In this post I want to show Exeter’s problem solving strategy. This is important, because it is SO different from how a problem like this is typically approached.
First off, the problem I am going to model is M1:21:11 [Math 1, page 21, problem 11]
11. Alex was hired to unpack and clean 576 very small items of glassware, at five cents per piece successfully unpacked. For every item broken during the process, however, Alex had to pay $1.98. At the end of the job, Alex received $22.71. How many items did Alex break?
In a typical Algebra 1 class we would try to get the learner to see the equation is:
.05(576x) + 1.98x = 22.71
In fact we try to get the learner to jump directly to the equation from the problem by deconstructing the sentences, and then solve the equation. x = 3, by the way.
Now, let’s see how Exeter expects and demands that ALL of the modeling problems are handled.
First off, we will be making a table. The headings in this table are mandatory and can not be short cut. The learners must label the table thoroughly so that it makes sense. Remember, this is the same problem as above. I am going to paste in my table all filled out, and then explain the essential elements.
Guess: # of broken bottles

# of unbroken

$ Paid for unbroken

$ subtracted for broken

Amount paid

Goal

Check

0

5760=576

.05(5760)=28.8

(0)(1.98)=0

28.80=28.8

22.71

no

5

5765=571

.05(5765)=28.55

(5)(1.98)=9.90

28.559.90=18.65

22.71

No

3

5763=573

.05(5763)=28.65

(3)(1.98)=5.94

28.655.94=22.71

22.71

YES!

B

576B

.05(576B)

1.98B

.05(576B)1.98B =

22.71


Okay, there we have. A decent example of what a modeling, problem solving solution would look like. At the beginning stages of Math 1, they would not demand the last row, the equation row. But quickly they would ask the learners to start generalizing their solution.
The guesses column are not set in stone. The guesses are going to be the learners guesses. They are going to guess whatever they want. I started with 0, because maybe he didn’t break any. Then I saw that was too high to my goal, so I figured Alex broke a few. Then I was too low, so I picked one in the middle.
Now, let’s examine what the columns mean. It is clear from the headings that each column has a very specific purpose and is clearly labeled. What are we guessing? We are guessing the number of glasses he broke. If he breaks 5, then he didn’t break 571. How do we get that, we subtract. Each column must have in it HOW they get the number, not just what the number is. And so on.
Notice that by the time the learner reaches the answer, they have worked several times the process, they know the multiplications, the subtractions, and they have the solution worked out. Where does the variable go? It goes into the spot where numbers change. What do we call the variable? Don’t care, use a letter that makes sense to the problem.
How do they start this process? The first problem that is a modeling / problem solving problem is M1:9:4. It looks like this:
Notice that they start by giving the table and even filling out the first row. The problem I worked above, didn’t have that level of detail. The learner had to provide it. That is the point.
EXETER MODELS AND LADDERS THE LEARNING UP TO THE LEVEL THEY WANT.
Yea, I shouted that. We have this impression that Exeter is so fabulous, that they don’t have to ladder or work with learners. We think that the learners just will magically go *poof* and be able to do all these things that we struggle with.
Guess, what, they struggle with similar things there as we do in our schools. It might be easier because of smaller class sizes, but the root problems are the same.
Okay, off my soap box.
The Algebra 1 activities have some problem solving activities, and they even are sneaky by giving a blank table with fewer columns than the learners need! The learner is pushed to make the table for themselves.
Think about this type of problem solving for special ed, or EL Learners. They have the numbers set up, they can see where the Letter for the Unknown goes, because it is the only number that changes when they are doing the problems. Wouldn’t this method help them out so much?!
Think about your average learner who struggles with parsing the language of the problem. If they work 10 or 20 of these as starters, as homework, as in class activities, do you really think they are going to stress about a word problem?
Nope, they are going to say, “Mr. Waddell, these are easy, can we move on to something harder?” And you know they will.
Think about the really advanced learner. They are going to resent the table after a short time, but they will go to the generalization much faster because of it.
Can you think of any downside to this method of problem solving? I can’t. I have done Algebraic Thinking’s “SOLVE” method, and other methods. None of them are as straightforward and easy to put together as this method. We could spend THOUSANDS of dollars on professional development on problem solving, and none of that money would come close to the success of just creating a table, labeling, and working it out step by step.
Guess and Check. That is what Exeter calls it. I call it just downright successful for every level of learner.