Why not a circle centered at the origin? Both the x = sqrt(1-y^2), same for y, coordinates are singular at +1 and -1, and hence difficult to evaluate by hand. Just try to evaluate sqrt(1 – x^2) near x = 1.

All you need for polar coordinates is a compass and a ruler to fix the radius, and blank sheet of paper.

The practical computer approach hides the difficulty in Cartesian coordinates by numerical evaluation of power series expansions of the circular functions (sin and cos). For graphics the pixel map is generally Cartesian and polar coordinates are mapped to it.

Your question brings out a nice collection of math, computation, and technology. Thanks to Dan for providing the forum.

Hi Dan, thank you very much for your answer.

Thanks for the questions, Katrin.

How much does the understanding of your students of coordinate systems improve by teaching them with the Pomegraphit method rather than teaching the regular way?

I haven’t run any specific study comparing this approach to introducing the coordinate plane with any other. I tend to a) draw inspiration from the Freudenthalian work on “progressive formalisation,” and also from the fact that I can always add the numeric coordinate plane. But I can’t subtract it once it’s added.

How do you arrange the transition from this method to using numbers?

I give them a single fruit on the coordinate plane. I ask them to “send a text message to a friend that precisely describes the location of that fruit.” They find it challenging. They rely on imprecise language. Then I add a grid to the plane and ask them to repeat the exercise. They feel something like relief. The grid is a tool to enhance their communication. They still tend to say things like “four left and three down,” which I need to help them formalize to (-4,-3).

It is difficult to understand mathematics, but it is one of the few ways students can see a large payoff from the struggle to understand. Using undefined variables as coordinates, displayed or not, is deception. Mathematics resides in the graph of the quantitative relation of measures of two fruit attributes, but I cannot see a way to measure or derive numbers for coordinates of undefined variables, and hence they are not quantitative. They are opinion.
Why do I think this is important?
Mathematics is a glimpse of both the difficulty and the possibility of understanding. Circumventing aspects that make it so leaves unique opportunities on the table. Students also need to see how the appearance of mathematics is used to lend undeserved weight to arguments.

Love it. Similar to David’s suggestion about variance, this probably deserves its own lesson.

I gave it a shot a long time in a lesson I regret for obvious reasons of sexism and objectifications, but I still like the illustration of the coordinate plane’s value. The dot plots don’t reveal anything all that interesting by themselves, but when you plot them against each other, the association pops.

Exciting title!

It takes you back to 60’s to see mathematics playing the star role in the real world. The issues it raises on the use of mathematics are still relevant in the application of social science to education.