Advanced #Countingcircles

I was reading @wahedahbug‘s blogging about #Countingcircles on the train home from work today and got so inspired I had to come home and write a blog post right away. Even though ostensibly her blog post was about elementary school math, I absolutely love the idea of doing Counting Circles with my students.

I’m always shocked by the fact that my students (MYP 4 & 5 Math Extended = US 9 & 10) can do complicated things and yet fall apart when faced with fractions. How is it possible that they can do things like prove that {2n + 1, n^2 + 2n, n^2 + 2n +1} will always result in a Pythagorean Triplet, but completely balk at 3\times\frac{2}{9}? And to convert 45 degrees into radians, some of them are literally using a calculator to divide 45 by 180 and then sticking pi at the end!

I want to integrate these Counting Circles into my class not only to work on number sense, but to work on more complicated things as well. In the beginning of the year of MYP4 we do a unit on Sets and Venn Diagrams. This is the perfect place to start off with Counting Circles. I’d like them to count by halves, but one time around the circle using decimals (ie 0, 0.5, 1, 1.5…) then using fractions not reduced (ie 0, \frac{1}{2}, \frac{2}{2}, \frac{3}{2}…), then reducing, etc. This will seem easy for an “advanced” class, but then I’d like to count by thirds but my idea is that it when a student gets to \frac{3}{3}, the next student will have to say “which is 1.” The following student will say \frac{4}{3}, with the student thereafter saying “which is 1\frac{1}{3},” etc.

I’d love to make this a sort of routine, because our second unit is Algebraic Expansion and Factorization and I’m really excited envisioning the Counting Circle of Perfect Square Trinomials. The first student will say “(x+1)^2 ” and the second student will say “which is x^2 + 2x + 1,” etc. We can also do this for the Difference of Two Squares — first student: x^2 - 1, ” second student: “(x+1)(x-1).” I like that it will be a routine, students can see the patterns, it really rams things home, and it forces that silent group who never say anything to actually participate in a safe way.

Third unit is Radicals and Surds. I am already picturing the first day going around the circle doing perfect squares. Second day will be surds (with the following student saying between what two integers the surd lies). And then the super exciting day where we do 1, \sqrt{2}, \sqrt{3}, 2, \sqrt{5}, \sqrt{6}, \sqrt{7}, 2\sqrt{2}, 3, etc! Yesssss.

I’m super excited to try this out…will report back in September!


About katenerdypoo

Middle Years and Upper Grades math teacher in the Netherlands teaching both within the Dutch national curriculum and the IB MYP and DP.
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