On Friday April 5th I went back to the Hartford Environmental Sciences Magnet School to present my second lesson to the students, this time about area and its relation to real-world context. The Lesson came again from the Eureka Math workbook and was titled “Area in the Real World”. The desired student outcomes, or objectives, were being able to determine the area of composite figures in real-life contextual situations using composition and decomposition of polygons, as well as being able to determine the area of a missing region using composition and decomposition of polygons.
I came to the classroom a few minutes early to set up the map of Hartford shown above and talk with Mr. Smith about the plan, where he informed me that he would prefer the students to use acute and obtuse triangles over right triangles. He also stressed the importance of them using the correct units in area, since many students had been forgetting to put the “square” units in their answers. The students I taught were the same ones as my first class, so it made it a bit more relaxing for me since I had already established a connection with most of the students. Mr. Smith had also already established a solid foundation on finding area in composite figures, my job was to reinforce and expand their understanding. In my opinion, the activity with the neighborhoods I printed out did a good job of that.
Learning from my past experience, I decided to focus less on teaching the subject matter and just introduced the lesson activity for the day. To promote equity and make the lesson relatable to the students, I centered the activity around the local Hartford neighborhoods, including the neighborhood of the school the kids attended, Behind the Rocks. I first asked if any of the students lived in Hartford and if they knew the neighborhood the lived in. I then asked if anyone knew what neighborhood the school they attended was in. After telling them the neighborhood they were in was Behind the Rocks, I explained to the students that the shape and border of this neighborhood, and all the other Hartford neighborhoods, was very complex and weirdly shaped, complicating our ability to find its area. I asked the students how we could find the area of these composite shapes, to which they answered by splitting it up into familiar shapes such as triangles, rectangles, and other polygons. I then wrote the conversion equation for the scale of the maps they were using from inches to feet and then from feet into miles so that the students would figure out the area in square mileage of the neighborhoods, and later the square mileage of the entire Hartford map I had put up on the whiteboard.
Before the students began their activity, Mr. Smith covered a few more points. I had forgotten to tell the students to estimate the borders of some of the neighborhoods since they were curved and not straight like the shapes the kids had previously been working with. Mr. Smith informed them of this and also told them to focus first on finding the area of the shapes before converting it into square miles. Mr. Smith also requested that the students create a chart with two columns like the one he had put up on the board. Using the chart, the students would create a row for each polygon they made in the neighborhoods, using the left side of the chart to show how they calculated the area and the right side of the chart to list the areas of these polygons.
The students each had an individual neighborhood for them to work on finding the area, some of them being more difficult than others due to the fact that they were smaller and had more edges. As opposed to the groups of three the students were in last time, they were in pairs, which was better since it allowed each student to have an individual partner to help them when Mr. Smith and I weren’t there.
Each table group I visited was able to split up the composite figures into simpler shapes, though some had more trouble than others. An example of something that gave a student some trouble can be seen above, where the complex edges made it harder for them to create a triangle or rectangle, though this problem was easily solved by them after I suggested drawing another line to create another shape. The problem that gave the students the most trouble was time, as they were able to find the area of all the shapes they created, it’s just the measuring and ensuing calculations to find the area that was time consuming.
I had planned doing a summative assessment of the students’ understanding of the lesson by seeing if they could correctly find the area of their neighborhoods. However, since class ended, many of the students were unable to find this area. This does not mean that they didn’t understand the lesson plan though. Instead of assessing a final answer, I did a formative assessment by checking the students’ work to see if they were doing the right steps to get the area of the neighborhood. All of the students were dividing the shapes into rectangles and triangles, both acute and obtuse. They were also using the rulers to measure the sides before plugging those measurements into the area formula of the specific shape they created, showing their work in the left column of their created chart. Most of the students weren’t able to get to the part where they added all of the smaller shapes’ areas together to get the total neighborhood area since they weren’t able to create enough shapes to fill the neighborhood in the allotted time. However, the students still knew that they would eventually have to use these numbers later, as they would write the numbers into the right column of the chart. Despite the shortage of time, some students were still able to find their area, and an example of one of these student’s work can be seen above. Above is also a teacher representation of a student’s division of a neighborhood, with the shapes they created highlighted in color. (Click to enlarge on both images)
When the time for my activity had ended, many kids were unable to finish finding the area of their neighborhoods, and thus we weren’t able to find the total square mileage of the Hartford area. I don’t have any video of my ending speech as my video cut off, but it consisted of me re-emphasizing the students’ ability to decompose composite figures into simple polygons and the ability to find the area of the composite figures by finding the area of the simple polygons. I thanked the students for being a great class and told them that even though they may not have finished, they all did a great job with the activity I presented them.
Much like my first lesson, I thought the second one went really well. The equitable situation of using Hartford neighborhoods got the students excited to do the activity. The only unexpected thing that occurred was how quickly the time went by. Before the class, I had worried that the lesson plan may be too easy for them and printed out extra copies of the neighborhoods for students that finished early. However, this was not an issue, as the neighborhoods were very complex and very few of the students figured out the area of the neighborhoods.
I think that my lesson plan taught the students how to split up shapes in a more advanced manner than they had been doing before. The complex shapes proved to be challenging for many of the students at first, however with some encouragement and help from Mr. Smith and I they were able to create the familiar shapes. An example of me doing this can be seen in the video above, though the audio quality is not the greatest.
Since the students showed an understanding of the activity, they were able to complete both lesson objectives of finding the area of composite figures in real-life contextual situations (Hartford neighborhoods) using decomposition of polygons and being able to determine the area of missing regions by finding the area of the simple polygons they created. The students showed their mathematical thinking in the way they divided up the shape as well as by creating the chart that Mr. Smith had asked them to. Any confusion was expressed by the students asking Mr. Smith or me for help, mainly on how to split up the shape or measure the side lengths. Some specific questions asked were “is this side [number] inches?”, “can I draw a triangle here?”, and “the area of a triangle is the area of a rectangle divided by 2 right?”. An example of a conversation between a student and me where I helped solve their confusion went something like this:
Me: Do you need any help?
Student: I don’t know how to split up this part of the neighborhood.
Me: Well you can always draw more lines to create more shapes, do you see any place where you can create another shape? Maybe right here, try connecting corners and see what happens.
Student: Ok, I see, thank you mister.
From the last time I taught my lesson, I felt like I did a good job working on the things I thought I did poorly on. I noticed that I had better posture and didn’t cross my arms as nearly as much as the first time I went. Prior to going to my lesson I had written a speech for my launch dialogue, and although I didn’t stick to it entirely, it definitely helped me better articulate to the students what I wanted them to do. I also went alone to teach my lesson this time, and as a result I didn’t have Kyle to help facilitate the activity. Despite this, I did feel more comfortable with the kids the second time coming.
If I had to redesign my lesson, I would have chosen to only pass out the simple neighborhoods to all the students since many were not able to finish. If the students were able to finish these neighborhoods, I would then pass out more complicated composite neighborhoods for them to complete. The ability to have actually found the area of the neighborhoods would have given the kids a greater sense of satisfaction. It also would have been interesting to see how the kids were splitting up the same exact image in different ways.
For my next lesson, I could work on trying not to spend too much time with certain groups, even if they need the help. Although I did make it around to all of the table pairs, some groups I didn’t spend as much time with because I spent a sizeable amount of time working with a different one. A way I could improve this is by first only spending a couple minutes at each table, no matter how much help they need, then circling back around to the pairs that needed help the most.