To begin today I ask students to determine how the words expand and condense are connected to logarithms. Students begin by trying to define the words. I am okay with students using their phones to find the definitions. Even with the definitions determining how these terms relate to what we have been doing is harder. Some students may look over their notes and the proofs from yesterday and determine the meaning.
Students share their ideas about the meaning and how it connects to logarithms. Once we have some productive ideas, I give students the second page of the bell work. Here, students use what was discussed to determine which side of the equation is expanded and which is condensed.
After a couple of minutes, I ask students if these equations look familiar. Some students realize that these are examples of the properties students proved yesterday. We then classify each side. As we classify I ask students to identify the property represented by the equation. I write the properties under the equations. The last problem uses two properties. At first, this confuses students. To show students the process, I will write out any missing steps in the expansion.
Yesterday students verified properties of logarithms. The last two questions on the activity required students to develop an argument. Today, I will give students a chance to see other students' reasoning. This is a good strategy for helping students to improve their own techniques. So, I have students share some of their proofs with the class.
As the arguments are shared, the students and I question the process. I ask other students to explain what was done by asking the following questions:
Once the last two arguments are shared, I give students a list of properties to put in their notes. Many students struggle with organizing notes. They write everything that is on the board. I make a list and tell students to identify this list as important. I say, "You will want to refer to this list as we work on problems."
I cover all but the first property to begin. Students write the property and then write an example of the property on their paper. Students work with others in their groups to write examples. Students are asked to share examples with the class if there is a property that is giving groups trouble. Once the properties are in students' notes, I am ready to for students to do some problems on expanding and condensing.
Expanding and condensing logarithmic expression is a skill often used in calculus. A complicated function can be rewritten as a logarithmic equation. Once transformed, the derivative of the function can be found more easily.
To help prepare my students for this, I give students some problems to expand. For each problem students work on the problem for a couple of minutes before I randomly pick a student to share what they have completed. If the student I pick is not done I have the student share what is complete and as a class we help the student finish the problem. As the work is shared the student either justifies the process or I ask other students in the class to explain the process.
I always ask my classes if a student has done the problem a different way. Students sometimes think they have done a problem incorrectly because it does not follow the process shared in class. When students are working I identify the different processes used and if a student does not want to share I ask the student to share the work.
During this lesson, I think it is also important for students to see how to reverse the process of expanding an expression. I discuss how condensing is working backwards from what we have just done. Students are given the following problems to condense. I let students work on the first problem for a few minutes. For students that are stuck I ask "What was the last step when we expanded? Can you imagine how to work backwards, just like solving a linear equation."
The first example is simple problem, so most students are able to condense the expression. Students who struggle generally are trying to do too much in one step. When students share their work I make sure that only one step is completed at a time. This helps students think about what is being done. I will also write the property we are using with each step. Some students do not immediately think about the possibility that these properties work both ways. Some think you have to work the property from right to left. Rewriting the property in reverse order will help students understand that we can look at the property both ways.
The second example is more complicated. Many students get stuck. Eventually, I will have a student share what has been completed we then work through the problem together. I remind students not to do too many steps at once.
As we end today I give students problems to work out of their book. I assign:
Page 241, #54, 56, 64, 70, 76, 78 from Larson's Precalculus with Limits, 2nd edition.
For a few minutes, I let the students work on problems on their own. I move to students that I identified as not understanding to review the material with them. I have these students do the first problem with me watching. I clarify any issues they have and then tell them to move to the next problem.