Last week, I bumped into Judith Seidel (@seidelj) at a lecture at the Museum of Mathematics. She suggested I explore a Cardioid Activity on Geometer’s Sketchpad — and just in time for Valentine’s Day too! A quick Google search yielded these resources which I forwarded to the Math Department:
1. A Geometer’s Sketchpad lesson plan:
2. A video tutorial with doable paper/pencil/ruler instructions:
3. More information about cardioids and the math behind them:
I’ve had a deep love and respect for Geometer’s Sketchpad since I was first introduced to it in 1994 as an undergraduate Math major (and aspiring math teacher) at Bryn Mawr College.
Later, I used Geometer’s Sketchpad during my student teaching stint at Strath Haven High School and again as a pre-Algebra/pre-Geometry teacher at The Dalton School.
Today in 6th grade Math at The School at Columbia University, Katie Klein (@KKleinNYC) and her associate teacher, Jazmin Sherwood, facilitated a great lesson on Fractals blending direct instruction, video, and self-paced sketching with and without technology.
1. Homework from the previous night was to watch the first 20 minutes of Fractals, Exploring the Hidden Dimension.
2. Here’s a link to beautiful photos of fractals found in nature: http://io9.com/incredible-photographs-of-fractals-found-in-the-natural-480626285
3. Here are instructions for drawing Sierpinski Triangles with paper and pencil:
4. Here are instructions for drawing Sierpinksi Triangles using Geometer’s Sketchpad on their laptops:
5. Here’s another resource for making other fractals with Geometer’s Sketchpad: http://www.gwinnett.k12.ga.us/PhoenixHS/math/GSP-website/17_Fractals(51-61).pdf
6. With additional time, students could explore fractals with Scratch or Snap (both are web-based block-based programming environments). Here are some links I gathered:
So, now I know that hexaflexagons are awesome! On Vi Hart’s YouTube channel she describes herself as a professional mathemusician at KhanAcademy. Here’s the description of her latest video:
This video is based on a true story. Arthur H. Stone really did invent the hexaflexagon after playing with the paper strips he’d cut off his too-wide British paper, and really did start a flexagon committee (which we’ll hear more about in the next video). The details and dialogue, however, are my own invention.
More info coming soon, along with patterns. But in the mean time you can find instructions and printable patterns on other parts of the internet.
Follow Vi Hart on Twitter at @vihartvihart.
Katie Hildebrandt (6th grade math teacher) and I just finished a 2-day unit on Tessellations using Geometer’s Sketchpad to rotate an equilateral triangle and translate a parallelogram. I can’t praise Geometer’s Sketchpad (sometimes shortened to Geo Sketchpad, GSP, or simply Sketchpad) enough. It is one of the few pieces of educational software out there that is entirely constructivist. You can actually learn math by using GSP. Just like classical Euclidean Geomtetry breaks down everything in the world to points, lines, and planes, so does GSP; You can quickly learn to construct, animate, and measure a range of sketches, from the simple to the complex.
In this brief unit, we verbally discussed how a vector is a geometric object (in this case, a segment) that has both a direction and distance. They notice the “di” in the beginning of each word. I also tell them that translating is the same as sliding, and both words have an “sl”. We talked about angles of rotation, indications of symmetry, interior angles of a triangle, and reinforced vocabulary: equiangular, equilateral, congruent, parallel lines, etc.
I always start off by showing them works by M.C. Escher. On the site, there is a link to a gallery of his symmetry drawings. I marvel at how Escher painstakingly drew his incredibly intricate and fascinating tesselations component by component. I imagine his pile of pencil stubs and eraser shavings, and reinforce for the kids how we can create infinite variations with GSP in a matter of seconds by clicking and dragging.
I posted something about last year’s activity here.
There is a great PDF with multiple activities put out by Key Curriculum Press (@keypress) embedded below or you can click here to download it. We use pages 7-8 for the translation activity and pages 9-10 for the rotation activity.