# Math Challenge: Can You Draw This?

## 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.

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.

1. 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.
2.

3. With a sharp hard pencil (2H or harder), lightly
1. construct three concentric circles with radii of 32 mm, 59 mm, and 78 mm.
2. draw a diameter across the smaller, inner (32 mm) circle that also intersects the larger, outer (78 mm) circle twice, with ticks.
3.

4. poke your compass into one of the intersections of the inner circle and its diameter;
5. without lifting the poked end of your compass, stretch your compass to the opposite arc of the middle (59 mm) circle; and
6. scratch an arc above and below (orthogonal to the diameter) the common centre of the concentric circles.
7. repeat the last two instructions after poking the other intersection of the inner circle and its diameter.
8. use your ruler to “connect” each intersection of the arcs and the common centre of the circles.
9. with your ruler set as just instructed, draw a diameter across the inner circle and two ticks intersecting the outer circle.
10.

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

11. poke your compass back into one of the two holes that you just made in the inner circle.
12. scratch an arc across the middle of the two 90° angles adjacent to the diameter you are “in”.
13. repeat the last two instructions for each of the remaining three intersections of the inner circle and its two diameters.
14. use your ruler to “connect” the intersections of opposite arcs and the common centre of the circles.
15. with your ruler set as just instructed, draw a diameter across the inner circle and two ticks intersecting the outer circle.
16. repeat for the other pair of opposite arcs.
17.

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

18. once again create arcs to bisect these angles following instructions 2i — 2l.
19. for each pair of opposite angles, with your ruler set as instructed, scratch a tick where the ruler intersects the middle circle (eight ticks total).
20.

21. starting with one tick intersecting the outer circle, connect that point of intersection with an adjacent point of intersection in the middle circle.
22. connect that middle-circle intersection with the inner-circle intersection along the same radius as the already connected outer-circle intersection.
23. 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.
24. repeat the last three instructions to complete the ring of eight squares.
25.

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

1. 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
1. the outer circle in one colour; Nguyen used green.
2. the horizontal and vertical line-segments (as you look on the figure) in another colour; Nguyen’s was dark-blue.
3. the diagonal line-segments in a third colour; Nguyen used light-blue.
4.

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

1. Check Nguyen’s post to compare your drawing with the original figure.

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

# Emulating Evolution by Involving Students

Evolution is one of those topics I really enjoy learning and teaching, particularly with open activities. The War of the Evolutionists Web Scavenger Hunt and Mock Trial, and the Evolution and Adaptation games are just some of my attempts to engage students in ecology and evolution.

Of these The Evolution Game, created by Simon Boswell and Phil Lewis, comes the closest to emulating the process and product of evolution. The rules and gates of the game are the processes of ecology and evolution; it is not a trivia game, a passive one, nor one like charades whose play has little semblance to evolution. As such The Evolution Game exemplifies that rare and powerful quality game that embeds students in their learning by forcing them to deal with and adapt to the “rules” or processes of evolution and ecology. It is however a long-term activity, like Risk, of both strategy and chance.

Published with permission

Just yesterday, I came across another great activity that comes close to emulating both ecology and evolution. The activity, created by Tyler Rhodes and featured by Scientific American, consists of two parts: a student-centered exercise and a technical exercise. The student exercise takes about an hour, or one period, to conduct. The technical one — creating a video — took Tyler, who claims to be expert enough to work efficiently, three months to complete. So feedback in the form of product is delayed, though formative feedback is immediate and embedded in the students’ own activity.

The idea is simple, if not elegant, and follows the same design premise as The Evolution Game and Bernie Dodge’s formula for game design that “Elegance = congruity between the forms of the game and structures within the content“.

Tyler drew a nondescript salamander-like creature and enlisted five independent groups of students (from five schools) to draw copies of this creature.

Once the students compared and discussed the new creatures, Tyler “exposed” the creatures to some ecological stress or change. The students had to vote which creatures perished based on the new ecology and the features of the creatures. According to Tyler, ninety-eight percent of the creatures perished (in a class of 30, where each student drew one creature, one creature survived). Tyler gathered the extinct creatures and repeated the exercise five more times with the survivors, each time with a new ecological event wiping out ninety-eight percent of the creatures. The six generations were kept or labelled apart.

The exercise illustrated branching phylogenetic evolution and coevolution — rather than the defunct linear evolution — as shown in the following drawing, where each “arm” of creatures came from a separate class of students, so giving five arms.

Click on the image to enlarge it.

What is nice about this exercise is that the students actively engage in, and embed themselves in, the process of natural selection by ecological change. They, being the active agents in both the creation and voting off of these creatures, were given the opportunity to experience and learn about the fundamental processes of evolution, much like The Evolution Game.

Tyler designed this process after a “Chinese Whisper” or “Telephone” game, where a message is passed from person to person and changes through mutation as it is delivered. In fact he presented it as a game. The message, however, was visual — the drawing of the creature — and the changing ecology affected the message. Tyler was specifically looking for a way to branch the creature evolution like a phylogenetic tree and his use of the “Chinese Whisper” or “Rumour” game enabled this.

Here is the final video.

The creatures in this video are those from the top-left arm of the Wheel of Life tree illustrated above. Tyler promises four more videos, for each of the remaining four groups of students. And he invites teachers to take his initial nondescript salamander-like creature, repeat his method and e-mail him facsimiles of the creatures created. If teachers take him up on the offer, he can bank, analyze and share some really interesting evolutionary results from the project. His conclusions should be interesting.

For more information on Tyler’s project, visit his blog, Evolution!, documenting his progress.

How can we emulate Tyler’s project for outcomes in our classrooms? Or have you done so already?