Mathematics 24x7

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Problem DT1:
You are spray painting a Jeep and need to cover a 170mm diameter circular spotlight with 48mm wide duct tape. You must cover most of the area of the spotlight using the least possible length of duct tape. You must cut the tape into three lengths, using right angle cuts, choosing the length of each piece. At least two corners of each rectangular piece of tape touch the circular rim of the spotlight. There may be gaps of uncovered glass, but the tape must not extend beyond the rim of the spotlight. What is the total length of tape that is required to cover one spotlight?
This is my attempt to put a geometric problem into a practical setting. Do you think that it is a good basis for teaching developmental math? If you were giving a problem like this to your students, do you think that you would simplify it or present it in different way? This originated from the "Filling a Circle with Rectangles" problem, with the added constraint of fixed width.
I also posted this problem on ELITE MATH CIRCLE on Friendster.com.

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I would not ask the answer. I would ask them to create a mathematical representation of the problem.
As result they should be speaking about "circle" and "rectangle" and not about "spotlight" and "duct tape".

Then, they should solve the abstract representation.
Then go back to the original situation.

I would ask to clearly separate the 3 steps. That way I could see what is wrong in their development.
Yes, present it differently.
Bring in the spotlight and a roll of duct tape. Set them on the table. Have students figure out how to cover the glass so as not to get any paint on the glass when spray painting the spotlight. Then discuss the what they come up with, as well as other possibilities, which might then include the parameters you initially had in mind.
Maybe even spray paint the spotlight and see if the solution works, how much paint was used and if there is enough left to paint the rest of the Jeep were you to bring that in; and how much more is needed. The more fun you can have with it the more students will become engaged and the more they will get from it, might even discover others ways of doing it. Problem solving has more to do with approach than the finish. Solutions are found in real life context, not in the limitations of abstract statements that are always a set up for the "right" answer.
Exam questions need to be precisely stated, but I'm not happy with this clause: "There may be gaps of uncovered glass, but the tape must not extend beyond the rim of the spotlight." It reads like a legal contract and many students have difficulty reading questions. It could be replaced by a diagram, if the purpose of the question is to test algebra. In practice the tape would extend over the rim, and the excess would be trimmed. I agree with Bradford that an open ended project is more engaging. There are other mathematically interesting solutions, including ones with radial symmetry, overlapping the rim, or cutting the tape into trapezia or triangles. Why duct tape and a spotlight? In America, duct tape is used for most everything. It has a standard width of 1 and 7/8 inches, or 48 mm. I searched the 'Web for an object that is 3.5 times wider, about 170mm in diameter. To emphasize the importance of optimization, specify a material more valuable than duct tape, gold leaf perhaps.
I would just ask...What is the least amount of tape needed to cover the spotlight?
Colin, I think this would be much better for an in-class exploration than as an exam question. Then you can set it up in the open-ended ways the others describe.
Christian Baune said:
I would not ask the answer. I would ask them to create a mathematical representation of the problem.
As result they should be speaking about "circle" and "rectangle" and not about "spotlight" and "duct tape".

Then, they should solve the abstract representation.
Then go back to the original situation.

I would ask to clearly separate the 3 steps. That way I could see what is wrong in their development.
An excellent problem, Colin. I like the way you have stated the problem in a way that many students can relate to.

I agree with Christian that the problem needs to be converted to a math model, but I think it should be done in a series of discrete steps instead of in one fell swoop, breaking the problem into simpler sub-problems.

One such sub-problem would be "what is the area of the spotlight?" From this, the concept of area of a circle comes into play. Many will have forgotten (or have never known) the formula for the area. Some side discussion could include the background for why the formula is true.

Another sub-problem involves the situation when one strip of the duct tape overlaps another and how some of the overlapping area must be subtracted from the total area covered by the tape.

I believe that postponing the math terminology as long as possible is helpful. Plunging into it right away causes the lights to go out of many eyes.

The problem statement needs some clarification. "At least two corners of each rectangular piece of tape touch the rim of the spotlight." Does this mean adjacent corners or can they be opposing corners?

Also, "There may be gaps of uncovered glass, but the tape must not extend beyond the rim of the spotlight." How many gaps and what total area can be left uncovered? If the area of the gaps is not defined, how are we to determine the total length of tape?

All in all an interesting problem that leads to decomposition into simpler sub-problems, but the problem statement needs to be made more clear (IMHO.) Thanks for sharing it with us.

Danny
I solved this problem using the geometry software Geogebra, using the Distance or Length function on the tool bar to calculate the length of each piece of tape. The total length of tape is: 9.02+16.31+9.02 = 34.35 cm = 343.5mm. The diagram was drawn with Inkscape.

I suggest deleting the horizontal measurements from the Spotlight diagram, and setting it as a problem. What are the lengths L1, L2 and L3 of each piece of tape and what percentage of the area of the circle is covered? It can be solved with the aid of GeoGebra, which has a toolbar function to calculate the length of each piece of tape (distance between intersection points), or by solving the equations for the points on a circle, using the equation x^2+y^2=r^2, with r=85 and y2=48/2 for the long piece of tape and y1=48/2+48 for the short pieces of tape. L1=L3=2*x1, L2=2*x2. The SVG and image files for these examples are available for download in the Box.Net panel on my 24x7.ning.com profile page: I created Spotlight.svg with the free Inkscape drawing package. I place these examples in the public domain. (My blog on Vox.com is published under a Creative Commons Attribution-Share Alike License.)
I find this math problem interesting because it involves common objects that students would encounter in their everyday lives. You are stating the problem in a way that most students could understand what you are getting at. All this is well and good but I think you are missing the most obvious objection that students may find with this question and that is why would anyone feel the need to put Duct Tape only on that portion of the Spotlight? Maybe this is because I work with much younger children but in my experience that is the question that many of them would focus on rather than the mathematics of the problem.
Evelyn, I accept your criticism. I began with an abstract problem, "Filling a Circle with Rectangles", and was trying to develop it in the direction of a practical problem. By changing the example to: cutting carpet for a circular room, we can see that the real world theme is usually covering a circle BEYOND the perimeter, not TO the perimeter, and trimming off the excess material. As Sue suggested: A more open-ended statement of the problem would let the students discover their own solutions. If we consider a gardener or painter rolling a circular area, we keep the constraint of a fixed width roller, and open up other solutions such as pulling the roller in a spiral path. I think of a spirally coiled mosquito candle or a vinyl record as practical applications of that principle.
This topic "Teaching Duct Tape Math" was a brainstorming exercise. I admit that it didn't work out, and I don't foresee classrooms of pupils trying to do math with duct tape. Do you have a better idea? Why not start a new topic on Danny's "Teaching developmental math Discussions"?

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