Math

Elmo, Four Fingers, and the Octal

If Elmo Ran the World…

I was watching a commercial on TV  last night and a puppet asked someone to give him a “high five.” He looked at his raised hand and seemed mildly shocked upon realizing he had only four fingers. That caused me to wonder. If Elmo were in charge of the world, would we all be using the octal (base 8) system? After all, I thought, the reason we probably use base 10 (or decimal) system is because we have ten digits on our hands. Why not include our toes and use the base 20 (vigesimal) system?  The Mayans used it. How about the binary system (base 2)? If it’s good enough for computers, why not for us? We’d only need to know two symbols, 1 & 0.


In my “research” into the base 10 decision, I came to learn that fingers (or toes) were probably not the sole reason for using base 10, but most likely it contributed to the final outcome. Some may think we use base 10 because of fractions and decimals. You can, however, do those in other bases, even though the term “decimals” refers to base 10.

Real World Examples

There are some “real world” examples of various base systems that surround us and our students:

  • Have you ever created your own web pages from scratch? Colors are represented using three hexadecimal numbers (base 16). This triad of numbers allows programmers to creatively use 16,777,216 colors by simply inputing three sets of two-digit numbers. Using #14C8C8 provides for this nice turquoise color. 
  • How about our clock? Seconds “spill over” to minutes using a base 60 system (sexagesimal). Minutes spill to hours using using sexagesimal. But then things change up a bit, as hours spill to days using base 24 or base 12 (tetravigesimal or duodecimal). Maybe students would better understand “time arithmetic” if they understood that regrouping from the minutes place provided 60 and not 10 seconds. How would your students handle 0:12:20 – 0:09:45= ?
  • Our customary measurement system includes hand-foot-yards (using base 3 or the ternary system). We don’t see the measurement “hand” that much, unless we’re measuring the height of horses, but it is equal to 4 inches. With some base 3 knowledge, it makes a little more sense.
  • Twelve eggs provides a dozen. A dozen dozens provides a gross. We can gather a dozen gross and we call it a great gross. These all use duodecimal system (base 12). Also called the dozenal system, base 12 is an efficient number system with more factor options than our more common number ten. Here’s an interesting article advocating tossing the decimal system in favor of the dozenal. There’s even a society of dozenal advocates who propose this new focus. Check out their website: http://www.dozenal.org
  • Finally, we can’t discuss “real world” number systems without presenting the binary system, with only two digits: 0 and 1. The very foundations of all things digital or electronic are built on binary. Why? Simply stated, they are built on transistors that are either on or off. Following digital logic, electrical currents flow through circuits and the presence of a high enough voltage returns a positive read (also interpreted as true or “logic 1“) and a low to no voltage signal returns a negative read (also interpreted as false or “logic 0“). When students see a bunch of 0’s and 1’s flashing across a computer screen or some geeky computer science guy wearing a similarly fashioned t-shirt, it will make sense.  Students will learn that 011111100000 actually means something. That’s 2016, by the way.

computer-code

Numbers in Base Ten

One of the main domains in elementary math is “Numbers in Base Ten.”  Why didn’t they just title it “Numbers”? Maybe that “base ten” part is significant. It at least implies that there are other number base systems. Maybe we should give them a little more consideration.

Here’s an example of how one teacher taught 3rd graders about number base systems. If you want to learn more about working with different bases, check out the Khan Academy series. If you still are struggling, you can always default to a base conversion calculator or just type your request into Google for instant results.

Reflective Questions (Or Include Thoughts In the Comments Section)

  • What would be the benefit of teaching students other number systems?
  • What would concern you about doing this?
  • How would you go about doing it?
  • Where would you see this in the developmental learning process of mathematics?

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