Day 9: Describing Pairs of Lines

Ok, I'm going to make a rule for myself and this blog. I'm not going to describe difficulties that arise from student behavior if I think it's unrelated to the modeling format. I say that because things appear to be going well in my two honors classes, and failing miserably in my regular class. 

Today's goal: get students to make connections between things they already know about lines (parallel, perpendicular, etc) and what those concepts actually mean. That might be my single biggest surprise teaching math - even kids who have been successful in their math careers generally have little to no idea how to explain math concepts without using math vocab. In other words, they can't apply math to non-obviously-math scenarios. 

I had 2 volunteers stand at their own starting points (taped on the floor). I then asked a random student from the class to make up a slope. Surprisingly, most responses came of the form "2, 6" (not 2 over 6), so I used that chance to reinforce the idea that slope is a number that indicates the direction of a line. I then asked the volunteers to walk in that "direction" with 'up' being forward and left/right being exactly that. I had backup volunteers stand at the original starting point and used a 2 meter stick to "connect the dots." 

1. Do the lines formed by these two points actually stop where the meter stick stops?
• So we know lines extend infinitely (written on board)
2. Do these lines intersect?
• Conclusion is that if you walk in the same direction as someone else who started at a different point, you'll never cross their path
• Therefore, same slope --> never intersect and we call this parallel
• Converse must also be true, if lines have different slopes, they must intersect
3. How many times will intersecting lines meet?
• This created some discussion. Some thought they might later on (with the infinite extension). I made a point to insist that lines are an imaginary construct - they don't bend around the Earth, they don't stop when the hit the floor, etc. 
• Conclusion is that they only meet once
4. What do you notice about the spacings created between the lines?
• Here I was trying hard to not talk about angles, especially avoiding words like acute & obtuse. I know they probably saw those words before, but we haven't discussed them as a class yet.
• Lead class to the idea that there are 4 spacings/angles, but really only 2 different ones
• Call these wide/thin or big/small
5. Do intersecting lines *always* make a pair of big/small spacings?
• If no, what's the counterexample?
• Here I used Geometer's Sketchpad to create perpendicular lines (didn't use that word yet) and measure the spacings (kinda had to call them angles at this point). 
• So it's possible to have 4 equal spacings. What must then be true about the slopes of the lines?
• Most kids would get the idea that the slopes are flipped. Had to point out that they're also negatives. 
• Define this as perpendicular

That's where we stopped. I did hand out U1 WS2 as HW, but didn't have time to work through examples, so I'm expecting tomorrow to be more oriented around work and maybe whiteboarding than discussion.