Category Archives: Geometry

Some of my favorite Geoboard activities


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.










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

Screen Shot 2014-11-16 at 4.17.04 pm Screen Shot 2014-11-16 at 4.17.13 pm







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.


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?


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.


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:, 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.”


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

Balloons +Tangerines

I totally stole adapted Sam Shah’s beautiful and inspired Related Rates and Balloons Lesson for my 8th grade math class.  I remade the graphic organizer to be a little friendlier looking for my middle school students, and to give them a chance to practice drawing with a compass and ruler.  They particularly liked the language of the degenerate circle.  From hence forth, that’s how we’ll be referring to all points.


After wrangling about the balloons not really being round, and about the fact that circles only exist in your mind anyway (you really need to enjoy these moments if you’re going to be spending time with 13-year olds), we measured the circumference, surface area, volume, and weight of balloons inflated with one, two, three, four and five breaths, and then calculated radius and diameter.  To connect with our study of functions and graphs from last month, we plotted and graphed our results and compared the linear and not-so-linear functions.  This was a great hands-on way of comparing rates.  I saw light bulbs go on for several kids when they realized that one function was getting bigger faster than another.

Two things I didn’t foresee:

  • Some of my kids are actually small enough that they couldn’t blow up the balloons.  Not a big deal, but interesting to remember.
  • Many kids wanted to inflate the balloons with more than 5 breaths.  I encouraged this, but somewhere around 8-10 breaths, my dollar-store balloons began to give out, and we were all a little on edge from popping balloons by the end of class.  Next time, limit to 5 breaths, or buy better ballons!


I like using the highly technical tangerine peel method to help explain the surface area of a sphere, (Thanks to Miss Quinn for reminding me of this!) though mapmakers might have something to say about trying to flatten the sphere in this way…

Concepts Of Circles And Volume Of Spheres Through The ORBEEZ Lens

It’s nice when there is the coincidence of a Common Core Standard, and the perfect experiential activities for your grade level.  To explore concepts of circles and volume of spheres, (CCSS 8.G) I adapted Dan Meyer’s Orbeez lesson.

The Orbeez marketing materials are so wonderfully sickening.  I made my Orbeez Orbinizer to match the color scheme.  Many of my students do better with a set of boxes like these to give them some boundaries, and I create an extra version with lines inside the boxes for a few.  Some do all of their work on graph paper, and staple it to the organizer, and one or two turn in much of their work electronically.  Good thing I’m easy.

My kids love being skeptical almost more than they love being sarcastic, and when I suggested that the marketers of Orbeez might be having us on when they claim that “Orbeez grow to 100 times their original volume…” they got all riled up, totally outraged, and were off and running.

What I Did

  • I asked them to work in pairs, played the Orbeez advertisement video, and put up the above image of the Orbeez materials (I blurred out the word “volume” and changed it to “size” so that kids would have to decide what the makers meant by “size.”  See below for the doctored image).  We brainstormed questions.  I fanned the flames of outrage, and then passed out the Orbinizer.  I ask kids to fill these out individually.  Even if they’re working in pairs, they each have to have individual writing as an artifact of what they’ve done.


  • Once we decided on the things we would measure, kids needed to strategize about how to go about this.  They thought of some ingenious methods of measuring the tiny beads; my favorite was lining up 10 dark colored Orbeez on white tape, measuring with a ruler, and dividing by 10 to get an “average” Orbee (Can that really be the singular for Orbeez?) size.  At a certain point, they realized that they needed to be able to calculate volume of a sphere to find the truth.  “Hey Nat, how do you find the volume of a sphere?”  Gotcha! Join me at the whiteboard…
  • Some direct instruction here.  There is interesting math in deriving the volume of a sphere, and if kids know the Pythagorean Theorem, they can wrap their heads around this – even if it did cause a little smoke to come out of their ears by the end.  I used THIS Archimedean logic as the basis for my discussion.  Even though my kids aren’t ready for calculus, they could understand conceptually what was going on here, and I think that it will help them to remember the ideas if not the formula.
  • I gave them the sphere volume and surface area formulas, and let them go for it.  They measured diameter, converted their measurements, and came to individual volume averages.  We averaged the whole class’ data as well, and got the Orbeez soaking for the next day.

Next Day

DSC03898 DSC04196

  • Kids got to class bursting to check on their Orbeez expansions.  Ensue fierce calculating.  Actually ensue “What was the formula again?” and, “Eww these things are GROSS.”  Then fierce calculating. Exclamations as they mostly confirmed the claims.  Some unexpected outrage that a few Orbeez actually grew BIGGER than they were supposed to.
  • Brought the class back together, and showed them how the makers had come to their claim.  The discussion on Dan’s blog made for very authentic experience for them.  We averaged the class’ findings, and sure enough our results came to almost exactly 100 X the original volume.  Usually the world is a little grungier, but it worked for us this time.  I’ll take it.
  • We settled down for some traditional work on a problem set about sphere volumes.  I suppose you have to at least expose the kids to what a textbook looks like.  :)


Orbeez Orbinizer

Orbeez Ad: Manipulated

There Can Be Only Five

These Platonic Solids – made out of playing cards – are one of the contributions my 8th graders are creating to exhibit during our Grade 5-8 Math and Art Festival next week.  The templates I used were from Jason at Mathcraft and George Hart.  Thanks to cheesemonkeysf for sending a Tweet out to share this resource.

Platonic Solid 1 Platonic Solid 2 Platonic Solid 3 Platonic Solid 4

Students had to articulate the characteristics of the five Platonic Solids, and describe in writing how their cards represented each solid.  Can you believe that there are really only FIVE of these?  It totally seems that there should be more!  My students and I spent some time looking for that elusive Platonic Solid #6.  I asked students to define in their own words each of the Platonic Solids.  These writings will accompany their displays.  I’m looking to challenge them a little further – if you have another idea or resource for a more complex card construction, leave a comment. (Thanks!)

For the few students who were ready for this, I had some reading available about the Duality of Platonic Solids, along with an inductive proof that “there can be only five” possible Platonic Solids (…wait is this where the idea for Highlander came from!?).


Triangle Centers and Student Empowerment

I’ve always found triangle centers to be highly intriguing.  The initial related mathematical questions are low-entry.  But the ideas can go really deep, and there are fun, and valuable visual and logic skills to exercise.

With my 8th graders, I began by giving out an Entrance Ticket.  We had practiced with the Pythagorean Theorem, finding missing side lengths, missing angles, and area, but we hadn’t explicitly referred to triangle centers before this.

The ticket asks them to plot the center of each triangle, but doesn’t give them any more specific instruction.  Most students wanted to know how to do it.  I was totally non-committal.  I pretended to be busy setting up the projector, and told them to hurry up so we could move on to the lesson.  I gave them a few minutes, and then collected the tickets.

Next, we visited Dan and Dave’s Interactive Ice Cream Shop.  This app asks you to decide which ice cream shop you should patronize depending on your position in the park, to describe regions of the park where you would decide on one shop or another, and then shows how your intuitive work compares to others’.  After getting over the idea of why anyone would possibly shop at a place called “Frozen Flavored Milk,” students definitely enjoyed this process.  (Thanks Dave and Dan for putting it together and sharing it so publicly!).  This is a great, low-pressure way to begin the exploration.  Click on the picture to visit their site.

Dave and Dan Ice Cream

Many of them now thought that this was the “right” way to find a triangle center, and wanted the entrance tickets back to revise them.  Their requests went totally unrequited.  We reviewed our tools for geometric constructions, including bisecting angles, and drawing perpendicular lines, and then headed for Geogebra.  Students created their own circumcenters (although we didn’t use the word yet – thus far, this is “Elsie’s method”).  Oh, and they had to pass the “drag test.”

VLUU L200  / Samsung L200            Appo 3

Then, we watched Vi Hart’s Infinity Elephants.  Kids love the Apollonian Gaskets especially, and to make these, we just needed to find the incenter.  “Wait a minute.  This is different.  Which one is the center? This one or the Ice Cream one?” I cruelly asked them the same question that they were asking me:  “Which method is right?  Which one is the real center?”  I act busy and move on to someone else.  Kids go to Geogebra, and create an incenter.

Now, I passed around some sheet metal triangles that I had lying around (detritus from my night job).  “Try to balance these on the eraser of your pencil.”  I let kids experiment for a few minutes, and then asked them to pass their triangle to the next group. Before attempting to balance this one, they had to mark a point on the steel.  How did they come to that point?  Back to Paper, pencil, and Geogebra for the centroid.

Construction 3Construction 2Construction 1

I’m still looking for an orthocenter hook.  The useful things I could think of are only useful if you’re already into the geometry (leave an idea in the comments if you have one handy), but my kids were into it by then anyway.  I just gave them another rule and asked them to construct bisectors.  As students constructed these, I wondered out loud if there was any difference between this center and the first one.

I asked them to complete the exit ticket.  Several still asked which one was the right one.  “Which one is actually the center?  No, really this time.”  I desperately tried to hand the power back to them by asking them the same question.  “YOU get to decide.”  That’s the whole point.  For me, it’s all about giving them the power to decide on their own definitions – based on their own criteria.

What exactly is the center of a triangle anyway?

Here are some extensions to have handy.  I used some of these over the next few days.

  • Given 3 cities, what city is at the circumcenter, centroid, etc?  Middle school kids love to find things like this on maps.  You can ask the question referencing ice-cream shops or balancing the entire state on a pencil eraser of that seems like fun.
  • How could you find the centroid of square?  A rectangle?  A pentagon?  Any irregular polygon?
  • In which of the centers is it possible that the center is outside the triangle?
  • Euler Lines – So cool!
  • This problem is from Cynthia Lanius, and was only peripherally related, but it was there and relevant at the right moment.  The Three Evil Dictators: To keep them from fighting, you must separate three evil dictators as far away from each other as possible on the surface of the earth. If you put one in Los Angeles, California, name two other places where the other two could be placed so that all are equidistant, and at maximum distances from each other. (Yes, you can put them in the ocean if you have to.)



Triangle Entrance and Exit Ticket