Math Photo: Pyramid Projection

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Naturally, the geometry of this simple piece of playground equipment caught my eye, but the shadows really sparked my interest.

The shadows are the projections of the edges of this pyramid, and they form a set of angles on the ground.  Notice immediately that the largest angle (the shadow formed by the “back” face) is the sum of the other three angles.

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There are many other interesting questions to ask, and relationships to explore.  What I was most curious about, however, is how accurately we could locate the sun in the sky based only on this information.

 

Demonstration of Linear Independence

vectors spanning the plane

I’ve put together a simple Desmos interactive that demonstrates the basic ideas of linear independence.

If two plane vectors are linearly independent, then every vector in the plane can be written as a linear combination of those two vectors.  Those two vectors span the plane.

By playing around with the sliders in this interactive, you can see how every vector in the plane can be expressed as a linear combination of the two original vectors.

Moreover, if you make the two original vectors parallel, they no longer span the plane.  That’s because the two original vectors are now linearly dependent!  Each is a linear combination (in this case, a scalar multiple) of the other.

You can see this Desmos interactive here, and you can find more of my Desmos-based demonstrations here.

Mashable: In Defense of Math

mashable artI had fun contributing to this Mashable piece, “In Defense of Math:  7 Reasons Numbers Rule“.

I, and a few other Math for America master teachers, were asked to pitch some ideas for a light-hearted refutation of the all-too-common “I hate Math!” refrain.

A couple of my ideas made it in, and I doubt it’s hard to figure out which were mine.  In any event, it should be easy to identify a couple that definitely weren’t mine.

A bit goofy, but all in good fun!

Writing in Math Class: Favorite Shape

favorite shapesI love giving short writing assignments to my math classes.  It’s a great way to get students engaged in mathematics in a different way, and it gives me a different window into how my students think and who they are.

Here’s a prompt I gave them recently.

Draw a shape that you like.  Write 1-2 complete sentences explaining what this shape is and why it appeals to you.

This simple prompt was something of an afterthought on a recent assignment, but as usual, the students surprised me with their responses.  Here are some of my favorites.

I like the square because of how organized it is.

The triangle appeals to me because it is the shape of things I love to eat, like a slice of pizza or a piece of cake or pie.

A rhombus is different from other shapes, but so alike.  It shows how a little change can make a new shape.

Circles appeal to me because they have infinite lines of symmetry.

This is a circle.  It has no corners and is symmetrical.  This shape is appealing because it feels open, and since there are no corners, there are no sharp edges to hurt you.

This shape is a circle.  It appeals to me because, as weird as it sounds, I want my life to be as perfect and as well-rounded as a circle.

A triangle is simple.  It has the least number of sides a polygon needs.

I like the square because I like things to be equal, not different.

I’m always smiling, laughing, and thinking after reading what students have to write.  There are lots of great reasons to get students writing in math class, so give it a try!  You can find more resources here.

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