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.
Saying there’s a lot going on in the Exhibit Hall of ISTE11 is gross understatement. Unlike my Aunt Debbie, I made it out of there without four tote bags overflowing with schwag. For me, the Exhibit Hall is about saying “Hi” to vendors I know and trust, avoiding the awkward hard sell, being wow’d by the next “must have” integrative technology, and summoning the joy of trick-or-treating with all the free candy offered on every table.
Below is an incomplete list of booths and presenters (whether old or new to me) I found compelling enough to write down in order to share with my faculty:
Pixologic http://Pixologic.com for 3D modeling software (export files to our 3D printer!) Sculptris – free, lite version ZBrush – full feature
EasyBib http://easybib.com Free bibliography composer Now let’s you make online notecards!
Parat http://paratsolutions.com Finally came out with an iPad tray to charge and sync 10 iPads. You have to charge it in their case, and they are demo’ing a prototype of the case (which I guess they’ll be packaging with the tray?)
Bretford http://bretford.com/apple PowerSync Tray for iPad – charges and syncs 10 devices, can be daisychained (but why buy 3 trays and not one cart unless you like the size of the tray) PowerSync Cart for iPad – charges and syncs 30 devices
UPrint by Dimension http://dimensionprinting.com UPrint 3D printing by Stratasys $15K prints up to 8x6x6″ $20K prints up to 8x8x6 $30K prints up to 8x8x12 $33K prints up to 10x10x12 $110K prints up to 14x16x14
iPevo http://ipevo.com iPevo2 document camera Chopstakes multitouch styli Cushi pillow stand for iPad
Epson’s Brightlink 455wi http://Epson.com/brightlink $1600 without software $1800 with RM Easiteach Unlike the eBeam, no additional equipment needed besides projector, surface, computer Infrared signal links from projector to pen tracks everything, 16:10 ratio throw Not multi-touch Projector has built-in wifi and speaker, ports for additional speaker
Livescribe http://livescribe.com Besides the Echo Smartpen, I was psyched to learn about Sound Stickers and Sticky Notes. Sound stickers – circles you can program with specific audio! Put them on pages of books and kids can tap it with pen to make audio play Sticky notes – sticky notes with dot patterns just like the notebooks They are partnering with http://Harvardsquareacademy.org – not public yet – and getting a community of minds from Harvard and MIT sharing content and lessons to give expertise to students in the classroom
Library of Congress http://loc.gov/teachers How and why to use primary resources, suggested tools for gathering primary resources
Lego Education http://LEGOeducation.us Jason Yount at Tech Support shared a ton of information http://tiltedtwister.com – can get the building plans and compiled code (Java) for Rubiks Cube solving robot http://tetrixrobotics.com for pics and plans of robots using tetrix pieces (claw design) Labview – industry standard software, $500-600 site license. Has help feature to assist you as you translate from RoboLab or Mindstorms into LabView WeDo – LabView for little kids – pretty basic looking programming language Renewable Energy Add-on and Activity Pack – need to purchase the new Simple Machines kit
Polyvision http://polyvision.com Eno Play turns Eno board (the whole board!) into a speaker! Fuse – document camera, web camera, and scanner in one
Vernier http://vernier.com Vernier is the developer of Logger and tons of sensors $1.99 Video Physics app – capture or import video with iPad (basketball, swing set, cars, conveyor belt) graph, share to Logger Pro and measure away
Doceri http://doceri.com Use iPad to control interactive whiteboard screen, write over and edit and play back
One More Story http://onemorestory.com Online library Child chooses a book and can experience it in one of three ways: either fully animated and controlled by computer, read by narrator, or read independently $1.20 per child per year – 100 student minimum $44 for a home license Will be available in the fall as an app
Sparkfun http://sparkfun.com Top reseller of Arduino projects Program with Modkit – upgraded version of Scratch (modulation code) They bundle an Inventor’s Guide As an educator – email education@sparkfun.com and get digital or paper educator guides with worksheets for teaching with Arduino!!!!
FableVision http://fablevisionlearning.com/fabatschool Fablevision is developing the software for the 3D printer made by Cornell University Prints models with hard plastic, silicon, latex, cheese whiz, frosting Printer will be about $1000 or less Comes in a kit to built Hopefully available to purchase by september.
Powerful Learning Practice Http://plpnetwork.com Sheryl Nussbaum-Beach (@snbeach) and Will Richardson(@willrich45) Year long, job-embedded PD versus workshops and keynote
2Bot physical modeling technologies http://2bot.com 3D classroom modeling, Uses a drill bit to cut through blocks of approved materials (foam, wax, wood) 12x12x2 blocks but separate “printouts” can be pieced together. Under $10K
I’ve been a fan of Geometer’s Sketchpad (GSP) since 1994 when I first learned about it as a Senior at Bryn Mawr College. It is among the best examples of educational software out there, as you actually learn the math by using the tool. I wrote a post in January about using GSP to do tessellations in 6th grade math, and Daniel Scher* contacted me about maybe collaborating to integrate The Dynamic Number Project and other great resources.
Last week, Daniel came to The School at Columbia University to meet a select group of teachers, technologists, and math liaisons. He’ll be showing them how to integrate The Dynamic Number Project and GSP into their curriculum, make constructions with GSP, and use the soon-to-be-launched iPad app of GSP! While the initial GSP app will only run interactive scripts, the next iteration will allow the user to make constructions. By getting the teachers’ hands on an iPad2 installed with the GSP app in June, we’re hoping to hit the ground running in September.
Daniel is a principal investigator for the NSF-funded project Introducing Dynamic Number as a Transformative Technology for Number and Early Algebra. Previously, Daniel was a program director at Best Practices in Education, where he specialized in educational technology applications and the adaptation of the Elkonin-Davydov mathematics curriculum for first and second graders. He investigated the epistemology and mathematics of Dynamic Geometry® while a researcher at Education Development Center. He is the author of Exploring Conic Sections with The Geometer’s Sketchpad and is the co-author of a geometry textbook and a precalculus Sketchpad module. Daniel received a BA in Mathematics from the University of Pennsylvania in 1989, an MS in Mathematics Education from Cornell University in 1993, and a PhD in Mathematics Education from New York University in 2002. (http://www.kcptech.com/pages/daniel.html)
I spent a few days this week with the 6th grade math teachers/students. At The School at Columbia University, 6th graders study Islam and Mecca as part of the grade-wide theme: How History Shapes my Identity. For the last few years, I’ve worked with the math teachers to show the kids how to design tessellations on the computer. Then, the students take their creations to Art class and build a physical model out of clay and/or paper. It is one of my favorite integrated projects. (All of our K-8 Themes and Concepts can be found here: http://theschool.columbia.edu/about/curriculum.)
Katie Hildebrandt, often confused from behind for one of her students (à laMacaulay Culkin), is an energetic and supportive member of the 6th grade team and has a natural gift for breaking down mathematical concepts for her students. Before Winter Break, we met and planned a 3-day mini unit for our first week back; It bridged a unit on solving equations with her next unit on the Cartesian Plane. We briefly went over how to make transformations and use specific menu options in Geometer’s Sketchpad, as Katie is one of those independent teachers that initially explores on her own rather than rely on my tutelage. Power to the people!
On the first day of the mini-unit, Katie led a class on reflecting a polygon over the x-axis and y-axis. Students explored the resulting coordinates and stated the formula as an algebraic expression. For example, reflecting over the x-axis means (x,y) becomes (x, -y).
The second day, Katie showed how to rotate a polygon 90, 180, 270, and 360 degrees around the origin. Again, students analyzed how rotating the figure affected the coordinates of the original shape. For example, rotating 90 degrees meant that (x,y) become (-y, x).
On the third day, I stepped in, and we talked about how a tessellation is a pattern of repeating shapes that do not overlap and have no gaps in between. I showed them some of M.C. Escher’s artwork, and we talked about how classic Islamic art would rely on geometric patterns rather than animal or human forms. Because of the nature of our curriculum, the students had similar discussions in Art and Spanish among other subject areas. I showed the kids how to use Geometer’s Sketchpad to build an equilateral triangle, alter one side, and rotate that side 60 degrees to create a new shape. Then we rotated this altered triangle 60 degrees 6 times to form a hexagon before we tessellated the whole hexagon.
For my tessellation activities, I use two online lesson plans that I located years ago: http://ww3.wpunj.edu/icip/itm/Lessonpl/sketch/rotate.htm – Triangle Rotations by Janet Mae Zahumeny of Roselle Park High School http://mathforum.org/sum95/suzanne/tess.gsp.tutorial.html – Parallelogram Translations by Cathi Sanders of Punahou School
I started exploring/playing with Geometer’s Sketchpad in 1994 as an undergrad at Bryn Mawr College. To this day, it remains my favorite piece of educational software. Not too long after, I learned about The Math Forum – an amazing resource for math teachers and students founded at Swarthmore College and now seems to be hosted by Drexel University. There is a great link about Exploring and Creating Tessellations: http://www.teacherlink.org/content/math/activities/skpv4-tessellation/home.html
Geometer’s Sketchpad resources: General Resources: http://www.dynamicgeometry.com/General_Resources.html Resource Center: http://www.dynamicgeometry.com/ Sketch Exchange: http://sketchexchange.keypress.com/ Workshop Guide: http://www.dynamicgeometry.com/Instructor_Resources/Workshop_Guide.html
Karen Blumberg 