Lesson Plan #2 Reflection

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.

Context

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.

Launch

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.

Activity

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)

Conclusion

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.

Reflection

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.

Improvements

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.

Lesson #1 Plan and Reflection

On March 1st, 2019, I went to the Environment Sciences Magnet School to teach my first lesson plan with Mr. Smith’s 6th grade class. The plan was to be based off of Lesson 20 in the class’s Eureka Math workbook, which was titled “Writing and Evaluating Expressions – Multiplication and Division”. The standards, or lesson objectives, required the students to be able to “use variables to represent numbers and write expressions when solving a real-world or mathematical problem” and “Write, read, and evaluate expressions in which letters stand for numbers”.

Lesson Introduction

Before I introduced the concepts of Multiplying and Dividing with variables and expressions, I asked the students if they remembered how to create and solve expressions using addition and subtraction, to which I received a resounding yes. With this understanding in mind, I skipped over a quick review of how to add and subtract with expressions and jumped right into my introduction of my lesson. Before class had started, I passed out a packet of the various CT transit buss passes with the prices labeled on them to be used with my lesson plan. My hope was for the students to become more engaged with the lesson with not only a hands-on object, but those objects being something that most of the students could recognize. I created a table with three columns, the number of tickets bought, the type of ticket bought (or the price of the ticket bought), and the total cost. The idea was that the students would realize that by multiplying the number of tickets by the type of ticket they would be able to get the total cost. For example, if a person bought 3 one-day passes (which were \$3.50), the students would multiply the two numbers together to get \$10.50 for the total cost. Using this table, I asked the students to fill in the table based on the questions I asked.

The first two questions did not require a variable and was meant to be a way to introduce the concept of expressions with multiplication. In the video above you can see one of these questions I posed to the students, and the creation of the table, followed by the solving of the table in the video below it.

After this, I introduced the concept of variables and asked the students to create the expression with this in mind. One of the students I called on was a step ahead and created the expression with the variable isolated. However the expression the lesson plan called for required the variable to be paired with a number, and the student easily created the expression in this form after I asked for it. The creation and solving of the table and expression can be seen in the videos above.

I didn’t know how much student understanding to expect before I presented this lesson, but many of the students were raising their hands to answer my questions and showed great understanding of the concept. I had originally planned on giving seven problems using the table, but ended up only doing five because the problems were too easy for the students.

Open-ended Problem Creating and Solving

After this introduction, I asked the students to think about how many times a week they rode the local bus. With the number of bus rides per week in mind, I then asked them which bus ticket would be the most cost-efficient ticket to buy. Kyle, Mr. Smith, and I walked around the class while the students worked together in their groups to figure this out.

Many of the students only rode the bus a few times a week, if at all. Due to this, it was quite easy for them to figure which ticket would be best, since they could just choose the cheapest one. Kyle and I tried to facilitate a more advanced thinking by asking them to compare the day passes against the ride passes. I would ask the students questions like “I need to ride the bus 15 times a week on 5 different days” to get them to think critically about which ticket type would be best for me specifically, since this situation would be more complex than the students’ personal experiences. The table groups each solved this in their own ways. Some groups would divide the price by the number of rides a ticket gave to see the cost per ride, and then divide the number of days a ticket gave to see the cost per day. Other tables created their own tables with expressions with the tickets they thought could potentially be the best for my situation, though in these scenarios the students sometimes left out the correct tickets from their calculations.

Conclusion

As the time Mr. Smith had given me to present my plan was ending, I called the class back to the front to summarize the lesson they had just learned. I explained how each person’s best price would be unique to them due to factors such as the number of rides he or she took a week, the number of days he or she rode a week, and his or her age. I then asked for groups to share to the class which ticket would be best for them based on their findings, as can be seen in the video above, before thanking the class for allowing me to teach them the lesson for that day.

Reflection

Overall I thought that the lesson plan went well, especially when the kids worked together in groups. While presenting my table to the class, I quickly realized that the kids had already seen these problems and had a more advanced understanding of the content. Due to this, I let the kids create and solve their own expressions sooner than I anticipated. I had originally thought that the activity would be a little difficult for the students, but their previous experiences in the lesson plan allowed them to apply more difficult knowledge to the activity, such as comparing expressions that the students themselves created. Even when Kyle, Mr. Smith, or I wasn’t at a table group, the tables were still trying to solve the expressions they created as well as experimenting with the tickets in ways that they didn’t think of from their personal lives. The open-endedness of the activity is what caused it to work so well, and I have to credit Kyle for this idea.

Although the lesson did go well, I am not sure that the students learned anything new. I discovered after my lesson that Mr. Smith had already taught the kids the content, so even though my lesson may have required the students to apply critical thinking to the concept, they already had a solid foundation of the lesson’s content before I came to teach. I realized that the students had already achieved both standards when I was presenting my table example to them.

When the students were working by themselves, there were some times where the students would be confused on how to answer some of my questions. When they were unsure of the answer, they would ask in a questioning tone if the expression they would create was correct or not. With some pushing into the right direction, it didn’t take long for the students to get on the right path and figure out the answers on their own. A lot of table groups had differing ideas on how to solve the questions Kyle and I posed to them. As said before, some created tables and others divided the ticket prices by the number of rides/days. A common theme in each group was the students explaining to each other how each of them thought they should create or solve the expression and questions I asked the students.

Improvements

As I re-watched my video, I noticed that I had my arms crossed nearly the entire time I presented my lesson, giving off a closed off impression. I also noticed that I would have poor posture and walked around the class “sloppily”. For next class, I am going to try to be more confident and present myself as more open to the students. Not only will this make me more professional, but it will also give me more respect and attention from the students both when I am speaking and when the students are working on something. Numerous times throughout my video, I would see that I had a hard time articulating the information or sentences that I wanted to, which is something that I was actually aware of during the actual lesson. To help with this next time, I’ll create a better “script” of the things that I know I am going to have to explain to the students. Also, being more confident in what I am teaching to the students will help, since I noticed that at certain times I, myself, would sound unsure of what I was explaining.

Prior to this, the only teaching experience I had was in informal settings, explaining how to do something to my friends or volunteering in an after-school program for elementary school kids where I would tutor some kids one-on-one in my junior year of high school (informally still). From what I remember from my volunteering experiences, a lot of the kids would stop listening to me when I tried to explain something to them, which would lead me to only tutor the same few kids that listened to me each day. Although 6th graders have a bigger attention span than the 3rd graders I tutored 2 years ago, I was able to keep the kids engaged more than I would have been able to then. It may have had something to do with me being the leader of the classroom, but regardless the improvement from the last time I had to teach something to a student younger than me was significant.

Next Steps Report

On Friday, February 15th, I went to observe Adam Smith’s math class at the Environmental Sciences Magnet School. The classroom tables were organized into groups of three, with some having only two kids (possibly because some were absent from school). The first class I observed were all separated by gender, and the second class was mostly the same, a few table groups were mixed genders.

Mr. Smith gave his lecture of multiplication and variables through the use of a power point and projector. There was a heavy focus on attempting to get the kids to treat variables as something different than both letters and variables. Kids would be asked questions throughout the lecture on how to do the problems and would raise their hands to answer these questions. As is somewhat expected, it was mostly the same kids volunteering to answer each time. The questions asked were mostly going from step to step on how to solve each problem instead of just asking for the final answer.

There would also be a time period to work on problems in a workbook that each student had. During these time periods, students either worked with their table partners, or alone if Mr. Smith requested it. Mr. Smith would then walk around the room to check on how each student was doing. At the end of class, an “exit slip” is given out for each student to do as well. This slip and homework was turned in before the students leave.