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Sunday, July 7, 2013

Pascal's Triangle, Binomial Expansion, Combinations

Looking at the probability unit (which in my classes will be the second unit we work on in the first trimester), I am trying to think of a way to show the parallels between binomial expansion, Pascal's triangle and combinations.  In previous years much of this was just given to the students.  (its only my second year teaching this and I took cues from what had been done before in many cases).

I'm considering a stations activity, but one in which groups are assigned to a station and work on it on a large sized whiteboard, and then without erasing a thing, they will be switched to another of the stations to see what has been done before and hopefully add to it.  When completed, the groups will post their whiteboards (with 2 additional ones whose use will be apparent in time) for a gallery walk (which is when those two other boards should be completed)

I have individual sized whiteboards that I purchased a couple years back and last year we got the school to get us Home Depot boards which we had cut down to 2' x 4' size for group work.

I see one group, being given the pattern for Pascal's Triangle.  (this should be a group of students who need a lower level introduction to the mathematics we will be doing).  Two other groups will split up the combinations (evens one group, odds the other).  Two other groups will work on binomial expansion of (x+y) with one group again doing evens and the other odds.  Ideally, weaker students will get the chance to rotate from the Pascal's Triangle station to both of the other problems.  Stronger students should be ok with only seeing Pascal's Triangle from in the Gallery walk.

I see the groups set up as 1. Pascal's Triangle (weaker), 2. Binomial expansion evens,  3. combinations odds,  4.  binomial expansion odds,  5 combinations evens

Thoughts:  I like the notion of being able to differentiate the work, with weaker students getting a view of all three procedures.  I like that I'm not just showing them these things and doing a "taah-daah!"  I like the idea of giving my strongest students the chance to practice distribution on a bigger problem, and have it not be busywork.  I worry that I'll just get people putting (x+y)^4 = x^4 +y^4 .  I'm thinking that showing the parallels for 7 rows should be sufficient and provide the students with enough practice of the skills I want.  I have a germ of an idea about having students "see" the differences between permutations and combinations using differently colored beads.  Maybe I can use something from that idea (which should come before this one) to flesh out the work being done by the combinations groups?

I have been trying to get the students to produce something personal to go along with these kinds of explorations (otherwise they don't remember the reason for the activity a week later)  This is still something I haven't really done on this idea yet, though I'm sure I'll think of something or something will be suggested.

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