In addition to working for the National Alliance on Mental Illness in Georgia, I am also a math tutor. I work mostly with students in Algebra and Geometry. Some of my students are in middle school and some are in high school. The talent levels are pretty much all over the map from those who lack some of the basic fundamentals to those who want to look ahead at what is coming next. So, sometimes I am using a number line to help a student understand subtracting negative numbers and sometimes I am discussing unit circles, derivatives, and integrals.
Today, I was working with one of my students who is having difficulty with algebra basics. He had a worksheet with four problems. The first two problems required the use of completing the square to solve quadratic equations while the second two problems required the use of the quadratic formula to solve similar equations. After explaining the process to complete the square and having him work on some problems, we moved on to the the quadratic formula. After working out a couple of those problems, I showed him that the same solutions can be found using either technique. Upon seeing that, he said that he didn’t understand something. I asked him what I could help him with and he said “I don’t understand why there are two different ways to get the same answer”. I tried to explain that the quadratic formula could be used practically any time while completing the square can be difficult if the “b” term is odd and if you don’t like fractions. And, if the “a” term is not 1, it can get more complicated. I could see I was not getting through to him.
Then I remembered that he told me he played soccer. So I tried this approach. If you have a 2 on 1 break, do you always use the same approach to try to score a goal? If you are the goalie, do you always play the 2 on 1 break the same way to prevent the a goal. If you are playing against a very good defensive team, do you play the same way as you would against a very good offensive team? It depends on the situation.
There’s more than one way to skin a cat. It depends on the situation. The trick is to be able to identify the situation and apply the best solution. Sounds like a good idea for life.