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.

4 George – Playing in Public

Remember those days when the snow first fell and the streets froze and you just couldn’t wait to get outside and play?

The first snowfall usually happened at night and, if it was early enough in the evening, you got to marvel at the many-colored flashes of tiny crystals fluttering to Earth and the multihued sparkle of snow twinkling in the building drifts.

Or maybe the first snow fell during the day and you inhaled upon seeing big fluffy white pillows sprinkling toward Earth, some melting but some also blanketing the yard or field or grove of spiring trees.

Perhaps it was long ago? Perhaps it was this year? Perhaps you were only a child or you rejoiced into adulthood?

Either way, you know you wanted to play.

Playing in Public

George Couros recently challenged his PLN to sketch people, particularly children, Playing in Public and document the process of sketching in a video.

The first to respond to this challenge (actually her response resulted in the challenge going public) was Malyn Mawby. Her terrific sketch and video set the bar high for future responses.

It also inspired my sketch.

This sketch features children playing in the snow: building snowmen, sliding on the frozen river covering the pavement and throwing snowballs from behind snow forts.

medium: pencil – HB2
time: 2 hours (draft and trace)
digitizing: 6 photos – 6 minutes
editting photos: 6 hours (see below)

The Concept

In the Northern Hemisphere, Winter is approaching and snow is beginning to fall. I found this in stark contrast to Malyn’s sketch of her girls playing on the beach. I also recognized that Malyn lives in Australia where Summer is just about to begin. What an opportunity, I thought, to illustrate that play is an all-year activity!

I was inspired also by the recent first snow that we experienced this season and my desire to build a snowman with my nephew. It will be too cold and the snow too dry to build a snowman when he does visit his uncle. So this sketch expresses a dream as much as anything.

I also recognized that sculpting of sand castles and snowmen and snow forts was similar enough to deepen the parallels between Malyn’s sketch and mine.

The children in my sketch are generic, pulled out of my head. I did not have anyone in mind, nor did I sketch from any photographs. I liked Malyn’s illustration of distance and collage in her sketch and tried to emulate these. However, unlike the children in Malyn’s sketch who are her girls over several years, the children on my sketch are contemporary. I think Malyn’s use of time to sketch her girls is personally nostalgic to her, so I did not want to replicate that effect and affect.

My decision to design the sketch as I did was strongly influenced by my own nostalgia: my memories of building snowmen and forts, of throwing snowballs and of sliding down an icy road, all in the name of play and fun. I remember also making snow angels, skiing cross-country and downhill, tobogganing, building quinzees, playing tag and hockey, and snowshoeing, but the sketch had only so much room.

Play is engaging in an activity to gain a state of flow. It does not necessarily need to be outside nor involve many people. Sketching, like the sketches Malyn and I produced, are also play, as is engaging in some work, such as wood carving or even math problems.

I write, paint and learn for fun. Everyone does something different.

The Process

The Sketch

I drew my sketch in three drafts. The first draft was just a small ghostly outline to develop my concept and place and frame subjects. My second draft was a full-page (8.5″ by 11″) sketch that probably was the better draft. It was more freehand and wispy, much like Malyn’s; however, I sketched it on a scrap sheet and the type on the opposite side of the sheet was clearly visible. I inked in the second draft and traced it systematically to produce the final sketch.

I am a mediocre drawer at the best of times and often rely on paint (oils) to bury my sketchy drawings. This time I did not have that luxury and, in Malyn’s words, my sketch is naive. (Yeah, there really is such a style; I was astonished when I learned this.)

Like Malyn, I have trouble with faces and hands. In my case, mitts served to bury hands. But my faces were a problem. I wanted to show enjoyment, so I needed to draw faces.

Yet, all the noses I sketched are gigantic. Additionally, the faces are masculine. Look closely at the girl sliding on the road. If she didn’t have hair, she would be another boy in a male dominated sketch! Insincere political correctness aside, I wanted to show girls and boys playing together, so I wanted some girls. That slider was a nightmare to sketch. It took some heavy editting just to imply she was indeed a girl and not some guy with a strange toque or wig on his head. Notice, no one else on the sketch has hair!

So sketching was definitely a fun and often comic challenge.

The Photographs

I took pictures of the third draft of the sketch at various intervals during its creation. A tripod and frame to place the sketch in would have helped with squaring, leveling and maintaining the same height for each photo. The photos I took came out with my sketch crooked and even trapezoidal. I ended up editting each weird product of my photography with Windows Paint, even though it took hours to do so.

In hindsight, I could have taken one photograph of the final sketch and deleted objects as I saved the editted sketch. This would have made an even smoother video, but it rang of cheating to me. I was rather enjoying sketching, photographing, sketching and repeating the process.

The Video

Below is the final video of my sketch. Enjoy.

I used Windows Movie Maker to create the video. The audio was clipped from Play in the Snow, a 1945 education film in the public domain and presented by Encyclopedia Britannica Films on Prelinger Archives. I used RealPlayer to trim and convert the film’s MP4 video into MP3 audio clips, so I could use these in my video.

The sound is a bit crude and out of step, but it carries the spirit of kids enjoying the Winter season.

Troubles with George

In his challenge, George forgot to mention that he wanted a silent video. So I spent hours adding audio to my video, only to get a tweet that silence works best. Fine, I can do that. George then also asked for a video in MP4 or AVI format. Okay, I’ll see what I can do.

It was not as obvious as it looked!

I was only following Malyn’s lead.

How do I convert RV and WMV into MP4 or AVI on my machine? I searched RealPlayer. I searched Windows Media Player. I searched Windows Movie Maker. Hmm! This was a problem.

I finally did discover how to save my original Movie Maker project as an AVI. The option was not as obvious as you might think. I also removed all the sound from the project. So, this is easy. Save. Check and test to make sure the new format took.

213 MB!

Are you kidding me? That is 40 times larger than the RV and WMV versions with sound. This is six photos stitched together in a video folks. What went into that 213 MBs?

Okay. Upload the silent video into George’s dropito.me box to go with the RV and WMV versions, with audio, that I already uploaded.

Forty two minutes later, the silent video is finally uploaded!

Watermark? What watermark?

I replay the silent video on my computer. No watermark! Perhaps it is a dropito.me thing?

So, I upload the 213 MB video into my Teaching Resources webspace. Another 42 minutes later, the silent video is ready again.

It works!

Troubles with George!

Reflecting on Fun

This was a fun project, even the part regarding the enormous silent video. Playing is engaging in any activity that leads to a state of flow. In this case, humour also played a big part. George can ask me to create another video for him any time.

Pi Day: Part Tau

There has been increasing buzz about Pi and Pi Day lately, probably because July 22 — 22/7 — is approaching. And though any web search can find this buzz, I would like to add a comment about Tau Day (June 28), which is adding noise to the excitement.

“Ya! Do tip: Laud Tau, at dual Pi, today!” “I prefer Pi”

Tau, having been recently replaced by phi as the symbol of the Golden Ratio, is a proposed symbol for the value 2*Pi, lauded (to borrow from the palindrome above) by those who think 2*Pi is a better measure in all ways than Pi.

This claim, of course, has created a heated but sometimes hysterical discussion, and post and video “war”, between two extreme camps: the Pi’s and the Tau’s. Add to this a third camp, those who argue for Eta (Pi/2 or Tau/4) as the standard, and a fourth, who promote Pi/4, and we have a mathematical event to behold.

Of course, none of this really has serious implications, since we all know that Pi/3 or Tau/6 ¹, the sextant, with its relationship to the equilateral triangle, radian, Babylonian sexagesimal system, Earth year and rational cosine value, is the real fundamental unit of the circle.

It seems we are confused.

This of course is all in good fun. Yet tell that to Tau Beta Pi and Tau Alpha Pi, the Engineering and Engineering Technology Honor Societies, whose society names are at risk at both ends.

It’s happened before

Nor is this the first time a math convention has been questioned. In fact, it is not even the first time a circular measure has been questioned.

Before there was trigonometry (the study of measures of closed three-kneed figures or triangles), there was circle geometry (the study, which we still have — but lacking the now analytic trigonometric part, of regular closed no-kneed figures or circles). In circle geometry, the chord is king and emphasis is on the geometry and measurement of the circle, line segments and angles. Everything was working really well.

Then came sine!

Sine, of course, is half a chord, chord/2. It seems weird to us now, but at one time sine was the oddity trying to replace the convention. Yet, when it did become convention, a new field of math was born.

In a history shamefully oversimplified, circle geometry split into two fields and, for the sine portion, analysis, ratios, angles and triangles became the emphasis, so leading to the title of trigonometry. Circle geometry continues to deal with geometry and measurement. And the chord is the usurp outsider.

And history we witness

So we have Tau (whole), and Pi (half), and Eta (quadrant), and Pi/4 (octant), and Pi/3 (sextant). I wonder what history will come of this.

Importance: A rose in the classroom

Does it really matter what constant we use as the base unit for circle measurement? They are just names. Some formulae will work better in some situations than others, and this will change with situation.

How we choose to deliver the concepts to students is far more important than what we call them. Truth be told, all systems should be taught interconnectedly with no mention of which might be the opinionated best.

The key is engagement and problem solving. Students need to understand how to use the math, why it works, where it is used and how it is used. It would benefit them if they learn their own formulae, and we help them “conventionalize” these to fit what other mathematicians do and say.

Changing of the conventional Pi to Tau, or Eta, or either of the other measures, might change the nature of circle and trigonometric math in ways we can not predict at present, but that will come in the future. Today we have these five fundamental units, which are all arbitrary, math-founded and related. Who is to say which is best?

By any other name, is a rose not still a rose?

More resources

Want more? Visit Benjamin Vitale’s June 28th Pi is wrong! Here comes Tau Day, watch Vi Hart’s Pi is (still) wrong, and read Bob Palais’s original π is wrong! which started the Tau, then two-pi, movement.

Then read Mike’s response to The Pi Manifesto, from the creator of Spiked Math Comics, then the continued debate on On Pi Day we eat pie. On Tau Day we eat Taoists? and ‘Tau day’ marked by opponents of maths constant pi, including its comments.

Given that we are rather gossipy creatures, most of the posts, discussions and video have titles that attack poor Pi and its Day, or Days (March 14 or July 22). But a few out there attack Tau as well. The Eta’s, Quarter-Pi’s and Equilateral’s just cling where they might be heard. Sounds like a school yard, doesn’t it?

This post actually started as an update comment to my Math Challenge: Pi Day, but the comment morphed into a post of its own, so I decided to make it so. If you are interested, please visit my Pi Day math challenge post.

¹ Given its Babylonian pedigree, perhaps we should call Pi/3 Sedis, which is six in Assyrian.

Math Challenge: All-digits Arithmetic

Dennis Coble, @DennisCoble, just tweeted me this challenge an hour ago.

Here’s 1 that might interest you. Numbers 1-9 all used: 3 digit number, plus or minus 3 digit number, gives another 3 digit number.

As usual, the problem is deceptively simple, as is the solution. However, students could be engaged in their activity of this task for a full period. 😉 And the ordering of student ability and success could be shaken. 🙂

Enjoy.