Geometric Puzzles at Asilomar

I'll be presenting a session on Geometric Puzzles at the Asilomar meeting of the California Math Council. (Saturday, Dec 2, Sanderling, 1:30pm The printed program says I'm in the middle school, but that is not correct. The app has the location right.). I will include material that I believe is relevant to teachers from kindergarten to tenth grade. Hoping to see some of you there!

Given that this is CMC-Asilomar's 60th anniversary, I thought it would be a good time to reminisce: this is nearly the same topic I presented in 1984. After rejections by several publishers, my first book (Pentomino Activities) had recently come out. It later got combined with two other pentomino books, and that combination in one big binder remained in print for about 30 years. It may still be available from mheducation.com, item # 0884883744. I highly recommend it. A new version, with fewer puzzles, can be purchased from Didax. It comes with an e-book version, so you can project any page from the book, and you can manipulate virtual pentominoes on the screen.

My pentomino obsession was followed by a series of puzzle books on polyominoes and supertangrams, all of which are now free on my Web site. After that, I moved on to other curriculum development projects, but I maintained an interest in a tool-rich pedagogy and a puzzler's ethic, both of which originated in this early involvement with pentominoes. The more observant among you may have noticed that my Web site logo is based on a pentomino P:

--Henri P

PS: I link to many of my geometric puzzle creations here

Puzzles for the Classroom

In my last post, I shared some generalities about puzzle creation. Today, I will zero in on the specifics of creating puzzles for the mathematics classroom. I will do this by way of analyzing some examples.

Multiple Paths

A characteristic of all classrooms is that they are constituted of students whose backgrounds and talents vary widely.  Offering multiple puzzles simultaneously can help, as it allows students to find their own way through the set, by selecting puzzles at the appropriate level of difficulty, and/or by pursuing partial discoveries. This addresses classroom heterogeneity, while having all students work on closely related problems. Here are some examples along these lines:

• Staircases: find sets of consecutive whole numbers whose sum is 3, 4, 5, etc.
• Egyptian Fractions: find three fractions with numerator 1, whose sum is 4/3, 4/4, 4/5, etc. For example, 4/5 = 1/2 + 1/5 + 1/10
• Make These Designs: find linear functions whose graphs create these designs.

All three activities allow the students to find their own path through them. They avoid a common pitfall of curriculum development, which is the hubristic belief that one is capable of writing a single sequence of puzzles that will work just as well for all students. This is a common failing of both traditional and contemporary curricula. For example, the consistently brilliant Desmos environment offers teachers and curriculum developers the ability to craft one-path-fits-all sequential lessons in the Activity Builder. The best Activity Builder lessons, such as Marble Slides, incorporate many excellent puzzles. This is vastly better than most supposedly "intelligent" educational software, which tries to eliminate the need for teachers and is based on reductionist and insulting memorize-the-algorithm-and-practice sequences. Still, one can hope that a future version of the Activity Builder will allow the creation of choose-your-own-path activities.

Features of Effective Classroom Puzzles

In addition to the availability of multiple paths, the above three examples also share other properties.

1. They are reversals of standard classroom activities. Instead of the mind-numbing request to "add these numbers", "add these fractions", "graph these equations", the questions are reversed: "find numbers whose sum...", "find fractions whose sums", "find equations whose graphs...". Reversal, in fact, provides a powerful mechanism for the construction of classroom puzzles: start with what you're trying to teach or apply, and reverse the question. Voilà! You've created a puzzle.
2. They are non-random practice of important skills. Drill is not necessarily a bad thing, but random drill is boring and thus can be counter-productive. In these examples, drill is in the context of an interesting overall quest, and thus much more motivating. Also, unlike random drills, it lends itself to reflection, discussion, and generalizing.
3. They are each a set of related puzzles, rather than one-of-a-kind puzzles that rely exclusively on "aha" insights. Therefore, solving some of the puzzles helps the student develop skills and intuitions that can then be applied to other puzzles in the set, and more importantly, contributes to their mathematical maturity. This also means that they provide an excellent environment for teachers to provide hints, and scaffold student learning. For example: "solving this easier puzzle will help you make progress on the one you that is currently frustrating you."
4. They are interesting to both kids and adults. I have used these in the classroom with students at various levels, and in professional development sessions for teachers, and found that they are just as engaging for all. This is in part due to their "low threshold, high ceiling" quality: all include simpler and more difficult puzzles. Moreover, they suggest additional questions, such as the creation of similar puzzles, or the generalization of results, or the need for a proof.
5. They involve significant mathematics and carry a substantial "curricular" load. They are about the math teachers and students already know they should teach and learn. Using non-math puzzles as a "change of pace" is a waste of precious class time, and gives students the wrong impression that "normal" math is no fun.

One cannot expect all these criteria to apply to every classroom-bound puzzle or puzzle set, but hopefully they are helpful guidelines for teachers and curriculum developers.

More Examples

Geometric Puzzles
As a young elementary school teacher, in the 1980's, I encountered geometric puzzles in Martin Gardner's books and columns. At the time, there were nice tangram-based materials for elementary school, such as a fantastic set of puzzles by EDC, but there was not much using pentominoes. I decided to create my own sets of pentomino puzzles, suitable for students. The key insight was that puzzles that did not require the use of the full set were much more accessible than the 12-piece puzzles discovered by Solomon Golomb and popularized by Martin Gardner. More accessible, but still interesting, and in many cases extremely curricular! I started with well-known puzzles from recreational mathematics, explored them on my own, and translated the fruits of that interest into classroom materials. This was an ongoing creative obsession over many years. You can read more about this work on my Geometric Puzzles page, though in fact this has infiltrated many other parts of my work as a curriculum developer. [Note to Northern Californians: I'll be talking about Geometric Puzzles in the Classroom at the Asilomar meeting, on Dec 2. See you there!]

Algebra Manipulatives
One of the features of the lessons I developed for algebra manipulatives in the 1990's involves a crucial re-envisioning their role in the classroom. The standard algebra tiles lesson is based on the idea that the tiles illustrate what is going on with the symbols. In my Lab Gear materials, I turn this around. Start with a geometric puzzle: arrange these blocks into a rectangle. Then interpret what you accomplished with the help of the rectangle model of area. This is more fun, more accessible, and in the end more effective. I also introduced a whole genre of perimeter puzzles (e.g. use an xy-block and a 5-block to create a figure with perimeter 2x+2y+2), and visual patterns based on these blocks (what is the 10th figure in the sequence? the nth?)

And yet more examples
I will not comb through my (freely downloadable) Geometry Labs and Algebra: Themes, Tools, Concepts to find all the puzzles they include sprinkled throughout, but I should mention my puzzle-based approach to geometric construction, which I presented in multiple blog posts and on my Web site.

Puzzles Throughout?

As you know if you've read this far, I'm a big fan of puzzles in math education. However, there is no one way: while puzzles are an essential ingredient in effective teaching, they are not everything. There are very interesting and fruitful explorations that cannot be described as puzzles. Still, even topics as dry as factoring a sum of cubes, or function behavior, or rate of change, can be turned into puzzles! Teachers, curriculum developers: stay alert to those possibilities!


--Henri

Stumped by Euclidea

I've really enjoyed solving the puzzles in Euclidea, a brilliantly designed app for iOS and Android. The basic format is "given this, construct that". You start with just two tools: a straightedge and a slack compass (i.e. a compass that does not remember the radius it was last set to). As you find useful and reusable constructions, such as how to drop a perpendicular, those become available as additional tools. The interface is simple and elegant.

For each construction challenge solved, you "get" a star. If you find a construction that uses the minimum number of steps, you get another star, and yet another one if you find an optimal straightedge and slack compass construction. Searching for these optimal solutions is usually more difficult than merely solving the puzzle, but it can be instructive. For example, puzzle 2.7:

 Given a line l and a point P on l, construct a perpendicular to l through P.

My first solution was based on standard techniques to bisect an angle, or to perpendicularly bisect a segment. However, knowing that there is a three-step construction using straightedge and slack compass got me thinking. Here is what I came up with:

       

Step 1: Draw a circle through through P, with center O not on l. It meets l again at a point Q.
Step 2: Draw a line through O and Q. It meets the circle at a new point R.
Step 3: The line through R and P is the desired perpendicular to l.

This could lead to a great lesson: why does this work? will it always work? To answer this question students need to know that the sum of the angles in a triangle is 180°, plus the isosceles triangle theorem and some basic algebra. If they have trouble, you may offer the hint: "when working with circles, listen to the radii!" Indeed, analyzing the figure below should lead to Thales' theorem about an angle intercepting a half-circle:

(All figures created in GeoGebra.)

Actually, such a lesson would work best after working through Labs 1.5, 1.6, and 1.7 in Geometry Labs (free download). The reason is that the labs give students practice solving problems of this type with actual numbers before tackling the general case. (These labs require circle geoboards, or circle geoboard paper.)

Anyway, back to Euclidea. I was not always able to find optimal solutions. My first failure in this regard is on 1.7, inscribing a square in a circle, given one vertex on the circle, in seven straightedge and slack compass steps. My best attempts required eight steps.

In any case, I was able to solve all the puzzles one way or another, until 10.6:

Construct a circle through P that is tangent to both sides of the angle

My first reaction was that this was merely combining and extending two problems I am familiar with. First, the classic construction of a circle tangent to two lines. And second: given a line l and a point P not on l, construct a circle through P, tangent to l. Both puzzles are part of the construction unit I assigned year after year to my geometry class, and the second is the underlying strategy for the construction of a parabola with focus P and directrix l.

And yet, I was not able to crack Euclidea 10.6. I did it in GeoGebra:

 

As you see, there are two solutions. I used the fact that the center of the desired circle is equidistant from P and each side of the angle. Therefore it lies on the parabolas with focus P and the sides as directrixes. So it must be at the intersections of these parabolas:

This was straightforward, as GeoGebra has a Parabola tool, but I still have no idea how to do this in Euclidea. The creators of the app claim this can be done in six steps (using any Euclidea tools*) or 11 steps (using only straightedge and slack compass.) If you figure it out, I would love a gentle hint, as Euclidea is not allowing me to proceed any further until I solve this.

* Available tools for 10.6: straightedge, compass (slack or rigid), perpendicular bisector, perpendicular line, angle bisector, and parallel line.

[Breaking news: The Euclidea developer gave me a big hint via Twitter. The approach involves dilation. I am embarrassed I didn't think of it, as I had used a very similar strategy to construct a tangent to two circles.]

--Henri

PS: for my thoughts on the mathematics and pedagogy of geometric construction, see this recent post, (and for more, follow the links therein.)