# Engaging students: Exponential Growth and Decay

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Chais Price. His topic, from Precalculus: exponential growth and decay.

How does this topic extend what your students should have learned in previous courses?

Every year in elementary through high school it seemed like I had some form of standardized test. These test typically consist of various problems, which include patterns and sequences of patters style of problems. I always found it more helpful when being introduced to more complex and intimidating concepts, to relate the general idea to something much more simplistic. When teaching a lesson on exponential growth and/or decay I plan on starting off the lesson with problems like:

These two patterns are pretty basic and finding the next one in the sequence shouldn’t be to difficult. This begs the question what if I wanted to find some enormous value for n. For questions d, a student can answer the question by drawing or counting but it will take some time. Or the student could find an equation that models the sequence of patterns. The equation would obviously be an exponential. From this point the teacher could discuss how these functions appear on the graph by simply observing what is happening in the sequence. In the first picture alone with the triangles, we only have 4 triangles shown and the first triangle is solid black. If we continue on, the next one in the sequence would represent basically our x values on a graph and the amount of triangles growing exponentially represents the y values. By using this previous knowledge the teacher was capable of relating a new concept with a much simpler approach.

How could you as a teacher create an activity or project that involves your topic?

Has anyone ever asked you if you would rather have a million dollars, or a penny that doubles everyday for an entire month? I heard this question probably when I was in high school. I am pretty sure that I picked a penny that doubled everyday for a month only because it was the least obvious and it seemed like a trick question. However this is an example of how only 31 days explode into a fortune. After the first week of doubling you only have a little over a dollar. In fact you really don’t start making any real money until about the middle of the 3rd week if you chose to have a penny double everyday for a month. It turns out that by the last day of the month you end up with over 21 million dollars. This is once again because the function is growing exponentially. The link at the bottom of the page has a story that uses this same idea about a raja from India who made a young girls request to have a grain of rice double everyday for a month. This story can be fun to read and engaging for the students as well. After the story is read, there is a calendar where the students will fill in each day the amount of rice given to Rani, the young girl in the story. This calendar has a few random days filled in so the students know if they are on the right track. This activity serves as an engage/ explore for more of an introduction to exponential growth. The students could graph this function of type some points into the calculator to see the function explode. Let x represent days and y represent the grain of rice each day.

How can technology be used to effectively engage students with this topic?

Dan Myers is a teacher who developed a style of teaching called the 3 act lessons, which incorporates multiple technology applications such as video recording, as well as imaging and photo editing. Each act is designed to teach a lesson like a movie divided up into parts. I came across this lesson of his which I think is awesome. In act 1, there is a 24 second video with these words at the beginning: “ a smaller domino can topple a domino that is up to 1.5 times larger in every dimension. “ The guy on the video explains that the smallest domino is 5 mm high and 1mm thick. This is all you are given. Then the teacher asks something to the class along the lines of “ If you wanted to topple over a domino the size of a sky scraper, how many dominoes would you need? “ This opens the door for students to both question and reason. Make a prediction and write it down. Have the students write down an answer they know is too high and one they know is too low. That is the end of act one. As we get into act to we need more information just like in a movie. Act 2 answers the question how many dominoes are present in the video. It also provides a data sheet that has the heights f several sky scrappers. This is a very discussion style lesson so in act 2 we would continue to promote discussion and questions. Then finally in act three we come to the conclusion. The man in the video had 13 dominoes and the biggest one was barely up to his waste. It turns out that if we were to keep adding dominoes that grew 1.5 times more than the previous one, the 29th domino would be as tall as the Empire State Building. That is exponential growth at its finest.

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