**Please, do not visit her post just yet**

Fawn Nguyen wrote a post on Friday that caught my attention.

She divided her math class into pairs, making one person in each pair a *Describer* and the other person a *Drawer*.

She then gave the *Describers* a figure the *Drawers* were supposed to draw at 1:1 scale and three rules designed to prevent the *Drawers* from seeing the figure, the *Describers* from seeing the *Drawers’* drawings, and either from using gestures and body language to signal information.

They were allowed to talk all they wanted.

I thought today I would be your *Desciber*. We can use the comments to this post to talk to each other.

**Exercise in visualization and communication**

This Challenge is an exercise of visualization, communication and knowledge. Visualizing and knowing what one is seeing (recognition: visualization, knowledge); describing this efficiently and effectively (oral communication); visualizing what is described (listening, visualization); connecting that visual to known shapes and images (knowledge); and efficiently and effectively drawing the visualized figure to scale (visual and haptic communication, knowledge) are foundational skills our students need to use and communicate mathematics both within and beyond the classroom.

I participated in a similar exercise Dr. David Pimm conducted in a Math ed graduate class. The *Describer* picked a part of a much larger and more complex figure than Nguyen’s figure and described it and its location to the *Drawer*. The *Drawer* visualized, located and identified, rather than drew, what the *Describer* was seeing. We were required to sit on our hands, a behaviour that would have made Nguyen’s *Drawers’* jobs much harder. But the *Describer* and *Drawer* in Pimm’s class had to confirm orally that the visualizations match (secret messages, discrete mathematics), a slightly different skill than Nguyen’s *Drawers* were learning.

Both exercises are simultaneously frustrating and engaging, like an addictive game one struggles in, cannot win and yet cannot put down. By design, the exercises target these emotions; one’s communication and visualization skills; and one’s knowledge. These are emotions and skills that a majority of their time the mathematician and the teacher — indeed all people — encounter and need to cope with. (For this reason, Pimm’s exercise was perfect for preservice teachers.)

Good luck. Enjoy this math Challenge.

**Ready, set, …**

There is only one rule here: **do not visit Nguyen’s post no matter how tempted you are to do so** until you have finished drawing your figure.

Feel welcome to comment me with queries, comments, your final drawing, instructions for a figure you find or design (please provide a URL to this figure, so others and I can find it, and yet not see it here), and most importantly your reflections (both experiential and critical) after taking up this challenge.

Draw a figure following these instructions.

- Use a ruler and a compass to draw this figure. Blank paper; a sharp pencil (2H or harder); eraser; and either coloured pencils (green, light blue and dark blue), another sharp pencil (HB or softer) or a pen (or three of different colours) may also help.

- With a sharp hard pencil (2H or harder), lightly

- construct three concentric circles with radii of 32 mm, 59 mm, and 78 mm.
- draw a diameter across the smaller, inner (32 mm) circle that also intersects the larger, outer (78 mm) circle twice, with ticks.

- poke your compass into one of the intersections of the inner circle and its diameter;
- without lifting the poked end of your compass, stretch your compass to the opposite arc of the middle (59 mm) circle; and
- scratch an arc above and below (orthogonal to the diameter) the common centre of the concentric circles.
- repeat the last two instructions after poking the other intersection of the inner circle and its diameter.
- use your ruler to “connect” each intersection of the arcs and the common centre of the circles.
- with your ruler set as just instructed, draw a diameter across the inner circle and two ticks intersecting the outer circle.

Notice, the two diagonals now frame four 90° angles.

- poke your compass back into one of the two holes that you just made in the inner circle.
- scratch an arc across the middle of the two 90° angles adjacent to the diameter you are “in”.
- repeat the last two instructions for each of the remaining three intersections of the inner circle and its two diameters.
- use your ruler to “connect” the intersections of opposite arcs and the common centre of the circles.
- with your ruler set as just instructed, draw a diameter across the inner circle and two ticks intersecting the outer circle.
- repeat for the other pair of opposite arcs.

There are now four diagonals that frame eight 45° angles.

- once again create arcs to bisect these angles following instructions 2i — 2l.
- for each pair of opposite angles, with your ruler set as instructed, scratch a tick where the ruler intersects the
middlecircle (eight ticks total).

- starting with one tick intersecting the outer circle, connect that point of intersection with an adjacent point of intersection in the middle circle.
- connect that middle-circle intersection with the inner-circle intersection along the same radius as the already connected outer-circle intersection.
- complete the square by connecting the inner-circle intersection and outer-circle intersection with the intersection in the middle circle that the unfinished square opens toward.
- repeat the last three instructions to complete the ring of eight squares.

The construction is complete; however, you might want to emphasize the figure by colouring or darkening its components.

- With a sharp soft pencil (HB or softer) or three pencil crayons of different colours, or with a pen or three pens of different colours, darken

- the outer circle in one colour; Nguyen used green.
- the horizontal and vertical line-segments (as you look on the figure) in another colour; Nguyen’s was dark-blue.
- the diagonal line-segments in a third colour; Nguyen used light-blue.

I stippled the squares and scumbled or hatched alternate rhombi to emphasize the three dimensional effect.

So did you do it? Did you get it right? What did you learn? I would love to know.

Send me your queries, your comments, your final drawings, your instructions for a new figure, and your reflections (both experiential and critical).

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