Calculator Boot-Camp and Solving Exponential Equations
Lesson 13 of 21
Objective: SWBAT use a calculator to effectively solve exponential equations
As the students enter, I ask them to write the name of the calculator AP they have downloaded for their IPAD on the board. Next, while I take attendance and visit any students who were absent from the previous day’s lesson, I refer them to this site.
Although this link is connected to 6th/7th grade mathematics, it is a good reminder for the students that the order of operations exists, and it matters how we type things into our calculator. To make this activity more challenging, I force the kids to work in pairs to communicate the problem to the other person verbally – who does nothing more than type exactly what their partner says into the calculator. Even if they can do the math in their head, I still make them tell the person precisely what to type. This makes them process the order of operations, attend to precision, and communicate effectively during the warm up. In the end, I emphasize how much of a shame it would be to mess up on the SAT because you misused the order of operations on a calculator!
As the students take turns doing this for the first five minutes, I take a look at the AP’s on the board and familiarize myself with any that I have not seen. In the future, I plan to recommend a few of the most user friendly versions to my classes on day #1.
A. (OPTIONAL) Logarithms Video: Brief History, Motivation, Explanation
After the warm up, I have my students watch the brief video above. It provides an excellent summary of what we have covered so far and cleverly ties in the historical roots and motivation for logarithms. When dealing with lower level classes, this video might serve as an excellent scaffolding component. It often helps kids to hear a difficult concept from multiple sources – as well as have the ability to pause and replay the conversation to slow the learning process down.
NOTE: In the past, I used to teach the students how to use the logarithm table. However, today I simply show the students what the table looks like and explain how our calculators have replaced the need for it. I would much rather spend the time on the historical significance of logarithms rather than how to use a clunky table!
B. (OPTIONAL) workshop for struggling students
See “Remedial Workshop” PowerPoint. Although I did not use this in my lesson, I hoped you may find it helpful!
In this section of the lesson I transition into a modern day example of an exponential function. Due to time constraints, I do not take the students through the derivation of the equation. There will be time for this in my upcoming lessons! I throw this in here because I believe that it is good for the students to see an exponential function in action before they try to tinker with its internal gears. The real life nature of compound interest also allows the students to focus on the reasonability of their answers, and gives them the confidence they will need to attack upcoming complex problems. By providing them with a frame of reference, compound interest also will compare/contrast nicely with exponential decay, as well as other exponential functions.
On the first slide, I take time to talk the student through the variables and what is going on behind the scenes in compound interest. I try to shed light on current interest rates, and often pull bank CD’s into our discussion. Typically, students are very inquisitive. Allow time for questions, and be prepared to provide honest answers if you are stumped!
Slide #2: The problem asks the students to perform a simple “plug and chug” operation. Students may need to be reminded that exponentiation must take place BEFORE multiplication! Hopefully, however, all of these issues were ironed out with the opening activity.
Slide #3: This example requires the students to solve for time. Since the variable is in the exponent, students must take the common log of both sides to solve. This is a GREAT time to make a big deal out of this as a teacher. The students can now solve equations with variables in the exponent – even when like bases can not be found!
Slide #4: This slide is meant to serve as a concept connector and discussion platform to slide #3. Essentially, it asks the students about the doubling time for a greater amount of money with the same compounding variables. I poll the students to see who thinks the greater amount of money will double faster, slower, or the same as the smaller amount of money. To create a mathematical discussion and debate, I ask the students to talk at their table and come to a “table consensus” – and report their consensus back to me. With students sitting in like ability groups, you will be surprised at how heated the discussion gets! Ultimately we solve the problem to find out that it takes on the exactly same time to double as the previous amount of money. Be sure to allow time for a 2-3 minute follow up discussion about how the students beliefs and assumptions have changed!
Slide #5: This time, the students are asked to solve for the interest rate. Typically, there are very few questions with this example and it can be conducted in small groups as the teacher rotates the room offering assistance. As students complete the final example, I transition them into independent work time on the homework assignment.
I am a big advocate of morning review sessions. Every morning before school there are no less than 10-12 kids in my room getting math help. I work really hard to make this happen, and sometimes offer donuts to any attendees. Although the students do not know this, I make note of who comes in for morning help. I am sure to praise the kids for their extra efforts!