It is the beginning of the year in Algebra 1. Some students have completed Algebra 1, but still have some gaps in their understanding. Others have never had Algebra 1. The homework problems last night were very "traditional" (think Dolciani) and asked students to write equations for various consecutive integer problems. Somehow we came up with the following equation as we reviewed the homework.
In other words, the difference of the squares of two consecutive integers is equal to the sum of the two integers. I had a student suggest a number for n and we saw that the equation was true. We then picked a different number for n and saw the equation was still true. This led to a student asking, "Will that always work?" At which point I got math-geek goosebumps (a proof in an algebra class!) and quickly expanded the left side and did some simplification and answered the question with a "Yes, it will always work." As I looked up, I could hear the crickets chirping in the silence and the deer-in-the-headlight stares of my students.
In my excitement, I had forgotten that my students didn't know what it meant to square a binomial, combine like terms or get a solution to an equation that was an identity. So, now I am going to rewind this lesson and start again.
Student: Will it always work?
Me: I don't know. How could we figure out if it always works?
Another Student: We could try different numbers.
Me: That sounds like a good idea. Turn to the person beside you and pick a pair of consecutive numbers. Then do the calculation of the difference of the squares and compare it to the sum. (Demo with two numbers suggested by a third student.)
Students work together and start to think that it will always work.
Me: Do a few examples show that something is true? We have only looked at a few cases. How can we know it is always true?
Student: You could look at lots of examples.
Me: But how many examples would be enough? Let's look at something simpler. Is 2(x+1) = 2x + 2? Always? How do we know?
And so on...You get the idea. The point of this blog is that I flubbed it up. But I recognized it and have thought about how I would do it differently. I can already tell by the questions students are asking this year that they have more background knowledge and more of a mathematical disposition than I originally gave them credit. This could make for an interesting year!