The Un-Puzzle

I’ve heard this said a thousand different ways:  a task does not need to apply to the real world in order to be engaging.  Dan Meyer’s version seems to be thrown around most often:  The “real world” isn’t a guarantee of student engagement. Place your bet, instead, on cultivating a student’s capacity to puzzle and unpuzzle herself.

Today is Homecoming Friday.  It’s tough to get students engaged today, as their minds are on the game and the glitter (oh, the glitter).

Here’s a very short video clip of the noise level in my Algebra class.  Crickets.

No, I’m not giving a test.  I gave them a puzzle called Quadruple Sudoku:


In short, besides regular Sudoku rules applying, the four small numbers are clues as to what goes into the boxes touching them.

And both classes, all period, the brain sweat was palpable.  Why, on such a wild, exciting school day would these kids be so focused and so engaged?  The answer I come up with every time is puzzling and unpuzzling.  unnamed (7) unnamed (6) - Copy unnamed (1) - Copy unnamed (2) - Copy unnamed (3) - Copy unnamed (4) - Copy unnamed (5) - Copy

By the way, Nrich has tons of these fun, puzzling, engaging variations on Sudoku.  Check them out.

A Twist on Old Venn

How many of you went nuts over the Google Doodle for John Venn’s 180th Birthday?  I have no shame in admitting I spent more than a few minutes messing around with it.

These not-so-modern overlapping circles of wonder have fascinated mathematicians, scientists, and even linguists alike.  When searching for rich tasks for my college algebra classes, I came across this new twist on the traditional Venn diagram:


This activity can be applied to all kinds of topics with the main task being to find an equation to fit into all eight of the Venn diagram regions.  Since we are working with systems of equations, I offered this challenge to my classes:

Can you find three graphs that all intersect and also each intersect one another at unique points?  Also, is there a 4th graph that does not intersect the first three?  


Out came the iPads and Desmos.  Here are a few highlights:

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Some of my observations during their work time:

  • A few of them assumed we were creating an actual Venn Diagram with Desmos. I made sure the expectation was more clear the next period.
  • Attention to precision was important.  Some students assumed that if the three graphs appeared to cross one another, their task was complete.  They were mistaken when I zoomed in to examine the intersection points.
  • Students assumed that if a graph did not intersect another in their viewing window, it didn’t intersect at all.  We had some good conversation about where graphs might cross as the x and y approached infinity.
  • Using sliders in Desmos makes this task more doable in one class period.
  • I wonder if they would be able to solve for their intersection points algebraically.

Side note:  these RISPs (Rich Starting Points created by Jonny Griffiths) are all available on this website, and are excellent starters for college level mathematics.  

If You Give Homework, I’m Talking to You

Maybe it’s because it’s Friday and this has been one action packed week, but I am FIRED UP.  I’m fired up about the amount of out-of-school homework we give our students, especially in math class.

Casey Rutheford had a great idea the other night.  He did a Twitter search for “math homework” and examined the results.  Go ahead and take a look for yourself.  You may not be shocked at all, but reading tweet after tweet of math homework making students cry should make you, as an educator, want to sob.   Additionally, with impeccably good timing, John Stevens gave us all something to think about in regards to the homework debate.  The entire post is worth every second of time you can spend with it.  He highlights the student voices in this conversation.  Those voices are the ones we often aren’t really listening to.  He reminds us that there is a whole child to develop, not just a math brain.

The big question I have for my fellow educators is:  are you taking the time to listen to your students’ voices?   Are you considering the education of the whole child, especially during the hours when they aren’t in school?  What purpose does the homework serve?  Is it really fulfilling that purpose?  Do we really feel students do better as a result of homework, or are there other factors that play a much bigger role?

I’m not saying don’t ever assign homework.  I just don’t think homework needs to be a knee-jerk reaction to the end of a math lesson.







Nrich – For What It’s Worth

One of my favorite problems (and the one I presented at TMC this year) is What it’s Worth? from Nrich.  To say I “like” this problem would be like saying Sarah Hagan “likes” interactive notebooks.  Clearly an understatement.

Anyway, here’s the prompt:



What I like most about this problem is that there are so many, OH so many, methods to solving it.  It is a FANTASTIC way to get students to focus on the pathways to the solution rather than the solution itself.  After the students figure out the value of the question mark, they go about discussing the numerous methods they used in order to arrive at their answer.  Furthermore, the problem includes 6 “beginnings” of solutions and learners then need to make sense of those as well as determine how a solution was reached along that path.


Left: New format for discerning methods. Right: Old format for discerning methods.

To my surprise, along with Nrich’s site updates, this problem has improved as well.  Rather than showing a written start to the problem, provided are 6 visual introductions.

This allowed for an incredible amount of discussion involving each method.  And even those METHODS broke down into different methods.  It was method madness (awesome madness).  0917141216-1

Welding Math and Metal – Day 2

If I needed to choose the most productive portion of most students’ week, Monday morning, first hour would be pretty low on the hierarchy of engagement.  I was undeterred because making sure we had the correct solution was important.

We discussed that the radius of the spool would decrease every time a layer of wire was used.  They began calculating the resulting wire as layers were removed.  This served as an excellent opportunity to introduce summation notation and a great practical use for the mathematics behind it.  It seemed like a much better option than to add up dozens of calculations anyway.


When we arrived at our correct answer (with the desired units) of 1.98 miles, the questions and estimating didn’t end.  They wanted to know how far they could stretch such a wire.  Would it go to the edge of our campus and back?  Would it go from here to the middle school?  Could you go all the way to the grocery store?

They settled on taking the wire, running it out to the edge of the soccer practice fields and then running it all the way to the middle school sign.  It ended up being, to the hundredth, the exact amount of wire we had, provided that someone would stand and hold the wire at the edge of the soccer field.  I loved the attention to precision. I also loved that they were so savvy with Google Earth.



The Welding of Metal Tech and Math

An early morning text from my brother prompted this tweet:

We estimated a little, but the Slinky didn’t yield much discussion until someone shared: I wonder how much wire is in one of those Mig welder spools!

If you are like me and have no idea what a mig welder is, here’s a photo:mig_welder

These things hold massive spools of 1 mm thick wire.


Luckily, the welding instructor was willing to part with one of these for the hour.  Unluckily, the only information we had was the  1 mm thickness of the wire and the spool’s weight of 44 pounds.

Our initial thought was to weigh a snippet of wire and then scale it to the entire spool, but I was sure that the welding teacher wouldn’t have appreciated the rogue math teacher messing with his supplies.   I praised them for the interesting method anyway.  They then began measuring:  height, diameter of the spool, diameter of the inner circle, diameter of the wired portion, number of wires going up the spool.  It was math-magical.

Here are their calculations:


Now, I realize that these are a little off because we needed to take into account that the circumference of the spool is getting smaller as we move inward.  But I was very pleased with their work thus far and their vigor in posing this problem and then working to solve it.  I’m excited about where we can go with this type of problem posing.  There are only eight students in this class, all boys that have metal in their bloodstream.  I am hoping that with these sorts of ideas, I can engage them in math that excites them more often.  Maybe I can even get them excited about my Slinky question.

Sitting in a Circle, Talking about Numbers

“I feel like all we do is sit in a circle and talk about numbers.   It doesn’t even feel like work.”

“This class is more exhausting than my PE class!”

“It’s nice to be confused and then un-confuse ourselves.”

These are words I’ve overheard from my college algebra students this year.  I couldn’t be more pleased with the strides they are making with my problem-solving framework.  I learned the hard way last year that you cannot just throw a problem solving scenario at a student and expect them to immediately persevere, even if they understand the underlying mathematics involved.  Having learned from my mistake, I sequenced the problems this year in a way that has worked to build on their Algebra problem-solving skills.  Furthermore, I’ve put them in groups of 3-4, which has helped tremendously in getting them to talk about their approaches.  Last year, while in pairs, the conversations didn’t occur as naturally as I had hoped.    Here are a few of the problems we’ve tried:

Multiplication Square C thumb (1) thumb


Additionally, we’ve used other Nrich problems such as Odds, Evens, and More Evens.

And to add some non-dairy whipped topping to this algebra awesomeness, my students are breezing through visual patterns and having some great conversations about them.  Credit here is due to their fabulous algebra 2 teachers who began visual patterns with them last year and let them struggle with them.  The result has been deeper connections and a more thorough understanding.