I had a wonderful DO NOW in my Advanced Precalculus Class that allowed me to have an insight that I haven’t had before.
I asked students:
Sum the following:
For some background, during our last unit, students worked on The Fistbump Problem, and they eventually figured out how to add 1+2+3+…+n.
I wasn’t sure if students would make the connection to that problem and the first sum. A few groups did, and rewrote it as:
And then they did the sum the knew how to (from the fistbump problem), and tripled it.
Others did the “pairing” method where they said 3+300=303, 6+297=303, 9+294=303, … but got a little stuck on figuring out how many pairs they had.
Regardless, they had the appropriate insights.
I love question (b) because even if students didn’t get (a), they should have been able to say the answer to (b) in relation to the answer to (a). So I threw up the answer to part (a) that students had provided, and said: “Okay, let’s say you wanted to quickly get the answer to (b) without doing an arduous process… What do we do?”
Groups had 30 seconds to think and talk.
Almost all saw the answer to (b) was 100 more than the answer to (a) — because you’re secretly adding 1 to each of the terms in (a) to get the terms in (b), and there are 100 terms!
These two questions gave me a whole new way to think about all arithmetic series… as “transformations” of the series 1+2+3+…+n.
If I want to add 5+8+11+14+17+20+23+26+29+32+35, I could do the following
First find out how many terms there are in the sum (in this case, 11)
Sum the series 1+2+3+4+5+6+7+8+9+10+11=66 (this is the one thing kids need to know how to do)
Triple the sum to get the correct “common difference”): 3+6+9+12+…+33=66(3)
Add 2 to each term to get the sum I want: 5+8+11+14+…+35=66(3)+2(11)
I have to think some more about this, but I really think I’m onto a new way for students to look at arithmetic sequences as a transformation of the most basic arithmetic sequence. The transformation is a stretch (multiplication by common difference) and a shift (addition of a fixed value to each term). Whether or not the new way gives students anything better or more useful than the approach I currently use with them is a different story. But I love that *I* had this new insight about something I’ve been teaching for years.
UPDATE: I forgot to mention that I showed it to another math teacher, who said, well, yes, it’s just:
which sort of took the neatness out of it — probably because of the math-y symbols. But then I realized that the thrill I got out of it was the conceptual insight, not the algebraic insight. And I figure students who might be able to discover this on their own will get the same thrill. If you spend a lot of time with the summation sign, what this approach might allow is have students discover the rules of using the summation sign.
I love one mathematical motto of class being take what you don’t know and turn it into what you do know. That undergirds lots of what we do in class. And that is precisely what this approach is doing!