For the second time this summer, I taught a version of my Visual Algebra workshop, this time as part of a summer institute at the Atrium School near Boston. (Earlier in the summer, I did this at Synapse School, in Silicon Valley, and wrote about it here.) Once again, I walked away from the workshop with some new ideas and understandings.

Because of the way the institute was structured, I ended up focusing mostly on the use of manipulatives in teaching algebra and pre-algebra. This meant mostly the Lab Gear, but I included a couple of sessions on using the geoboard. In particular, we went through the activities from *Geometry** Labs* about area on the geoboard, which lead to a geometric understanding of simplifying radicals. (See Labs 8.4 to 9.5 — free download.)

A key lesson in that unit is the search on the geoboard lattice for “tilted” squares that have as many different areas as possible.

Of course, the slopes of the sides of a geoboard square are opposite reciprocals of each other. For example, in the above example, the slopes are 2 and -1/2. Let’s call the positive slope the “tilt” of the square (2 in this case). If the sides are horizontal or vertical, we’ll call the tilt 0. In their search for geoboard squares, students often structure their exploration using the tilt as an organizing principle.

Workshop participant Stephen Bonnett came up with a fun extension for this activity. I asked him for a write-up. Here it is, very slightly edited:

What I envision as a fun challenge is to encourage students to maximize the area they can enclose within a standard 11 by 11 geoboard in some number of

non-overlapping squares, each of which has a different tilt. They could start with 2 squares, and see that they actually can’t make a 9×9 and 1×1, because they have the same tilt. Then you can make it harder by requiring 3 squares, 4 squares, etc. You can ask whether there is some number of squares where it would be impossible to fit that many squares on the board. (I think that answer is 7). In my work, I found total areas of 62 for 5 squares, and 63 with 6 squares.

Blog readers: can you do better than Stephen? Of course, you should not mention his numbers to your students. The largest areas they obtain will stand as the class record.

I don’t know if there’s much math significance to this exercise, but one thing I like a lot about all these hands-on tools is that there seem to be virtually limitless extension problems possible for students who get the original task quickly.

Well said! And in fact, this leads me to another insight I had as part of this workshop.

A question that comes up frequently when I introduce activities involving manipulatives is “Some of my students don’t like manipulatives because they feel they don’t need them, as they already know the material the activities are supposed to teach.” My usual answer is that sometimes those students resist because they have been praised for their mastery of memorized symbol manipulation techniques, and they are intimidated by the visual aspect of working with manipulatives. That is indeed often true, but there is a deeper issue. If you use the manipulatives *only* as a way to provide a geometric model of the algebra, some students will pick this up quickly, and will feel that doing a lot of work in this environment is tedious.

This is why it is so important to include a lot of challenging / interesting / difficult work with the manipulatives, with the puzzle-maker’s aesthetic I try to incorporate in my lessons. For example, with the geoboard: how many points can you put on the geoboard so no three are on a line?; find a shape with a given area; find squares with all possible areas; and so on. And likewise with the Lab Gear, with pattern blocks, with the circle geoboard, with the Geometry Labs template, with electronic graphers, etc. Such activities are interesting to a wide range of students and, frankly, to many teachers. This makes them an important ingredient in the tool-rich pedagogy I recommend. They help build your strongest students’ interest, and set up an atmosphere of exploration and collaboration where all can thrive.

— Henri