Sunday, January 13, 2013

Prepping Classes Today: Algebra Skills for Calculus

My Calculus I course last semester went very well, and I'm excited about the changes I made. (Some posts on it are here, here, here, and here.) I won't be making any major changes this semester, but I plan to have lots of fun tweaking what I came up with in the fall. Today I'm making sure all my handouts of the first unit are ready. (Classes start tomorrow.)  

Sam Shah has written extensively about his Algebra Boot Camp. I asked for his complete list a while back, and have sporadically played around with it. Now I'm firming up my own handout on the algebra skills needed for our first unit.

For him, it has worked best to review the algebra skills just before they were used. I want to review them just after the students have seen why they'll be needed, so they have more context. They've done some of this so many times, I think my students need to see a reason for re-learning these skills. So I'll be handing this summary out later this week or early next week, after we've begun to use some of these skills in our exploration of the derivative. Thanks, Sam.

The section and problem numbers listed come from our official textbook, Briggs, Calculus: Early Transcendentals. We'll also be using Matt Boelkin's open source text, Active Calculus, quite a bit for the first unit.

Algebra Skills needed for Understanding the Derivative (unit 1) 

Most calculus textbooks have a review chapter before they begin exploring calculus. I have found this to be ineffective. We will review the needed algebra as we go along. You have already seen a few algebraic topics come up. If you aren’t completely solid on these, do what it takes to get solid.

• Determine the equation of a line given two points, or a point and a slope, or a graph of a line (1.2 #11,12),
• Find the average rate of change over an interval given a function or a graph of a function (2.1 #7,8),

• Approximate, using two points close to each other, the instantaneous rate of change at a point, given a function or a graph of a function (2.1 #9-24),
• Explain clearly why the procedure you used gives an approximation of the true instantaneous rate of change,
• Clearly express what is happening to an object given a position versus time graph,
• Sketch a velocity versus time graph given a position versus time graph,
• Evaluate f(x+h) for any given function f(x) (1.1 #28),
• Expand the expression (x+h)n using the binomial theorem,
• Rationalize the numerator when it has a subtraction of square roots,
• Simplify complex fractions,
• Construct the formal definition of the derivative by modifying the definition of slope,
• Apply the formal definition of the derivative to simple polynomials and to simple square root functions (3.1 #11-38).

Next up for this class, some exercises for the skills not covered in our text.


  1. This is brilliant. I liked Sam's way of reviewing, but I think you are correct. Students will see the value so much more if the skills they need are explicitly pointed out after they are needed for the class work. Students at this level (HS or college) are able to assess for themselves what they need to do.

  2. Love the list, and I am also using active calculus as a text this semester!

  3. Yesterday I realized we needed 'simplify complex fractions' on this list, so that we can do the derivative of y=1/x from the definition. (I've edited above to add that.)

    I use mostly Boelkins' Active Calculus for the first unit, but I didn't like his materials as much for later units. I've organized differently, and I work through many of the ideas differently. I wrote lots of posts in the fall as I was thinking it through. 2 each in July, August, and September, plus one in November. Let's talk by email, and I can show you where I am now. (I'll post a summary afterwards.)


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