This week's problem is about a periodic graph.

So, #1 is all about proportional calculations. #2 requires creating a graph; I can imagine students using the points from #1 as the basis of their sketch. #3 is about the circumference of a circle. So this problem is accessible to middle school students. Students don't have to know anything about the sine curve or equation to approach this problem. But if this problem is introduced after we have studied linear equations and transformations of those lines, I would start with enrichment stations ... from this desmos file.

*The desmos file is an amazing collecting of applets and interactive spreadsheets. Students manipulate a, b, and c (period, amplitude, and vertical shift) in the sine equation to match the movement of an engine, roller coaster, and scientific data (daylight hours). The website says it is for upper math students and that students should be familiar with sinusoidal curves. For this lesson, I don't think students need that understanding. In our studying of linear functions, we would have noted that there are other functions that make other shapes or designs. And that playing around with the parent equations of those functions you can transform them.*

So ... first I would show a picture of a ferris wheel and/or a video of a ride on one. I will invite students make notes about what they notice and what they wonder. In groups, students would share, and then we would share out in class ... making a list of their thoughts. I will invite students again to continue thinking about their ideas as we do some exploration activities.

Next, I would set up exploration stations (at least the engine and the ferris wheel). I would invite students to explore, discuss, and develop ideas about what is happening. I would have some guiding questions.

Then ... when we would go back to the textbook problem, just the picture and the seconds needed for a revolution. "What are questions we might ask and answer from this picture? Work with your group to develop 3 questions. At least one must include a graph. Answer your questions on a separate sheet. Be sure to label you work carefully."

When groups are finished creating and answering their questions, I would collect those and sort them by types of questions. I may create a few question cards myself depending on the groups' response to the task. In our next class we would work through some of the student developed questions. Last we would return to their noticing/wondering to discuss any ideas we had not yet explored.

http://www.mathdemos.org/mathdemos/sinusoidapp/sinusoidapp.html

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