• Completing the Square

    Here’s what I’ve used in class this week, mostly to good effect.

  • Implicit Differentiation Examples

    Here are some example-ish tasks I used to introduct implicit differentiation.

    First, a warm up, to help students notice and remember relevant things they had already learned about the chain rule:

    Then another little warm up, just to understand what the problem we are trying to solve (what is the slope of the circle?) actually is.

    Then, the worked example. I left the last step blank and put analysis questions on the side, because I really just wanted to make sure everyone noticed everything. (I wasn’t really trying to get students to articulate a generalization at this stage.)

    This was followed by an opportunity to apply this new approach to a new problem.

    On a following day, I gave another warm up, again helping students remember things they have already seen in algebra:

    It’s product rule time! Here’s a fun little implicitly defined curve:

    Same type of example provided again here:

    Then we did this comparison between two derivative computations. And then it was time to do more conventional practice.

  • Comparing Explicit and Implicit Differentiation

    Here was the original problem from Calc Medic:

    The thought of comparing two strategies made me think of the Compare & Discuss project from Jon Star, Bethany Rittle-Johnson, and Kelley Durkin. I took the answer key from Calc Medic and rejiggered it into a comparison between two worked out solutions:

    Here were the analysis questions I asked, which could probably be improved. Students noticed that both Lea and Marco started by multiplying both sides by y^3. I asked as a follow-up whether Lea and Marco had to perform this step to complete their techniques. (I think Marco could have used the quotient rule. Lea had to put the equation in explicit form.)

    I think it’s helpful to see implicit/explicit differentiation side by side. We need to think of implicit differentiation as another technique that does the same sort of thing as our other derivative-finding strategies. People may think comparison is mostly for spotting differences, but it’s also about coming to see deep similarities among those differences.

  • Mistake+Example for finding the distance of a trip

    This mistake came via Bowman Dickson:

    Here is the example/mistake that I came up with for sharing with students:

    I’m trying to make clear what’s right, but an important part of this is students coming to understand why the procedure on the right doesn’t work. The point is that the slopes of the lines don’t match up, so the two-triangle path is longer.

    These analysis questions set a teacher up to talk about that — or at least that’s what I was going for!

    I would follow this up with another similar (but different) problem.

  • Faded Practice with Equations of Lines

    I took these images from the Kuta Software worksheets (actually the answer keys) and am turning them into some faded practice for writing equations of lines.

  • Finding the Constant of Proportionality

    Looks like I took the table from the lesson summary from one of these lessons and wrote an example activity around it.

  • Solving Equations that have fractions

    Here are two worked-out strategies, in the spirit of Compare & Discuss — a great site for worked example activities.

    I followed this up with some faded practice:

  • Inscribed Angle Mistake

    Here’s one of those mistakes that crop up every year. When I hit one of those, I try to “call it out” in a mistake analysis activity where we talk about why it’s wrong and what is less-wrong.

    It is empowering and good practice to immediately use the new idea to solve a new and related problem.

    From my notes, it looks like I then followed up with this very solid practice activity, which I learned about from Jo Morgan’s Resourceaholic.

  • The Product Rule

    Nothing especially groundbreaking here, though maybe there’s something interesting in the use of examples to cover this territory.

    The next step is a bit slower than some people would go, but I’ve had students correctly guess the product rule from the following example, which is kind of fun.

    I present the rule itself in the context of an example. I wanted to use functions that were really different from each other so that it would be easier to see the structure of the rule. The tradeoff is my students were less familiar with the derivatives. Maybe next year I would insert a polynomial example before this one:

    Part of me hates the quotient rule, I just want to use the product/chain rules for everything. That bias comes through a bit in this line of questioning.

  • Intro to Right Triangle Trig with the Tangent Function

    I start my unit by asking students to compare the steepness of ramps. We don’t start the unit with a worked example, but after I have explained that the tan table translates angles into slopes (gradients), I do present something like this.

    Once students can use the tangent function to solve a steepness comparison question, I toss in missing side problems.

    I kind of like the way I’ve included the tan table itself in these examples. Looking back, I also like how the “slope ratio” and “solve” are separated visually and with labels. I’m really trying to make clear that there are two distinct steps.

    Here are the analysis questions that go with that example:

    This is followed up by practice:

    I think it can be useful to see how I expand out from the example and start giving students steps to solve on their own. Here is the next activity in my sequence:

    Finally, once students are reasonably proficient using tan(x) to solve triangle problems, I introduce sin(x). Here is an example I use:

    And here are the analysis questions and the application problems: