Category Archives: Graphing

2017 Desmos Art Project

 


Our results were  so successful last year that we made only small changes to this year’s Grade 10 graphing/art project.  I made some small changes to the guidelines and the rubric to simplify and clarify things for students, and as always, the files are below in case they might be useful for you.  I’ve also included a pdf of this year’s student work in case some exemplars would be useful.

 

A few reflections from this year…

  • This is my sixth or seventh year doing some version of this task, and it was nice to focus on improving student work rather than improving the project
  • I am super proud of this year’s student work.  I had worked with many of these students as 8th graders, and it was very gratifying to see their growth over this longer time.  Some students who were not my strongest in Grade 8 did really impressive work this year
  • This project is time consuming for students, especially those who decide on ambitious piece.  Interim deadlines are key, and I think I will add one more next year to keep students on track so they are not in a panic just before the final deadline.  Although students were really proud, this did take a little of the joy out of the last few hours of work.  (But I think was in part because they had a big science test on the same day as this project was due)
  • Some of the more ambitious work included shading using inequalities, which were restricted by complex functions.  The functions were correct, but the software was a little buggy in rendering these, so they didn’t always look perfect.  A couple of students were a little frustrated by this – the one and only time I’ve seen students frustrated by Desmos
  • Drawing with graphs remains a powerful way to motivate practice and students’ interest in understanding how equations relate to functions
  • Asking kids to commit to re-creating something forced them to be purposeful and deliberate in every choice

 

Here are the guidelines and student work.  I also give students some sentence frames along with suggested vocabulary to help them with their reflective math writing.  Find me on twitter, by email, or in the comments below to continue the conversation!

Desmos Art Project Student Work 2016-17

Desmos Drawing Project Guidelines and Rubric_Revised

Desmos Drawing Project Reflection Assistant

 

MTBoS 2017: My favorite… tool for teaching transformations

My favorite tools for teaching transformations from parent functions are the Desmos Marbleslides. This is the first year that I have been able to use these activities to cement our learning across function families in our Algebra 2 classes. While these aren’t exactly skill and drill practice, they do seem to give students similar opportunities to do the repetitive work that is needed to build procedural fluency.

Just a few of the reasons I love these marbleslides…

  • They are consistently motivating, fun, and engaging
  • There are opportunities for creative solutions
  • They present open problems with multiple solutions, battling the idea that all math problems have exactly one answer that is in the back of the textbook!
  • As a teacher, I am always interested in and surprised by student solutions – very different from much of my grading
  • Students demonstrate perseverance through these challenges – they really want to come to solutions, and will keep working until they succeed

I made my first custom Marbleslide for students to practice transforming absolute value functions. My activity is basically an exact copy of the Desmos team’s work, but with Absolute value equations. The custom activity was very easy to build, and I am turning over some more creative ideas to explore now that I have done this.

I am pretty sure that part of our success with our understanding of transformations has come from the course map this year. We are basing this year’s sequence of topics around families of functions. We began with an informal study, just looking at shapes and appearances of graphs, and what kids of situations might be modeled by different function types, and have been adding formal analysis of each family with each new unit.  Starting with this big picture has given students a framework to fit each family into – they are connecting what is similar and what is different as they dive into each new kind of function.

It has been amazing to see – we have just gotten into trigonometry, and by the time we got to the sine function, kids were so comfortable with shifting graphs around the plane that I didn’t need to do any explicit instruction – they knew to play with the constants to get their graphs to shift in different ways, and with very little prompting from me, they argued out the differences between period and amplitude shifts.

I am excited to see how these understandings will transfer to the Desmos Drawing project this year. Last year’s students set a pretty high bar, but this year’s 10th graders are already demonstrating a deeper understanding – and 3 months earlier.  Stay tuned!

Shifting Populations

I’ve been wanting to write and think about this for a while now, and with the end of the school year, I finally have a little time to reflect on and share some more work from this year’s Algebra classes.  Population shifts make for potentially compelling and authentic math modeling tasks.  Last year, I had great success  with grade 8 students, who compared linear and exponential models of the population growth of developing countries, and made predictions for the population of their chosen country in the year 2050.  This work was adapted from Kyle Moyer and Zack Miller(@zmill415)‘s Booming Populations project.  I wrote up some reflections on that project last year.  This year, I collaborated with some colleagues to adapt this work for my current grade 8 students, and to extend this work for my grade 10s.  The rubrics and guidelines are at the bottom if you can adapt them for your use. :)

Computer Work

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.

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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.

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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.

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-updated

 

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.

BOOM

“The UN experts disagree about what the future will hold, so we figured that if we wanted answers to our questions that we would need to become the experts.”best_global_math_big_marker_logo.001

I was fortunate to catch the rerun of an excellent Global Math Department presentation by Kyle Moyer and Zack Miller(@zmill415).  They presented their approach to curriculum and instruction, which focuses on project based learning, and integration in the math classroom.  They included a description of their “Booming Populations” project, designed to study and compare linear and exponential functions by examining population trends and predicting the population of a country in the year 2050.  The materials they designed are well thought out and put together, and I decided to adapt the project for my Algebra 1 students in Cyprus.   This was a rich experience for my students for many reasons.

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I used a gallery walk format to build background knowledge and pique interest, and there was quick and solid engagement.  Students were fascinated by the world population trends, and were especially hooked by the leaps in population size over the last century as compared to the rest of history.  This example of exponential growth was both attention getting and highly understandable.   There was built in choice.  Students were allowed to choose a country – and I can’t overstate how much of a difference this makes for them.  They picked a country that they had some interest in or connection to; a family connection or a place that they had visited or wanted to visit, or just a country that they wanted to learn more about.  Choices ranged from China to Greenland to Peru to North Korea, allowing for deep comparisons of statistical trends, modelling validity, and evaluation of source data.

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The work was easily and naturally differentiated.  Students who were approaching mastery could plug numbers into a slope-intercept equation and into a standard exponential formula, and those who were ready could really push the nuances of their models.  I even had a few students dabble with quadratic models (…and this was before we had covered quadratics in class).  Advanced students could keep on adding complexity and depth to their predictions by taking into account more pieces of information – demographics, political stability, or even global climate change (will the Maldives still be around in 2050 or will the islands be underwater due to rising sea levels leading eventually to a zero population?).  And this was naturally self-paced as well.  Very few students reached a “stuck” point, where they needed to wait for the teacher to tell them where to go next.  Over the four weeks that we worked on this, I used a combination of discovery-based lessons and some direct instruction to help students build skills to be successful in this project.

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Students were asked to examine and compile population data for their country from 1960-1990, and to create linear and exponential models to study this data.  They then created a model to predict the trends that they would expect from 1990-2015.  After comparing this model to the actual population numbers, students committed to one type of model to predict the population of their country in the year 2050.  They were required to complete a written analysis, and to present their analysis and predictions to an audience including a “panel of experts” at our “2015 Population Summit.”  Knowing that they would be presenting this work publicly lent gravitas to most of what they did – they were invested in understanding and being able to explain the math that they used, and to justify the decisions that they made in creating their models.  They learned to harness the power of spreadsheets to help them to organize their data and to create graphs – a really great skill for them to practice.  The public nature of this work forced them to make accurate graphs, and to consider carefully decisions about scale, and how to best communicate data visually.

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This was definitely some of the best learning that I have been able to orchestrate as a teacher.  Every student achieved the basic learning targets, and most exceeded the standards.  Students were comfortably using vocabulary like linear vs. exponential models, initial condition, growth factor vs. growth rate, and I heard many arguments between students who were invested in defending the mathematical choices that they had made.  This project found that sweet spot between just enough structure to keep everyone on track, and enough freedom to allow students to make decisions and to own the work.

While I shamelessly use and reshuffle ideas from books and from the MTBoS, I nearly always have to tweak and remake the materials for my students.  The language, design, or content have to be customized to meet them where they are, and to give them just enough information to succeed without giving them so much that they don’t have the chance to do their own thinking.  The materials that Zack and Kyle have so openly shared (THANK YOU Zack and Kyle!!)  are as close to ready-made as I have found.  I made some minor tweaks to the guidelines and formatting, but used almost all of their work.  Their approach to teaching math is very well articulated, and their Global Math presentation is very much worth watching in its entirety as well.  Their use of “playlists” to help students self-direct is especially interesting.

I am hoping to develop this into a more interdisciplinary and comprehensive project for next year, and perhaps something that could be a staple of the 8th grade curriculum.  My goals for our math program include building inquiry into the math class process, and creating connections between math and other content areas, and I am especially interested in feedback on ways to leverage these things.  Please do throw your ideas in the comments.  If you’d like to see some student work or reflections, just drop me a tweet or an email.  While student presentations were strong this year, I will make sure to add in more rehearsal time for them to practice next time – especially when they request that the panel of experts ask hard questions.

“Hello and welcome to the 2015 AISC population summit. In our 8th grade Algebra class, we have been looking at world population trends, and thinking about what will happen going into the future.  The UN experts disagree about what the future will hold, so we figured that if we wanted answers to our questions that we would need to become the experts.

 Each of us chose one country to study. We examined our country’s population changes since 1960, and created graphs and mathematical models to help us predict what the population of our country will be in the year 2050.

 We compared a linear model and an exponential model, and decided which one we thought would make a better prediction for our specific country. We did some basic research into our country’s history to give some context to our math models.

 We hope that you enjoy yourselves, that you learn something, and that you are willing to ask us hard questions and give us critical feedback.”

BTW: The Desmos Penny Circle is of course a perfect companion/ follow up to this activity.

Epic Poodle Bungee Jump

My Algebra students just completed a project using linear equations to predict how many rubber bands it would take to give their poodle a thrilling, yet safe bungee jump from the school balcony.  This activity was adapted from NCTM’s Barbie Bungee activity.  I do appreciate the opportunity to discuss with students the cultural stereotypes and gender assumptions perpetuated by Barbie, but in this case, using these stuffed poodles allowed us to focus on the math and on the fun. IMG_1735

Students were asked to string together one, two, three four, five and six rubber bands, and then make a prediction based on their measurements.  The object was for their poodle to get as close to the ground as possible, without actually landing on its’ head.  They predicted how many bands they would need for the first balcony, and if they survived the first jump, the poodles made the jump from the second balcony.

Check out the students’ reactions at the 1:20 minute mark and the 1:36 minute mark – in my experience, this is not your typical reaction to getting the right answer on a math problem!  And look at the celebration from the winning team at 2:23!  This engagement lent leverage into our reflective learning process.

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I often struggle with how much structure and guidance to give students with a project like this.  I want to set them up for success, but it is important to me that students have the opportunity to think things through for themselves, and to have the opportunity to get the wrong answer.  My students are used to me telling them that the best thing they can do is to make a mistake …and then to go back and figure out where their mistake was (Thanks to Jo Boaler for sharing her convincing evidence of how the brain grows by correcting errors in our thinking!).  My preferred method is to give students a compelling task with very few suggestions, and to share tools and advice only when they are requested.  If an activity is choreographed well, they tend to ask the right questions.  And if they totally screw up their model, it is an opportunity to go back and see why.  If we can convince them to care, then when they make an error, they will want to go back and see where and why they made mistakes.

My students know that we learn math socially in our classroom, and that I always leave room for sharing of methods and thinking, but they are also used to two times when the class needs to be silent – at the start and the completion of tasks.  Everyone needs space to generate their own thinking and ideas  – and the room to come up with their own first guess (I have been convinced by Dan Meyer and others of the engagement generated by asking students for an initial guess).  We also make sure to reflect individually on our own ideas and performance, and this is a regular part of our class as well.  Everyone is expected to look back to see how well our plans and our models worked.

Of course, I include myself in this reflection as well.  I think that this one went well.

Who Are You Calling Math Face? (How DO you turn that frown upside down?)

Awesome idea + powerful and GREAT tool + Sweet math wall = this year’s math faces.

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Thank you Fawn, thank you Desmos, and thanks students for your cool ideas.

My students created their own “math faces” through graphing with linear equations, quadratics, conics, and a few trig functions.  They used Desmos.com (an online graphing calculator) to explore different facial expressions, and were asked to articulate the equations behind each feature of their graphs.  They practiced transforming linear and parabolic equations, and learned about restricting domain and range.  Students’ manipulations made subtle differences in their facial features as they figured out how to move eyes up, down or around a line of symmetry, or how to change amplitude or period of a sine function to make a thicker mustache.  I’ve done this project a couple of times now, and I am surprised each time how motivating creating a face can be for some students.

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Desmos has created this beautiful classroom interface, which allows the teacher to see all of the student work at once, or filter students by things like who’s used an inequality or which students have restricted the domain or range.  This gives me an instant formative assessment where at a glance, I can easily target my advice or questions for students.    But even more importantly, students can see each others’ work, and share ideas in real time.  We spent about one hour as a full class on this, and then students were asked to complete a math face of their own over the next week.   I did this project early this past Spring with both Algebra 1 and Algebra 2 students.

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I could see some students struggling with the transformations, but caring enough to work through the struggle.

Screen Shot 2014-03-26 at 9.39.59 AMI am very grateful to have discovered this task.  In my experience as an art teacher, differentiation is often natural and practically effortless.  A student can attempt to draw a portrait whether it is their first try or if they have been practicing for years.  There are different conversations I can have with students depending on his or her experience, but every one of them can approach the task  – and right from the start is set up for success.  Math teachers don’t often have this built in differentiation.  So often, our lessons are targeted to a highly specific set of procedures, for which students must be in exactly the right place in the Algebra sequence.  The low entry, high ceiling aspect of drawing with graphs makes it something we can return to with students again and again.

Mountain Climber Problem: A Nice Follow up to Graphing Stories

I have a new job this year: teaching Algebra 1 and Algebra 2 at an Expeditionary Learning High School.  So far, it is an excellent place; strong and present leadership, dedicated colleagues, and a mission of rigor, relationships, and relevance.  I am very lucky.

Student Graphing StoryLike many Algebra teachers tuned-in to blogland, My students and I worked on some graphing stories to begin the year.  We started with Dan’s Site, and graphed some of the actions (stories) included in the video work there.  Students then created their own actions and short videos (we have 1-1 iPads here this year! Hurray!).  I hope to produce some of these videos with accompanying graphs, and publish them here.  It would be great to have them for next year’s kids to use as well.

After our video analysis, we turned to some work from The Language of Functions and Graphs, an excellent (if somewhat dated) text.  Though the problems are not visually seductive and compelling for students in the way that the graphing videos grab their attention, they are very provocative in terms of the math thinking they demand – and like it or not, we live in a world where students need to practice with traditional looking materials so they’re not caught off guard when taking standardized tests or in their next math class where the teacher may have a more traditional approach.  I found an electronic version of this text HERE, and modified some of the activities – for example, I removed the text but used the graphic from page 42 and asked students to write a “story” depicted by the action in this graph.

The Mountain Climber

As a follow up to this graphing work, I asked them to grapple with a problem involving a mountain climber.  My teaching partner reminded me of this problem, but I’m sure I’ve seen it before.  If you know where it came from, please let me know so I can give credit where it’s due.  Here is the problem:  A climber leaves base camp at 6AM one day, climbs up to the peak and arrives at 6PM.  The next day, she leaves the peak at 6 AM, and begins to climb down.  At a certain point on the trail, she notices that she was at exactly the same spot at exactly the same time on the previous day on the way up.  The question is: What are the chances of this happening?  I asked for initial guesses, and there was unanimous agreement that this would be highly unlikely.  Students talked about this for 5 minutes or so, and as no one was graphing, I suggested that creating a graph might be one way to examine this problem.  After another few minutes, I strongly suggested graphing.  It is so difficult for students to connect yesterday’s work with today’s sometimes – I need to work on this!

Mountain Climber GraphI love this problem because the answer becomes totally clear when you make a time vs. elevation graph – and the answer violates nearly everyone’s expectations and leads to a surprise!  Many students got stuck in their initial guess, and even when we went over together what the intersection of the two lines implied, they tried desperately to draw a version of the graph where the two lines didn’t intersect.  When they figured out that even skydiving down wouldn’t work, some resorted to teleportation.

As a nice reminder, the whole Junior class hiked up a mountain to end the week, and at least 6 students brought the problem up to me during our hike.  It was cool that this one stuck with them.

UPDATE: I added some of the materials I used below in case anyone would like them.

Graphing Stories 2-Create your own

Graphing Story Entrance Ticket

Graphing Story Exit Ticket

Body Graphing

Here’s another wonderful activity I learned at the Dana Hall Math Conference.  I’ve written about more activities I learned there HERE and HERE.

Set up a Cartesian plane with x and y-axes wherever you have outdoor space at your school.  You can use chalk if you have a blacktop, or string, or cash register receipt rolls.  Label the axes, and label the scale on each one.  Ask students to choose a spot on the x-axis, (and make sure that some choose negative numbers).  Take a portable whiteboard with you (or you could prepare some paper ahead of time), and write a function of x (e.g. y=2x+3).  Ask students to “act out” their x-coordinate on the plane.

Body Graphingps2

My students got the hang of this quickly, but they also appreciated the challenge.  You can make this a mental math practice by giving targeted, or more complicated equations.  Or you can give them paper or whiteboards on which to calculate.  It worked well to illustrate functions – whenever one student was not “in line” with the others, it was obvious that something was wrong, and there was rich discussion about how to fix the problems.  It also showed in a memorable way where linear functions got their name (That was my exit ticket question for the day).

I had really good success with this, and the kids very much appreciate getting to go outside.  I recently did some professional development with an expert on brain development, who suggested that kinesthetic activities like this really cement concepts in kids’ memory as well.  An outside location was important because we could effectively illustrate how quickly some functions grow (compared to others) by allowing real distance for exponential functions.

I came up with an extension on the fly that seemed like a good idea, but didn’t work out so well.  I’ll share it in case someone has a thought about how to make it work better.  I asked one kid to come up with a function, and bring the other kids to x,y coordinates that fit their function.  Then I asked other students to try to identify the function.  It seemed like it should have been rich, but it fell sort of flat.  Please share any idea that might improve this.

There are definitely other variations on this lesson floating around, including this example, specifically geared toward slope-intercept form.

In a related post, Michael Pershan writes a nice post about functions HERE.  I especially love the yarn.  There is something wonderful about the aesthetic sensibility; about how that slightly sad piece of yarn is elevated by giving it “function” status.  While we’re on yarn, check out the yarn work from Minneapolis artist, Hot Tea.  I’m not sure how yet, but I’m confident that these artworks will have a classroom use at some point.