This week’s PreAlgPoW, which asked students to find a pair of integers that would satisfy four equations at once, had a lot of guess and check solutions. That got me wondering… What do we want to know about guess and check solutions? What can we learn from comparing and studying different guess and check solutions? Are there different ways to guess and check? Are some more elegant or effective or efficient?

Another thing I got to wondering was, if I were a classroom teacher and the many students who submitted guess and check solutions were in my class, how would I orchestrate the class discussion on the solutions? Thinking about Peg Smith and Mary Kay Stein’s 5 practices for orchestrating math discussions (you can read more about them on Christopher Danielson’s blog), I wondered, “Which students would I select? In what sequence would I ask students to present their work? What connections would I help students make?”

Here are six solutions I selected that I thought made an interesting guess and check story:

First I looked at all the equations to see which one would have the least possible answers. That equation was 3Xquadrilateral divided by 8Xcircle. I knew this was the best one to choose because I just needed to find the ricipricol, which was 8 over 3. After I figured out the answer to that equation I just plugged the rest in to each equation.

I got my answer by finding the factors of 48 and testing a pair of factors for each equation until I got to the last number sentence and realized that I could’ve gotten my answer by just looking at the two numbers on the fraction.

I got the answer by looking for number that can be subtracted by each other that will equal -5. I found many answers but then I was stuck with one answer, 3 and 8. So I checked it with the other questions. They all matched. So I finally checked it with the first question. Then it was correct. The answer would be 3 and 8.

For this problem I first knew that for the last problem, the numerator and the denomenator had to be the same number. To start I used the LCM of 3 and 8 which was 24. When I plugged in 3 for the circles and 8 for the quadrilateral, it solved all of the problems so I knew it was right.

I got my answer from guess and check. i took the problem below the first and tried Circle=1 and Diamond=6. i did that because 1-6=-5. i put it into the top and saw that it did not work. i tried C=2 and D=7, but it did not work either. i tried C=3 and D=8 and it worked on the first, second, third, and fourth problem.

I notice that the circle is larger than the quad. First, I substituted 2 and 3, which made no sense. Then I tried 10 and 15,  which still didn’t work. then, I looked at the bottom, and found the obvious answer. Circle is 3 and quad is 8.
(1*3)(2*8) makes 48.
3-8=-5
8sq+3sq= 73.
3*8
—–  = 1
8*3

Below are my thoughts on one possible sequence and connections story. How would you order these samples? What connections do you notice?

I would start, in this case, with the student who communicates the least, to avoid having his or her thinking overshadowed.

I got my answer by finding the factors of 48 and testing a pair of factors for each equation until I got to the last number sentence and realized that I could’ve gotten my answer by just looking at the two numbers on the fraction.

I notice that the student described a systematic strategy of listing factors and testing them all. It sounds like they may have been frustrated by only later having an insight that could possibly have saved them a lot of testing.

Next I would invite the person who seemed to be the least systematic:

I notice that the circle is larger than the quad. First, I substituted 2 and 3, which made no sense. Then I tried 10 and 15,  which still didn’t work. then, I looked at the bottom, and found the obvious answer. Circle is 3 and quad is 8.
(1*3)(2*8) makes 48.
3-8=-5
8sq+3sq= 73.
3*8
—–  = 1
8*3

I notice they kept track of whether their answers made sense, and showed the calculations confirming their final answer. I noticed they guessed low numbers, then higher numbers, and finally moved to focusing on one specific clue. Both students now have mentioned the last clue, so I would invite this student up to talk about the last clue:

First I looked at all the equations to see which one would have the least possible answers. That equation was 3Xquadrilateral divided by 8Xcircle. I knew this was the best one to choose because I just needed to find the ricipricol, which was 8 over 3. After I figured out the answer to that equation I just plugged the rest in to each equation.

It sounds like this student focused on the last clue on purpose and it’s neat to hear their reasoning for that. The next student might make us wonder about some assumptions:

For this problem I first knew that for the last problem, the numerator and the denomenator had to be the same number. To start I used the LCM of 3 and 8 which was 24. When I plugged in 3 for the circles and 8 for the quadrilateral, it solved all of the problems so I knew it was right.

Why is the LCM an important idea to point out? How is this student’s work different from the previous student’s? Is there anything that might have tripped up the previous student if this problem had been a bit harder? I would hope students might notice the slight difference, but if not I might ask questions like “what did you mean by ‘to start?’” and “did you think 3 and 8 were the only pair that would work for the last equation?” And have both students address the question, “how many solutions can the last equation have?”

Once the last equation has been explored a bit, I’d like to hear about this solution:

I got the answer by looking for number that can be subtracted by each other that will equal -5. I found many answers but then I was stuck with one answer, 3 and 8. So I checked it with the other questions. They all matched. So I finally checked it with the first question. Then it was correct. The answer would be 3 and 8.

And this one:

I got my answer from guess and check. i took the problem below the first and tried Circle=1 and Diamond=6. i did that because 1-6=-5. i put it into the top and saw that it did not work. i tried C=2 and D=7, but it did not work either. i tried C=3 and D=8 and it worked on the first, second, third, and fourth problem.

I’m interested in whether the students did the same things and described them differently, or if they had different processes. I’d be curious to hear from the audience which they found easier to make sense of. I’m even more curious to learn more about the choice to focus on the second equation.

Some themes I hope might emerge:

  • Looking at guess and check, we’re often curious about where you decided to start guessing and what you guessed first, next, etc.
  • Learning about what you guessed helps us know the ways in which you were systematic, and maybe even helps us think about the structure of the problem.
  • Looking for shortcuts and efficiencies is valuable, and different people see different advantages to doing the problem different ways.
  • Assumptions happen in guess and check. What ways of explaining your work make assumptions visible? Why are assumptions important?

Eventually I’d like my students to consent that they will tell us about different guesses they tried, any thoughts they had about where to start guessing, and how they know they are right in the end (including how they know they found all the possibilities).

What are your goals for the guess and check strategy?

Some “Integer Images” links in case you are interested:

Some more “Five Practices for Orchestrating Mathematical Discussions” links: