After completing the study of trigonometry, I have reached the section of Pre-Calculus that is a smattering of review, extensions and seemingly random concepts. They aren't actually random, in truth, I sat down with the AP Calculus teacher (who is also teaching one section of Honors PreCalc with me) and we went through the textbook choosing topics that students are weak in when they reach Calculus. We are still figuring out how to connect them into a coherent curriculum, but at least I can point to my Math Practice Standards posters as one constant throughout the year.
So, when I got to inverse functions I put out a call on twitter for interesting ways to study inverses. If students had already learned about them last year I didn't want to go over everything as if they knew nothing, but also couldn't assume they remembered all about inverse functions. Hedge to the rescue! She came up with an awesome introduction to inverse functions that I tweaked slightly to fit my needs.
It turned out that students didn't remember (or learn?) that the graphs of inverses were reflections over the line y=x, so this was the perfect combination of review and new. There were also many choices in how students wanted to complete the assignment. The only definition I provided was that an inverse "undoes" a function, so outputs get turned back into inputs. (Someone want to add inverse to the vocab list with a better definition? Please?) This meant that some students inverted the tables, graphed, then figured out the function while other students inverted the functions, made new tables and graphed. The difference in method provided an excellent opportunity to start talking about restricting domain, the biggest challenge of this unit.
The second day we practiced taking inverses of functions using inverse operations. Most students did well with this, except for a couple students who wanted to do everything in their head and inevitably mixed something up. There really is a reason I ask students to show their work, and it isn't because I love to torture children. Mixed in with their practice problems was the interesting case of f(x)=-x+3. It is its own inverse! So of course we discussed this anomaly and yet again alternated between graphic and algebraic representations until they were convinced that they had sufficiently described all possible cases where a function is its own inverse.
After introducing the term one-to-one and practicing a few domain restrictions we played a 'game.' Each student received a number cube (and I told them not to call them dice, because then we must be gambling! and they laughed). I projected 6 functions called f(x) and 6 functions called g(x). Each function had an inverse on the screen, but they weren't matched up. Some students immediately started matching functions (next time I'll make them harder to match), but the goal was to practice function composition as well, so I told them to keep that information secret. Each person rolled their cube; the partner on the left determined f(x) while the partner on the right determined g(x). Then the partner on the left found f(g(x)) while the partner on the right simplified g(f(x)). Finally, the pair compared and if the functions are inverses they decided if a domain restriction is necessary. You may have noticed it's not really a game at all, but it involved dice so students thought it was fun. They did tired of it before completing all 36 pairs (shocking, right?) so we finally matched the pairs and listed all of the domain restrictions.
The best part of this unit was the result: 5 perfect scores out of less than 40 tests and 17 students aced at least one standard. One of the students who didn't do well came after school to remediate and when I talked him through one example he exclaimed "This is easy! Why didn't I get this before?" and his friend said "Maybe because you were on your phone..." We all laughed, then he proceeded to get a perfect score on the retake. It's amazing what they can accomplish when they focus.
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