Calculus: Finishing up Our First Unit (Meaning of Derivative)

All three of my classes seem to be going great this semester. Students seem to be working hard, and diving in enthusiastically. I'm very happy.

Last semester I figured out the order I thought made sense for Calculus, and now I get to refine it. Students seem less nervous this semester. (I asked if it made them nervous to not be working from the book, and quite a few raised their hands. I reassured them that we would use the book more when we get to the second unit. I guess they're trusting me more.) I think the students last semester eventually saw that my plan was working, and maybe they've passed the word along.

I've been feeling cranky because I haven't been able to feel caught up after my lovely little vacation this past weekend. Perhaps taking the time to write about this wonderful class will help me get over my crankiness. I've just finished making the test I'll be giving tomorrow. I feel like we've done everything we can in class to get ready. Many of them will still fail parts of it, but they'll get their second chances, and hopefully the retakes I offer are teaching them how to study. (I'm looking forward to seeing how my fall Calc I students do in Calc II, and how my fall Calc II students do in Calc III. If the ones who would have failed without retakes actually do well in the next class, I think that's evidence that they've learned how to study more effectively.)


Unit 1
This first unit was on the meaning of the derivative. Standard textbooks usually have one or two sections on this, and so it gets considered for under a week. We have spent 10 class sessions (over 12 hours) on it. Mostly, we worked from activities in Matt Boelkins' Active Calculus. (I also had them read the first 6 pages of Morris Kline's Calculus text, which gives a great start on the history of calculus.) We used all the  activities in Boelkins' section 1.1, plus activities 1.8, 1.10, preview activity 1.5, and activity 1.12. Students wanted more problems like these from activity 1.10 ...

... so one of my students pulled together graphs from other parts of the book, reversed some of the original graphs, and gave us 8 more problems, including these ...

I hope this work helps them understand the idea of the derivative better.

A few links from this morning...

I have a stash of dozens of links that I want to share, but I never have time... This morning, as I caught up on my Google Reader backlog, I kept a few windows of coolness open, and I now have a few minutes to share. (Please let me know if any of these links are useful to you.)


Games

• Math Munch shared some games and things. I want to try the Loops of Zen when I have some playtime, and I want to watch the video on 3-color cellular automata. (They got some of the links from Casual Girl Gamer, whose blog I'd like to check out too.)

Resources for Teaching

• I've made some good use of the visualpatterns.org site (set up by the fabulous Fawn Nguyen) with my pre-calc class. Chris Shore's reminder about the estimation180.org site made me realize that might offer my students some good challenges too.

Empowering Ourselves & Our Students

• Math makes a lot of people feel bad. If you have have ever felt oppressed by math or math class, you might be interested in following or joining the Liberation Math class starting now.
• Nepantla, living in tension: I scanned this post, and want to read it more slowly later. I want more detail on this idea.

89 interesting Physics Problems

I found this on Patrick Honner's blog. The pdf cannot be saved or printed. So I need to save its address. Thus this short post.
 

This pdf gives the first 89 problems from the book 200 Puzzling Physics Problems, by Gnadig, Honyek & Riley: http://catdir.loc.gov/catdir/sam ples/cam034/00053005.pdf  (No solutions included.)

Patrick says, "The problems here are simple to state, but seem to get at profound mathematical and physical ideas."

Math Teachers at Play #58...

... is up at Let's Play Math!

Denise starts it off with a puzzle:

A Smith number is a [composite, that is, not prime] integer the sum of whose digits is equal to the sum of the digits in its prime factorization.
Got that? Well, 58 will help us to get a better grasp on that definition. Observe:
58 = 2 × 29
and
5 + 8 = 13
2 + 2 + 9 = 13
And that’s all there is to it! I suppose we might say that 58′s last name is Smith. [Nah! Better not.]

• What is the only Smith number that’s less than 10?
• There are four more two-digit Smith numbers. Can you find them?

Then she adds sweet little riddles and jokes. The posts she links to look great. I'm looking forward to spending more time with them.

Calculus: Mastery Tests (SBG)

As I've mentioned before, I don't care for the name Standards-Based Grading (SBG), but I like most of what people are doing and thinking that goes by that name.

However, I've noticed that most people seem to assess very small, detailed standards, and I want to assess broader topics. Maybe that's because high school teachers have more time with their students than I do with mine?

I've moved away from my old tests .. "on chapters 2 and 3", and have moved to something between that and the usual SBG. I test after each of my units. Each test is composed of mini-tests on each topic, and students may retake the mini-tests one by one.

Here's another handout I made today:

Mastery Test List

There are 15 mastery tests listed below. You will have more than one chance on each of them. On the mini-tests marked R, you may do retakes in my office. The others will be available during a later test.

Problem-Solving
Each test will have one problem-solving problem. You must get this problem right at least once during the semester. (There will be a few of these on the final for people who haven’t yet gotten one.)

Unit 1: Exploring the Idea of the Derivative
• Definition
• Tangent R
• Graphs
• Rate of change (from a table of data)

Unit 2: Derivatives of Polynomials, Trig functions, Products & Quotients; Graphing
• Polynomials R
• Periodic functions with products and quotients R
• Graphing

Unit 3: Exponential Growth, Compositions & Implicit Functions, Optimization, Related Rates
• Basics R
• Implicit (including related rates)
• Optimization


Unit 4: Limits, Anti-Derivatives, and Area
• Limits
• Anti-derivatives R
• Area
• Position, Velocity & Acceleration

Unit 5: Volume
(No mastery test, but this will appear on the final exam)



I believe my students are learning how to study because of the opportunities they now have to retake tests. I think this policy is making a big difference in their learning.

What do you think?

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.

San Diego Event: Math Poetry Reading

Are you in San Diego? Please join us at ...

 

A Reading of Poetry with Mathematics

5 – 7 PM    Friday, January 11, 2013

Room 3, Upper Level, San Diego Convention Center  San Diego, CA
sponsored by the Journal of Humanistic Mathematics

at the Joint Mathematics Meetings

Poetry reading organizers are Mark Huber, Gizem Karaali, and Sue VanHattum

Open to the public.

 

 

[Snagged from JoAnne's blog, Intersections -- Poetry with Mathematics.]