Natural Logarithms

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Students will be able to use e and the natural logarithm to model continuous growth.

Big Idea

This lesson highlights one of the most famous, and useful, numbers in mathematics.

Warm Up and Homework Review

5 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Natural Logarithms which asks students to do some dimensional analysis.

I also use this time to correct and record the previous day's Homework.  

Continuous Growth Investigation

20 minutes

The goal of this lesson is to extend what the students learned about finding interest into continuously compounding interest.   The number e makes a lot more sense when students have a basic idea of where the number comes from. 

We start by revisiting the formula for finding compound interest.  I didn’t put in the interest formula into the PowerPoint as this will depend on where the class took it in the prior lesson.  At this point, we are going look at a simpler version so we can focus entirely on the differences that come from different methods of compounding the interest.  We are going to set the initial amount to 1, the interest rate to 100% and the time to 1 year.  Depending on my students, I will either present this simplification to them or guide them to it by asking things like “’What could I do to make this problem easier so I can just focus on the compounding portion?” 

We use this simplified version of the interest formula to find interest compounded yearly, quarterly, monthly, weekly, daily, hourly, every minute and then every second.  It's important not to spend too much time on this section.  I may assign students or groups of students to find parts of this rather than everyone do everything.  It becomes obvious once the work is compiled that the values are moving towards e.  We then introduce e as the limit that is approached as we get closer to compounding continuously.   

This is the point where the students are introduced to the value e.   The investigation leads a clear path to this number.  Students look at e like it's some sort of unknown creature.  I always try to emphasize that it is a number just like 5.  I ask the students why they think we use a letter to represent e rather than the actual number (Math Practice 2).  This will bring up the fact that the only representation we will be able to give of e as an actual number would be rounded off and therefore not precise.  It also helps to relate e back to π.  They are familiar with π by this time.  

*Please note that this investigation was inspire by  

Applications of Continuous Growth or Decay

20 minutes

Next, we look at a list of some of the many uses of e in continuous growth or decay:

-Interest, appreciation and depreciation


-Rate of dissolving

-Cooling and heating (Newton’s Law)

-Radioactive decay

Here, I introduce the continuous growth or decay function.  I chose not to have the students derive this formula. Instead, we are going to relate it to the general exponential form that they already know (Math Practice 7).  This relationship is important to their ability to actually put the equation to use.  It was a challenge in writing this unit on exponentials to pick and choose among the amazing amounts of good models.  I chose this rather than focusing on something like half life or doubling time since it flowed so nicely with the last lesson on compound interest as well as covered the natural logarithm in a way that naturally flows.  I have taught curriculums before where natural logarithms are just stuck in at the end and students have not feeling for their place or importance.

The remainder of the lesson covers different applications of the continuous growth formula.  We will start with a continuous interest problem.  Next, we move to population growth.  Finally, we look at radioactive decay(Math Practice 6). 


1 minutes

The Home Work has nine problems that allow students to get more experience with the continuous growth formula.  The final problem may be a bit of challenge so it can be used as an extension if desired (Math Practice 1).  

Exit Ticket

5 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

This Exit Ticket ensures that they have a decent understanding of the continuous exponential growth function.