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Creative Commons. Fraction comparison for 4th Graders. They’ve been working a lot with representing fractions as circles and as rectangles. They’ve done some basic addition with fractions. Most aren’t generally able to find equivalent fractions. What mistakes do you expect to see in the class set? Make a prediction! In the comments, would you please answer this question: Which mistake most surprised you? Kid 1 Kid 2 Kid 3 Kid 4 Kid 5 Kid 6 Kid 7 Kid 8 Kid 9 Kid 10 Kid 11 Kid 12 Kid 13 Kid 14. Before the iPhone, man had rocks. The first true multi-tools were of course rocks. Take the chopper, a crude tool chipped from the volcanic stone nephelinite. First used in Tanzania some 1.85 million years ago, it was the stone-age equivalent of the smartphone. Grant me the imperfect comparison: For thousands of years, the chopper served most of humanity’s needs. It was used to hunt, to dig, to build and butcher.

No, we don’t use our smartphones for those things exactly, but they are similarly indispensable. As our needs and desires have increased in complexity, so too have the number and nature of our tools. Perhaps the best place to appreciate this is the third floor of the Cooper Hewitt Museum in New York. Click to Open Overlay Gallery So What’s a Tool, Anyway?

In the broadest sense, a tool is anything that helps get stuff done. One beautiful example is the ceramic penicillin vessel created in 1940. Of course, tools aren’t purely utilitarian. It’s an interesting point to remember. §. Small Cubicuboctahedron – Robert Webb This is the small cubicuboctahedron, as drawn by Robert Webb’s Great Stella software. It looks simple enough, but it conceals some interesting mathematics. For starters, the yellow pieces are actually regular octagons which are mostly hidden from view. The three shades of yellow are parts of three octagons, but there are also three more, each parallel to a plane containing a red square. Thus, this polytope has 3 kinds of faces: • 6 red squares, • 8 blue equilateral triangles, and • 6 yellow octagons. Don’t be fooled by how the octagons cross each other.

As you trace out a small loop traversing all the faces that meet at a vertex, crossing edges and ignoring false edges, you traverse first a square, then an octagon, a triangle, then another octagon… and finally you return to where you started. So, this polytope has regular polygons as faces, and every corner looks like every other: more precisely, its symmetry group acts transitively on the vertices.