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.

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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.

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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
  11. Add important unit vocabulary
  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

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Callisto, 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.

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Amit, Grade 10

This year’s work was definitely the strongest yet, and I owe the major improvements to my reading of  Nat Banting’s post, which extended this project to another level for my students. In the past, I have asked students to create a graph, which had features of a face or a building or a plant.  This year, I asked my grade 10 students to choose a graphic, photo, or work of art, which they had to replicate using only equations.  I asked that they choose an image that was meaningful to them for some reason, and then helped to guide them to something that was challenging, but that they could accomplish – a natural moment for differentiation, built in to the process.  In the earlier versions of this project, students had been motivated by trying to make their face look angry or happy or sad, but they didn’t have a specific place where their equations had to end up. Asking kids to commit to re-creating something forced them to be purposeful and deliberate in every choice.

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Ilyas, Grade 10

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.

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Anastasia, Grade 10

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.

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Karim, Grade 10

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.”
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Marianna, Grade 10

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

 

Put Your Marks Where Your Mouth Is

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…)

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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.

Problem_Solving_Rubric_Grade_10

PDF: Problem_Solving_Rubric_Grade_10

Problem Solving Protocol_Grade 10

PDF: Problem Solving Protocol_Grade 10

Problem Solving Protocol_Grade 10_Blank

PDF: Problem Solving Protocol_Grade 10_Blank

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.

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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.

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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.

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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.

Bullseye

This is a short reflection from a lesson focused on solidifying understanding of linear and absolute value equations with Grade 8 Algebra 1 Students.

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I created a game, based on the Green Globs software.  I’ve never actually used the original materials, but it looked like it would be a highly motivating activity, and being on a tight school budget, I decided that since I wouldn’t be able to make the purchase, next best thing would be to use Geogebra to make my own materials.  I called my game “Bullseye.”  I bet that the original version is slicker and more complex than mine, but it worked pretty well for us.

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Here is what a “game board” looks like.   The basic idea of the game is that you need to write equations which, when graphed, hit the green dots.  Your team scores points based on how many green “orbs” your graph hits.

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I grouped students in pairs and gave them whiteboards.  I handed out the rules, and projected the game board.  Students had 2 minutes to decide on their best two equations.  At the two minute mark, we called “markers down,” and students held their equations in the air.  We entered them into Geogebra and calculated their scores for the round.  I also stole the scoring from the Green Globs people: for each equation, 2 points for the first orb, 4 for the second, 8 for the third, etc. doubling for each additional orb.  Asking them to work in pairs was key.  They were forced to talk and argue about the best two equations to choose.

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Students who spotted the absolute value equation in this one ran the table!

The “Expert” games included “Shot Absorbers.”  If your graph hits a shot absorber, you don’t score any points.  When these were on the board, I also allowed inequalities, but you might want to allow piecewise functions or Domain or Range restrictions if that’s where you’re at.

My 8th grade group this year is by far the most competitive group with whom I’ve worked.  They are just dialed-in when they are competing against each other (There is a total ruckus in the room when we play Grudgeball!).  I have to admit that I am not much of a gamer.  I don’t really play games, and I’m not a very competitive person.  But we need to adapt to the group that we have.  These kids are really pretty good sports.  They desperately want to win, but they are also good losers.  Even though Nathan Kraft has decided that it is potentially destructive to his classroom culture, it just works for my kids.  And as long as I have them playing in pairs or groups, at least there’s collaboration in addition to competition.

Here are about 12 game boards along with my instructions.  These could be very easily modified to work for quadratics or whatever functions you’re studying.  Let me know how it goes if you try this out, or if you have ideas for improving the game.

Bullseye Game Files and Instructions

UPDATE (2/7/2016): Of course several better versions of this activity surfaced quickly.

Generating questions

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

betterquestionsMy 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.

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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.

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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

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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..

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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!”

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How do you create suspense and anticipation in your classroom?

2016 MTBoS Blogging Initiative: The One Good Thing Glow

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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

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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

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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!

The Thirsty Crow

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Not sure which one I saw first, but I got the idea for this lesson hook from at least two teachers: Jensilvermath, and Pam Wilson. Both are creative educators, and generous online colleagues, who share their ideas, resources, and materials.

Screen Shot 2015-12-03 at 7.33.57 PMOne of Aesop’s Fables tells the story of a crow who comes across a half-full pitcher of water in the desert. He cannot reach the water until he figures out that by dropping pebbles into the vessel, the displacement causes the water level to rise until he can quench his thirst. Using this narrative as our lesson hook, students were given a cup full of marbles, and a graduated cylinder partially filled with water. They were asked to predict how many marbles they would need to reach 2000 mL, and then how many more until the water overflowed.

I have found that giving too much structure can take some of the life out of a task, but not enough structure, and students flounder. In this case, I asked them a direct question, but did not suggest any methods at first.  As we were right in the midst of linear equations, my assumption was that they would jump right to dropping their marbles into the cylinder, creating a scatterplot, find an average rate of change and line of best fit.  But students always surprise me.  They asked for an extra graduated cylinder to do some experiIMG_4298mentation, and pulled out the scale to start weighing marbles.  They traced the cylinder base to see how many marbles fit in that circle.  As we had more than one color, it was important to them to see if the lighter blue marbles were consistent with the dark blue – something I hadn’t even considered.  One group even qualified their prediction with the caveat, “…if the ratio of light blue to dark blue marbles is consistent with our sample, then
this prediction should hold.” What a nice expression of understanding. Reminder to self: always give students as much freedom as possible. Let them run until they really need help.

The students who dropped their marbles into the cylinder one at a time collectIMG_4312ed data points as the water level rose. They created scatterplots of this data, and calculated an average rate of change. Next, they used this information to find an equation for a line of best fit, which helped them to make a confident prediction about how many marbles they would need to bring the water all the way to the top. We took out enough marbles to test their predictions, and added them to the cylinder until the water level reached 2000 mL and then until it overflowed. Cheers and groans for the accuracy of their predictions.

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Creating ways for students to create mathematical models and make predictions is one of the most important opportunities that I can set up for them.  These types of tasks help students to connect the math from their classroom to questions that they will come across in the real world. Even if they will not need to calculate the number of marbles to overflow a cylinder, they will almost certainly need to use similar problem-solving skills, and equally importantly, they will have to decide what math skills they need to apply to novel situations.  Students react very strongly when they see the “answer” to this type of task – very different from how most students react when looking up the answer in the back of the math textbook.  Even reluctant mathematicians couldn’t help but look closely as we counted the last few marbles out!

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I used this video to introduce the Crow and the Pitcher. It’s short but gets the point across. I shot a video of our cylinder, and edited it into a 3-Act format while I had the supplies out.  I think that if you can get your hands on some marbles and a vessel, you may as well do this hands-on, but in case you don’t have a bunch of marbles handy, or if video is your preferred medium, I’ve published the materials below for you to use.  Did I give enough information in act 2 or did I forget something?  Please do let me know if you use any of this, and how it goes …and don’t forget to check out the Action Version…

Thirsty Crow Act 1

Thirsty Crow Act 2

Thirsty Crow Act 3

Thirsty Crow Act 3 Extended (Includes Action Version!)