## Des-Man and the Best Report Ever.

I was thrilled when I saw that Fawn Nguyen’s Desman idea had been picked up by Desmos because it was the perfect thing to assign to my MYP5 Extended (10th grade) class as a Criterion C: Communication project. Brief background — MYP Mathematics is assessed on four criteria, one of which is communication, and usually involves some sort of report writing in math.

We had just finished a unit on functions, including domain, range, tests for whether something is a function, and tying together the various types of functions we’d learned so far (linear, quadratic, exponential, modulus) and relations (circles, ellipses) and transformation geometry (not to mention inverse and composite functions and function notation). So I decided to have my summative assessment be the creation of a Des-Man and an accompanying report.

The students were given the following assignment:

Follow the first pair of instructions to begin to create a Des-Man. You do not have to use the two things the program starts you off with; it’s just to give you instructions on how to restrict domains and ranges. Be creative in your creation of Des-Man. Use lots of different types of functions (play around and explore the various tabs to see what is available to you; think about how to transform functions as well to get more interesting designs). You must restrict the domain/range for at least one function. (Note: You do not have to put everything into y= form to get Desmos to graph it!)

Write a brief report outlining how you have made your Des-Man. Your report should include:

·     Your Des-Man with any necessary annotations.

·     A list of the functions you used and if possible identify them by type of function. You should identify what role they play in your Des-Man as well.

·     A list of things you used that are not functions, with an accompanying explanation of why they are not functions.

·     An explanation of when you restricted the domain (and/or range), how you did it, and what it means to do so.

·     If you used transformations, explain what you did.

The students loved this assignment. And what I loved about it was that it was so open-ended and allowed them to use their creativity as well as showcase their deep mathematical knowledge through the written report. Writing reports is integral to the MYP mathematics curriculum, but I have a feeling it’s not a common thing for most teachers/students (in fact, whenever we get new students they are almost always surprised to have to write detailed reports for my subject). But I can’t stress enough how great it is to get students thinking in this manner.

Here’s one student’s Des-man and the first two paragraphs of her essay as a sample, where she shows that she understands the formula for a circle completely and how its formula relates to translations:

“I started by making the first circle with $x^2 + y^2 = 40$. Using only $x^2$ and $y^2$– instead of including numbers such as $(x - 2)^2$ and $(y - 2)^2$ – made the center of the circle at the origin. Using 40 on the other side of the equal sign meant that it had a radius of 6.32 which seemed a reasonable amount. It is not a function because it does not obey the vertical line test. If a vertical line was drawn through the circle, the line would hit the circle twice.

Then, I drew the eyes with the equation $(x - 2.75)^2 + (y - 2.57)^2 = 1$. The ‘x-2.75’ and ‘y-2.75’ in the brackets would ensure the centre of my circle was at (2.75, 2.75) and the number after the equals sign would ensure there was a radius of one (radical 1 is 1). To get an identical circle in the same position on the other side of the face I reflect it over the y-axis.  To do this I multiplied x by a negative making the following equation: $(x + 2.75)^2 + (y - 2.57)^2 = 1$. The negative sign in the brackets with ‘x’ became a positive. Again, this is not a function because it does not obey to the vertical line test.”

Another student, with just a quick excerpt from her report where she showcases her knowledge of the components of a simple quadratic:

$y = 0.5x^2 - 4 \{-4.7 \leq x \leq 4.7\}$ — This parabola forms the face outline on the graph. The 0.5 in front of the $x^2$ makes the parabola wider, as it is less than 1, and the -4 at the end moves the vertex down 4 units. The domain is restricted so that the parabola stops before the head is too big.”

(And yes, that is a cosine graph in her face, which we did later that term. Love that the explored fully without even knowing what they’d get!!)

But my favorite report is from a student who created a very simplistic face and then perhaps realized that part of the grading was on being creative. He wasn’t creative with his face, but boy was his report creative. I present to you (the non-math) excerpts from my absolute favorite math report I’ve ever read:

The des-man that was created on this wonderful afternoon was heavily inspired by Jony Ive. The Apple-esque des-man retrieves its beauty from its simplicity…

…The circle is split by the y-axis pure for design reasons. If it were not to, the iFace would not look symmetrical and clean. The numbers are also chosen for specific reasons, as they are rounded, which adds to the simplicity of the calculations. For example ten was chosen as the value for the radius of the face, so that the x-axis would be tangent to the circle. As Steve Jobs once said, “A product should not only look good from the outside, but also on the inside, showing the love we have here at Apple for our products.”…

In conclusion, a simple yet beautiful des-man has been created. Even the (tones of the) colors, red and blue were carefully chosen and required a lot of discussion and evaluation. At the end of the process, one can enjoy a product that was beautifully designed and carefully crafted to ensure the best possible quality for the iFace. “Simplicity is the ultimate sophistication.”

Literally the best thing I’ve ever read in my life. Thank you Desmos & Fawn for an inspired project! If people are interested in hearing more about writing reports for math, I’m happy to spread information because except for the hassle of grading essays, I think they’re really wonderful.