their reflections:

• “I was quite surprised that I could replicate a drawing by using graphing.  If somebody asked me to do it last year, I would say that it is a “mission impossible.”   However I was able to do it.”
• “As my piece of art, I chose the logo of the football club Barcelona because I am a big football fan and FC Barcelona is a club worthy to be recreated through the use of quadratic equations in vertex form. In addition, the logo was an appropriate challenge for me, containing easy and smooth curves but also difficult shapes, like letters or circles. When the project was assigned, I was skeptical that it was possible to recreate an artwork, just by using equations. But now that I am done and a proud owner of a recreated art piece, I strongly believe that it is possible (obviously).”
• “I found out that desmos is a really good tool to practice and sharpen your understanding on any equation and in my case it was the vertex form of a quadratic. Desmos allows you to experiment and find new ways to fix the problems or even work more efficient in order to surpass the problems in the first place. I am proud of the detail and sharpness of my work in general. I tried really hard to make the whole piece smooth and detailed. In order to do so, I zoomed in a lot and by doing so, I identified minor mistakes and was able to fix them.”
• I chose “Pumpkin Pepe” as the subject of my project because it provided the right level of challenge for me and it was really fun to do. Overall, I really liked this project because it solidified my knowledge of graphing equations and has made me more comfortable using parabolas. I found that my understanding of quadratic equations really improved while I worked on this project because before, I wasn’t sure which variable shifted the parabola which way, but now I understand.
• “I found that my understanding of parabolas and linear equations really helped me improve, and made me more confident during my work on this project. At first parabolas seemed to not make any sense to me, but now I feel like I really understand the way they work. Now I have the capability make connections with all these equations in the real world.”

Here are the project guidelines, the rubric, some peer editing forms, the “reflection assistant,” and a .pdf, which has a range of student work.  My rubric borrows from the I.B. Math Internal Assessment Guidelines, as one of my tasks as a grade 10 teacher at my school is to do some specific preparation for the I.B. program in grade 11.  Thanks in advance for any feedback on this project, and on the guidelines and rubric.

Desmos Drawing Project Guidelines and Rubric

Desmos Peer Feedback

Desmos Drawing Project Reflection Assistant

Desmos-art-project-student-work-2015-16-updated

I was struck recently when reading Robert Kaplinsky’s post, Why Are You Using That Problem?.  In this piece, he articulates 3 different reasons that we might choose to use a particular problem: to introduce a concept, for productive struggle, or for problem completion – each of which has its own value.  Kaplinsky convincingly argues that we should be purposeful not just in which problems we choose, but in why we choose a particular problem.  I have been considering his thoughts in reference to productive struggle, and specifically in how to best to support students when we choose a problem with this as our goal.

I have always talked about encouraging and nurturing a classroom culture in which it is safe to take risks, and to experiment.  When I introduce a novel problem, I tell students that my expectation is that they try – that making an attempt is what I value.  My marking scheme always always includes credit for getting started, and for each step of a problem.  But in some ways, my marking scheme has been generally geared toward a sequence of steps leading toward a correct answer.  In awarding credit for specific answers, I have been communicating to students that I value those answers.  If I want students to put value on productive struggle, I need to demonstrate that that is what I value – and by extension, that is how they can earn marks.  (For now, I’ll leave the question of whether earning marks in general is productive at all…)

I worked with Dr. Andreas(@DrAecon), my excellent colleague in the chemistry department, to create a set of guidelines and a rubric to support students in becoming independent and confident problem solvers.  In our guidelines, we tried to give students some explicit ideas to try when they are presented with a novel situation, and in our rubric, we tried to express criteria that values the process more than the answer.  We want students to make valid arguments, to justify their reasoning, to persevere in problem-solving,  and to feel that they can take risks, including taking a path that might not lead to a correct solution.  I hope that our guidelines and materials communicate these values.  I want to be explicit and transparent with students about this.  We’ll need to set aside time where we are not focused on learning specific content, but instead are focused on becoming strong and reflective problem solvers who know that they can earn respect and grades with this focus.  I want to put my marks where my mouth is.

Here are the rubric and problem-solving guidelines, based on Polya’s four steps.  I’ve included both Word and PDF versions, and a blank version, which is intended for students to use as an organizer for their work.  These are written with 10th grade students in mind, but I plan to create simplified versions of these for middle and elementary school students.  Please let me know if you can use these, and if you have any suggestions for improvement.

# There Can Be Only One (Marker)

Observing a student working on a whiteboard is the best way that I’ve found to get immediate insight into his or her thought processes.  Perhaps because of the impermanence of the medium, students act much more freely than when working on paper.  They are more willing to take risks and to potentially make mistakes.  Even when writing in pencil on paper, the act of erasing is slower than it is on a whiteboard – it seems like they can think more quickly and freely on the whiteboard, which leads to a more fluid thought process, and less barriers between their thinking and their writing.

I’ve been a huge advocate of students’ use of whiteboards since I began teaching.  One of the very first things I do when I move to a new classroom, is to cover as many surfaces as possible with whiteboards.  Asking students to stand up and work in a visible way has the immediate effect of increasing sharing of ideas and showing thinking in a public way.   And it’s fun and they just really like it.

John Orr’s whiteboarding protocol in his recent “My Favorite” post (Week 2 of the MTBoS 2016 blogging initiative), has filled in a piece that was missing from the work in my classroom.  When he groups kids at the whiteboard, he gives each group only one marker.  Every few minutes, he calls “marker switch” and whoever has the marker has to give it up to someone else in their group.  Sounds simple, and I know that I have read about this somewhere before (I think maybe in Henri Picciotto‘s blog but I couldn’t find the reference), but I never thought that it would have the profound effect that I observed when we tried this.  When the kids all had markers, some would inevitably be drawing instead of  mathing,  most would be doing their own thing, and they would occasionally talk to each other.  Providing only one marker forced communication and collaboration in a way that I just hadn’t anticipated.  If they wanted to express their idea but it wasn’t their turn to draw, they had to argue for it verbally.  I can’t recommend this strategy highly enough, and it has had a very positive effect in my class.

A related, but maybe non-mathy aside: I was an art teacher, both at the college and the community level for years before I got into math teaching, and have continued that work along with teaching math.  Most of my life as an artist has been focused on making sculpture, but I did some animation and installation work for a couple of years, which involved a technique I learned from studying the South African artist, William Kentridge.  The process involved making a charcoal drawing and taking a photograph of the drawing.  The drawing would then be erased and/or altered slightly, and then photographed again.  This process was iterated again and again and again (this must be related to my interest in fractals…).  The photographs could be played in a sequence, which gave the illusion of motion.  Here is an example of an animation of some flying bats, which I used as a projected component of an installation piece.

For me, this process of animation was extremely freeing.  I was not afraid to make marks on the page because I knew that whatever I did would be erased soon.  There would be a record of the act of making the drawing, and all of the pieces would come together to form a whole, but each individual drawing would only be seen for fraction of a second.  I think that kids experience something similar to this when white-boarding.  They are more inclined to take risks and just try things because there is no danger of permanence.

Try giving them only one marker!  And let me know how it goes.

# Generating questions

MTBoS blogging initiative, week 3!  This week’s prompt focuses on questioning.

My colleague, who teaches the grade 6 and grade 7 math courses at my school is in training to run a marathon.  He has put together a training program for himself, which includes a schedule of endurance-building, and he has been collecting data with a GPS watch.  As he examined the data, he thought that this might make a rich exploration for his students and we have been working together to set up a project  for them.

Here is what the raw data looks like

We started with the driving question: How long will Mr. Feutz take to complete the Limassol marathon? and then we began by brainstorming questions together.

• How long will Mr. Feutz take to complete the marathon?
• How many steps will he take to complete the marathon?
• How many calories will he burn during the run?
• What percentage of his overall time will be spent moving?  (Compared to taking breaks)
• What will his average heart rate be during the marathon (In B.P.M.)?
• What will be the shortest/longest mile time, and what is the range between these?

We tried to analyse which questions are actually interesting, and what might we be able to ask kids to do with them.  While running, he found that he was constantly doing math of one sort or another.  How much further will I run today? When will I arrive back at home?  Things that he had genuine curiosity about, and questions that math gives us the power to answer.

We set up a graphic organizer, and decided to ask our driving question directly.  Here are some of the kids’ initial responses.

They always manage to think of something that we haven’t anticipated (…which is why we love kids!).  “Will you be listening to music, because if it’s like… Taylor Swift, you wouldn’t be as inspired as if it was like… hard rock.”  Students were given a graphic organizer and asked to write their first guesses.   As they acquire more information, they will refine their estimates.  Part of the beauty of this work is that they will get to actually test their prediction, and compare their answer to what actually happens.

Students will revisit this project over the next weeks, and will be asked to refine their work.  They have already studied unit rates, and are moving into work on ratio and proportion next.  We are hoping that more questions will arise as we continue this work.  My favorite so far is, “Given a start time, time spent running so far, and a map of the run, can you figure out where Mr. Feutz is now?”

Here is our Graphic Organizer – totally open to your critique and suggestions.  What questions can you add to our list, and how do you come up with your project questions?  We would be most grateful if you share your curiosities or strategies in the comments.

Related:

# Create Suspense – MTBoS Week Two: My Favorite

For week two of our blogging challenge, we were asked to write about one of our favorite lessons, games, resources, tools or strategies. It was tough to pick one.  I have so many excellent resources and tools, that as I reflected on what to write about, it made me once again realize what a great time is it to be a math teacher and just how lucky I am.  What a hard but awesome job, and what a generous and sharing community we have.

I really like keeping students in suspense.  If I can set up a situation where students want to know what’s coming next, that often translates into engagement and the desire to learn.  When you watch your favorite TV show, and it ends on a cliff-hanger, you make predictions and you think about it in between episodes.  You are connected to and invested in the story, and you can’t wait to see what happens next.  I want my classes to have at least some of this kind of anticipation.

I also like to create some public presence for math in my school, and I try to create a bit of suspense around this as well.  Typically, a few days before we publish our work on the math wall, I will put up a provocative question, or something to generate interest. This week I just put up a funny title with a big question mark, and listened for the buzz..

As a culminating activity of learning about graphing linear equations, I asked Algebra 1 students to create “math faces” through drawing with graphs. They used Desmos (…it was very tempting to write a series of “my favorite Desmos” posts – everything those guys do makes my classroom better!) to create their works of mathematical art, to practice transforming linear equations, and to solidify their understanding of domain and range.  I ask students to make sketches ahead of time to ensure that they are purposeful in manipulating their equations.  This is an activity
conceived of by the incomparable Fawn Nguyen, and one that I use every year.  I have written about it before as well.  This kind of task gives all students an easy entry point, but allows for real complexity for those who are ready. This low entry, high ceiling aspect of drawing with graphs makes it a rich and motivating activity that we can return to with students again and again.  Although the Des-Man activity is not currently available through the teacher dashboard at Desmos, I have heard that it is getting a make-over and that they will be bringing it back again.  Each time, I am amazed at how motivating this activity is for students.

After a couple of days, we published our process and our results on this year’s “Sweet Wall of Math!”

How do you create suspense and anticipation in your classroom?

# 2016 MTBoS Blogging Initiative: The One Good Thing Glow

As a math teacher, it’s really fun to work with students who come to my classes already loving the subject, students who have already mastered the content from last year’s course so they are ready to dig deeply into novel problems in our courses, and students who feel really good about themselves as math thinkers.  Doesn’t this describe most of our kiddos?

The reality is that a large part of my job ends up being working to alleviate the trauma that students bring with them to Algebra 1 or Algebra 2.  They have known since second grade that they are not a “math person.”  They feel that they are not as good at math or as fast at math as others around them.  Many of their mathematical experiences have left them feeling inadequate, and math has been a place where their self esteem has been eroded.  Not a big surprise that they have trouble accessing the beauty of the subject.

One of my grade 10 students showed up this year with all of the marks of earlier trauma.  She was reluctant to speak in class or even when I worked one-on-one with her.  When she did answer a question, it always sounded like she was asking rather than answering (…y’all know that “I have no idea if what I’m saying is right” tone of voice).

This week, she had a perfect score on her linear systems and inequalities assessment.

And this wasn’t an easy test.  I always include some questions that ask kids to synthesize and apply, and to recall ideas from earlier work – typically, I have very few 100%s.  In fact, hers was the only perfect score in the class.  Although I generally avoid comparing students to each other, I couldn’t resist sharing this with her.  You should have seen her trying to pretend that she wasn’t beaming!  I wrote a note the her mom, to share how proud I was of this effort and of her success.  When I saw her mom the next day in the school lobby, there were tears in her eyes while we spoke about this.  She said that her joy had nothing to do with the score.  She didn’t care about the test score, but she could see and feel the difference in her daughter’s confidence and sense of self.  I couldn’t agree more.

What a wonderful way to begin my Thursday.  The One Good Thing glow stuck with me all day!

# 2016 Blogging Initiative

I am participating in the 2016 MTBoS Blogging Initiative.  I am doing this in part in to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS

I’ve just dusted off my “About Me” page to include the schools at which I’ve taught, and I’m excited for the next month.  You, too, could join in on this exciting adventure. All you have to do is dust off your blog and get ready for the first prompt to arrive January 10th!

# Lies and Collaboration

I actually enjoy the puzzle-like aspect of exponent rules, and simplifying radicals.  For me, there is something satisfying about learning ways to manipulate numbers and letters – probably why I love algebra so much.  But I am tuned-in enough to my students to know that many of them don’t find the same satisfaction from doing this work just for the sake of intellectual exercise.   And since calculators came into vogue, it’s been harder to justify the need for rationalizing the denominator or expressing the square root of 50 as 5 times the square root of 2.  But we are tasked to follow standards that often include these kinds of skills and it has been helpful for me to turn this into exploration or game learning as much as possible.

I did some mining of the MTBoS for ideas to teach rules of exponent arithmetic and came across this post, which includes a nice exploratory worksheet from Andrew Stadel.  He describes a similar issue with contextualizing exponent rules for middle schoolers – one of the really great things about our online community are these moments where we are reminded that we are not alone.  He asks his students to find the mistakes in the equations, to explain where the author went wrong, and to find the correct solutions.  He used a bunch of the common misconceptions found on mathmistakes.org to help students to catch themselves in the common errors.  Very nicely done.  This would have been a good lesson as is.

Then I remembered the Bucket O Lies protocol from Nora Oswald at Simplify With Me.  Nora manages to gamify math like no one else that I’ve seen.  She manages to add entertainment even to potentially dry topics like this one.  I combined Andrew’s worksheet with Nora’s idea to make a bucket-o-exponent lies.  I printed the 3 worksheets, cut them out into individual problems, folded them up, and put them into buckets (or baskets).  Voila! Drama and Motivation.  In pairs or threes, learning happened.

Of course, I hammed it up with the students.  There’s nothing like telling teenagers that someone is trying to get one over on them to motivate them.  This has worked well for me in the past, especially when it came from advertisements.  I riled them up by acting outraged that someone had created this whole set of math material, which was full of mistakes!  (Actually, I blamed Andrew :) ) …Lies I tell you… these baskets are FULL OF LIES!  Let’s find the mistakes so we can write a self-righteous set of corrections back to this author who was deliberately spreading bad math.

They quickly saw through my act, but it was enough.  They were already motivated in spite of themselves.  Andrew’s worksheet was just enough for everyone.  I started by coaching the groups who needed help getting started and moved to pairs who were making mistakes with fractional exponents.  For my honors group, I added a few more examples with rational exponents.

Thanks Andrew!  Thanks Nora!  Our generous community is the Best!