I used http://mathbitsnotebook.com to research how to partition segments, and developed my notes for the lesson from here. I used the first method they list - using similar triangles.

Here is how my lesson developed throughout the day:

__1st period__: My students were wonderfully patient. They helped me realize that seeing the graph for each example would benefit some of them. (I didn't include them in the notes handout to save space. Next year I know that I cannot afford to omit coordinate axes.) In calculating the rise and run of the segments, I would ask, "how many spaces are there from _____ to ____?" Not all students are able to grasp how that works with positive and negative numbers being mixed together. We realized that direction of the rise and run matters. (The website didn't show an example of partitioning a segment with a negative slope.) My students struggled with understanding the concept at first, as I struggled with my explanation. The first example was rough, they asked many questions throughout the second example, and then were confident for the last two examples.

After this class, I decided that I needed to have some way for my students to graph the line segments. I printed out a class set of coordinate axes, and slipped them into the ticket holders I purchased to use as dry erase boards.

__2nd period__: This class is my advanced class. They progressed through the examples much like my first period did, but being advanced, they didn't need the graphs like my first period did. This information will help me plan for differentiation next year.

__3rd period__: This is my most challenging class. (On this particular day some of my heavy hitters were absent, making class run much more smoothly overall.). By now my explanation was much better, but there is still plenty of room for improvement. These are the students that really benefited from the graphs. This class also advanced through the examples as my earlier classes did. In this class, we also discussed the possible shortcut of graphing the line segment, finding the x-coordinate, and then using the graph to determine the y-coordinate.

Although teaching this topic our first day back from break was challenging, I really enjoyed the process of learning to teach the topic better for next year. I haven't had such huge epiphanies lately because I've been so comfortable with the topics I've been teaching. I look forward to seeing how the changes I'm making to future topics effect how I teach and how my students learn.

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