Math lesson reflections

Our lesson on logarithms did not go as planned. Our original plan was as follows,

1) Ralph do a short history as a hook
2) Write up some of the rules on logarithms
3) Derive multiplication and division laws for logarithms
4) Have the students do two quick questions to review the logarithm rule blog(x) = log(x^b)
5) Have an engaging question for the class to do

For 15 minutes this was way too ambitious of a time-line. The history went a bit longer than expected taking up ~50% of our time, not giving much time to work on the lecture portion and student centered activity.

The big issue from what I gleamed from the comments was that there was not enough activities for the students. So how would I change it for the future?

I would cut the history to 2 minutes max and quickly write up the basic rules of logarithms, such as for log(0), log(1), when base is the same as what is being logged. For this type of presentation, we can assume that we are just doing a review of the rules and the students have already learned it before.

I would cut out the derivations for the multiplication and division laws. In fact, I don't think I'd include those laws at all for this presentation. For 15minutes we need to keep it extremely short. What I would do is write a function on the board, starting with y = x and ask the class, 'without using a calculator how would I graph y = log(x). The purpose of this activity would be for the students to recognize that when x <= 0, there would exist no y. When x = 1, y =0. When x = 10, y = 1 etc. I would then get them to do the same graph but with a different base instead of 10 and see how the different graphs compare.

I could then give them an equation such as y = x^2 or x^2 - 3 and we could repeat the above process. They can check their results using a graphing calculator after wards. The purpose of this activity would be for them to recognize that a log of a negative number or 0 is always undefined and a log of 1 is always 0. Also, to graph it without a calculator, the students would have to recognize how to choose easy number to work with when dealing with logarithms. For example, if you were working with a base 2 logarithm, you would want to solve for when the inside is 2^n.

Example, for y = log(x^2 - 2) [base 2]
If y = 1, then x^2 - 2 = 2. Solving for x you get 2. Next you do for y = 2, then x^2 - 2 = 2^2 = 4. Solving x you get root(6). It's even easier to graph the logarithm if you were t first graph y1 = x^2 - 2 first, and then look for key locations on y1 such as when y1 = 0,1,2,4,8,16, 1/2, 1/4, etc.

That's all I have to say.