Draw a polygon following grid paper lines. No crossings, no holes — in other words, a polyomino.Now try to inscribe a square in it, with all its vertices at lattice points on the perimeter of the polyomino. Here are two examples: Conjecture: it is impossible to draw a polyomino that does not have such… Continue reading Inscribing Geoboard Squares in Polyominoes
Tag: Geometry
Proving Pick’s Formula
Pick's formula is a way to find the area of a geoboard polygon by counting interior pegs and boundary pegs. Students can discover the formula by doing some experimenting under teacher guidance (see Geometry Labs 8.6 or Algebra: Themes, Tools, Concepts 4.12.) I have used this in the classroom for decades, because it is such… Continue reading Proving Pick’s Formula
Another "K-12 Unsolved" Problem
In a recent post, I mentioned a problem posed in 1917, which remains unsolved and which lends itself to use in K-12 education:Consider an n by n lattice. Is it always possible to choose 2n points in it so that no three points are in a line? Today, I present a related unsolved problem I… Continue reading Another "K-12 Unsolved" Problem
Summer Workshops
I'll be teaching four workshops in June, at the Urban School of San Francisco's Center for Innovative Teaching. I just posted the info here. Some changes from past years:I've broken up my Geometry workshop, which used to be three or four days, into two chunks, two days each. The first (Hands-On Geometry) will be based… Continue reading Summer Workshops
Asilomar report, Part 1
Once again, I had a great time at the Asilomar conference of the California Math Council. Here are some notes from the first two sessions I attended.Scott Farrand (of Cal State Sacramento) and UC Davis's Rick West's presentation "Diophantine Equations Can Hide Geometric Surprises" was a fun way to start the day.Think of two whole… Continue reading Asilomar report, Part 1
Geoboards and Dot Paper
If you are familiar with my curricular creations, you know that I often use the geoboard as a microworld to introduce interesting problems and important concepts. This is in line with my call for a tool-rich pedagogy. (A geoboard is a square lattice pegboard on which students use rubber bands to create and investigate geometric… Continue reading Geoboards and Dot Paper
"Enrichment"
During my first ten years as a teacher, I worked in elementary schools. In addition to team teaching my own class (grade 3, then 4, then 5) I was a math specialist for grades K-5. The basic idea was that there was "normal" math (a lot of arithmetic, textbook-based), and there was "enrichment" math. I… Continue reading "Enrichment"
Unexpected ratio
Triangle ABC is equilateral. MA/ME is the golden ratio! (George Hart showed me this result some time ago.)Vynce Montgomery pointed out that therefore, this could be a way to construct a pentagram. To respond to this challenge, I hid everything but MA and ME, and proceeded from there:--HenriPS: thinking about this a few days later,… Continue reading Unexpected ratio
Crowd-Sourcing
Well, not much of a crowd, because not many people read this blog. Still... I created four worksheets of sorts, each one consisting of a short title and images of a dozen circles. Each circle is divided pizza-like into slices, as in the example above. The four titles: Angles, Fractions, Time, and Percents/Money. Here is… Continue reading Crowd-Sourcing
Straightedge and Compass
Back when I was a high school student, I had mixed feelings about compass and straightedge constructions. On the one hand, I liked the geometric challenge, on the other hand, I hated the physical challenge of working with an actual compass. Maybe 20 years later, I had exactly the same experience as a high school… Continue reading Straightedge and Compass
Henri's Math Education Blog 