I encourage students and teachers to explore multiple ways to think about and solve problems. I believe that teachers should not necessarily hold back from questions that professional training suggests “belong” in another course or require skills beyond what we think might be required. Being able to struggle with a problem that is just beyond our reach gives us opportunities for joy, inspiration, creativity, exploration, and mathematical insights. By sharing “stretch” problems with young people, sometimes I learn (or relearn) strategies that my mind might not have seen because my teacher experiences and math studies have trained me away from considering some approaches.  

I had a fun time with my fifth-grade daughter a few evenings ago exploring this problem from NCTM’s “Figure This!” (http://figurethis.org/challenges/c58/challenge.htm) —a few years before her first formal algebra class:

 My daughter dove right in. After a couple of minutes, she had her first answer. Her first attempt—which I would not have considered—was strategic guess-and-check, a super strategy given the math she knew. Her first guess was that the yellow fish weighed 5, making the blue fish from the first scale 1 and the green fish from the second scale 2. But combining those two made 3, not the 5 needed by the third scale. She lowered her guess for the yellow fish to 4, recomputed, and had her solution. 

That’s when I reminded her that a good answer to every math problem always does two things:  

(1) It shows that the given answer is correct; and  

(2) It shows that there are no other answers.  

My daughter knew that she had a good answer but wondered how she could know whether any other weights would work. A teaching moment! 

I asked what would happen if she combined the two scales. She said that the last two together would have one yellow, two green, and one blue fish and would weigh 12 in total. Removing from this the first scale’s information left two green fish weighing 6, which meant that one green weighed 3 and that the other two fish weights could be found from the last two scales. She was now convinced that there was only one solution. 

As a final approach, we combined all three scales to get six fish—two of each—weighing 18 total, from which she knew that a triad of one of each weighed 9. Removing the fish shown on the first scale from this left one green weighing 3. Removing the second-scale fish from the triad left one blue weighing 2. Removing the third-scale fish from the total left one yellow weighing 4. 

My daughter thought that it was really cool that all three strategies worked out to give the same answer even though they approached the problem differently. Joy! And I was amazed that we had solved some decent algebra without ever resorting to formal notation. Inspiration! 

CHRIS HARROW, casmusings@gmail.com, a National Board Certified Teacher, is the mathematics chair at the Hawken School in Cleveland, Ohio. He blogs and presents nationally on the educational uses of technology and on Computer Algebra Systems (CAS) in precollegiate mathematics. He is also the recipient of a Presidential Award for Excellence in Mathematics and Science Teaching.

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