Here are two of them: I used this to help kids wrap their minds around basic integral Calculus applications, and this one reviews some algebra skills from earlier this year, plus introduces the necessity of going in between algebra and the graphing calculator sometimes. The problems are not ground-breaking, but I think they've definitely helped to break up the skills practice, so I'm happy to share them if they might be useful to someone else.

From earlier months of this year, one thing that I did that totally helped with teaching Related Rates is that I first taught implicit differentiation with respect to time, and formally assessed students on this skill, prior to starting Related Rates word problems. (Sorry if this sounds obvious; it wasn't that obvious to me last year, teaching Related Rates for the first time!) Here is how I introduced implicit differentiation, using the analysis of non-functional relationships as a premise. After this, I had the students do some pure skills practice in converting geometric formulas to differential equations with time as the domain, before introducing my scaffolding for related rates problems and the many related problems I took from Bowman last year. I felt really good about this sequence of skills this year, because I noticed that it really made the problems more accessible to ALL students (as in, by the time they got to the word problems, they were really only focused on parsing the word problems process, rather than simultaneously struggling with the algebraic skills of differentiating implicitly). I recommend trying this, if your students get baffled by Related Rates problems.

I am also trying to place a general focus on vocabulary and communication this year. I've been doing this in all classes by giving the kids a list of essential questions at the start of a unit, and then having them journal their responses to those essential questions throughout the term. For example, for our current term in Calculus (which is short, only about 5+ weeks), I gave the kids the following questions. The questions are a mix between related rates (which we did at the start of the term) and intro to integral Calculus.

*How are "implicit differentiation", "chain rule", and "related rates" all related? Illustrating this with a simple algebra example may help to clarify your thinking.*

*Take one of our Level 2 or Level 3 Related Rate problems from class and explain/describe, step-by-step, how you are able to find the missing information.*

*What is integral Calculus? Describe a couple of situations where this concept is useful.*

*Choose an exponential function of the form f(x) = a*e^(x - k), by assigning values for a and k. Estimate the area underneath the curve of f, from x = 0 to x = 5, using a total of 10 rectangles. Show both left-hand sum and right-hand sum, and draw labeled diagrams to show what your numbers mean.**Show, step-by-step, how you would calculate the enclosed area that lies between two functions f and g, where f is a quadratic function of the form f(x) = ax^2 + bx + c and g is a trigonometric function of the form g(x) = m*sin(n(x - k)) + p, where the value of n is not 1. You get to choose the parameters a, b, c, m, n, k, p to start, but make sure n is not 1.*

Students have shown a varying degree of enthusiasm about the journal assignment, even though I have been doing it since the start of the school year and explaining periodically its purpose. Part of the purpose of this journal is to get them to record, in their own words, examples and explanations to important concepts, so that they can have a succinct set of notes for future years. Another purpose is for me to see what they write periodically, so that I can informally gather information about common misconceptions for the topics that we have finished learning, and clarify them with the class. As it turns out, however, the naturally reflective students are thoroughly utilizing the journal to dialogue with me about their understanding, and the rest of the kids see it as a drag to have to keep revising their explanations until the end of the term, so the work that I receive is kind of a mixed bag in terms of quality. It has been a somewhat tough sell, but one that I think is important, because from time to time, students would comment on how they notice that by answering questions in their journal while learning the concepts (instead of putting it off until the end of the term), their understanding improves in real time. Do you do something like this in your classes? How do you drum up enthusiasm for such a revision-based assignment?

That's it for now! My Geometry students are wrapping up their 3-D project, which is very interesting as per usual. They have some really neat designs this year, which I might share at some point. Algebra 2 kids are knee-deep in thinking about the domain and range of different function types, and thinking about transformations on the various functions. It is nice to hear them go, "Ooh, ahh..." as they realize that they can connect information from different types of functions. Not much to write home about, but a productive time of the year nonetheless!

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