I am trying something new for UNL Math Day contest prep this year. I have dubbed this week "Geometry Week."
It's like Shark Week... only BETTER.
I prepared a packet of every Geometry question that has appeared on the PROBE I since 1990. I am allowing the kids to work in whatever groups they want, whether it's individually, pairs, small groups, or large groups. Below is one of the first questions kids worked on today. A couple kids approached me during class to ask if they could make shadows on the Promethean board. Awesome. Here's the question:
1991 UNL Math Day Probe I Exam
First, the kids asked for a cube. They are using a wooden cube I keep in a plastic container of manipulatives in my desk drawer. Next, they asked for a flashlight. Add that to the list of random objects I need to have in my classroom. I apologized for not having one. Then, they asked if they could use the projector to try and make the different shadows on the Promethean board. Brilliant idea, kids. Do it.
During our bowl team practice last night, we ran into a 'tough' problem.
The problem appears in a contest designed to allow competitors 30 seconds to answer. I am wondering whether anyone reading this blog can find a 'fast' way to do this problem. I will summarize the work we did together in the classroom and see if someone has a more efficient method. We began by re-writing the line in slope-intercept form.
We know this line with slope -1 must be tangent to the unit circle. To help the kids visualize the problem (and because my drawings on the dry erase board aren't always to scale), we constructed the setting in Geogebra.
The slider c is tied to the red line. The red line in the photo is obviously NOT tangent, but we can use the slider to move c to a value that gets pretty close. We can empirically find a value of c that gets the line close to being tangent to the unit circle, but there should be another way to obtain a more precise result through other means. We can also tell there should be two possible values of c per the picture below.
We then began trying some algebra to see if we could find some relationships. Here is what I wrote on the board in typed form:
This didn't seem to be leading us anywhere. We abandoned that work temporarily to return to the graph and began making constructions. See the work below.
We know the height of both of the two colored triangles has to be (√2)/2, so we can infer the value of c that works for the tangent line entering the first quadrant is 2√2. Then by inspection, we can see the c value for the tangent line entering the third quadrant is the opposite, -2√2.
Does anyone have a different approach they would like to share? Or maybe a suggestion on how to speed this process up so the answer can be obtained in under 30 seconds?
An administrator told me about a problem from the world outside school he faced while building his home last year. He had gone to the post office to confirm the location in which he wished to place his mailbox was acceptable to the U. S. Postal Service mailbox guidelines. The administrator poured the footing and ordered the stone for the mailbox. A few days later, the mail delivery person informed the administrator the mailbox placement was incorrect. As a result of a miscommunication, the Post Office instructed the administrator to move his mailbox, causing the administrator to lose his considerable fiscal investment in the mailbox. The issue has to do with the route the mail delivery person drives each day. Below is a map of the neighborhood in question.
Exhibit A: The neighborhood. Mailboxes appear as an "X." The house in question is drawn in red. Street names have been suppressed to protect the innocent. Red ovals indicate either a cul-de-sac or a potential entrance from a main road into the neighborhood.
Some background information: mail delivery personnel prefer not to get out of their vehicle. Many mail delivery vehicles have the driver's side door on the right hand side, so a person could infer the mail delivery route using the condition the mailbox must be on the right hand side of the delivery vehicle.
The administrator came to our classroom Thursday. He served as a guest speaker for twenty minutes, drawing the annotations on the Google Maps image. It reminded me of a press conference format; the students asked questions about relevant and irrelevant information in an effort to fully understand the problem.
This problem is the perfect follow-up to the "Chomp the Graph" activity on graph theory. Students can investigate routes using directed paths. Students can compute distances by coordinatizing the image and applying the scale which appears in the lower left. What the students really want to know is whether it is possible to optimize the mail route in a way that benefits the administrator. My students will work on this problem next week in the computer lab. I am eager to see where the investigation takes us.
In my collection of problem resources, I have a book full of MENSA problems. One particular puzzle caught my eye a couple years ago.
Exhibit A: An innocent looking trapezoid, just waiting to be cut up.
Forgive the distortion caused by the scanned page. The intent of the question author is to offer what appears to be, for all practical purposes, an isosceles trapezoid. Showing this problem to students seldom leads students to uncover what the author of the book claims is the solution.
Exhibit B: "THE SOLUTION"
This problem struck a chord with me. I wondered whether it would be possible to find other solutions. I wanted my students to investigate the possibilities and, if there were no other ways to do this, to rule out cases with mathematical reasoning. Several of my students came up with what appeared to be a solution, but it seemed like an accident of distorting the sketch a student made. I challenged the students to rigorously show whether or not the solution works in all cases, or if there were certain dimensional restrictions or conditions under which the students' solution worked. One of my students went on to model this problem in Geogebra.
Exhibit C: A student models a solution in Geogebra. Is this solution a special case?
We should be careful about the language we use in class as math teachers. Had I told my students to find "THE" answer, they may not have uncovered the possibilities of the colored trapezoid above. My students went on to write inequality statements under which the second solution exists or does not exist. When I posed the problem to my students, I asked for "AN ANSWER," a far cry different from "THE" answer.
Amazing how definite a difference an indefinite article can make.
One particular problem type that appears periodically on the UNL Math Day PROBE I involves tying an animal to a building with a leash. The student then must make determinations regarding the area in which the animal can roam. Below are the problems that have appeared on the PROBE I since 1990.
1994 PROBE I #18
A square shack 30 feet by 30 feet is in the middle of a large field. A goat is tethered to one corner of the shack by a chain 60 feet long. She cannot get into, onto, or under the shack, but can graze anywhere else she can reach on her chain. What is the area of the portion of the field she can graze?
2001 PROBE I #2
A dog is tethered to the corner of the outside of a ten foot by ten foot building by a leash which is 14 feet long. How much area (outside of the building) can the dog roam?
2002 PROBE I #25
A dog is tied to the corner of a ten-foot by 20-foot shed on a rope having length 60 feet. Assume the dog starts out as pictured below and winds his way around the shed counterclockwise as far as he can go. What is the total area (in square feet) swept out by the rope?
With my students, I approach the three problems in similar ways. I want students to successfully obtain the correct solution. Next, I want them to write the area in which the animal can roam as a function of other features of the diagram. What matters here? The length of the leash? The dimensions of the building? The shape of the building? The direction of the rotation? Writing a function gives the student an opportunity to work towards solving for all possible cases and unearthing the features of the setting we can treat as invariant.
Geogebra provides a great medium for modeling this problem. A fun exercise involves asking what happens to the area in which the animal can roam as the length of the leash exceeds the perimeter of the figure. Modeling the overlap becomes a challenge quickly. Jerel Welker, a math coach in Lincoln Public Schools in Lincoln, NE, has used the most recent of the three listed problems as a wonderful professional development opportunity for middle school and high school math teachers.
What other features of these problems can we augment to challenge our students?