Referencing what students had proposed before (using A, B, C, and D to indicate the vertices of the rectangle) I labeled the blue cars vertices. I also label two lines. I asked students to consider how they could label the graph so that they could identify and distinguish between each car (the three rectangles). Many of the students struggled with this task. As I walked around to different groups I kept asking how they could label different things on the graph to help identify which car was which.
Eventually, students started to label either the cars or the vertices of the cars. We discussed why labeling was important. Besides simple identification, the process makes you think about what you are looking at and what you consider to be important features. Labeling is an early stage of evaluation and analysis. Forcing students to think about what to label and how to label helps them to better see and understand what they are working on.
I had groups present their labeling ideas. Some good ideas and ways to label came out from these presentations, including making use of subscripts. One group used ABCD for the vertices of each car but then included subscripts to identify if it was car 1, 2, or 3. This way they could not only identify the car but the corresponding corners for each car. I liked the thinking that went into this labeling.
After we had everything labeled, I asked students to identify, using proper notation, 3 segments, rays, lines, and angles. I also asked them how they would label the plane on which this graph sat. This was their homework assignment.
The next class, I started with looking at the results. Again, many students were stuck or struggled. Students worked in their groups while I put up the five categories on the board. I asked students to write up on the board what they had come up with. Much of the notation was missing and students were simply writing letter combinations.
Once we had several entries under each heading, I asked students to reference their graphic organizers for notation and comment on or correct what they say. Students slowly started going to the board to add arrows or missing lines above the letter pairs.
Soon we had corrected the notation and students started asking questions, such as what happens if you reverse the letters for a ray or does it matter which two points you pick on a line to use as the label. These were really good questions and discussions that showed students were resolving issues they had with notation.
At this point, the graph was getting a bit messy. I was trying to move one of the cars and getting lines or points caught in the selection. I told students we should move to working with a simpler model at this point so we could more easily focus on specific characteristics.
Again, what I noticed was the class did not have any issue with looking at a graph that had fewer lines and items on it or focusing on a figure not drawn on a graph. The transition seemed to be more natural because of where we started.
I am now transitioning to covering what may be considered more traditional geometry topics, such as the angle addition postulate, which is where I went first.
I used measuring angles formed between fingers and then had students measure the angle formed by their thumb and pinky finger. I should have emphasized to students we are creating a model of our hand by having segments represent fingers. As it was, students drew outlines of their hands and measure angles from these outlines. The sums were off quite a bit in some cases but students were still getting the idea of adding angles together to find the angle of a bigger angle.
I went through some definitions and then we worked through some examples. As we progressed, students were fine as long as the values given were numeric. As soon as values were given as expressions, they froze.
I asked them what the angle addition postulate stated, which they were able to tell me. I wrote this out and asked for each angle, what was the measurement given. I wrote these out under the generic postulate and then they started to understand. Of course they struggled a bit with solving the resulting equation but most were able to work through their issues either on their own or with their group.
I will continue working through the first introductory topics in this vein, referring back to the parallel parking problem as an anchor and reference.
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