At this point I realized that we should probably do an example using the formula. I crowd sourced values for the coordinates and this is what we got:
The issue of x1 - x2 versus x2 - x1 came up naturally and one pair of students said that it didn't make a difference as they had tried it both ways (yay!). So I asked why. It took a little digging but we got there eventually...
A student also said that their calculator gave a different answer which led me to talk about simplifying roots a little (this is in the grade 11 curriculum in Ontario).
Then we moved on to finding the midpoint.
It is amazing how "trained" our students are - we just worked out the distance formula, so I must need to use that! About half of the class worked out the distance between the two points and divided that by 2. I asked what they had found and whether that was the midpoint which caused them to stop and scrunch up their faces a little. I let them work on it a little longer before I put this up:
Some immediately made the connection to finding the axis of symmetry given the zeros of a parabola.
We consolidated:
They worked through a simple example and a student wrote their solution on the board which we, as a class, improved. The mathematics was all correct, but it looked like magic numbers on a page. I stress the fact that solutions should be written such that anyone in their class can follow from the beginning all the way to the end.
Then they tried this one:
There were many different approaches used to solve this (two shown above) along with many students having no idea where to even begin. My response to them was "You have a midpoint formula - which parts of it do you know?" or something like that. We will definitely have to do a few more practice questions like this.
It was interesting to note that a number of students found the difference between the values and added/subtracted that to the midpoint to find the endpoint. I think I will try to incorporate that into a homework set somehow.
They got the last 10 minutes to work on this solving linear systems review in preparation for tomorrow's quiz. They also got this for homework.
When I did midpoints a couple of weeks ago, I tried to do it by investigation also, for the first time. I had them think about what was halfway between two numbers on a number line, which we then extended to horizontal and vertical lines in the plane, which then we extended to slanted lines. They were able to see the method right away, which was good, but I'm trying to decide if it was too heavily scaffolded (I did something similar with distance the following day). Do you think because you did quadratics already and they have found the axis of symmetry before, that that helped?
ReplyDeleteI am not sure that it helped, but doing midpoint consolidated finding the axis of symmetry, if that makes sense.
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