Distance and Midpoint formulas

 Distance and midpoint can be difficult to teach to students, especially ones who do not live where there are distance signs or city blocks.  Out here everything is one hour by air, or about 20 minutes by boat, or maybe an hour by snow machine.

The town has two stop signs I think but no one really pays attention to them.  So when it comes time to teach the distance formula in Geometry, it becomes very difficult.  Even the city does not have any real city blocks so I can't even use that as an example.

Often times, I bring in a map of Anchorage, the closest city to me with real blocks, and copy a part of it onto paper with a grid over it.  I do this by making a copy of the map, then making a second copy of a grid from an overhead on top of it.  Voila, I have a wonderful grid.

I have the students use a marker to label the left bottom corner as 0,0 assuming map is only the first quadrant.  Then they label the lines as 1 to the end on both the X and Y axis so we can create ordered pairs to use in calculating the distance. 

The first time I have to find two intersection on the map before discussing how we'd find an accurate distance between the two points.  They often suggest we count blocks but I ask how do they count accurate blocks since they've suggested we go down and over or diagonally?.  This leads to a nice discussion on how could they do it.

Eventually, I lead the discussion to the Pythagorean Theorem and its application by having them create a triangle from those two points.  this leads to how it can be applied with points, on to the formula itself.  It helps them see how the formula works.

We do this for a few places in Anchorage in terms of going from Walmart midtown to Freddy's near the airport.  It really gives them something to relate to.  Once they've got it down, I move them to a regular grid with points to have them practice the distance formula.

As for midpoint, I get out a ruler for them to measure the distance between the two points on a general sheet of paper.  I ask them to locate the midpoint on the line and then ask they explain their method to me.  I get a lot of "Duh's" from them but it works well.  I extend this to a coordinate grid by having them find two points on the line, connect the points with a line and then they measure the line to find the midpoint.  They mark the midpoint on the line and find the coordinates.

I ask them how we might accomplish this without using a ruler, or pencil and just find a formula for the midpoint.  Eventually, they get to the formula which we add to their notes.  This can take one or two periods to complete but they have a better understanding of the concept rather than just knowing the formula.

Please let me know what you think.  I hope you have a great day, I'd love to hear from you.