Introduction to Exponential Functions - Rational Exponents
Lesson 1 of 21
Objective: Students will be able to extend the meaning (and rules) of exponents to include rational exponents. They will activate prior knowledge and applications of exponents from their introductory study in Algebra I.
As is the case when rolling out most new units, activating prior knowledge is critical in this lesson over rational exponents. All students can obtain a positive feeling of success at the start of the day if the beginning of the lesson is carefully rolled out. To do this in a creative fashion, I created a fictional twitter “person” named Eddie Exponent. As the students enter the room I have them tweet Eddie with hash tags about their current knowledge of him. (This may need to be modified if twitter is not allowed in your school building, or you do not currently have the supporting technology. However, it would be just as easy and engaging to make a PowerPoint slide with Eddie Exponent and have the students put sticky notes around him about their prior knowledge. I also highly recommended www.polleverywhere.com for a similar effect) Although it sounds “cheesy”, high school kids love it – and it activates prior knowledge!
After seeing the twitter, poll, or sticky note results I ask one or two students to summarize the list of “knows” and classify the feedback into CHARACTERISTICS, RULES, or OTHER piles. I ask the two students who are leading to gather feedback from the class as well. I DO ZERO TALKING AT THIS TIME, and simply stay out of the way. Usually, all of my classes are able to come up with the complete list of exponent rules from Algebra I, however, you may need to fill in any gaps. Expect this entire process to take 6 or 7 minutes. Although it may be tempting, especially if you have not tried this in your classroom before, do not assign the task of summarizing the list to only high achieving students! I have found that in most cases they do not turn to the rest of the class or advice and input as much as other students. That is, until they are “coached-up” and catch on to the level of expectation for the collaboration!
After activating prior knowledge of exponents, the teacher should prompt the students to analyze the graph of y=2^x (usually, I put this graph up on my projector screen so that the students do not need to take time to graph it – there will be time for that at a later date). One by one, I ask students to come up to the graph and place a sticky note on the points of the graph that correspond to the twitter activity responses. Students will place sticky notes on 2^0=1, 2^1=2, 2^2=4, and so on. I would advise you to randomly hand out exponents and place them at their location on the graph of y=2^x. Most of the time, I give the sticky notes out at the door and ask the students to hold on to them until the appropriate time in the lesson. With my graph, I went out to 2^8 and it worked out really well. (if I would have gone much farther, however, I would have been through the ceiling - which is a great visualization of exponential growth!) Once the graph is created, the wonderful discussion begins!
Driving Question(s) we will ask students to answer:
If the graph of y=2^x is continuous, which it appears to be, how do we account for the “gaps” between sticky note points? What types of exponents satisfy the function within these regions?
Resources: www.polleverywhere.com - requires use of student technology for responses, projector, sticky notes
The Heat of the Lesson
A) Thinking Groups
As we ease into the middle of the lesson, I ask the students to chat with their thinking groups to discuss what type of exponents are present between the points we have already identified (the sticky notes) on y=2^x. Typically these thinking groups almost unanimously come to the conclusion that “the exponents in the gaps must be fractions”… this kicks to door open to the mathematical punch line and jump-starts the remainder of our study! (Upper level students may also be able to extend the fact that irrational exponents must also be located between our integer locations.)
As a powerful example of what is taking place during exponential growth, I next hold up a piece of paper for the students to see. I ask the students how many times they think that I would have to fold the piece of paper to give me a thickness that would be enough to make it to the moon (and instruct them to hold their comments until further instructions – I didn’t the first time I did this and the students immediately wanted to shout out their answers, haha!).
Try having them discuss the idea with their thinking groups and reconvene with their final guesses. Then (after a drumroll) reveal to the students that it would take ONLY 47 folds (obviously 'theoretical' folds - I mentioned that Myth Buster's TV show has tried this) to make it to the moon. To wrap up the activity, connect the data to the graph: Imagine how high through the ceiling our output would be at 2^47 – this is exponential growth at its finest! This type of growth is SO IMPORTANT but SO DIFFICULT for human minds to grasp. We will take a financial look at its importance in the coming days.
(In the back of my mind, I know that my favorite exponential growth lesson is coming soon! I hook the kids with an easy "paper-folding" example, and will look even deeper into exponential growth in the upcoming days. In each case I really emphasize that LINEAR GROWTH is easy for most human minds to graph, but relatively FEW have an understanding of the power of EXPONENTIAL GROWTH. It really gets their attention once I mix in dollars and cents!)
B) Guided Practice/Skill Building
Students should be able to see that rational exponents are a combination of a power and a root. They should also be able to use the existing rules of exponents (our prior knowledge piece) to extend these concepts to rational exponents. I do an example or two on the board prior to giving the assignment. (Since the mathematical symbolism will be much easier to see and understand I a video, I will upload the link to this in a future reflection.)
Resources: Video coming soon!
End + Homework
A) Homework Time
Distribute assignment and allow time to work/ask questions. I typically assign problems out of our course textbook for this lesson, because it aligns nicely. However, you could easily create your own.
Optional “Workshop” for struggling students
Typically a handful of students need a more significant review of exponent laws from Algebra I. I empower (and occasionally need to remind) them to attend this “mini concept review lesson” to get extra support.
B) Lesson Closure – is that all?
Return to the sticky notes graph that we started with and have the students collaborate briefly with their thinking groups over the following question:
Now we can account for A LOT more of the points of y=2^x…but we still can not account for all of them… Why?
IRRATIONAL NUMBERS CAN BE EXPONENTS TOO! (if desired, throw up a slide or diagram of the types of numbers to show that the rationales do not account for every type of exponent that we can have.)
Resources: Mini-whiteboards for additional examples. In this lesson, I elected to utilize our school’s math textbook for the homework assignment. It aligned nicely to the exponential properties we studied in the lesson.