Teaching Exponents – Why Rabbit Trails Matter

Anyone that has ever been in a conversation with a friends knows how rabbit trails work.  You start talking about one thing and then you are reminded of something else and after a few minutes you are talking about something entirely unrelated to what you originally began with.  This can be frustrating at times but in an educational setting, tangents matter.

One of my favorite things about homeschooling is the organic flow and natural progression of learning.  I have a very loose schedule that I follow on those days when I’m not feeling particularly inspired, and more often than not, the inspiration comes with just that little nudge (I’ll share my schedule in the following weeks).

So it was a few days ago.  I wasn’t “feeling it”.  So I looked at my handy dandy little schedule and since it was a Wednesday, it was our day to study a famous Mathematician or Scientist.  I pulled out our copy of Mathematicians are People, Too and opened up to the chapter on Archimedes.  We picked up where we left off and read about his discovery of exponents.

That is when inspiration hit.  Lucy is just starting to get into exponents in math while Emma and Spencer have no experience with them.  I pulled out my white board and let the Spirit guide.

“Ok guys.  What is addition?”

Blinks and questioning stares.

“Addition is really just a way to count REALLY fast.”

Light bulbs turning.

“So if addition is just a really fast way to count, multiplication is just a really fast way too…”

“ADD!”

“Right!  Now if multiplication is just a really fast way to add, EXPONENTS are a really fast way too…”

“MULTIPLY!”

“Excellent!”

I showed them what an exponent looks like and what it means when you see one.  We practiced with some simple examples like four to the second power, two to the fourth, and one to the one millionth.  The kids each took turns writing the exponents that I told them to write on the white board.  I also showed them the fun trick when the base is ten (write one, and then how ever many zeros the exponent says).

“Why are exponents important?”

Blinks and questioning stares.

“What was Archimedes trying to do when he discovered exponents?”

“Counting sand.”

“Right! Can you count that high?”

Predictably, my kids starting shouting out words like bazillion.  After reminding them that there is an actual order to numbers and you can’t just make them up, we got back to the subject and I explained to them that another point of exponents is, not only to calculate more efficiently, but exponents also give you the freedom to write numbers that are far bigger than you would ever be able to write otherwise (i.e. grains of sand or stars in the sky).

“Ok, one last thing.  Some of these exponents have nicknames.  Four to the second power is also called ‘four squared’, sixteen to the second power is sixteen squared.  So ANYTHING with the exponent as two, is whatever the base is SQUARED.”

Nodding heads in understanding.

“Do you know WHY it’s called that?”

Blinks and questioning stares.

“Well, lets figure it out.”

I drew three dots on the board.

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“What number do these dots represent?”

“Three.”

“So if I want to draw three to the second power, how many dots do I need to draw?”

“Nine.”

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“What shape does that make?”

“A square!  And what number, multiplied by itself makes the number nine?”

“Three.”

“Right!  So three is the ‘square root’ of nine because three times three equals nine.”

“Ok, we know that works for three squared.  Should we test it to see if it works for other numbers as well?”

We experimented with a few more numbers and verified that anything squared created a square when drawn.

“Yup,they all work!  And any number with a whole number square root is a SQUARE number!  So 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are all SQUARE numbers!

Seriously, you’ve never seen kids so excited about exponents.  Since they were so into it, I continued.

“Do you want to know the other exponent number with a nickname?”

Vigorously nodding heads.

“Lets work with three again.  Three to the second power is three squared.  Three to the third power is three CUBED.  Why do you think we call it that?”

Lucy pipes in at this point, “because the answer makes a cube?”

“Exactly!  So what is three to cubed?”

“27.”

“Right, but we can’t draw that the same way we drew three squared, can we? So let’s get out the linking cubes to test this out.”

So we got out the linking cubes and put 27 cubes in front of each person (I was glad we had so many).  First we linked three cubes together.

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Then we made three squared.

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Then we counted together as we put each piece on our creations

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Sure enough, 27 makes a perfect cube.

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I was tempted to continue experimenting but I knew that if I kept going they were going to lose interest, so I called it a day for our family lesson and the older kids dispersed to work on their individual studies.

This is why I don’t make detailed lesson plans.  If I spent weeks coming up with a plan I would feel obligated to stick to it and I would miss the opportunity to be flexible when inspiration strikes.

I admit that while this exponent lesson may be an old hat to you, it wasn’t to me.  It was directly the result of the Spirit speaking to me when I was ready and open to inspiration.  I was fortunate enough to have recently read about “triangle and square numbers” in The Beginner’s Guide to Constructing the Universe.  Figuring out the connection between square and triangle numbers and exponents came to me when I was describing exponents to the kids, and cubed was a small mental step from there (I only wish I could have shown them the fourth dimension as well… time to brush up on my tesseract trivia).