I’ve been playing with the sequence of my Alg2/Trig/PreCalc-ish class over the years. I’ve mostly been traditional, teaching Trig in the spring in a large chunk of maybe 8 weeks or so. This year I tried to mix things up, set up more spiraling, and find a more ideal balance. One of my goals at the start of year was to find a unit that was challenging and “new” enough to students to get them ready and focused on my class…and I think I found a sequence that works for me!
I started this year with a 5-6 day “unit” on Factorials, Permutations and Combinations. I thought that this was a nice introductory topic that both didn’t require a whole lot of previous knowledge and also avoided repetition/review of concepts from other math classes. (At my school, they don’t see anything with factorial, P, or C until my class.) It has been an amazingly improved first week/first unit, and here’s why:
- My class is special. Starting with factorials, a pretty much new concept/notation to my students, made my class “feel different.” It’s not just another year of more algebra stuff. Students felt that they were actually learning a new topic, and they worked very hard to learn the material.
- Anyone can do this. Along with #1, this topic is both new but attainable for anyone who puts in the effort. I snuck in some initial discussions on growth mindset with this.
- SBG. Having a topic that is not a repetition of anything in Alg1 helped me to explain Standards Based Grading and my quiz/quiz-retake policy in a way that students totally got. They now understand what a Learning Target is.
- Abstract Thinking. Talking about n, n-1, n-2, etc, allowed us to have discussion about what these terms mean (in the sense of previous term…setting them up for success with sequences) and work in the abstract. This is the first class where students focus a bit more heavily on generalizing/abstracting with variables and not just working with the concrete (plugging in numbers).
- Proof. I can introduce proof with relatively simple topics, setting students up for success later as we do some proving throughout the course, especially with trig identities! Today, we proved that nCn = 1 always and nC1 = n, something the students understood from the a few concrete examples and could make sense of the relatively easy proof. Students were evening thinking of their own extensions, like does it work for P?
- Graphing Calculator. Working out n!, Permutations and Combinations can involve a lot of multiplying. Isn’t it nice that the Graphing Calculator already has these functions?! Having this unit at the start of the year and teaching these calculator functions made having a TI-84 (or some such graphing calc) imperative! Students are no longer slow to get their calculator and develop the routine of bringing it to class.
- Grouping Expressions. Solving problems like word arrangements might involve a solutions such as 9!/(2!3!). When students see this written as a fraction, they don’t write the parentheses, but when they put it in the calculator without grouping the denominator, they get the wrong answer. This was a great time to review the importance of Order of Operations and correct grouping in the calculator!
- Debate! Lastly, I usually start the year with a week or so of logic because it’s a natural way to introduce my debate strategies and protocols. However, after a few days of learning P and C, I introduced the debate protocol (claim + warrant) as a way for us to debate whether a problem is a P or C or neither (starting the discussion of “none of these” being an answer from the start of the year!
So with this one unit I was able to set the tone for the year and stress a lot of routines and common errors that have come up over the years. We now have a starting point to refer back to each time one of these topics comes up!