In precalculus, our first chapter is full of minutia…important minutia but minutia none the less!! So one of the biggest challenges is how to expose my kiddos to these ideas, make them stick while keeping it interesting. So a matching activity seems to work the best. They had 15 minutes to sort, match and produce their poster. Lots of great clarifying discussions while completing the activity!
We started our study of exponential and logarithmic functions yesterday. Because much of the book work is a repeat of last year’s experience, I like to add depth and breadth to their experiences this year. Thus, we study the logistic function. As an introduction beyond seeing the equation and graph in Chapter 1, I use the Logistic Function Exploration by Paul Foerster. (This link will get you to the worksheet indirectly.)
Today, students came in with their attempts. Although #2 didn’t ask for the exponential equation for the first 6 points, we used it to review solving a system of exponential equations…easy-peezy. Then we tackled #3…lots of groans and exclaimed, “that problem was impossible,” and “I couldn’t do it.” So I played “Vanna” and the kiddos talked me through the process; used color to help connect the process some. When we used the natural log to solve for “b,” we talked at the “why” behind the canceling; great review of log and exponent properties!! One student asked if they could use the logistic form that used the denominator of 1 + aBx instead of 1 + ae-bx; this gave a great opportunity to talk about the fact that B = e-b .
Also, we didn’t use the calculator until the very end. Wanted to re-emphasize with them that they CAN solve any problem using algebra and algebra properties, and that the calculator helps us get a number we recognize, even though it isn’t exact. Such an important skill for calculus, don’t you think?
We also looked at #6 which gave them the initial value and the point of inflection. We drew the sketch and then found the equation for that logistic function. Also asked what other point they “know” is on the graph without knowing the equation. Then they have some practice problems for homework.
Yesterday I had my precalculus classes tell me those areas they needed to review before our big test on Thursday. We used a Table Talk type activity in which each student in their group of 4 wrote done any topic they wanted reviewed (they had about 5 minutes for this silent, individual brainstorm) and then the group determined the key 3-5 areas their group had in common. I then used their posters to create a 10 topic review. As I picked the questions, I tried to weave in topics listed on the posters that weren’t the top choices but easily tied into the top 10. How do you review for assessments with your students?
Today my precalculus students were introduced to Parametric functions. At the beginning of class, we graphed a parametric equation by hand, discussing the resulting ordered pairs and identifying the independent and dependent variables. Then we used the power of technology and dynamic representation, via the Nspire, to graph the same function on the handheld. We traced the function, compared to the table values, explored what the t-interval and the t-step controlled, and switched the functions around in the hopes of deepening their conceptual understanding of a parameter. Students in their groups did additional exploration problems and applications. They were intrigued! Although I would have loved to do a kinistetic activity, I just couldn’t think of one that was powerful enough. I’d love to hear how you introduce parametric functions!
Our text introduces 12 Families of Functions early in the year, viewing them through the Rule of 4 lens: numeric, graphic, analytic and verbal. My students have studied the Families of Functions for three years and know 8 of them (linear, quadratic, cubic, exponential, logarithmic, square root, reciprocal and absolute value) pretty well. I didn’t want to bore them going through these same functions again while adding in 4+ 2 ( logistic, greatest integer, sine, cosine, tangent and constant functions) to their repertoire. So why not a card sort? I had an old activity using 8 parent functions from Algebra II that included the name, the equation, the graph, the domain and range; so I added the 6 new ones and a partial table of values to get at the numeric representation. The methods of organizing was very interesting and telling…but even more enlightening was the discussions about how to match up equations, names, etc. of functions they had not seen before. This activity took about 30 minutes, but every group eventually got the correct matchings. All groups then glued their grouping in an organized way to a poster paper. How do you teach these functions and their characteristics to your students? FYI: I love my magnetic clips…makes posting student work a breeze. Definitely worth the $1 per 8 clips at the dollar store!!