Try tracing the figures on the following page without lifting your pencil from the paper, without retracing any lines or crossing any lines. Some figures you can trace and some figures you cannot. Euler, an early mathematician, studied figures like these and came up with a rule so that you would know which figures could be traced and which could not. He found it helpful to categorize vertices as odd and even, depending on the number of line segments coming from each vertex. Investigate figures to see if you can come up with Euler's rule for classifying figures as traceable and those that are not traceable.
The class was measuring and classifying angles and naming polygons. We were also working on looking for evidence in student work to begin self-assessing problem solving. I had the students thinking of themselves as investigators looking for evidence in their work, as well as other student work. This problem seemed to go along with our investigative work from a different angle. Instead of looking for evidence, they needed to produce the evidence.
This task allows me to determine which students can experiment, record results and look for patterns when what they are looking for is obscure. A student will need a way to organize the results of their experimenting carefully so they can begin to see patterns. All students started to trace the figures I gave them. Most students at this point only recorded whether the figure was traceable or not. Some realized that a chart that kept track of the number of odd and even vertices probably would be helpful since that was clearly important . It was mentioned in the problem and in our discussion of the problem I emphasized the importance of odd and even vertices. As students came up with rules, some tried out their rules on figures they drew and tried to trace. Two, 45-minute class periods This task can be used on units that study famous mathematicians, art projects and puzzles. Be sure the students understand odd and even vertices and that this knowledge is needed to come up with the rule. I let kids (Ha! I really did not have much choice - the investigation lends itself to discussion of the traceable and untraceable figures.) discuss which given figures were traceable and which were not so that everyone had the correct information to work with. If students came up with a rule that was not correct, I often tried to make a figure that would contradict their rule, so that they could go back and look at their data again.
- *Extra copies (or tracing paper) of the figures
- Transparencies (with dry erase markers) so kids can trace shapes without altering them.
- Graph paper (Some kids might want to organize their chart.)
*Some kids trace so many times over one figure that they do not have a clean copy to work with. If the figure has zero or two odd vertices, the figure can be traced. You start at one of the odd vertices and end at the other odd vertex.
A Novice paper would be one where the student has evidence of tracing the figures, but really does not see how odd and even vertices help in solving the problem and does not have a strategy that would help make a rule.
An Apprentice paper will have evidence that the student traced the figures correctly and has taken a stab at coming up with the rule. There is evidence that they are thinking about odd and even vertices, but cannot quite come up with a rule that works for all the shapes. They may come up with some rules that work with some figures, but not all figures.
A Practitioner sees that if a figure has zero or two odd vertices it can be traced. They may or may not try some figures of their own to verify their solution. I have also considered work to be a Practitioner level if the student indicates a figure with less than three odd vertices is a solution, even if they do not see that a figure cannot have less than three odd vertices.
An Expert paper shows that a figure with zero or two odd vertices can be traced. They also show a deeper understanding of odd and even vertices by explaining why this rule holds. They show that they have also tried the rule on other shapes. An Expert may also make mathematically relevant observations that demonstrate higher level thinking skills.
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