Sabrina Goldberg is the 7th grade Math Teacher at The School at Columbia University. She has a classic project she initiates every year with her students:

Box Project

A rectangular box is made by folding an cutting an 8"x10" shet of paper. The Flap EF equals the height AB of the box.

EF + ED + DC + BC + AB = 10"

DC + CR + RS = 8"

Parameters:

The 8"x10" sheet of paper costs $4.00.

It cost $0.02 per inch to cut the paper.

You can sell scrap paper at $0.06 per square inch.

It costs $0.20 per inch to tape up the corners.

You can sell the finished box for $0.30 per cubic inch.

Make such a box with a depth of 1.5 inches.

If you make cut AB = 1.5 inches, find:

volume of the box

cost of the cuts

value of the scraps

selling price of the box

profit

surface area of the box outside, not including the flap

Build expressions above using AB = x inches

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.

Daniel Scher is a principal investigator for The Dynamic Number Project. He works to create the iPad-compatible constructions for the Sketchpad Explorer app. Sketchpad Explorer is the iPad companion to Geometer's Sketchpad software program - both are published by Key Curriculum Press (@keypress on Twitter).

Daniel will be coming to The School at Columbia University to help me help the 5th grade teachers integrate Sketchpad Explorer into their math curriculum. He's psyched to watch/evalutate how kids learn with the app versus the software versus without either, and I'm excited to get a full grade of teachers to use a tool that I consider to be an industry standard for math educators. I love Geometer's Sketchpad, and I recently gushed about Sketchpad Explorer on Scholastic's Best in Tech Today.

Upon opening the Sketchpad Explorer app, there is an animated proof of the Pythagorian Theorem.

On the bottom right corner is a book icon. This takes you to many options, one of which is to visit the Sketch Exchange community site. From here, you can click Sketch Exchange sketches tagged with ipad, and then you can choose dynamic number in the tag cloud. These sketches open up with the full Geometer's Sketchpad software program as well.

There are four pages of sketches to choose from. I initially chose Balance Scale: Solving for Unknowns Part 1. When you choose a sketch, further down the page you'll find links to download activity notes, worksheets, and the actual sketch with the .gsp suffix. I was super excited to note the CreativeCommons license (!) of Attribution Non-Commercial No Derivatives which translates to: Give me credit, don't make money off it, and don't alter it. I love Creative Commons almost as much as Geometer's Sketchpad. :)

I had a bit of fun playing with this screen trying to figure out the value of the star and other shapes using the fulcrum, and yes, I think I'm smarter than a 5th grader.

Download the Sketchpad Explorer app here: http://itunes.apple.com/us/app/sketchpad-explorer/id452811793?mt=8