Pyth On

Mel Bochner, Pythagoras (4)
from wikipaintings

Arithmetical Design (quite a fun tumblr) posted this beauty today...

I thought that this was something that screamed to be dynamic. Off to the GeoGebra Cave, old chum!
















The sketch started with a right triangle, and then the regular polygon tool to make the squares on the side. I wanted the triangle connecting the next squares to be similar to the original, so I made the side of a square to be the new hypotenuse, rotated it by one of the non-right angles then used the perpendicular tool to make the similar right triangle. Finally, I constructed  the first two additional squares.

Clearly too much work to repeat in the dozens. To use the Create New Tool command you select item or items in the sketch. Then select the command from the tools menu. My first try I forgot that I would need the points to make subsequent squares. Delete the bad tool from the Tool Manager. (Can also rename there if you're trying for something more pythy than Tool 1.)

When I had the squares and vertices selected, the second step of the Create New Tool dialogue was to determine the inputs. GeoGebra will select some ancestors to start, but you can modify the inputs. In this case, GeoGebra selected my first two free points, which doesn't suit. I wanted the inputs to be the the endpoints of the hypotenuse. At the last step you select a name and can attach a custom icon if you're being tricksy.

Once I had the tool it was quick to construct the spirals, and then aesthetics like a coloring scheme and positioning. From the GeoGebra color dialogue you can click the plus, which brings up an RGB color input. (For those times when you need beige, 255-245-235.)

I was going to stop there, but decided that people needed to be able to make their own spirals how they wanted, so added a checkbox to go back to the beginning. (If you make something send me the pic and I'll add it to the post.) Sadly the new points show up with labels - I don't know how to turn that off. Maybe if the labels are off before I make the tool? Tried that and it works!

Here's the finished sketch at GeoGebraTube: teacher page or applet. Sadly, the custom tools don't seem to show up in the HTML5 mobile applets yet.

Bochner has several mathematically influenced paintings, as well as the first three Pythagoras painitings. Check them out at wikipaintings.

Exponential Potential

It really struck me listening to Shawn Cornally in this week's #globalmath session on SBG (click on the Recording tab) how his perspective as a physics teacher leads him to approach his math lessons as experiments. Starting with an experience that makes us want to model or makes modeling useful is definitely the start of some of my favorite math lessons. (While I'm writing this he tweets: "Leaf-Blower soccer went over *really* well today in physics. (vectors, f=ma)")

Starting exponential functions with my preservice teachers, I love to use this lesson adapted from a 5th grade Math in Art lesson. (From my pre-blog webpage.) The idea is the multiplicative patterns present in a Sierpinski Carpet.


(Also in Word format if you want to edit.)

One of the interesting discussions in the initial exploration is the 9 or 17 issue. 9 squares if we count the number of squares as distinct shapes, 17 if we unitize to the smallest level square. For algebra students there's some good opportunities for equivalent expressions, regression and even deduction of function rules. This is a good opportunity for sharing how recording how you're getting your answer can be more powerful than recording answers. The 17, for example, is 1·9+8, then the next level is 9·(17)+8·8. But later, most write it as \( 9^n–8^n \) - which can lead to a pretty neat binomial expansion. Maybe even more interesting and accessible is the 9=1+8, so the next step is 73=1+8+8·8, and gives them a way to generalize this pattern besides recursion.

Once we get to the design your own carpet, there are so many new patterns to find. Here are some samples from this week:

Note that these last two aren't really Sierpinski patterns - but they still raise interesting patterning questions that are extensions of what we already noticed.

Probably easy to see why this is one of my favorite lessons. I also have seen the power of adding in places where students who have not traditionally been strong in math class can do amazing work.

The next lesson to follow up this one has some other opportunities to gather/generate multiplicative data.



It always strikes me how even college math majors find things to be surprised about in this data. Especially the penny balancing one. This class made some neat displays of their data - but I haven't taken the pictures yet. (Didn't know I would be blogging this, as I thought I already had! Maybe I was thinking of the quadratic simulations?) I'll add them at first opportunity.

First Opportunity:

Next we'll look at modeling this data symbolically using technology, and asking questions that raise the need for logarithms. Since, of course, every exponential data set is logarithmic when seen through the looking glass.