# 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.

# 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.

# Math: Let Students Earn Their Weight

In a post he published yesterday, John Golden offered two geometric problems he had included in a math final for his preservice math teachers. Please visit his post and take a crack at his problems.

What I really liked about these problems was how open they were. The first problem, giving a circle geometry diagram, challenges John’s students to “Figure out some of the missing information in the diagram”. The second problem, also giving a diagram — this one of a series of lines, segmented only by the diagram edge, asks students to “Find more angles”.

At first, I thought this was a rather vague way to offer problems, particularly on a final exam where marks are summative and presumably weighted more. Then, I gave it some thought. I am not sure what John had in mind, but having questions which ask for “some” or “more” answers leaves the onus of how much work the student does and what weight the problem has on a test on the student. A way to assign a mark to such questions is to make “Each correct answer worth one mark”. Then all one has to do is make the test worth out of so-many maximum marks (less the total possible to get), order the questions so the “big” ones come first and voila … one has a test that the student weighs.

But what about students who answer only some questions thoroughly and skip others? One could make each question worth a minimum amount of marks, or make all questions equally difficult or multi-outcomed. There are likely other ways to ensure all outcomes are covered by the student.

It is an idea, one that just occurred to me as I read John’s post. What I really like about it is that it is truly open. The student cannot prejudge how much a question is worth and allocate effort accordingly. The grade the student gets is tied to the value the student, not the teacher, applies to each question.

What do you think? Do we herd our children with our judgement of worth? Should we be teaching them to do this intrinsically?

Oh, and by the way, my next post will show my solutions to John’s questions.

# 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.

# Math Challenge: The Irritating Polynomial Factoring Problem

Every math week, I plan to provide a math challenge which could take you seconds to hours to solve, assuming you don’t cheat by using technology to solve it for you.

Today, I offer a time filler I occasionally use while substituting a junior or senior high math class that has “completed” all of its work, usually before I even enter class. I learned this problem from Blaine Dowler of Sylvan Learning. Usually I reward the first person who solves it, and usually it is the “weakest” student who does it.

The challenge is: If the product of two numbers is two and their sum is five, what is the simplest sum of their inverses?

I would love to hear how you solved this challenge.

Good luck.