The title may make this post seem fictitious or as if I am a Buzz Lightyear fanatic

but I am really not. As I was thinking about exponential equations I was awestruck by the size which powers of 10 can represent and the difference that adding one zero makes to a number. Thus the thought came to my mind, what if thinking exponentially could take you to infinity and beyond…

We have been talking about trying to make a lesson plan in my Math teaching course for exponential equations. I recently watched an extremely thought provoking video of the World in powers of 10 narrated by Morgan Freeman on viewing life in powers of 10. The video was very interesting to me in its simplicity, but the number of things I could imagine a student noticing wondering about the video was intriguing! The video starts in a small town in Italy where Galileo lived with the diameter of one small circle with diameter of 1 meter and multiplies it by 10 over and over until the circle encompasses the whole Milky Way Galaxy. Then the video shrinks down by negative powers of 10. As I watched the video I thought about how a teacher could form a lesson that began with this video or incorporated it in the middle.

Here are some ideas and questions I came up with to facilitate a discussion and get students to engage in problem solving:

Based on Common Core Math Standard CSS.MATH.CONTENT.HSF.LE.A.1.A Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

I could have students come up with a linear function based off something they notice in the video and compare it to the exponential pattern displayed by the Cosmic Voyage. Also, they could ask their own questions and seek to find solutions based off of what they know about mathematical questions. Some great questions students could answer are: How many times would a ten meter diameter need to be multiplied by to be equivalent to the exponential power of 10 that encompasses the whole Milky Way Galaxy? How would you create an equation to model the growth represented in this video? .

Another option would be to lead a class discussion in which students propose how they think about the patterns presented in the video. It would be interesting to see how students would want to model the growth in the diameter of the circle linearly or exponentially or some other way. I could let students break up in groups and I could represent their thinking on the way the diameter was changing in the video by modeling their thinking on the board. Then I could make connections between student responses to explain the exponential relationship presented in the video.

A third intriguing option would be to have students go outside and create their own exponential pattern using a different base unit, assuming the concept of exponential was already introduced. Students could use a measuring tape and start small and model their functions growth in a table and see what the exponential growth looks like with a tape measure. Students can make observations about their models and share them with the class up front or by writing them on the board, or by explaining them to me and I can write them down for them. This sharing with the class will facilitate collective learning and growth from the students understanding of each others models and observations.

Students can gain much from talking out loud or writing down what they notice and wonder from a short video or in class activity. Following each of these activity options I would have students reflect on what they learned to help them develop a deeper understanding of exponential equations and encode the information they have obtained in to their long term memory. I loved this video and hope that I can use it in my classroom someday to help students to understand and be able to relate the difference between an exponential relationship and a linear one.

The whole activity makes me wonder whether students would better grasp math if teachers always had a single activity which we referenced back to in order to help students consolidate their thoughts? I speculate that students would benefit immensely from rich activities like this that facilitate deep student thinking. Space is a vast world that we have hardly explored in comparison to all of what is out there. I think having students recognize the enormity of exponential expansion or decay in comparison to linear increases or decline will greatly benefit students. The success behind this activity lies on having students step into the hula hoop and see the expansion in powers of 10 out to the galaxies and the decay in negative powers of 10 down to the smallest of atoms. Students benefit greatly by seeing math in the world around them and this activity provides that opportunity to students.

The options for a lesson on exponential growth based off of this awesome video seem vast, so let me know if you have another lesson or activity idea and comment below!

Infinity is an intriguing concept, and the paradoxes that come with it make for puzzling discussions. I like connecting it to exponential growth and it’s rapid increase. Saw a cartoon recently about walking a graph to the moon. If each step is 1 on the x-axis, how many steps on 2^x to get to the moon? Jupiter? Pluto? Bonus: fits easily in a classroom.

This post could just benefit from some consolidation at the end. What have you gotten out of these speculations, or what does it make you wonder about?

Other C’s +

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This sounds like a great topic to talk about. I like that it starts off with a video because it can really grab student’s attention. Also since you’re talking about powers of 10 you could throw in some discussion about scientific notation since that always uses a number multiplied by 10 to an exponent. It all sounds like a very intriguing lesson and discussion.

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In my teacher assisting placement, we watched a very similar video. The students were amazed at how much measurements later in the sequence changed compared to the initial few – this goes for both directions, on both the macro and micro levels. I would second what Dakota mentioned as well. It demonstrates very easily the need and usefulness of scientific notation, both in terms of efficiency and ability to compare numbers quickly.

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