Clapping Games and Multiples

Last week was the first week of the new school year at da Vinci -- and my first year as a math teacher here. This year, I am teaching Common Core Math 6 and Math 8, as well as a small section of high school Geometry for eighth graders who took Algebra last year before the school switched completely over to the Common Core math class structure.

For sixth grade, we start with factors and multiples, so as I gathered getting-to-know-you activities for the first week, I had those topics in mind. For eighth grade, we spend the first few months on algebra, so I wanted to find patterns we could talk about -- especially non-visual patterns, since we will do plenty of those already.

I've been connecting with other teachers on Twitter, and someone retweeted a link from @iPodsibilities to this video with this clapping game:

"Aha!" I thought, "Non-visual patterns! and factors and multiples!" So in Math 8 and, especially, Math 6, we worked with this game (through about 2:02 on the video) over the past few class days in between things like learning the bathroom pass system and passing out textbooks.

The students were very focused in learning the game -- I was amazed how many could do the whole thing after only a few times through it, and even those who took longer (like me) were willing to keep at it till they got the hang of it. Perseverance, hooray!

Eventually in all the classes we described the patterns in "Sevens" something like this:

1. slap slap slap slap slap slap slap
2. slap clap slap clap slap clap slap
3. slap clap snap slap clap snap slap
4. slap cross slap clap snap clap slap

We first explored questions like these:

When we do "Sevens," what do the four patterns have in common?
What is different among the patterns?
Which one is hardest, and why?
Where do you see repetition?

Students in every class noticed that each pattern has more different moves than the one before; each pattern begins and ends with a slap; and patterns 1-3 repeat some moves in the same order. (Note: We did only the basic "Sevens" game, not the extra pattern mentioned at the end.)

Then we started exploring what other numbers besides 7 would give the same kind of behavior for these four patterns if we kept repeating the same moves in the same order, especially the beginning and ending with a slap. We said if this happened, the number "worked" for all four patterns. For instance, we tested 9:

9 with pattern 1: slap slap slap slap slap slap slap slap slap (works)
9 with pattern 2: slap clap slap clap slap clap slap clap slap (works)
9 with pattern 3: slap clap snap slap clap snap slap clap snap (doesn't work, because you didn't end with a slap)
9 with pattern 4: slap cross slap clap snap clap slap cross slap (doesn't really work, because you stopped in mid-cycle as you repeated the moves)

Students in every class found at least one other number that "works" for all four patterns. Can you? We also talked about what kinds of numbers "work" for the second pattern and why (hint: half of the natural numbers work).

Today, for the sixth graders, I reproduced a way of writing out the four patterns that students in some classes came up with:

1. slap slap slap slap slap slap slap
2. slap clap slap clap slap clap slap
3. slap clap snap slap clap snap slap
4. slap cross slap clap snap clap slap

Then I asked them why they thought I did the underlining the way I did, why I wrote the last slap in red, and what ideas they could come up with for the kinds of numbers that would "work" for each pattern.

To my delight, all three sixth grade classes explored these questions thoroughly and in every class, someone eventually mentioned the magic word MULTIPLE... as in, "Pattern 3 will work for any number that is a multiple of 3 plus 1." This led very nicely into a review of what multiples are, which sets us up well for this week's work, which was one my main goals!

We touched very briefly on why numbers that "work" for Pattern 4 also work for Patterns 2 & 3, but that part was hazier for them... which is OK, because after we study common multiples it will probably make more sense.

There was a particularly great math moment in Period 5 when Melody came up with a mind-blowing procedure for finding numbers that "work". She noticed that 7 works, and 13 works, and 25 works. Then she decided, and started proving to herself, that in general, if a number works, you can double it and subtract 1, and you will get another number that works. Therefore, for instance, 25*2 - 1 = 49 works. I could see that the numbers she was coming up with were all multiples of 6 plus 1, so I agreed that each of them worked, but it wasn't till after class that I sat down and proved her method would always succeed.

If you've had a few months of algebra, give the proof a try! (I'll probably sic my Geometry class on this one soon.) Numbers that "work" can be described as 6n + 1, where n is some natural number. Show that if you double any number that works and subtract 1, you'll get another number that works. Isn't that an awesome discovery?