In years past, I’ve usually taught Midpoint and Distance on the same day or at least on consecutive days. After a reminder of some brain research last fall – how our brains store information by similarities but retrieves information by differences – I decided to try things in split them up this semester – hoping to lessen the confusion students often face (do I add or subtract with midpoint/distance formulas?). Again, this confusion stems from teaching a procedure without paying close attention to in-depth student understanding.

I chose to introduce “Midpoint with Coordinates” the same day we were working with segments, bisectors, midpoints of segments. No bells/whistles here – just the basics

I gave students a grid index card and the points A(2, 1) B(8, 11) and C(8, 1) to attach to their INB RPH. Simply starting with locating the midpoint of the AC and BC. But also asking them to compare/contrast the coordinates of ACE and BCF each time.

Finally, asking them to locate G, the midpoint of AB. Walking around the room, it was quite fun watching the various strategies. The great thing was asking students to share their different strategies. One used rise/run, several “counted diagonals” from A and B until they got to the middle, one used the midpoints of AC and BC and traced up from E / over from F until he found where G was located. After discussing methods using the graph, a student stated “*I just added my x’s then divide by 2 and added my y’s then divide by 2.” * When discussing how the coordinates were alike/different, a student asked “Isn’t that, what C____ did? Just averaging the x’s and averaging the y’s?”

So, I never actually gave them the “Midpoint Formula.” Awesome. Of course, we went on to practice the skill a few times. I also chose 8 questions from Key Curriculum’s Discovering Geometry (did I mention, I *LOVE* this book?!?!? And have since the mid-90’s!) – that required a bit more thinking beyond skill/drill. Two questions that led to some great discussion today was:

Find two points on segment AB that divide the segment into three congruent parts. A(0,0) and B(9,6). Explain your method.

Describe a way to find points that divide a segment into fourths.

But in class, I offered another – **what about if I need to divide it into fifths?** Students worked individually, pair-share – then class discussion. Quite different approaches. I loved it.

What was even better, a student asked, *“But the examples we’ve used all have an end point at the origin. Will it still work if the endpoint is not at the origin?”* Aaahhhhhhhhhhhh! That’s music to my ears! Wow. Wow. I love it. I love it. I love it.

This is a nice little open question to share with your students. It definitely allowed me to see student understanding of the task by their work / responses / discussion.