## Powers of 2

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn explains that the meaning of “2 to the power of 4” is a product of four factors of 2: $2^4=2\cdot 2 \cdot 2\cdot 2.$

This is the beginning of exponentiation and it is easy to understand—the power (in this case, 4) tells us how many factors of the base (2) there should be.  I usually tell my college students tongue-in-cheek that, “Mathematicians are laaaaazzzzzy. We came up with the notation $2^{30}$ because we got tired of writing all 30 factors of two out!”

There are a few issues to watch out for when introducing a child to exponents (some of which I mentioned in the video):

• At this level, after students understand multiplication, powers of 2 are no harder or easier than learning addition or multiplication facts.
• However, it is a new operation and one that is easily confused with multiplication.  The statement, “2 times 4,” sounds a lot like, “2 to the 4th.”  Furthermore, the processes for evaluating the expressions are are similar but for powers we use multiplication instead of addition: “2 times 4” is describing a number of addends while “2 to the 4th” is describing a number of factors.

For these reasons, you (as the parent or teacher) should be very deliberate about asking, “What is 2 to the power of 4?” to help your child/students understand that you are asking for something very different than “2 fours.”

• Definitely stick with “powers of 2” until students are comfortable with the meaning of exponents and can confidently tell you what $2^1, 2^2, \ldots, 2^{10}$ are.  Remember, you are trying to help students learn the meaning of the operation, not memorize a bunch of numbers. (If you feel you can move on, go to powers of 10 next, and then to powers of 3.)

Once they are comfortable moving back and forth between $2^7$ and $128$ the real fun can begin—learning the properties of exponentiation like $a^m\cdot a^n=a^{m+n}.$  Don’t worry about this for now; Autumn and I will show you how easy and pleasurable it is to learn some of these properties in another video.  For now, just concentrate on learning the powers of 2 themselves (up to 10 or 12).

• Finally, it is much easier to remember the powers of 2 if you use “tags” or “pins,” i.e., memorizing a couple of easy-to-remember powers of 2, and using them to quickly figure out the rest.  I recommend: $2^5=32$, $2^8=256$, and $2^{10}=1024$.  Then one can quickly find $2^{11}=2^{10}\cdot 2= 1024\cdot 2=2048$, which is just an easy double.  You will see the “story” Autumn and I used to remember $2^8=256$ in the video.

Check back soon for a new video on the properties of exponentiation.  In the meantime, enjoy learning the powers of 2 with your child/students! CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636 Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.
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### 2 Responses to Powers of 2

1. elivesey2013 says:

Autumn is doing a great job. Her work with the powers of two is wonderful. I am sharing these videos and written commentary with teachers as well. Keep it up Scott!

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2. Val Byrnes says:

I would of loved to hear Autum describe how she came up with the solutions.

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