Introducing Exponential Notation and Exponential Relations

In Ontario, exponential notation is introduced in grade 8 in order to pave way for topics such as area of a cirlce and Pythagorean Theorem. As students reach secondary, exponential notation becomes increasingly more important as more complex functions are introduced in each course. Recently, I had a number of teachers emailing me for an interesting 3 act style problem to introduce exponents. While Double Sunglasses by Dan Meyer and Dark Side by Jon Orr are two great tasks involving exponents, I’ve found that they can be pretty difficult for students in grade 8 to wrap their heads around when they are just being introduced to exponential notation.

I think many are familiar with the Wheat and Chessboard Problem and some have even taken their own spin on that to make it more relevant to current times. Dave Bracken from my previous school, Belle River District High School used to tell students a story that went something like this:

I was speaking with your parents last night and they were all complaining that you weren’t doing your chores around the house. After some brainstorming, we came up with an offer for you. If you do your chores every day for the next month, your parents will give you $10,000 or a penny on the first day total, two pennies total on the second, four on the third, and so on. Which would you pick?

So while the story alone will lead to a pretty cool introductory task, I thought I would try to put something together that might help with the delivery and hook.

Task #1: $10 or a Penny a Day

Here’s a good starter to lower the floor and get kids talking. My expectation is you shouldn’t require more than 10 minutes for this quick minds on.

Act 1: Introduce the Problem

Show students this video.

This can be some interesting discussion because the video does not give you details for how long. When would one be better than another? What if this was for the rest of your life? Does that change things?

Act 2: Give Some Details

Show students this video.

Or, here’s a screenshot:

Now, students are given the question:

How long will it take before the penny a day is your best option?

Act 3: Show the Answer

My expectation is that it won’t take students too long to realize that there are 1,000 pennies in $10, so you may not even need to show the Act 3 Video.

Or the screenshot:

Task #2: $10 or One Penny on Day 1, Two on Day 2, Four on Day 3, etc.

Now, the scenario gets a bit more complex. However, I would suggest using this task again more for discussion and not spend much time on it.

Acts 1 & 2: Introducing the Problem

Even though there is an Act 1 video (here), I might recommend jumping straight to the question by showing the Act 2 video.

Or the screenshot:

So students are now confronted with this question:

How long before the “penny doubled” option is best?

*You may want to reiterate that students are getting one penny on the first day, then the original one penny turns into two pennies on the second day, it turns into four pennies on the 3rd day, etc.*

Act 3: Show the Answer

Again, students will probably figure out pretty quickly which option is best.

Now, show the Act 3 Video.

Screenshot:

In just over a week, students will begin to benefit from the penny doubling option.

Task #3: THE MAIN EVENT

As mentioned previously, tasks 1 and 2 were created as a way to get kids thinking, talking and discussing. Now, we move on to the new learning goal we intend to hit: introducing the need for exponential notation.

Act 1: Introducing the Problem

Start off by showing the Act 1 video.

Screenshot:

In Act 1, students are asked:

Would you rather take $1 Million or the “penny doubled” for a month?

*Again, reiterating that students are getting one penny on the first day, then the original one penny turns into two pennies on the second day, it turns into four pennies on the 3rd day, etc.*

I’d try to let kids get going on their own at this point, even without showing Act 2.

Act 2: More Information (Optional)

If some students are struggling to get started, you may choose to give students the visual provided in the Act 2 video.

Act 3: Show the Answer

Check out the answer here.

Screenshot:

Sequel: Is There A Better Way?

Act 1

Show students this video.

Or, the screenshot:

Act 3

After a short amount of thinking time, we introduce the idea of exponents and exponential notation for the first time using a direct instruction approach starting with this act 3 video.

Screenshot:

So, give it a shot and let me know how it goes in the comments!

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About Kyle Pearce

I’m Kyle Pearce and I am a former high school math teacher. I’m now the K-12 Mathematics Consultant with the Greater Essex County District School Board, where I uncover creative ways to spark curiosity and fuel sense making in mathematics. Read more.

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