More Puzzle Extension: Expressing n as the sum of consecutive integers
This is a puzzle for you, and a description of what I actually did with some kids.
How many ways can n be expressed as the sum of consecutive integers?
The descriptions will come from actual classroom experiences from this Fall. The class is an elective “Combinatorics” with mostly seniors. Almost half are enrolled in calculus, some in precalculus, a few in trig, and some are not taking other math and are using this elective as their last math credit. The text I use is Mathematics of Choice by Ivan Niven. It’s from the MAA’s New Mathematical Library series (mid-1960’s) which includes about 30 titles, including Yaglom’s 3-part Geometry, which wonderful puzzler Tanya Khovanova blogged about last month.
The course meets 4 days each week, one of which I reserve for games or contests. Of the teaching part, it is a mix of lecture (which always includes some dialogue with the students) and “problem solving” in which I pose a problem that I believe will take a while (“while” being loosely defined) and the students work in small groups.
I introduced ‘Polya-style’ problem solving ……at the beginning of the term, and enforce it in three ways: 1. the regular problem solving classwork, as I just described, 2. problem “write-ups” after Polya, preferably with explicit reference to planning, mistakes, replanning, and some sort of “looking back” discussion, one due every 2 or 3 weeks, and 3. A non-routine problem on each test, which must be attempted or solved using the same problem-solving style. There is an unfairness here: how can I ask a contest-style question as a test question? I handle this by 1. manipulating expectations – running through steps at least as far as devising a plan and carrying it out, even if the plan does not lead to a solution, has a reasonable chance of earning half credit, and 2. by manipulating points. The exam has 90 points of more routine work. The puzzle/problem is worth 20 points. Is it wrong to allow students to score over 100? Maybe. But I have not very confident mathematics students who are now willing to attack problems, even when they don’t know what to do. I claim that this is real payoff.
So, that’s the backdrop. The math is both serious and different from what they do in regular classes. The kids are a mixed group. The style is problem-solving.
Foundation – How many factors does 72,000,000 have?
Three weeks earlier I posed “How many factors does 72,000,000 have?” After allowing them to begin creating lists, or not create them because it just seemed to annoying to bother, I interrupted and pointed them towards examining smaller numbers.
Before any could make progress, I began a whole class discussion, organized around a chart on the board. I prompted them to examine 2, 4, 8, 16, 9, 125 and other powers of a single prime. It took just a moment for several to discover and share with all that has k+1 factors. I forced one of them to tie it back to the “+1” in the answer to “How many numbers are there from 10 to 20?” — in this case they were counting from 0 to n.
I played the same game — my prompts, their work — with a load of semi-primes. This time a few kids were even faster discovering that pq has 4 factors. I got them to list the factors for pq: 1, p , q, pq and I organized the list in a square, set them to work on 12, 20, 30, 100 — and you could hear the generalized rule emerging at table after table, like popcorn popping. Then a lull. Then some kids remembered we had a question about a particular number (72 million), but getting the answer was sort of anti-climactic. The big work was already done.
Setting the stage: How many ways can 1000 be expressed as the sum of consecutive integers?
The week before the big problem, they attacked this. Some nice insights jumped out. For example, several groups noticed that if one solution was 198 + 199 + 200 + 201 + 202 that another must be -197 + -196 + … + 196 + 197 + 198 + 199 + 200 + 201 + 202, and that the answers would pair, and the number thus must be even. We had some nice discussion of whether to count 1000 itself as an answer (I said yes), and how to search. They noticed that dividing by 2 was useless, but led to a sequence. I pushed them to see that also led to a sequence.
The discussion went well. Also, for those of you who like this sort of thing, two groups finished in what I considered too short a time, so I lied and told them I thought that there answer was too small. Now, they know I lie, but they also know that I know a lot, so they have become more used to responding, “we think we are done because….” which I consider a good thing. I don’t want them to stop because I say enough, but rather because the mathematics suggests that they have finished.
The big day: How many ways can n be expressed as the sum of consecutive integers?
I let them move into the groups they were used to, and then I pulled a surprise. I removed one friend from each group, and moved them into a “foreign” group. I was in a room with large board space, and I labeled areas for data, for insights, and for a solution. I gave them the problem, and I engaged myself in conversation with a colleague or busy work; it was my intention that they could have up to almost a full hour. And I watched the board.
The data that was going up, numbers with the number of ways they could be expressed, was accurate, but not very interesting. The groups that occupied themselves with getting lots of data apparently did not spend much time looking for patterns. The insights that went up seemed a bit more interesting. One group wrote that could only be written as the sum of consecutive integers in 2 ways. And then a couple of groups started muttering about odd factors. There were about 20 minutes left. I ignored that good progress seemed to have been made, and announced that all the “foreigners” should go back to their natural groups.
Now, during a problem solving session I might interrupt to get groups to share progress so far, or even just general approaches, or plans. That way lagging groups get to participate further along in the process. But I did not want to facilitate this time. So my ‘refugees’ played the same purpose. The groups that had made little progress received a messenger, and the refugees who had not made much progress joined groups that were ahead.
And then we were almost done, and could not finish. They all were fixed on the odd factors. Twice the number of odd factors. And they were right. But they had, collectively reached the end of their mental effort. In the last 10 minutes I told them that I would give them a grade for the day, one grade for the whole class, but to maximize it they needed to tell me how we could know, looking at n, how many ways it could be expressed as the sum of consecutive integers. I lifted the restrictions on where they could sit. They coalesced into two large and one small group. And they fell short.
When we looked again, the following Monday, it took less than five minutes for them to make the jump.