Midtown Middle School had been chosen to host the first ever, Annual Mathematical Olympics. Teams of students from all across the state will gather and test their mathematical skills as they vie for the grand prize trophy. Ms. Maple's middle grades math class has been charged with the task of selecting a design for the grand prize trophy. They know that they want it to be a thing of beauty, able to stand beside the sports trophies in any school's awards showcase, and make the winning math team proud! They also know that they want the trophy topped with a replica of a typical middle grades mathematician. They are coming to you for help since they want to be certain that the replica is correctly scaled and looks just right atop the trophy.
Using what you know about body proportions and trophies you have seen, please submit a plan for the trophy so Ms. Maple's class can choose a winning design. Be sure to let the class know about the decisions you made in coming up with your design. Who knows, maybe they will learn something from you.
This assessment task followed several measurement lessons where students practiced linear metric measurement by measuring themselves, making "half-sized me" cutouts on newsprint and "micro-me" sketches on index cards. I wanted to see if students were internalizing the skills needed to use ratio and proportion to create scale models. They had to scale their body measurements down to a figurine that would fit atop their trophy. Further, they needed to scale the entire trophy down to fit on 8 1/2" x 11" paper. Students should be expected to use prior knowledge gathered from the "micro-me" lesson to create a scale model figure for the top of the trophy. My students spent a fair amount of energy designing the actual trophy based on observation of school sports trophies, consideration of the size of the school's trophy case, and their thoughts of what a "typical middle school mathematician" looks like. Most students spent between three and four hours working on this task. We devoted two, 45-minute classes to this task, and then kids worked during independent study time to complete the write-up. This task could easily be incorporated as a part of a unit on sports. With the Summer Olympics coming up, many teachers are planning such units. Kids could design a trophy for a particular sport they are studying. You could also do this at Academy Awards time, requiring students to create an "Oscar®"-type award for the best performance in a problem-solving event. Students need basic skills in creating scales, ratios and proportions, prior to undertaking this task. Borrowing samples from the school trophy case helped students get some idea of what a "Grand Prize" trophy might look like. We talked about the fact that any props on the trophy also needed to be made to scale, e.g. pencils and calculators held by the figurines.
- Measuring tapes/meter sticks
- String
- Newsprint paper
- Graph paper
- Calculators
- Samples of trophies
Most students used a ratio somewhere between 1:12 and 1:16 to scale their body measurements to the mathematician on the trophy. They then reduced the actual trophy by half or one third to fit it on the paper. Some students drew the actual trophy size, although this would not be a very impressive trophy!
This student seemed to understand that the task was to create a trophy. However, one wonders what "guy" was measured. Why is the scale 26 cm if the "guy" is now 6.5 (not 6.75) cm after he was reduced to "half-sized"? There are no labels on the representation to offer any clues. The scale is improperly written. There is not enough evidence of mathematical reasoning for the reader to understand what was done in solving this task.
The student gives a scale for downsizing the trophy to fit the paper, but does not address the scale for creating the mathematician replica. This student selects the trophy size based on what is easy to draw, rather than what would make an impressive award. There is evidence of mathematical reasoning and understanding in the reference to halving to create a scale and the correctness of the scale 1 cm = 2 cm being equivalent to half. The representation has labeling, perhaps too much, but the scale is not properly identified. The student uses minimal mathematical language in this piece.
This student understood the need to downscale a middle school person to an appropriate size for the trophy top. There was also an understanding of the need to further downscale the entire trophy to fit on the paper. There is sound reasoning behind the choice of model for the replica. The student used the 1:16 scale to downsize the props on the trophy, too. This student makes the connection from this task to the last one done in class and utilizes skills from one to apply to the other. The explanation of the solution is clear. An appropriate mathematical representation is used on page three. There is good use of mathematical language throughout the solution.
This student took seriously the charge of representing the typical middle school mathematician and actually measured several students and found an average height to use as the model for measurements. The student gave thought to the relationship of the figurine to the overall trophy size and incorporated this into the design. The student does an excellent job of explaining exactly what was done at every step of the way. There is no need for the reader to infer how and why any decisions were made. The mathematical language is precise and the mathematical representation is carefully done and accurately labeled, with all parts appropriately scaled. It is obvious that this student was engaged and had invested much thought, time and energy, as well as mathematical knowledge in completing this task.
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