Transversals, KenKen, and the Art of Making Puzzles

I’ve taught Geometry for four years now, and this year I’m completely writing my own course notes, essentially a textbook. I burned the boats, so to speak. And yes, it is because I’m a crazy person. But, also the books cater to these state standards which can be so very mind numbing. if I were to read through the TEKS (Texas standards for you outside the good nation), I would see dregs like this:

“111.41.C.5  Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to: (A)  investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.”

If the numbers at the beginning didn’t get you, the inane droning of vocabulary might have. But you know what? Puzzles in Geometry are badass. Kids like to do them, I like to assign them, they’re stretching, and they actually match up with class standards.

In the Beginning

Sometimes I start with low-level, accessible puzzles like this one from KenKen.com. They have to use some logic, some elimination, but nothing so taxing that they’re spent.

image

What I’ve noticed is that the “easy” (in quotes because it is banned in my classroom) problems generally only have one difficult spot – once a particular square is filled in, the rest of the puzzle becomes rapidly finished. Old Windows users, think of Solitaire when all the Aces through the 8’s are already moved to the winning area – from that point on it’s just a click party.

Back to Geometry

In that light, when I see puzzles like this… they’re a little on the soft side.

image

There’s not really any conundrum. The only way I can really have fun with a puzzle like this is by letting all the students solve for angles 2, 7 and 8 and then telling them they’re wrong (unless they’ve explicitly assumed two lines are parallel). So it’s off to Google (or Bing if any sponsors are reading this) to find better puzzles.

From the Discovering Geometry page we can see this one is an upgrade:

image

This puzzle is interesting to look at. Most of the students can solve these pretty quickly – they’re accessible and they can check themselves by looking at groups of angles that form circles. They feel great when they finish, but they’re not the most cognitively demanding puzzles. We can reinforce some vocabulary (what’s the relationship between angle f and the angle with 64 degrees?), but the only other meta question for me that really comes up is: Which angle is the easiest to find first? Or, how many angles does it take to get to angle c?

In my estimation, there are only a couple of points where kids can get productively struggle – probably around getting angles d and c.

So I keep searching, when I come across this:

image

Ok, here we have something interesting. If I was in my first year teaching this class, I might explicitly give them some of the angles (say, angle b is 40), and ask them to solve for the others. If I’m really devious, I might try to sneak in some Algebra components (what if angle f equals 4x and angle a equals –8x +20??), but I’ve been hearing some powerful arguments against that.

Now that we have a diagram, and I’m in a safe place in terms of my job, we can start to ask a different type of question, one that I’ve never asked before…

How many angles must you be given in order to find all the other missing angles?

I’m no Christopher Danielson, and I don’t know much about Hiele levels of proof in Geometry, but this is the kind of question that I’d like to give to students more often. The discussion was fascinating. About half of the students were convinced that you only needed one angle. The other half were convinced you needed two angles. Even though we haven’t used the word “similar” in class once, after talking with others, most students became convinced that angle a and f were congruent, as well as b and ace, and acb and e.

I can still play the devil’s advocate pretty well – even if the entire class was convinced the number of angles needed was “two”. I can simply give them angles a and f and say “gotcha suckers”. 

Yes, I have academic standards to which I am accountable. But this underlying level of communication, this “System 2” thinking where students move from instinct to a prolonged, effortful attempt at problems is my main focus.

I haven’t done it yet, but this problem is going to be on their next assessment. I haven’t even begun to work it out for myself because I’m interested in how students approach it.

“What information do you need in order to solve for all of the numbered angles in the figure below?”

image

Other thoughts:

How  does one make a good KenKen puzzle anyway? What makes one “easy”, “medium” or “hard”. I share with you the rabbit hole of information I’ve come across, or I could say – have you ever tried to make one?