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The typical explanation for modular arithmetic is calling it by another name, "clock-arithmetic", and comparing it to the way the hour value of clocks "resets" every time it has passed midnight.

This is great illustration in Europe. However, in America, notation for time presents the following problems:

  • The numerical hour resets after 12 hours, rather than after a full day
  • You have to deal with AM/PM
  • Time goes from 1 to 12, rather than 0 to 23

For these reasons, I fear using the same illustration with American students would cause more confusion than clarity.

Is there a simple illustration that would show modular arithmetic, that would be more familiar to American students?

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    $\begingroup$ As a response to your main assumption(s) about American students: It is sometimes referred to as clock arithmetic in the United States, too, and I have not heard of confusion arising around AM/PM or the 1 to 12 feature. If you use the term "Army time" or just the "24-hour clock," then students should adapt without an issue. Whether the clock itself is an ideal teaching tool is not for me to say, but I'm confident that this model can work about as well with US students as with non-US ones. $\endgroup$ – Benjamin Dickman Oct 23 '14 at 6:31
  • $\begingroup$ Modular arithmetic is nothing more than finding the remainder in integer division--I assume this is still taught, but maybe I'm wrong. Finding real world examples outside of integer theory is hard for me to come up with though. $\endgroup$ – Jared Oct 23 '14 at 8:19
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    $\begingroup$ I've never noticed the "clock arithmetic" as a problem with students if you are referencing an actual physical circlar analogue clock, which I presume are the same in the USA as they are here in Australia. When you do a calculation like "$10 + 4$", the $10$ is a place to begin and the $4$ is a distance to go (just like with ordinary numbers). If you start at the $10$ position on your clock and go $4$ hours forwards you end up at the $2$ position, so $10+4 = 2$. No student I have used this with has ever had a problem with it. $\endgroup$ – DavidButlerUofA Oct 23 '14 at 12:49
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    $\begingroup$ It starts at $0$ since $12=0$ $\endgroup$ – Steven Gubkin Oct 23 '14 at 12:53
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    $\begingroup$ Well yes @StevenGubkin, it doesn't really matter, you just have a different name for your identity. But the 0 isn't written on the clock is what I was getting at, which might possibly cause students issues comparing to standard modular arithmetic. $\endgroup$ – DavidButlerUofA Oct 23 '14 at 13:08
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What about using it to calculate the day of the week for some future date? I'm just spitballing here but something like...

"Today is Tuesday. Jacob knows that his math test is going to be in 17 days. What day of the week will his math test be on?"

This type of problem is very accessible and can be solved without explicitly using modular arithmetic. I would imagine this to be a good warm up problem just to get students thinking about "wrapping around" and similar concepts. After discussing, you could then ask

"Today is a Friday. Janet knows that her mother's birthday is in 241 days. What day of the week will her mother's birthday fall on?"

which is easy with modular arithmetic, but would be tedious to do week by week. I feel like these are good, contextual examples that might actually be useful for students every once in awhile. Also, the intuitive way of solving these problems, where you would divide by 7, find the remainder, and add that to the current day, is modular arithmetic, which may make it stick more readily. Also, the modular base is a small number, 7, which I have found helps students to understand mod operations easier.

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I've always preferred the module 60 argument in seconds per minute. This is closer to the actual modulo arithmetic than the 12 or 24 hour clock. We might say things like:

30 minutes and 14 seconds

but would never say

30 minutes and 60 seconds

we would say

31 minutes (and 0 seconds)

This same idea also works with minutes per hour.

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I think you could build a lively unit around percussion timelines:

"Perhaps the most quintessential timeline is what most people familiar with rockabilly music dub the Bo Diddley Beat, and salsa dancers call the clave son, which they attribute to much older Cuban and Latin American music."


  ToussaintFig2


Godfried T. Toussaint, "The rhythm that conquered the world: What makes a "good" rhythm good," Percussive Notes, November Issue, 2011, pp. 52-59. (web link)

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    $\begingroup$ As a followup to this, that author has an entire book out with essentially the same title. $\endgroup$ – kcrisman Mar 29 '19 at 12:24
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Casting out nines is a good way to check arithmetic calculations.

Most American students are unfamiliar with casting out nines, but it is very easy to teach the method.

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  • $\begingroup$ The answer would be improved by explaining what is "casting out nines". $\endgroup$ – Tommi Mar 20 '19 at 6:17
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I like the application of angle-measure. You have a few different systems to choose. You can discuss radians mod $2\pi$ or degrees mod $360^o$. But, I suppose you could do some silly thing and invent pizza slice angles where $8$ slices gives you a full circle. So, you can either go $7$ slices or $-1$ slices to get to the same cut. These would be equal mod $8$ slices since $7-(-1)=8 \equiv 0$ (mod 8 slice).

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  • $\begingroup$ incidentally, if the students don't know what angle-measure is, my application has an added benefit of helping to inform. $\endgroup$ – James S. Cook Oct 23 '14 at 12:56
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How about the mechanical odometer of a car (or any other mechanical measuring device that exhibits rollover---restarting to zero after reaching the maximum reading)?

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  • $\begingroup$ A typical car odometer might rollover after 99,999.9 miles. $\endgroup$ – Joel Reyes Noche Oct 23 '14 at 5:06
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    $\begingroup$ And by that you mean "A typical car odometer might be mod 100,000". $\endgroup$ – IQAndreas Oct 23 '14 at 5:14
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    $\begingroup$ It's a good idea. The only potential problem with that illustration (or anything with modulus of a power of 10) is they might see it as "ignoring" or "trimming off" the leftmost digits. $\endgroup$ – IQAndreas Oct 23 '14 at 5:18
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    $\begingroup$ Odometer in my opinion is better for teaching kids about binary numbers stored in computers. $\endgroup$ – Ali Caglayan Oct 23 '14 at 16:02
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One potato, two potato, three potato, four! The game is mod however many kids are playing. Board games with spaces around the edges work too, though they have much larger mod values.

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