I tell my students all the time that they should not expect to produce solutions on their first efforts like those that are published. There was almost ALWAYS messier work at the start that was used to “get a handle” on the problem’s essence.

I, too, did some numeric work to start the problem. After I had a grip on the numeric, I turned to algebra. I found it interesting that you solved it by comparing equal distances while I tackled (almost) entirely from Michelle’s perspective.

Thanks for the lead on a good problem, and thanks for the props.

]]>I should have made note of Michelle’s speed dropping out in the penultimate step. What I didn’t put in my write-up was some initial playing around in Excel, basically solving the problem numerically. It was apparent then that Michelle’s speed was irrelevant once I had found 2/3 m/s as the walkway speed.

]]>How much time I spend in this algebra varies depending on the level of the class. Exponential and log algebra is very useful in some scenarios, but I think teachers err when they drive to algebraic mastery in first encounters. I spend a couple days on this algebra and then move on to other topics and applications. But I make sure to keep spiraling back to the topic to reinforce the ideas. This is strongly supported by brain research which says that learning with time to forget and then re-exposing actually has a stronger influence on long-term recall. When I return with this particular lab 1-2 weeks after initial exposure to exponential algebra, I reinforce the algebra AND the sense of play and pattern discovery in math. Exploring again with logarithms reinforces the initial algebra, and creates strong, EXPLICIT parallels between exponential and logarithmic algebra–a point missed by many sources. Students have to see the connections, and they have to see them over time.

Arithmetic aside, a second goal for all of my math classes centers on math as the science of patterns. When you notice something unusual happening, it is powerful to be able to explain WHY. For this scenario, almost every student has been disturbed by the fact that vertical stretches have no ability to shape-deform an exponential. This runs counter to everything they’ve learned about transformations: scale changes (not dilations) alter curves.

]]>Thanks, Dennis. Your note reminds me why I emphasize clear writing and understanding of audience to all students (not just AP).

For the AP, students need to understand that their graders are absolutely not the same as their teachers. A teacher knows what students have learned and probably knows what was mean when by unconventional notations or phrasings. Unfortunately, that assumption can never be made for AP graders. Those who grade AP responses need to be convinced that a student knows what he or she is doing in a very precise way. They don’t know the students, and they should not be expected to give any credit for even marginally vague explanations. It is the students’ jobs to write clearly. It is our job as teachers to help them learn that skill. It’s a good habit for any writing that will be (or might be) read by someone unknown to the writer.

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