# Playing with a proportional reasoning activity- art/math/learning

I’m always trying to think about how I can bring activities together thatachieve the goals of getting students to use math, things like: measure, gather data, use the data to design something or predict an outcome, and have some kind of application that might be engaging to them. As I was walking my dog this morning, […]

#### Written By rodaniel

On September 3, 2010
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I’m always trying to think about how I can bring activities together thatachieve the goals of getting students to use math, things like: measure,

gather data, use the data to design something or predict an outcome, and have some kind of application that might be engaging to them. As I was walking my dog this morning, I started thinking about (don’t ask me why, I just did) having students draw a human figure. It’s a skill that is hard for me, but it’s not difficult for artists, because they have explored and memorized the expected proportions for “average” people. Check out Sherri’s blue sausage people if you don’t believe me. When I talk with Catherine or Sherri about drawing a person’s face they start with identifying the basic shape of the face, and then find the mid-line and half-way point.

Interestingly, the half-way point is approximately where the eyes go. In fact here are some guidelines for drawing a human face:

• Eyes: The eyes are always halfway down, between the top of the head and the bottom of the chin. They are also an eye-length apart. This means that, however long you decide to make the eyes, there will be that much space between the eyes as well (yes, break the horizontal distance into thirds; who knew artists did so much rational number thinking when they drew).
• Nose: The rest of the face underneath the eyes is divided into thirds (breaking the bottom half vertically distance into thirds or sixths). At the one-third line will be the bottom of the nose.
• Mouth: The next third, or two-thirds of the way from the eyes to the chin, should be the mouth. The mouth’s edges should be in line with the middle of each eye. To check this, put your pencil on the middle of one of the eyes. If the lower part of the pencil touches the outer corner of the mouth, it is aligned correctly.
• Ears: The top of the ears line up above the eyes, on the eyebrows (It doesn’t say in this set of guidelines, but I wonder what the ratio of the top of the eyebrow is with the overall face? Hmmmm…).  The bottom of the ears line up with the bottom of the nose.

Here is a sketch of a face with places to record measurements for all kinds of different distances. (Note the image doesn’t necessarily reflect the estimates above so that students need to use their own data to identify those ‘averages’.)

With these two ideas (drawing bodies/faces & measurement) in mind I think it is interesting to have students gather all kinds of measurement data (see my previous post on Measurement in the POS for KY), identify the means of the different measurements and use the measurements to create ‘average people’ or the ‘average face’.

This activity is a simple concept, not overly exotic, can be pulled off in relatively short amount of time, and opens up other opportunities (which is the part I think has great potential for extension/connection, but alas a post for another day).

When working with middle school students there is often a concern about being different. Obviously, this old man isn’t too concerned about fitting in anymore, but an alternative to measuring their own faces is to have students gather pictures of other people’s faces and measure those. One thing I like about this idea is that students can measure the attributes lots of different ways- measure from a hard or paper copy, measure electronically in the paint application on the computer using pixels as units (see next month’s blog post about specifics), measure using metric or English standard (or both if you want to look at the data from a bivariate perspective). It’s pretty easy to imagine students taking their own pictures to measure and makes sense to me, but both options are great ways to get students measuring.

I know this isn’t an original idea because artists have been doing this for centuries, but I think it is time mathematics teachers got in on the action.