GEOBOARD ACTIVITIES FROM A PROFESSIONAL DEVELOPMENT WORKSHOP

GEOBOARDS

We chose to spend the entire day with geoboards to give a little impression of how many different things can be accomplished with the same “manipulative”.

Activity 1:

Work with a partner. Partner A makes a pattern, using one "geoband" (=rubber band), without letting Partner B see it. They then put their geoboards together back-to-back and Partner A describes the pattern in such a way that Partner B can reproduce it. It should be a strictly verbal description--which is made easier to accomplish if Partner A holds up the geoboards using both hands!

When they have finished, they switch off and repeat the activity.

Why this activity? The prime reason is that communication and articulation are being recognized as key ingredients for mathematical understanding and security, and have long been undervalued. This activity provides an opportunity for communication with the added feature that the results, successful or otherwise, are instantly apparent, with no outside arbitration needed. This latter characteristic is one that I look for whenever possible--if the activity provides its own feedback, the students learn to rely on themselves more and the teacher less, which is a highly desirable state of affairs.

Another attractive feature is that the activity can be adjusted up or down in terms of the ability levels of the students. The very young could be told to make a shape that touches only three pegs. The hot shots could be permitted use several geobands.

This also provides an excellent springboard for introducing coordinate systems (they will arise naturally among older students and need only be broadcast to the rest of the class) and symmetry. The latter turns up almost automatically, since some patterns will be reproduced straight and some in mirror image. Geoboards are excellent for studying symmetries (for instance a geoband stretched down the center provides a good axis of symmetry to reflect through).

Activity 2:

A swift one. I made a pattern on my transparent geoboard and showed it for 45 seconds while everybody looked without touching their boards. Then I gave time to try to reproduce the pattern. Everyone got one more peek of about 10 seconds and then we checked the results. It was a definite challenge, but one that a lot of people met. The second time around even more people did.

My original take on this was that the basic virtue of the activity is sheer pure build-up of visual skills, as it is nicely non-verbal in its requirements. On the other hand, over the years I have discovered how much is added by a discussion of how people go about "learning" the pattern. People are often astounded and enlightened by what they hear. After doing a second one, I have frequently had people attribute their success entirely to having adopted someone else’s method.

Activity 3:

Work on your own, but confer with neighbors.

First instruction: "Surround a 2 by 3 array with one geoband."

This is an intentionally ambiguous instruction, but thrown in as a consciousness-raising device: our instructions are often ambiguous, because when one knows what one wants it is easy to fail to recognize alternate readings of what one says. It therefore behooves us as teachers always to bear in mind that the student who does something unexpected and "wrong" may be doing it as a result of a legitimate, or perhaps almost legitimate, interpretation of our instructions.

With the instruction amended to "Surround an array two units by three units", the activity can proceed to its next step which is to use a single geoband to divide the enclosed region in half. After having done it once, record that method and do it again another way...and another,...

I included this activity partly because it is a good model of a non-test assessment opportunity. It is a question to which every student can provide at least one solution, but it is open-ended and provides opportunities for the more advanced student to slip in some quite sophisticated tactics.

Activity 4:

Everybody made a one-geoband shape. Then I chose a criterion and found three patterns with it and three without – at least I tried to. Without saying what the criterion was, I put the yes’s together and the no's together and then everybody had first to decide whether a seventh one belonged with the former or the latter and then to defend their decision. When we got my criterion more or less sorted out, other people came up with different (and generally considerably superior) criteria to use.

This activity has lots of virtues. An obvious one is that it is easy to get the class thoroughly involved, and to remove the teacher from the center of the activity altogether, so that the class is acting independently. Another is the richness of the possibilities for content--convexity, symmetry, right angles, area, and far more.

Activity 5:

Again everybody made a one-geoband shape, this time with no crossing of geobands. They then transferred the patterns onto dot paper. We then posted the figures in columns, with each column made up of figures with one particular number of sides. Since they all started at (more or less) the same level, this produced an instant bar graph.

Clearly the leading virtue of this activity is that it leads into bar graphs, and leaves openings for all of the good bar graph questions: which one are there the most of? how many more of those are there than of the one that has the least?...

There is another virtue which is not unique to this one, but which is nonetheless worth pointing out: there is no right or wrong. All figures are equally acceptable, and it is the very variety that makes the graph interesting.

Activity 6

We began with a brief review of area. We defined a unit of length to be the distance between two contiguous pegs, so that a unit of area was the area surrounded by four pegs in a square. The challenge then posed was to make as many different shapes as possible each having area 2 square units. This produced an ever-more-interesting sequence of shapes, and provided a lead in to a series of area worksheets taken from “Dot Paper Geometry”.

Activity 7 (or maybe 6.5) : Pick’s Theorem

This was intended as the day’s final “Hurrah”, but owing to a general state of physical and mental exhaustion, it came out more like the day’s final “Ja, so?” We’ll give it one final moment of glory to start the next workshop then let it glide peacefully to rest.

Pick’s Theorem allows you to find the area of a geoboard figure on which the rubber band never crosses itself just by counting the number of pegs the rubber band touches and number of pegs inside the figure. What makes it fun is that knowing the existence of such a theorem, by looking at enough examples one can figure out what the theorem has to be. Doing so does, however, require energy!

The theorem itself is:

The area is equal to the number of pegs in the interior, plus half the number of pegs the rubber band touches, minus one.