Class begins with student receiving the homework and learning targets for the unit. I do this for every unit. I briefly review the learning targets with the class. While I am dealing with attendance, my students use graphing calculators to graph the functions on the board. Students work together and help each other remember the key features asked for in the bell work problem.
When I finish my bell work tasks, I begin to circulate, encouraging students to help each other as I move around the room. I am keeping track of concepts students are struggling to remember, since those will be the topics we will focus on when we discuss the problems.
After about 5 minutes, I will have students put up the key concepts they found. I then begin reminding students of the concepts they have forgotten. I usually have to review the concept of End Behavior. I write the first part of the End Behavior statement (as x approaches infinity then f(x)....) on the board. This statement is how the Algebra 2 teachers discussed end behavior last year, so just writing the beginning statement makes some students say, "Oh yeah, I remember now."
If students continue struggling with the End Behavior, I will pull up the table for the exponential function on the calculator. It helps many students once they see the numbers. The pattern of getting close to zero (very close), without ever reaching zero, is observable in the table.
Once we determine the end behavior, we will switch our attention to describing Asymptotes. To begin I ask students to explain what the term asymptote means to them. If students cannot give an explanation, I ask them to use their phones to find the meaning of the word. This surprises some students, but others are glad to have the freedom to look up information on the phone. After a students reads the definition, I write the definition up on the board. We look at the definition and use literacy strategies to understand the definition (i.e. breaking the definition into chunks, rephrasing the definition into student language). Once the definition is found we then identify the asymptotes for these functions.
I now give each student graph paper and asked have them sketch the graph y=3^x. Students are required to label 2 points on the graph. After y=3^x is graphed, I have students graph y=log of base 3 (x). Students use their calculator to graph this function. Students are also asked to label 2 points on the new graph.
After students have worked for 2-3 minutes I show the students the I made graph. students come to the board and label points on my graphs that they labeled on their graphs. Most students graph (0,1) and (1,3) on the exponential and (1,0) and (3,1) on the logarithm.
What do you notice about the two graphs? Some students notice that the functions reflect across y=x. I draw y=x onto the board graph for other students to see.
What do you notice about the points you plotted? Here I want students to see that the x and y have switched.
Knowing that the graphs reflect across y=x and that the x and y are switched in the graphs what can you say about the 2 functions?
Once the students realize that the functions are inverses, we generalize the key features of exponential and logarithmic functions. As we state the features I focus on connections between the two functions such as "Compare the domains and ranges. Why are they switched?" or "How are the asymptotes related?" By understanding how the key features are related students will be able to describe a logarithmic graph by seeing the an exponential graph or function.
I continue developing the connection between an exponential function and a logarithmic function by giving students another problem. I ask students to use the graph on the exponential to sketch the logarithmic function. After graphing the logarithmic function students write the equation for the logarithmic function. Students use the first problem to help with writing the logarithmic equation.
The class discusses how to write the equation of the inverse and how to sketch the graph of the inverse. If students are struggling I have students work another problem using y=7^x.
As we finish the lesson, I ask the students to think about how they can use an exponential function to graph a logarithmic function. I write a logarithmic function on the board and we brainstorm strategies for the best way to graph the function without a calculator.
Some students want to rewrite the function as an exponential. Students make a table for the exponential function or they plot some of the points for the exponential function on a graph. By now, these students know that they need to switch the x- and y-values to get coordinates of the graph of the logarithmic function.
This closing activity reinforces the connection between exponential and logarithmic functions.
For tomorrow's lesson I need students to do a exploration at home. An Interesting Exploration of Numbers demonstrates how the number e can appear in many different places. I have the students begin the the activity at the end of class. Students need to complete as much of the task as they can before the next class.
In this activity the students are going to reason numerically. If they are successful, then they will see a pattern. It is difficult, but, they may discover that dividing a number into sections of e yields the biggest product. This fact is one among many reasons why e is an amazing number. It often pops up in unexpected places.