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.

Inspiring the Next Generation

I have some interesting news to share.

The Grade 10s in one of the schools where I sub began their poetry unit in English this week. I subbed for them yesterday.

One of their tasks yesterday was to write a poem in one of the forms they had already learned, then share these with the class. There were some very reluctant students; they had a low opinion about this sharing business, particularly their contributive involvement in it.

I decided to break the ice by sharing one of my poems. And I had access to two: those I published in my writing blog, which you can alternatively link to through the Write Group wiki.

The poem I chose to share was Van Gogh and the Moon. It was a hit, particularly when I explained to the kids that the poem was an in promptu (five minute) response to a writing prompt in the local writing club.

So, yes, I got a chance to plug the Write Group as well; I told the kids that students from the school were part of the group, which peeked more interest.

But more importantly, it got each of the students to open up and share some of their poems, not just those they wrote in class yesterday, but those they had access to through their iPhones and other devices.

It was a perfect marriage of teacher and student sharing, technology (I used the Smart board; the students used their devices), and encouragement and modelling by example.

It never ceases to amaze me how well these teachable moments go when the teacher releases control and opens up to her or his students. (Of course, it also never ceases to amaze me how badly such moments go as well at times. There is a definite case for timing and thoughtful and responsive judgement here.)

These students have everything to be proud of. They have incredible imaginations, and a deep and active appreciation for written communication.

Moments like these remind me how much I love teaching, and learning with, these students.

Periodic Troubles and Screencasting

Screencasts and concept illustration

It is interesting how posts develop in this blog. For the last two weeks I have been working on a screencasting post to celebrate my first screencast, which I posted on my YouTube channel. But I have had a little trouble with some unexpected construction scenarios in GeoGebra for another screencast I wanted to add to the post.

And just yesterday, I was asked to teach several classes of an introductory Periodic Table unit. My students consistently (every year) have trouble with key concepts in this unit, and I always end up wanting for some videos to illustrate the otherwise arm-waving concepts they have difficulty with.

So, with this Periodic Table unit in mind, last month I participated in a Moodle course on screencasting, my notes for which you can find on my PD page. The way I see it, there is no video out there that will address my students’ confusions better than a video that I make myself — unless of course my students make their own videos. And I do realize this video will likely not serve the specific needs of other students and other teachers, as it will just miss the mark and fail to address exactly what other teachers need to elaborate.

I already have several resources for this unit: a couple of web pages on introductory molar chemistry (conceptual only) and periodic tables, which are chalk full of links, and a post that complements these.

But I want to see if there is more out there. So I started a tweet, and realized there was no way I could say in one tweet what I wanted. So I started this post. Interesting how these posts develop.

This post is more of a request. My students are just getting their first exposure to the Periodic Table. They have trouble with:

1. valence, families or groups (periodic table columns) and periods (periodic table rows) — essentially, basic electron configuration principles, Pauli’s Principle, Hund’s Rule, et cetera, and
2. energy states and electron exchange during ionic reactions. Much of the problem with this lies in the insistence of schools on teaching kids to “orbit” electrons around the atomic nucleus, thereby confusing orbital and electron configuration understanding.

Do you know of or have any great short activities, concise and clear resources, or videos that address and illustrate these concepts?

I would like to expose my students to others’ works and explanations, in addition to those I will be making this week and weekend. The resources and activities need to be short since I am a substitute and for the most part will be following the lesson plans of the teacher I am subbing for, but I would love to expand my students’ learning base.

I appreciate any help anyone offers and will proudly credit social networking and individuals for contributions. Knowing my kids, they will be thrilled by the reaching out and the resources that come in.

Thanks in advance from my students and me. I will post a follow-up to this request and let you know what resources were used, how they were received and how they helped my students’ understanding.

Inspiration Against a Lost Generation

This video serves two purposes. First, it is a creative way to communicate a profound and inspiring message for everyone, by juxtaposing opposing tones via a literary twist — a winding and unwinding technique common to folktales. Second, it is a reminder to me, and perhaps to many of you, that this is what my generation felt and talked about when we were in high school and graduating. I find the echo of my thoughts juxtaposed against this video nearly a quarter of a century later rather interesting.

There are so many parallels, some of which we recognize right away, some of which we forget until we are reminded.

For some reason, this video reminded me of my generation’s movement to curb pollution and yet the nearly simultaneous increase in vehicle turn-over and layers of packaging around otherwise small items. Today, our kids and students are still moving to curb pollution, though with a narrower scope — less, if any, emphasis on all the forms of pollution, including light, noise and odor pollution, and more emphasis on pollution that perpetuates and aggravates global warming. There is much happening in the World that they have little notion and control of, as was true for us. However, as we became more pollution, waste and recycling conscious and active, I wonder what their generation will accomplish.

What parallels do you see between the priorities and ideas of your students and those you had when you were their age?

Blog Bling and Creative Commons

Want some photos, clip art, videos, music, sounds, polls, inframe-content or other bling on your blog? Not sure where to find it and if you can legally use it?

The Commons

Last week, I attended the CEATCA 2012 Teachers’ Convention. And, like the last one, I left with some information to share.

The convention this year had a couple of Creative Commons PD sessions. I am a strong advocate of respecting copyright and attribution. This is largely because I am a teacher and I want my students growing up mindful of and reverent to their own and others’ thoughts, actions and creations. I also am an artist and writer and care about others’ intellectual and artistic properties.

Pete MacKay of 2Learn.ca, in his CEATCA Convention PD session, A Picture Is Worth A Thousand Words But Is It Free To Use?, summarized the Creative Commons, public domain, Copyright Zero and copyright licenses and how they should be dealt with. There were a few points that I would like to share with you.

1. These licenses apply to everything. If you didn’t make it, someone else owns it and you need to determine how you can use it. In Canada, according to Pete, fair use is no longer in effect. It never was like fair dealing in the U.S., but now we don’t even have it. I am not sure how I feel about that. It will have severe repercussions in areas of research.
2. Even if you get permission to use material, the creator still retains his or her rights. You do not own it. Yeah, old news, but news that is often forgotten.
3. You do not have the right to broadcast what you purchase or what you get through Creative Commons. This includes mood music before a presentation. This is not news, but it is worth rementioning. (By the way, I now have permission to use all parts of that 50 second video.)
4. Share-alike means if you use somebody else’s work as a part of something you make, what you make must also be share-alike. You can not copyright your work while it contains share-alike components. This is an interesting slant on what I understood share-alike to mean.
5. If you play music over an image that is protected by no-derivatives, you are breaking the law. But if you insert that image into a silent slideshow and in no other way alter the image, you are not. Okay! That one still baffles me.
6. Any looser license you apply to any of your work, such a Copyright Zero, is irrevocable due to the grandfathering effect.
7. YouTube has a three strike policy. First strike, you get a notice, but you don’t have to do anything. Second strike, you have an ad place over your video (but these can be closed anyway). Third strike, you are “encouraged” to buy iTunes. There is a fourth strike somewhere in there too: you have to watch a copyright video. (This information was provided by Dave Mitchell, who discussed copyright in relation to YouTube in another PD session of the convention1.)
8. To quote Creative Commons Search, “Do not assume that the results displayed in this search portal are under a CC license. You should always verify that the work is actually under a CC license by following the link.” In short, trust only the raw source of the material.
9. If you want to discover how an image is licensed, you can reverse search it.
10. You can also cite material properly in current APA and MLA format.

So, bottom line, why should your students and you adhere to copyright laws? Well, how would you feel if someone took credit for your thoughts and work? And what if the material you are proudly exhibiting on your site was stolen by the person you got it from? Who gets the blame? Copyright protects and respects everyone, including you.

The Bling

The PD sessions on copyright that I attended at this year’s convention inspired me to publish and share my go-to image-and-media resource page with you.

This list of 150+ search engines, tools, resources and licensing explanations for enhancing your posts and adding that deeper dimension to your content is where I go when I create media on my blog. I offer it openly in hopes that you find it useful in your image and multimedia adventures. Just respect the copyright wildlife and you will do well.

Have fun with the resources.

The Algorithm in the Code: Building an Inquiry Tool

A couple of days ago, I posted a Math Challenge posed by David Wees some weeks ago. The code emulated Euclid’s Algorithm of Coprimes and GCFs.

First analysis

Analysis of the code reveals that, when a=0, b=b and, when b=0, a=a. However, a reaches zero at the code’s onset, while b does the same after the code runs through scenarios when b≠0. This implies that one of the two values reaching zero is key to the code and the quantity of the other value when this happens is informative.

Tabulating the difference between possible values of a and b within an arbitrary range of integers might illustrate how b=0 is reached. This process falls in the Planning and Implementation steps of David Coffey’s thinking-stage charts. Here is my table for mapping the “moves” of the code within the range of -9<a<9 and -9<b<9.

Playing with b≠0

Notice that a>b below the a=b or 0 diagonal. So, for instance, the difference between a=6 and b=4 is 2, found in the bottom-left triangle of the table. In this triangle, according to the code, the difference a-b equals a. So, now a=2 and b=4.

Repeating the process using a=2 and b=4 produces a difference of 2, this time in the top-right triangle. The new difference b-a equals b. Now a=2 and b=2, which produces a difference b-a of 0. Since b-a = b, the code ends with b=0 and a=2.

Cases

There are six distinct cases where the code returns unique case results.

Case 1: a = b

When a=b, the code returns their common value. Why? As shown in the example above, the step after a=b is b=0 and the value of a is returned. This value is that when a=b.

Case 2: either a = 0 or b = 0

A starting value of a=0 returns b. A value of b=0, returns a. This is a rule built into the code. But what would happen if the rule were not followed?

Let’s take our b=0 and a=2 example beyond termination. Continuing the while-loop produces a difference a-b of 2, in the bottom-right triangle. This difference returns a=2 and b=0, exactly where we started.

What if a=0 and b=2? The difference b-a returns b=2 and a=0, another recursive repeat.

So, a=0 returns b and b=0 returns a. If a=b=0, zero is returned (in agreement with Case 1).

Case 3: a < 0, b < 0 or both < 0

When either a or b or both are negative, the code never resolves to termination (except when a=b, Case 4). In fact, the greater value iterates to infinity in steps of the lesser negative value.

Let us try a=3 and b=-2 (we could easily have tried a=-2 and b=3). The difference a-b returns a=5 and b=-2, which in turn returns a=7 and b=-2, then a=9 and b=-2, ad infinitum.

a=-3 and b=-2, on the other hand, returns (a=-3,b=1), (a=-3,b=4), (a=-3,b=7), again ad infinitum.

Case 4: a = b < 0

Contrary to Case 3, when a=b<0, the common value of a and b is returned, in agreement with Case 1. Ignoring this, Case 3 is followed; however, there is no condition that rectifies the ambiguity of which direction, toward infinite a or infinite b, the map should follow.

Case 5: +a and +b share a common factor

When a and b share a set of common factors, the greatest of these factors is returned, as per the a=6 and b=4 example which returned 2, the greatest common factor of 6 and 4.

Case 6: +a and +b are coprime, or relatively prime

When a and b do not share a common factor, 1 is returned, since 1 is the only natural number that is a divisor of both.

Let’s map a=3 and b=8. As you can see from the table below, 1 is returned.

The analysis of cases weaves over and through David Coffey’s thinking-stage charts’ Analysis through Verification stages.

Interpretation and Pedagogy

I was introduced to the formal Extended Euclid’s Algorithm via induction within a discrete mathematics university course. It was taught to me as a means to learn modular mathematics, so not much emphasis was placed on explaining the Extended Algorithm nor the induction. In fact, given this challenge posed by David Wees, or perhaps more so the table derived from it, the manner in which I learned the Extended Algorithm was probably the worst possible.

David’s challenge and the table offer great entry tasks into the study of GFCs, coprimes, Euclid’s Algorithm and several branches of mathematics that build from them. Before the Algorithm is even named and formalized, students get to explore its mechanisms and formalize their own rules based on their mapping activities. Once they master the code and table, they can learn the corresponding Algorithm schema with emphasis on matching the items of the schema to the mapping on the table and the methods in the code. Then the Algorithm can be named and its uses illustrated.

For those students who do not know code, the teacher can interpret the code with them and offer scaffolding afterward. The code is probably easier to understand than an instruction list, if instead of treating it as code, the teacher treats it as an outline of process. Notice the subtle difference here between instruction (do this) and process (this is how this works).

The table doesn’t just determine GFCs and coprimes, it illustrates how greatest common factors and relatively prime, or coprime, numbers are calculated. It also illustrates why negative integers do not produce finite results, except where a=b, and why a=0 returns b and b=0 returns a.

One question that might remain is what the table and code return. In the case of positive integers, the returns are obviously GCFs or 1. Interpretation can determine whether the initial values are coprime or related by common factor. But what does the return of a when b=0 and the return of b when a=0 mean?

Quite simply the returns are the divisors of the numbers being analyzed. So, if one of those numbers is zero, it stands to reason that the other number is a viable divisor of zero. For instance, when b=0 and a=3, the return of 3 signifies that zero is divisible by three. Arithmetically, when a=b=0, infinity or undefinable should be returned, since conventionally no number “can” be divided by zero. This is the one flaw in this code and table.

In order for the constructed table to be a viable tool for learning Euclid’s Algorithm, it should be printed out or created with non-erasable ink and the mapping should be done with pencil and eraser. The table can be used several times then to build literacy, mastery and fluency of Euclid’s Algorithm.

Do you have any tasks that engage students in active learning of the outcomes, content, skills and concepts you are teaching?

Math Challenge: Do you know what algorithm this is?

David Wees came up with this challenge. Determine what algorithm this code emulates. You will find the answer more informative if you create a table to see the pattern of moves the code makes. The table can then be used to introduce the algorithm formally to your students. Or better yet, get them to build their own tables from the code. Try a range of integers to test the code. What patterns exist?

I will post my answer in a couple of days.