|A visual representation of √5 and √20|
The reason activities built around lattice points are effective is that they build on student strengths. Work with whole numbers and discrete environments is more accessible, while still carrying the same conceptual load. And it does not prevent one from following up with real numbers and continuous environments — in fact it makes it easier. Some of my best geoboard lessons appear in both Algebra: Themes, Tools, Concepts and Geometry Labs. (See below for a list of lessons and labs about fractions, similarity, slope, area, distance, the Pythagorean theorem, and simplifying radicals. Many of those activities fit the criteria for "rich activities" which I articulated in a recent post.)
Why am I bringing this up now, you ask? Well, I recently attended an awesome meeting with mathematicians and math educators. Our charge was to choose 13 unsolved math problems that would lend themselves to a short lesson at each level from Kindergarten to 12th grade. The idea is not that the children will solve those problems, but that they would work on a version of one of the problems that is appropriate to their grade level. Moreover, in order to make this attractive to a maximum number of teachers, and to make it happen in a maximum number of classrooms, the idea was to select problems that were "curricular", in the sense that they fit the curriculum teachers and students already believe they should be teaching and learning. I was of course excited by this concept, and I am pleased with the progress we made at the meeting.
Much to my delight, three of the unsolved problems we discussed lend themselves to treatment on geoboards and dot paper. Whether or not they make the final list of 13 unsolved problems, I cannot recommend them highly enough to teachers who are already using geoboard lessons and labs. (And, really, to any teachers.) Stay tuned: I will introduce them one at a time in upcoming posts.
PS: where to find my geoboard and dot paper lessons
- in Algebra: Themes, Tools, Concepts: 1.12, 2.12, 3.12, 4.12, 5.12, 6.12, 7.12, 8.3, 9.2-9.A, 9.9, 9.C, 11.2, 11.3, and 11.A
- in Geometry Labs: 8.4-8.6, 9.1-9.4, 9.6, 10.1, 10.2
These links are organized better, and others are added on this page.
See also my conversation with G.D. Chakerian on the subject of the use of the geoboard to introduce the Pythagorean theorem.
PPS: A version of my tilted squares lesson which leads to the Pythagorean theorem is also available as a "formative assessment" unit for 8th grade from the Shell Centre. I have no idea whether they created this from scratch, or whether the idea reached them directly or indirectly from my materials, but I wrote my lessons a full twenty years before they wrote theirs. You saw it here first! :)