# Backwards, Upside Down or Inside Out?

## Objective

SWBAT express exponential equations as logarithms. SWBAT use technology to solve exponential equations expressed as logarithms.

#### Big Idea

Take your students to Nome, Alaska as they learn to use logarithms to find out if the Great Mercy Race was successful.

## Set the Stage

20 minutes

I begin this lesson telling my students the story about a Diphtheria outbreak that threatened the people of Nome, Alaska in 1925 and that the rescue inspired the Iditarod Dogsled Race!  I stop before divulging the end and fill in some additional information about diphtheria and the mathematical considerations for a disease spreading. Here's the lesson hook, I challenge my students to team up and figure out whether or not the diphtheria outbreak claims all the young children of Nome before the serum arrives (MP1, MP4). This is not a short introduction to logarithms, but it has been worth the time invested since students are engaged in the true story.

As students take on the challenge, some may need support knowing where to start.  I prompt with questions such as, "What do you need to figure out?" or "How much time do the children have?"  As teams come up with an answer I ask them to be prepared to share their math and their thinking.  When everyone is done or after about five minutes I randomly select teams to present, while the remainder of the class asks questions and critique's their work. (MP3).

We continue until every team has a chance to present, then I ask if anyone had any difficulty with this problem.  Typically someone comments about the large numbers and the fact that they had to either guess-and-check the exponent or estimate from a graph.  At least one student will ask whether or not the children were saved and I respond by asking them what their results showed.  I won't give the result of the mercy run until later in the lesson so that my students are confident in their answer because they understand how they arrived at it, not because I confirm it.

Summarizing this part of the lesson I ask if anyone has ever used logarithms - most often seeing some blank stares as students ask what are logarithms?  This class discussion then leads to connections about inverse functions  and I do a whole class demonstration using logs to demonstrate.  My educreations video also shows how to use logarithms to solve this challenge.  We then practice with more exponential problems in the next activity.

## Put It Into Action

25 minutes

For this pair-working section of the lesson I use an assortment of real world exponential problems that can be solved using logarithms, as explain in my video.  I keep the Nome outbreak posted on the front board for reference and have students work with their front-partner.   After distributing the worksheet I check for understanding with fist-to-five then tell students they have about 15 minutes to complete these the work.  Some teams need a reminder that they need to solve the problems by hand until they isolate the variable before they can use their calculator to find logarithms.  (MP1,MP4)

While they're working I walk around offering encouragement and redirection as needed.   Although students know the lesson is about exponential functions and logarithms, some might struggle with writing the exponential models.  To help them I might prompt with something like "Remember the parent function for exponential equations?  How is this problem like that parent function?"  Other students will want more direction for going from exponential to logarithmic form, so in most circumstances I encourage them to work together using the example on the board as a reference so they can build their understanding of the relationship between exponents and logarithms. (MP1) The exception would be when both members of a team are really struggling with the process, in which case I might walk them through the first few steps using another example problem.

After 15 minutes or when all the teams are done I ask them to exchange papers with the team to their right, with the team furthest right passing their paper to the far left.  I remind my students that checking another team's paper means they need to do more than just mark the incorrect answers.  They are also responsible for writing the correct solution including any work or explanations. (MP3) This means that if they can't explain the correction themselves they need to ask me for an explanation they can copy to the paper.  When we've covered all the problems I ask if there are any other questions, then have the teams put their names in the upper right corner of the paper and return it to the original writers so they can look it over.  I give another few minutes for everyone to review their own work and ask questions then collect these papers so I can see how well my students are understanding the lesson.

## Wrap It Up

5 minutes

To close this lesson I bring the class back to the diphtheria outbreak in Nome and I generally have several students who are interested in the end of the story.  I suggest that first they look at their original calculations about whether the serum got there in time and now that they understand solving using logarithms , I give them a moment to check over the problem independently and see if they are still (or more) confident about their answer. (MP6) When they are all ready, I share the good news ending, that the mushers got the serum there in time to forestall any further deaths!