In previous years, I've taught related rates in calculus class by the book. As in, we worked through the example problems from the textbook together and then students individually worked on the practice problems. After working through some examples, I would give students some more challenging problems as an assignment. Recently, a teacher reached out looking for some new ideas for teaching related rates. We brainstormed some ideas of how we could make related rates more engaging and hands-on. We decided to create some stations where students could experience and take measurements of related rates in action. It turns out, this is probably the reason that related rates problems are in our textbook. While doing some research to prepare this activity, I found an article titled, "The Lengthening Shadow: The Story of Related Rates" by Bill Austin, Don Barry and David Berman. This article is from Mathematics Magazine (Feb 2000, Vol. 73, No. 1, pp. 3-12). In this article, the authors state that related rates problems originated in the 19th century as part of a reform movement to make calculus more accessible. By observing changing rates, students would be able to measure concrete examples and discover their relationships. Joshua Bowman discusses how he grounded his teaching of related rates using observations in his blog post Using calculus to understand the world. Related Rates StationsFor this activity, we decided on four stations: Station 1 - Blowing Up a Balloon. Blow 5 big breaths into a balloon. After each breath, measure the circumference of the balloon and calculate radius and volume. How are radius and volume related? Station 2 - The Sliding Ladder. A metre stick is sliding down the wall. The bottom of the metre stick is moving away from the base of the wall at a constant rate. How fast is the top of the stick sliding down the wall? How are they related? Station 3 - Building Fences. Build several “fences” (rectangles made with multi-link cubes) such that the length is twice the width. Put them in order of size. Measure each rectangle’s length and width. What is the rate of change of the area? How are perimeter and area related? Station 4 - Driving Cars. Two toy cars are traveling at different rates in perpendicular directions. How fast are each of the cars travelling? How fast is their distance apart changing? How are they related? After taking measurements at each of these stations, students drew a picture and created an equation to relate the quantities to each other. We used implicit differentiation to determine the relationship of the rates and then we tested our equations with our collected data. Reflection and ResourcesThe activity went a bit long for one class period. Next time I would either split the activity up over two class periods or reduce the number of stations to three. This will allow more time to consolidate the learning at the end of the lesson. As Tracy Zager says, "never skip the close." If you're interested in giving this activity a try, below are the files I used: Related Rates Stations Google Slides Related Rates Recording Sheet Update - 13 AprilAfter doing this activity a few times in classrooms, I decided to reduce the number of stations from 4 to 3. In a 75 minute period, students were able to complete the three stations with a few minutes at the end to consolidate the lesson. I also changed the set up for the driving cars question to make it a bit more interesting. Here are the updated files that I've been using for 3 stations and a different problem for the driving cars problem. Related Rates Stations Google Slides Related Rates Recording Sheet Nova Scotia Mathematics Curriculum Outcomes Calculus 12 A3 - Demonstrate an understanding of implicit differentiation and identify situations that require implicit differentiation Calculus 12 B14 - Solve and interpret related rate problems EL
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