Teach 180: Know Your Scale (Day 137)

When I am preparing my lessons and have an "Ah Ha!" moment, I often work that "Ah Ha!" moment into the next lesson.  This blog describes such an event.  Today students were working on areas bounded by curves.  One such problem had students finding the area between y = x² and y = x² + 3.  I was working with an individual student and was having trouble showing the area of interest with Desmos. After a minute or two, I got the correct graph by creating a compound inequality to graph the region between the curves and then using a compound inequality to restrict the domain.

After class was over, I realized it would be good to have all my students create this for themselves in desmos.  I still had the page open in desmos and decided to use desmos to find the area between the curves with a definite integral.  At first I was confused.  How can the area be 3?  There are way more than 3 squares shaded.  Is Desmos not recognizing something I typed?  Did I do something wrong with the set up?

Since I was really tired, I decided to work on something else for a while and come back to it.  I often tell my students that staring at something won't make you understand the problem.  Sometimes we need to leave a problem and come back to it later with a fresh set of eyes.  That's what I did.  Based on the title of this blog entry, you may have figured out what I was missing.  What is the scale here?  Each square is not 1 unit long, but 1/2 unit long.  This means that the area of each individual square is 1/4 of a square unit.  Each group of 4 squares is 1 square unit!  Now the answer made more sense.  When I present this to my students tomorrow, it will be interesting to see how quickly they note the scale in the graph.