Real Number Exponents

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Students will be able to further extend the meaning of exponents to include irrational numbers and get their first taste of an exponential function.

Big Idea

This building-block lesson includes a touch of technology, collaboration, and critical thinking as the students further familiarize themselves with the rules of exponents – but more importantly, extend these rules to irrational exponents

The First Pitch

12 minutes

As class begins and the students are finding their seats, have them place their sticky note at the appropriate output on a large coordinate grid – this may be but is not limited to: a projector screen, whiteboard, or smart board.  (This can also be the same grid used in the previous lesson – a great way to activate prior knowledge!) 

Opening directions could read as the following: 

Welcome!  Please evaluate y=2^x for the value of x that I have given you.  Then, place your sticky note at the appropriate output on the class graph and take out your assignment from last night

I usually link this into my course agenda, but if your school does not have access to this technology then you could easily accomplish it through a simple PowerPoint slide.

(All of this takes place as the students enter.  These directions should be written or projected on the board so that the expectations are clear for all students.)

After all students have had an opportunity to grapple with and position their sticky note, be sure that there are no outliers on the graph that deviate from y=2^x.   If a few of the plotted points (sticky notes) do deviate from the graph, analyze the potential causes for this happening.  I DO NOT LET MY STUDENTS USE CALCULATORS DURING THIS ACTIVITY.  I force them to estimate the location of the irrational exponents based on their knowledge of the curve from yesterday, and the integer results that they are familiar with.

Next, prompt the students to discuss the following question in their thinking group and be prepared to have one person from the group share out:

How have we now (already) extended our properties of exponents without even learning a lesson?  What type of numbers have we now included?

Allowing all groups to share out is key.  All students should feel safe to wrestle with the complexity of the mathematics and share their perspectives of the question.  If you would like to try something new, and your students have access to technology, have them type their responses into a poll that you can create for them.  This is fast, easy, and allows students to be comfortable sharing their results.  Results can be easily projected real time as they are entered by the poll users.


I LOVE to include an element of surprise in my lessons any chance that I get.  When I am pulling up the poll website, I quickly flash up a website that is awarding $1,000,000 to anyone who gets a perfect March Madness Bracket - - acting surprised, I stop what we are doing with exponentials and draw the kids’ attention to the $1,000,000 bracket challenge.  I emphatically propose to them that we should shut down what we are doing this week and start filling out the brackets! (this lesson falls during the basketball frenzy in March) I excitedly comment:  "Surely if we divide up the task of filling out all of the possible bracket combinations, then we will be guaranteed a correct bracket... and a MILLION BUCKS!" 

 (I have NEVER had a class that doesn’t get SO EXCITED about this opportunity.  Usually, they want to get started right away.  I even go so far as to have printed off a NCAA bracket or two, which I then place on top of a big ream of paper from the copy room – this makes it look like I have printed hundreds of brackets... when really I wasted no ink or paper at all.) 

 Once I have the students emphatically hooked, and assure them that I will split the money with them… (usually this takes some convincing, and a few kids even make me write a quick contract!)  …I begin explaining that we must divide the possible brackets equally between each person.  I start by drawing the specific example of a 2 team bracket on the board, and talk about the possible outcomes of the game.  The students easily see that in a 2 team bracket, there are two possible outcomes.  Now, when we extend this to a 4 team bracket, I have the students quickly draw out the possible outcomes until they are able to determine the how many are present in a 4 team bracket.  It usually only takes the students a couple of minutes to find that there are 8 possible outcomes.  I chart all of this on a spreadsheet and inform the students (if they can not see the algorithm on their own) that an 8 team bracket will have 128 possible outcomes.

I then commission the students to find the mathematical function that models the possible outcomes of a bracket.  This takes just a short time, but the students are usually able to see that 2^(2-1)=2  and 2^(4-1)=8 and 2^(8-1)=128… This portion of the activity connects nicely to M.P. 7 and M.P.8! 

THEREFORE, we can generally say that if “x” number of teams are assembled in a bracket, then there are 2^(x-1) possible outcomes.  AHHHH!  The mathematics of exponential functions shows up!  Usually the kids begin to catch on that it was a planned distraction that I have created with the $1,000,000 bracket challenge.  However, it has a deep mathematical connection to exactly what we are studying!

So, now that we have the function, I have the students utilize technology to evaluate how many brackets our class needs to fill out.  It turns out that 2^(64-1) = 9,223,372,036,854,800,000 possible brackets!  What a great way to illustrate exponential growth!  EVERYONE IN THE WORLD could fill out a million brackets and it still would not be even close to how many we need to account for every possible outcome. 

(The kids are really disappointed when they discover that they can’t win a million bucks – but they have a story of exponential growth that will last a lifetime!  They also gained valuable experience wresting with a wide variety of the math practice standards.) 

Resources:  bracket challenge link – these are all over the internet in March!   I have attached a Fox Sports link

Lesson Middle

15 minutes

After the students have had an opportunity to complete and summarize the opening activity, then the stage is set to move forward with the lesson.  If the students are struggling to explain that now we have extended our knowledge of exponents to all REAL numbers, have them return to the whiteboard to retrieve their sticky notes.  Next, in their thinking groups, have the students place them into RATIONAL and IRRATIONAL piles.  Refer to the number diagram from yesterday to illustrate that we now know that exponents can be extended to include all REAL numbers – rational and irrational.

Now that the overarching meaning of exponents has been extended to include all real numbers, exponential functions can be defined.  (However, will hold off for this important discussion tomorrow!)  At the very least, the sticky note graph clearly illustrates that we have increased the frequency and proximity of points for the graph of y=2^x.   

I have found that irrational exponents can be rather intimidating for many students.  For this reason, I have elected to scaffold around how irrational exponents can be used with our existing laws and rules of exponents – in other words, ease our way in.  I regularly use this strategy in my classroom because although I want to challenge students with the mathematics, I want them to feel confident at the start of a unit.  Please see attached examples and the corresponding assignment. 


End + Homework

15 minutes

A)  Homework/Collaborative time

B)  Lesson Closure:  EXIT SLIP PowerPoint

Please see the attached exit slip.  This is particularly important in this lesson because so much prior knowledge is required about the rules of exponents – and the students are sometimes hesitant to speak up when they know a concept is review.  Although it is an “exit slip”, the Define a function in your own words question will also serve as great data for you to see what the students know about functions.  This will be particularly important as we transition to the Common Core and students are pipelined into a new curriculum – functions are analyzed at a new depth!  I look forward to sharing a reflection piece to this component in 3-4 years to see how student answers have changed since the adoption of the Common Core.

Resources:  see attached PPT - exit slip completed on a half sheet of paper