We did not get through all I intended today to allow some students who wanted to watch the inauguration that opportunity. But we did address a couple of more questions from the 2-minute reflection students had completed. You can look back to the previous post to see the original task.
We addressed the two blue questions in the after lunch class. Why can’t you multiply the numbers by each other? Well, lets see. Again, as I did with another class, I asked them to add two numbers that would give us 18. We graphed our responses, then graphed the equations x+y=18. And likewise, give me two numbers that would multiply to give us 18. We graphed our responses along with xy=18.
When we added the equation to the product set, students were caught off guard with what they saw. WHY is there a graph in the third quadrant? Will that red curve ever cross the y-axis? Doesn’t it eventually get to the axis? Again, just attempting to address their question, by looking at a couple of horizontal translations and introducing them to that boundary line called an asymptote, led to even more wonderings. Which is what math class should be about. As long as they were on task, I continued to go with their questions. Only 3 students were not interested, who would likely have been off task no matter what I chose to do, so I made the decision to keep going with the majority’s curiosities.
Another student asked about our statement “x cannot be in the denominator” but yet when we find rate of change with a table of values, we compare y over x. Hmmmm. Good question. So I gave a table of values, asked the student to talk me through finding rate of change. When we wrote our ratios, what values did we use? Not the actual y and x values, but the change in y over the change in x.
The entire class really reminded me that we can say something with an intent, but what our students hear is something else…how important communication is, how important is it that we allow some time to process and clarify their misunderstandings.
Finally, we addressed the question, we’ve been told x’s exponent must be 1 in the linear function. We’ve seen greater than 1, but if it is less than 1, can it still be linear? Let’s see. Go to y=, type in x and choose an exponent less than 1. What do you see. Share with your neighbors. So, how would you respond to this question, students?
To me, this was one of the most productive two days I’ve had in this class. Students were engaged because we were addressing their questions. I’m not sure I actually answered their questions, but I provided them with some examples that allowed them to answer their own questions.