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Elementary Problem Solving — Making Generalizations

October 1, 2007

I have recently acquired Marilyn Burns’ excellent book 50 Problem-Solving Lessons (Grades 1-6) and have been using it to give me some ideas for the two elementary groups I work with.

The older of the two groups is for kids who have mastered things like addition and subtraction with regrouping (aka, carrying and borrowing) and are ready to move on to things like multiplication, division, fractions, etc. (Many of these things are introduced at some level even in the younger group, but this upper elementary group is where they really concentrate on these types of skills). I’d estimate that the “grade levels” represented in this group span about 3rd to 5th grades.

With this group, I tried an activity called “Making Generalizations” from the Burns book. In this lesson, the idea is to write several lists of four consecutive numbers and ask students to find things that are always true of each list. For example, if you subtract the first number from the last number, the difference is always 3.

Since this was my first time working with this group, first we talked in general about what kind of properties numbers might have. They can be even or odd. A number can be described as a multiple of another number. They can be prime or composite. (I did have a couple of kids in this group who knew what prime numbers were, and explained this to the rest of the group.)

We had a side observation that 4 = 2 + 2 = 2 \times 2 = 2^2 so we had a short discussion of the fact that those were really all different ways of saying to do the same thing.

As I wrote the lists of numbers, we talked about what consecutive means.

Then I had the kids break up into groups of 3 and work on finding generalizations about the lists of four consecutive numbers. Here are some things that they came up with:

  • subtracting the first number from the second always leaves 1
  • adding the first and last numbers always results in an odd sum
  • adding the first and last numbers always results in a prime sum (it wasn’t true, even with the 4 or 5 examples we were working with; the one counter-example was 39)
  • the sum of all four numbers is always even
  • the sum of the first and the last number is the same as the sum of the second and third number (setting the scene for a simplified discussion of the Gauss method of adding lists of consecutive numbers)
  • the sum of the first and third numbers is a multiple of the second number (we figured out in discussion that the multiple is always 2x)

The next session, nobody could remember that C-word for numbers that are all in a row. (I wasn’t really surprised about that. I find that learning the vocabulary of problem solving takes a lot of time.) I reviewed the last two observations above, and introduced the notion of average. We figured out, together, that the average of four consecutive numbers was always halfway between the two middle numbers. We figured out that the average of an odd-numbered list of consecutive numbers was always the middle number. Using both the formula for average, and using manipulatives to show that we could transform all of the numbers into the middle number by moving a few counters, we observed that we could add up a list of an odd number of consecutive numbers by multiplying the average (middle number) by the number of numbers in the list. (Since many of these kids don’t know how to multiply multi-digit numbers, and certainly not decimals and fractions, this early in the year, we stuck with low numbers and odd-sized lists.) So, for 8 + 9 + 10 +11+12, we laid out those numbers of counters in piles on the floor, showed how we could take a few from the larger piles to put on the smaller piles and end up with five piles of 10, and that there were therefore 5×10=50 counters all together.

I then handed out 100 charts and had the kids outline four 3×3 squares, strongly suggesting that they put at least a couple of them near the top of the chart where the numbers would be easy to work with. I think asked them to find things that were true of all of their squares, and that they thought would always be true of any such square. (In retrospect, I should have probably chosen the squares, or at least the first two, to make sure they weren’t all starting on numbers of the same parity.)

I was pleased that some of the kids added up the rows using the method we had just discussed!

Some of the observations they came up with:

  • the number below any number is 10 more than that number, but the number to the right of it is only one more (general observation about the 100 chart)
  • the ones digits of the three numbers in the same column are all the same (another general 100 chart observation)
  • if you take 1 away from the last number in any of the rows and add it to the first, you get the middle number (showing understanding of what we had just discussed)
  • subtracting the top left corner from the top right corner leaves 2
  • subtracting the top left corner from the bottom right corner always leaves 22
  • bottom right + bottom left = twice the bottom middle number
  • the sum of the second row is 30 more than the sum of the top row (and the same for bottom/middle rows)

I’ll have to ask their regular teacher if some of them know how to divide by 9, as I’d like to have them find the average of all 9 numbers in the square (preferably without relying on a calculator to do the math) next time we meet, as I think they will find the result pretty cool.

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