Before beginning the sprints, it's a good idea to very quickly review the basics of logarithms. I always begin by asking, "What is a logarithm?", and my students quickly catch on that the correct response is "A logarithms is an exponent!" From there we'll look at one quick example of conversion from exponential to logarithmic form and vice versa. I'll also ask what it means if the base is not indicated ("It's base 10!") and what "ln(x)" means. That was new yesterday, so I only expect a couple of students to quickly recall it. For the rest, I'll remind them that it's a logarithm with the number, e, for its base. I try to do all of this in a rapid-fire question and answer format - if you know the answer, just call it out!
With that warm-up, it's time for log sprint 5 and log sprint 6, with a minute or two in between for reflection and informal discussion.
This series of lessons has a strong focus on abstract mathematical concepts and skills development. It's all very important, but it's also important to give the students a chance to see how the mathematics may be applied in the real world.
This population growth problem is intended to be done rather quickly. I'll put the students in small groups of 3 or 4 students assigned more or less randomly, and then hand out Exponential and Logarithmic Models 1. As they collaborate to solve the problem, I'll circulate to make check for understanding.
The first two parts of the question look back to the previous lessons on creating exponential models, and I may need to remind some students of what we did then. To answer the third question, I expect the students to express the answer as a logarithm and then use their calculator to approximate its value.
Finally, after about 10 minutes, I'll ask one or two groups to present their solutions to the class for a brief discussion.
With a quick reminder of the two properties of logarithms that were discussed in the previous lesson, I will hand out Properties of Logs Practice 1.
Students will work individual at first, but after about 5 minutes I'll allow them to work in small groups if they'd like. Along the way, I'll be moving around the room to check for understanding. I'll give one-on-one help to those who need it and also make note of students who might be good at explaining things later.
These problems are intended to do two things. First, they provide some simple practice putting the two properties of logarithms to use. Second, they lay the groundwork for the next property that we will discuss. In fact, as students approach the end of the problem set, I'll ask some leading questions such as, "Did you notice any patterns in the nine expansion problems? Perhaps you should take a closer look at numbers 6, 9, and 10." (MP 8)
I'll call for everyone's attention, and then ask for students to read their solutions to the problems one by one. Along the way, I'll check for agreement (and check my answer key), and we'll stop to discuss any difficulties.
When we come to the end, however, I'll ask, "Did any of you notice any patterns that might help us answer the final question?" I already know the answer to this question, but I want to give some of these students the chance to volunteer. On the other hand, if no one was able to see the pattern, I'll have to draw their attention to it more carefully.
The property I have in mind, of course, is this one: log(a^x) = x*log(a). They've seen a couple of specific examples that lead to this conjecture.
Now, it's time to ask the most important question: Why does this work? Is it always true?
As a class, we will develop the proofs that are outlined here. Be careful not to give too much away; I know from experience that students are able to develop the proofs on their own if given the time and opportunity. (Ideally, I will be able to call a student to the board, and then I will move to the back of the room to merely observe and to offer encouragement. Any questions directed to me, I will bounce back to the rest of the class to answer.) (MP 1 & 3)
Once the proof(s) have been articulated, I'll return to the front of the room to summarize and point out what should be written in notes.