My last post was from about a week ago bragging about my Pre-Calculus students getting excited about studying exponential functions and log functions. Well last week went even better. They left me at the front board in awe of their skills. I have taught Pre-calc now for 8 years. And Trig College Math (a class between Alg 2 and Pre-calc) for the six years before that. Logs have always been difficult for the students to grasp. So this year I wanted to do something different, I needed to change how I presented the material. It was a risk that paid off huge dividends.
After the lesson on the passing the drug test problem, the students were invested on this unit. Tuesday I went in and had them graph 5 different transformations of y = 2^x. We used all our previous information on how different transformation changed the graphs. Because this was about the fourth time we talked about transformations, these five graphs were easy for them to understand and manipulate. Then we looked at all five, noticed similarities and differences, identified the domain, range, and asymptote. I tried not to get in their way. I asked the questions, they made the connections. Then we graphed 5 variations of log base 2 of x. It just so happened (deliberate on my part) that each log graph was the inverse of the exponential function they just previously graphed. They immediately realized the graphs, in fact, were inverses. Everything we just studied about inverses was still holding true within this new set of functions. Then we flip the note sheet over and fill in a chart with log form vs exponential form. NO stopping them. I threw 10 hard evaluate these log expressions by rewriting them into exponential form. Students were defending their answers to others in the group and in the class. They were more than ready for their homework assignment.
Wednesday - two questions... that was it. They only had two questions on the homework. Well on to log properties we go. In years past, it was the ol' I tell you the properties, you then immediately manipulate the properties with variables. This year, oh no we are changing it up! Evaluate these... I have them 5 or 6 examples of logs being added together immediately followed by the log of the product of the arguments. Do you see any patterns? Any special relationships showing up? And darn it, if they didn't arrive at the property themselves. The lesson continued into Thursday with a discussion on how the change of base is going to be antiquated because of new technology. But if they had an old school calculator, here you go. Throw in the variable now that they are ready for them and give them a 10 question true false quiz at the end. Lively discussion followed the grading of the quiz. Who selected true and why? Who selected false and why? Did any of you change your mind? Why? What's the correct answer we can all agree upon? Bell rings.
And here we go for Friday: Remember last Friday I asked you about passing a drug test? Who remembers the plot? What was that equation you came up with again? Remind me why you all wanted .75 instead of .25 again. Excellent. Thank you for telling me the equation again. Our task today is find out exactly when your body will have 1mg left. Oh crap, they are not listening to me anymore!!! They have whipped out their calculators and are scribbling on their papers. Within a minute I have answers from nearly every group. I stood at the board basking in the beauty of students solving exponential functions without ever being given the procedure to do it.
WAIT A SECOND!?!?!
How did you all solve that? I haven't taught it yet. And here come the explanations of what they did. I pretended not to believe they could solve such problems and gave them another. Less than a minute again, they had an answer. Well alright then. Today's topic: solving exponential and log functions. Level: apply Assessment: solve this equation (where they had to condense the log expressions before solving where there would a quadratic that would require the quadratic formula with two irrational answers one of which was extraneous). We got through about half of the examples I wanted to. I'm not in a hurry. They are understanding it like rock stars! The bell rings. Three students run to the front with their assessment question completed and want to know if their answers are correct. I didn't even finish the lesson let alone ask them to attempt the assessment question. And these three students are not my top students. In fact, one rarely speaks to me.
I left my class that day so proud of my students. They left feeling enabled and empowered about math.
Over the weekend, I reflected on why I think they are doing so well. I feel it's because I took the time to build the concept and lay the foundation of that conceptual understanding before I gave them the procedure. The procedure came from necessity, not because it was what I told them to do. I had to slow down to allow them time to discover it. But I don't feel I will need to spend 3 days of review before the unit test like I have the past 14 years. I will give them at least one day of review. But I think that is all they need at this point. Tomorrow we will finish the solving lesson. Tuesday we discuss logistic curves and their purpose in the real world. Wednesday, we will review.
And I would like to defend my class for a brief moment. Yes, it is an honors class. But this class has struggled the most of all my Pre-Calc classes ever. One third came from the honors track, one third from the traditional track, and the last third skipped a year of math and came from a traditional math class. These are not kids who love math and can't wait for the words to roll off my lips. But they love being in control of their learning. I have become more of a facilitator than I have ever been instead of being a straight out lecturer. And I am loving it too. This may sound like I am bragging, but how can I not? They are amazing me every day. I have to brag.