Some of my favorite Geoboard activities

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I love using math manipulatives.  For some kids, using visual logic, or a hands-on approach can help to remove barriers to understanding, and can take some of the intimidation and fear out of learning about difficult concepts.  Using Algebra tiles to teach polynomial multiplication and factoring quadratics help to reinforce an area model of multiplication, and completing the square – well it does make more sense if you actually complete a square.  (And if you get the lab gear designed by Picciotto, you can complete the cube too!)  I’ll post about my experience with these sometime soon.  For today, here are some ways that we have used Geoboards in my Algebra classes.

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My Algebra students had a blast using geoboards to find triangles and squares.  The Math Forum has put together a set of very rich ideas for using geoboards, and we worked over a few days to discover line segments, triangle area, and practice with the pythagorean theorem.  I asked students to use the geoboards to find solutions, and then to translate their ideas into pictures on dot paper by drawing all of their solutions.

I had some challenges handy for some students who were ready. Here are some extensions to the challenges posed by the Math Forum, which my students grappled with.

  • How many different squares can you find in a 5 pin x 5 pin board?
  • How many different triangles can you find in a 5 pin x 5 pin board?
  • How many total squares can you find in in a 1 x 1? …2 x 2? …3 x 3? How about nn?
  • How many total rectangles?  IMG_1322

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There was some beautiful and deep thinking going on, and this is a highly differentiable activity.  Students who are at the concrete level could physically count squares, while students who were ready to generalize could find the cubic (I guess that’s not too much of hint), which describes the number of squares in an area.  Teachers, feel free to drop me a note or tweet at me for my solutions.  In an effort to keep the problems from being Google-able, I haven’t included them here.

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Geoboards are excellent to physically see how  slope works as well.  For next year, I’ve ordered the 11 x 11 pin versions.  Stay tuned!  How have you used Geoboards in Algebra class?

 

Student Created 3-Act Math

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When will the world population reach 8 billion?

I integrated a bunch of 3-Act math tasks into my Algebra classes this year, and I love the spirit in which these can be presented.  3-Acts give math teachers the language of drama and storytelling, language often reserved for writing or drama class, revealing information bit by bit to students, and keeping them hungry for more through a regular dramatic format.  In my experience, this has been motivating for students, and this motivation has led to leverage for convincing students to care about mathematical rigor.  Thank you to everyone who creates and shares the work to make this possible, including 3-Acts, for the benefit of myself and my students.  MTBoS rocks, and it’s a verdant time to be a math teacher.

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How do slow and fast speeds compare in “Piano Tiles”?

For one of our final projects this year in my Algebra classes, I asked students to design their own 3-Act tasks.  Even though they had seen a number of these, and were familiar with this format, there was mixed reaction to this assignment from students (…and mixed results).  Some created really great work, but some still resent the idea that they are expected to be creative in math class.  Lots of students have a pretty narrow definition of math, and it’s really hard for many of them to shift in attitude – in spite of  my persistence this year in presenting tasks that required critical and creative thinking.  Lots of them have long since defined themselves as a certain kind of math student and have become accustomed to being taught procedures, and repeating them back on demand.  Just for context, while my school doesn’t totally track classes, in general the highest achievers are not in my sections, and I work with many concrete thinkers.

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What are Leroy’s chances of survival?

3-Acts are hard.  They are demanding on students, and they require rigor and precision, synthesis and critical thinking.  And this is a tough job for the teacher as well.  We need to craft the lesson in such a way that students actually feel a need for the math skills we want them to practice, and then make the right tools available and accessible at the right moment. I think that overall, I did have some success in shifting student attitudes in general towards math this year, and I think we created some good work together.  I think that just like me, students would get better at creative thinking in math with practice.  It would be good to try this assignment mid year, and then again at the end of the year.  During our share, they definitely enjoyed viewing and solving each others’ 3-Acts (some in spite of themselves).  And there is value in the final math class experience of the school year being so positive. Their 3-Act subjects ranged from estimation to frisbee to World of Warcraft.  Check out a few of their projects HERE or at the top of the page, and let me know what you think.

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How many mattresses?

BTW, I asked the following question on my end of year student survey: “What project/lesson/assessment have you learned the most from this year? Why?”  48/62 student responses included positive references to the 3-Act problems.  Here are a few excerpts:

  • “3 part problems- more realistic then normal word problems- feels like the math I know will be useful…”
  • “The 3 act problems. We did a lot of them and each time I could learn more of it. Culminating with creating one myself really helped to practice it even more.”
  • “3 act work because we never know what it’s going to be when you walk in class,”
  • “…you found something I was Passionate about and taught me how to make it in to a fun learning Experience.”

An aside: I’m heading out for an adventure, and will be teaching at an International School in Cyprus next year!  Even though I am excited about my new position, I am super sad about leaving my AWESOME school in Portland Maine: Casco Bay High School.  My colleagues were inspirational, demanding.  My math colleagues, and the junior team (BTW, check out the amazing 2014 Junior Documentary Work HERE) are world class educators, every one of them.  Derek Pierce, the school principal is a truly exceptional leader; supportive, inspiring, and kind.  I have been incredibly lucky to work for and with him.  Derek and my colleagues at CBHS encouraged me to take risks, and to push myself as an educator and a person, and helped me to encourage students to take risks.  Without their support, this kind of work would not be possible.  Thank you to everyone at CBHS! CBHS Staff Further Reading: I’m pretty sure that the 3-Act rubric I found HERE came from a Math Forum problem solving session.  I wonder how their students did?

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.

Intensives Week: Fractal Geometry!

Thank you Math establishment… for designing a curriculum that requires us to keep plowing through material.  For connecting teaching salaries to student test results and for keeping so much pressure on our class time that it feels like all we can do is skill and drill.   For creating a culture in which students think that math is only a series of formulas that they need to memorize, repeat, use on the test, and promptly forget.

Thanks!  No really… Thanks!  Because you’ve set the stage for kids’ minds to be blown when they are given real opportunities to study interesting mathematical questions.  This is a long post, but I had such a good experience, that I wanted to share details.  Skip straight to the bottom if you want to take the materials and run.  I promise I won’t judge.

DSC05551In January, and again in March, my Expeditionary-Learning High School dedicates a full school week to “intensives.”  Students choose from a list of course options and engage in in-depth study for five days straight.  This January, I led a study of Fractal Geometry.   A few kids signed up because they were interested in the subject already; more had math credits to recover.  But in spite of themselves, they had a blast.

Although this was far from a skill and drill study, it was rigorous work.  I’m all about learning targets.  This study had to be differentiated, as I had a group of students from 9th -12th grade all working together.  I teach college art as well as middle and high school math (I’m thinking of getting my 7-12 art certification), and as an art teacher, differentiation is often easy and natural.  Ideas and techniques become more sophisticated, but a beginner or a master can engage with creating a portrait.  The umbrella of Fractal Geometry cast a wide shadow, and allowed for many points of entry.

I’ve been especially interested in the current conversation in math education about what makes a problem “real world.”  Dan Meyer has distilled some of the more interesting arguments HERE.  I think that Fractals make for “real world” math study within multiple definitions.  Self-similarity is everywhere around us and is easy to spot – it’s actually hard NOT to see it once you’ve identified the property, and kids were pointing all over the place “…look a fractal!”  In his book, Fractals Everywhere (2000), Michael Barnsley puts it well:

Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same.

Kids were seeing fractals in nature, and in special effects, but they were also highly engaged with the abstractions that only exist within the math.  I had multiple kids emailing me with discoveries late into the evenings.

I’ve included all of the materials and plans below in case they might be useful for you.  Here are some (not so brief) highlights from the week:

DSC05585We’re in Portland, Maine, so I began with a local version of the coastline problem: How long is the coast of Peaks Island?  I printed maps of Peaks, and asked kids in pairs to use progressively smaller and smaller rulers and to try to answer the question.  We collected our data, recorded it on a chart and examined the results together.  “Wait a minute.  The coast gets longer and longer depending on the ruler!”  I let this sit in the air, and just moved the discussion forward by asking questions without adding my two cents.  Showing is so much cooler than telling!  It felt poetic to begin our study with the problem that brought Mandelbrot into the spotlight.  We visited his famous paper later in the week.

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You can see the care put into this visual exploration of iteration!

Next, we began an informal study of the Koch snowflake.  Students drew the first few iterations, and we generated a whole bunch of questions.  I nudged the kids who were ready to examine what happens to the perimeter and area as this shape is iterated.  Again, I was deliberate at this point about not doing any direct teaching.  I left their questions unanswered, which helped to build drama and curiosity.

We spent the afternoon of the first day with our anchor “text:” the excellent NOVA special, Hunting the Hidden Dimension.  Even for NOVA, this is a really exceptional resource; challenging, but highly understandable.  It’s worth purchasing a copy for the better resolution, but you can find it on youtube HERE.  I let kids know that their knowledge would be assessed through a piece of writing about one aspect of fractals, and gave them a note-catcher to record observations and questions.  I let them know ahead of time that everyone would be expected to share some learning, and I’ve found that a graphic organizer like this helps kids to remember to record their thinking while they watch a video.  We stopped the film about halfway through to share our first key learnings, and repeated this process at the end.

I heard some really beautiful comments amongst kids at the end of the day.  “I see math totally differently than I did yesterday.”  This is why I am really grateful to all those who tried to ruin math for these kids.

DSC05531On the second day, we delved more deeply into the Koch snowflake.  We learned about sums of infinite series, and the paradox of an object with finite area but infinite perimeter.  Wait.  WHAT?  Yup!  Minds blown again.  I introduced Sierpinski’s triangle, along with some more “mathematical monsters,” (they loved this historical context), and we made some charts to examine some of the properties of these classical fractals.  Work ranged from concrete counting of triangles to algebraic generalizations of nth terms to summing infinite series.  Differentiation in math can be so tough, but this was a place where it really came together, and I felt like most students were really working toward their personal best.  After a really rigorous morning, we spent the afternoon creating original works with two excellent sites: recursivedrawing.com, and Fractal Maker Express.

Day three, we looked through a “fractal library” I’d put together (I had a range of books; Fractals: A Graphic Guide was the one that really resonated in a broad way), and then played the chaos game.  HERE is a very nice electronic version, but I think that this is very effective if you print triangles on transparencies, pass out dice, and let kids play manually.  The chaos game has a truly unexpected result, and this comes together well when you stack the transparencies.  I asked kids to make predictions (in writing) every few minutes, and monitored carefully so that when each one got it, they didn’t spoil it for the others.  It was entertaining to see kids noticing the drawings of Sierpinski’s triangle all over the room.  “Wait!  It’s that triangle!”

Clint Fulkerson's Division-1

Clint Fulkerson’s Division-1

I made a connection with a local artist, Clint Fulkerson, who uses the logic of fractals to create some really cool art.  He was generous enough to let a herd of kids into his studio, and he was articulate about how his work uses properties of self-similarity and recursion.  He rode a nice line between a casual attitude, and rigor in his work; one thing I hope the kids take away from this week.  Check out his work HERE.  Clint has a show at the Portland Museum of Art’s Family Spaces, up through this July.

Kids were really inspired by Clint’s work, as well as his work ethic.  We spent the fourth day working on some individual and group projects.  Some kids created a paper version of the dragon fractal, while others studied strange attractors and the Mandelbrot Set.  We watched some excerpts from The Colors of Infinity, Arthur C. Clark’s 1995 fractal documentary.  Wow, documentaries have come a long way in 20 years (…and special effects, thanks to fractals!).

DSC05607Friday was for finishing up loose ends, publishing, individual reflections, group debrief, student write-up for the newsletter, and presenting our work to the school.  One of the best moments for me was when one of the students opened up our presentation by saying, “I thought that fractal geometry was going to be dry and boring, but it was actually ridiculously interesting.”

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Honestly, it’s hard to say how much of the week we spent doing rigorous math – it depends on how you define this.  And I had the advantage that this week did not have to stick to the confines of “Algebra 2.”  But I truly believe that any time spent on what some might consider “not mathy,” was more than made up for by their contagious interest and excitement about the topic, which gave me the leverage to dig deep.

Here are some materials and resources I used or created.  Send me a note if you’d like a more comprehensive resource list.  We use standards based grading, so you’ll see that the language matches “meets” and “exceeds.”  I’m hoping that the school runs this intensive every year, so please leave your thoughts or additional resource ideas in the comments.  Thanks in advance!

Fractal Entrance Ticket Day 2;  Fractal Entrance Ticket Day 3;   Fractal Entrance Ticket Day 4;   Fractal Exit Ticket Day 1;   Fractal Exit Ticket Day 2;   Fractal Exit Ticket Day 4;   Fractal Intensive Evaluation Form;   Fractal Intensive Learning Targets;   Fractal Vocabulary Blank;   Hunting the Hidden DimensionNotecatcher;   Peaks Island Tabloid;   Sierpinski’s Triangle;   Student_Newsletter_Writeup;   The Chaos Game;   The Koch Snowflake

The Beauty of the Jigsaw

Using a jigsaw is nothing new, but sometimes we hit upon the right format at the right moment.  I’ve had trouble drumming up interest in algebra word problems in the past (to be honest, I sometimes have trouble maintaining my own interest), but this jigsaw worked really beautifully.  My Algebra class includes a number of English language learners, and it seemed important to spend some time discreetly on dissecting, analyzing and solving word problems.

DSC05195We divided into four teams: mixture, work, systems of equations, and distance-rate-time problems.  Within each of these were three levels of difficulty.  Each team was responsible for learning their problems well enough to solve at least level one and level two problems, and for teaching at least level one problem-solving to another team.  For me, the work problems and the systems problems are the most straightforward.  The D-R-T’s are a little more complex and the mixtures seem to cause the most problems — something about moving between ratios and percent just confuses the heck out of them.  I was able to be strategic about who was assigned to each group.

We use a block schedule, so I typically have 60 or 90-minute classes.  We used the first 60-minute block, along with the first 15 minutes of the next block to become experts.  Students then moved back and forth between their “teacher” and “student” roles until they were able to solve level one problems independently from all four areas.  On the quiz, they were expected to solve level one’s from all four areas, and level two’s from at least three areas.  They could exceed the standard by solving level threes.

Screen Shot 2014-03-29 at 3.50.22 PMI did have some related activators, which helped to make connections and drum up initial interest.  For systems, I used some of Don Steward’s Cryptarithms, followed by this Ghost Whisperer Crystal Ball.  I was turned-on to this by Yummy Math, which has a nice lesson HERE.  The cryptarithms were surprisingly engaging and allowed us to practice place value in a way that felt like puzzles.  The Crystal Ball is just a blast.  Kids simply couldn’t believe it and were convinced that the machine must be listening to them, or that I was somehow involved in a conspiracy.  I highly recommend this as an activator or a stand-alone.  It demands a nice little piece of algebraic reasoning.  Dan Meyer’s Playing Catch Up is a 3-Act that goes well with these as well.

You can find these problems in any Algebra text.  I used some free ones from Kutasoftware, along with some I modified/wrote-up myself.  Here is the quiz I put together, along with some CCSS correlations in case they might be useful.

Word Problems Quiz

Word Problems Quiz With Answers

Word Problems Unit CCSS

Making Math Public and Visible

I had a very visual and enthusiastic group of grade 7/8 math students last year.  They were only too glad for an excuse to do some visual math like our super fun math and art show.

This year has been a little different.  My group is not naturally enthusiastic about math, and has needed some encouragement – and 11th graders require a different kind of nurturing than do middle-schoolers.  I’m definitely still learning.   I’m hoping that making some math visible will help to push things forward for my students, and help them to take their own work more seriously.

Math Wall

Here is the start of our Math Wall.  I put up 4 x 8 sheets of soundboard ($12/sheet), and covered it with paper to create a 14’ wide x 8’ tall bulletin board.  This is in the hallway outside of the classroom in which I teach 2 of my 3 sections of Algebra.

Fly or Drive

I put up our current project (…something I adapted from Dan’s Travel Lesson, and from Mr. Ward’s excellent essential question), with some teacher notes.  But I hope that my voice is eclipsed soon by student work.  Once there’s student work published on the wall, I hope that they get some public feedback.

Guess the Number

I’m also going to include a regular estimation or problem of the week, or post an interactive game on the Math Wall.  I’m hoping to nurture a culture of scholarship and connections with people outside of math class.  I’m starting with a great number guessing game I originally found HERE.  Hopefully it will get competitive.  Stay tuned for a report!

I’d love more ideas on how to leverage public space into student engagement and investment.  Thanks in advance for sharing your thoughts.

One of my favorite number activities is the 4 4s

I heard a colleague say recently, that math was good for three things:  “…making predictions about the world, making models of the world, and because math is beautiful.”  One of my favorite number activities, the 4 4s is low entry, and high ceiling, and the mathematical context by itself gives us a place to explore the beauty of numbers and relationships.

Mind your 4s

I created this floor to ceiling white board last year in my classroom with 4 x 8 paneling. Kids LOVED being able to draw and work on this scale!

I remember doing this myself as a student, and I was so glad for the reminder when I came across the Four 4’s activity at CAS Musings.  In a nutshell, the problem asks students to use four 4’s and any operations they can think of to get to each target number – I asked them to solve for every integer from 1 and 100.  The basic arithmetic operations +, -, ×, ÷ along with exponents, roots, decimals (4.4 or .4), concatenation (44), percentages, repeating decimals (.44…), are all allowed – and some funky ones are necessary (just try to get to 73 or 77!).  Depending on the level you’re teaching, you might include more advanced operations.  This was easily differentiated as well– I previewed the activity and assigned some specific numbers to students who needed quick success or more challenging work.

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There are many ways of getting to each number, and the multiple solutions leave room for kids to create and to DO math. Depending on the needs of your group, you could do this in a competitive or a collaborative way – for my group of 7th and 8th graders last year, collaborative worked better.  I think that my 10th and 11th graders would benefit from the competition this year, so I’ll add some structure and a prize this time – I’m thinking of awarding points for the most complex solutions as well as the most elegant solutions.  I’ll allow students to work together, and make sure to honor and highlight different ways of getting to each solution.

4s DetailLast year, this worked really well for reviewing and cementing proper notation, order of operations, factorial, multiplying exponents, and general number sense.  There were great student conversations.  My notes included snippets like “…wait.  Dividing by .4 is the same as multiplying by 2.5!”  I’ve included the worksheet I used below.

If you’re looking for a beautiful follow up, Fawn has of course upped everyone’s game with Foxy Fives

MATERIALS

MIND YOUR 4s.docx

MIND YOUR 4s.pdf

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

31 From 25

This is a quick one that I wanted to record here to remember, and to share.  Before we made our playing card Platonic Solids, I asked students to do a little number exercise with their cards, that I adapted from Sarah’s First Day Activity.

Playing Card Array

Their ticket to begin building the solids was the completion of this challenge: Create a 5 x 5 grid of cards, in which every row and column adds to 31.  We decided as a group that J, Q, and K were worth 10, Aces were worth 1, and all of the other cards were worth their face value.  The groups who began with some ideas of symmetry got to the answer much more quickly, but I had one group who just bulldozed their way through until they made it by sheer force of will.  We debriefed as a whole group, shared our different problem solving (…or bulldozing) methods. The kids were totally surprised to find that their solutions were not unique.  A nice extension would be to count the solutions.  How many ways are there to solve this array?

Next time.

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.