Another visualization task

Here is another interesting setup, shared in Tom Steinke's OAME 2017 session.  I have seen circles around squares and circles inside squares, but not circles and squares sharing a baseline.

I think a student could do a lot of productive reasoning about what the situation would look like prior to using the interactive sketch.

1. What does the rectangle look like at the extremes?
2. Will the side be greater than or less than the diameter?
3. What fraction of the side length of  the square do you think the diameter will be?

Opening the sketch and dragging the point, without showing the measurements, is interesting.  How close is a student's guess about where the position of the drag point must be for the rectangle to be a square?  How close is there estimation of the relationship between the square's side and the diameter?

Lastly, they could choose an arbitrary diameter, like 100, and see if they can verify the result. Surprisingly, everything comes out nice and evenly with narry a square root of two in sight!

Mathematical Poetry?

This is not as deep as Kaz's stuff, but comes from Harold and Kumar by way of the College Math Teacher,

I’m sure that I will always be
A lonely number like root three

The three is all that’s good and right,
Why must my three keep out of sight
Beneath the vicious square root sign,
I wish instead I were a nine

For nine could thwart this evil trick,
with just some quick arithmetic

I know I’ll never see the sun, as 1.7321
Such is my reality, a sad irrationality

When hark! What is this I see,
Another square root of a three

As quietly co-waltzing by,
Together now we multiply
To form a number we prefer,
Rejoicing as an integer

We break free from our mortal bonds
With the wave of magic wands

Our square root signs become unglued
Your love for me has been renewed

I probably should have saved that for Valentine's Day, but I have something else up my sleeve for that.

Once again mathematical proficiency is associated with embarrassing lameness - sigh.