Trying not to brag, but I must

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.

Who are you and what have you done with my students?

It has been far too long.  And I really wish the blog initiative was still kicking my butt to write at least once a week.  I have much to still do tonight, but I must force myself to reflect.  Here I go... and before I start, I need to say my Pre-Calculus kids are still shocking me this year.  Remember, they were all in their seats before the bell rang on the first day... back to the story at hand.

I had a second opportunity to see Steve Leinwand again with some more members of my department.  I was reminded about the following:

Evaluate 16(.75)^x  for when x = 8 in the exponential function 

versus 

You take 16mg of a controlled substance at 8am.  Your body metabolizes the drug at a rate of 25% per hour.  If you have to pass a drug test at 4pm with less than 1mg, will you pass the test?

Steve goes on to tell the story of how the Romain gymnast lost her gold medal because her coach didn't understand this problem.  Well, I just so happened to be starting the chapter on logarithms in three days from this afternoon conference.  Heck yeah, I'm so stealing this activity and having the kids get down and dirty with this function.

Three days go by and it's Friday... new unit day.  I pose the problem, give the students 10 minutes to come up with an answer.  Oh, and they had better be prepared to defend their answer and explain how they arrived at it.  They didn't even blink.  Bam, 15 minutes later I have kids telling to multiply by .75 not .25 each hour.  Paraphrased from students: Well Mrs. Berg, it's because it's too much work to keep subtracting the 25%.  You could, but this way was easier.  (HECK YEAH! - trying to keep my excitement PG here)  And no this person doesn't pass the drug test at 4pm.  Almost, two more hours and they would have, but not at 4pm.  Formula?  Sure I wrote a formula for it.  16(.75)^x.  (me in my mind - HOT DAMN - sorry for the French).  Oh, I said at first they would pass the drug test because I just kept subtracting 4.  But then I thought that was too easy, so I tried another way.  

I could not plan for my students to astound me this way.  They literally took over the conversation.  And I hadn't even gotten to the good part of how this was a true story.  They could google it when they got home, but a gymnast from the Syndey Olympics lost her gold medal because she took some over the counter cold medicine and the coach miscalculated the length of time on the banned substance.  They were eating in the palm of my hand.  

Then I said I had another real life story problem that involved exponential functions.  Are they ready?  They were ready as long as this one wasn't a sad story.  I then told them about buying our first family vehicle.  We were offered two choices $12,000 at 8% interest or $18,000 at 2% interest.  Which should my husband and I take?  I told them how we argued over it.  He wanted the lower interest rate.  I wanted the lower amount.  I just so happened to have my graphing calculator in my purse (just like Batman would wear his utility belt).  I worked the numbers.  We took the higher interest rate.  And then the bell rang.

I had assigned about 8 story problems.  They all groaned when I said that.  However, today, they came in without questions on those problems.  I asked, any questions your groups couldn't answer?  They said no.  The homework was actually easy and they liked doing it!  Ok, who are you and what have you done with my students?

Today we tackled graphing both exponential and log graphs.  Easy peasy, lemon squeezy.  Well, I hope.  Those buggars made the connections they needed to on their own.  I had very little to say today other than How do you know this?  Please, convince the class.  Do you agree with Suzy?  Who can help John out?

Tomorrow we pursue the properties of logs.  My goal:  Shut my mouth and let my students take over the learning.  After a small pop quiz.  1 story problem and 2 graphs... know or don't... show me.

Now, on to my other things I must attend to tonight.  Felt good to brag and post again.  Far too long I tell you.