The only prep I have as part of a team is geometry. I haven't taught geometry in five or six years, so the order and content emphasis has changed slightly. I am relying on the team to communicate what will be assessed and then I am adapting to take a more inquiry-based approach.
As part of our week of preparation and professional development prior to students arriving, I participated in a district training on teaching math. A couple of things I walked away with were some concise ways to communicate expectations and behaviors. I incorporated these into my first lessons with great effect.
I also thought that using parallel parking may be a way to convey geometric ideas that students could better appreciate. As 10th graders, either all have driver permits or are taking driver education classes with the goal of obtaining a driving permit. So, driving is on their minds and something they can relate to.
Unfortunately, in my search for materials to use in the classroom, I came up short. There are some very nice inquiry into parallel parking. Unfortunately, these lessons are oriented to in-service teachers and the math goes well beyond 10th grade geometry. For example, parallel parking can be used in teaching transformations. The lessons I found use the transformations to convey matrix representation of the transformations and applying these representations to calculate the image points of a pre-image. Really good mathematically connections but not 10th grade level.
As a result, I have had to improvise as I go along. I am trying to leverage the parallel parking idea as an anchor investigation while rolling in the geometry that the rest of the geometry team is teaching.
Below is a summary of what I have done the first two weeks.
Day 1: The first day was really an orientation day. I started off with a brief introduction of myself and my philosophy. I told the class that it is not about the math that you learn but about how you can learn to be a better problem solver through logic and reasoning. I used my own experience as a math major going into business and using my problem solving abilities to solve all sorts of different problems that businesses face.
Next, I went through the class expectations (picked up from my day of professional development).
Class expectations:
- Choose to be PRESENT
- Choose to be ENGAGED
- Choose to be an ACTIVE LISTENER
Learning is a process, not an event.
I also went through behaviors associated with different work aspects: as an individual, as a duo, as a small group, and as a whole class. These were presented during the professional development day as "what do mathematicians do." The reality is that most of these students will not be mathematicians, but they will work in some capacity. I modified this to extend beyond being a mathematician to what should you do when you are working in different environments.
I created signs that I posted front and center in the class for easy referral. The signs posted are:
When working as an Individual:
- identify characteristics
- connect to other situations
- explore
- check
- re-work
- try different options
When working as a Pair:
- brain-storm
- listen and critique
- question each other’s methods
- identify things that make sense
When working as a Group:
- collaborate
- corroborate
- divide work
- divide tasks
- verify work
When working as a Whole Class:
- listen
- question
- comment
With that gone through, I started to practice some of the behaviors.
I started off with a variation of "The Name Game" that I use in my discrete math class. This variation was used in my study hall class and I decided to borrow it. Students had to share their name and a fact about themselves. As usual, each person that follows needs to also repeat every else's name and fact.
This was a whole class activity, so I was able to emphasize that if you never hear the name you won't remember the name. I asked all students to think of the fact they were going to share before we started. Again, the idea is that they can now pay attention to the speaker rather than focusing inward on what they will say.
After going about a third of the way through the class, I asked students toward the end how they felt about remembering all those names and activities. Of course, they were feeling a little nervous. I suggested that taking notes may be an effective way of helping them to remember. The students sat frozen for a minute and then rushed to get their paper and pencils out. I used this as an opportunity to mention how hearing, speaking, writing, and seeing help to capture the information and bring multiple senses into play. I told students they could take the notes but at their turn they had to turn their notes over and recite from memory.
We proceeded through the entire class. As this point, I had students count off and re-arranged them into new groups. I then went through the class and recited their names and facts. I got stuck toward the end but then re-visualized the order in which I had learned the names and was able to complete the task. I communicated to the class what I was doing so they could understand how material learned in one order but now presented in another order (such as on quizzes and tests) could be re-captured.
To wrap things up, I had students consider the classroom behaviors and what those behaviors look like to them. This was their homework assignment.
DAY 2: I checked to see who had written down their ideas. I referenced the signs and asked students to pair up and discuss, and then discuss in groups of four. Finally, we did a share out as a class and got agreement to what expected behaviors should look like.
Next, I took the class off-guard by asking them what the wanted to learn. There was some discussion at their tables and the class came up with the following list of what they want to learn:
- problem solving (in an organized and understandable way)
- shapes, dimensions, length, width, and volume
- planes, lines, segments - how they intersect and are placed on planes
- story/word problems
- relate to real world
- why use geometry
I committed that I would teach them these things but also prepare them for common assessments and the possibility that at semester they may be transferred to a different class.
They also wanted to learn about me, so I opened up the class to questions and proceeded to answer all the questions they had about me. There were questions about my past work, my liking of teaching and the school, personal hobbies, pets, vacations, books read, and so much more.
Once we were done, I had a little bit of time to finally introduce the parallel parking problem. I began by asking who had a driver's permit or was working toward obtaining a driver's permit. All but one or two students raised their hands.
I then wrote down parallel parking and asked why it was called parallel parking. Students came up with informal ideas about parallel lines in describing the situation. I asked students how they would explain to someone how to parallel park. The students hemmed and hawed a little on this one, not sure how to explain the process. Finally one girl said she learned it in driver's training. That you back up until the center of the parking car is lined up with the rear bumper of the front parked car and then turn the car to a 45o angle.
Whoa! You need geometry to park your park.
I asked students to consider how much space you would need between two cars in order to parallel park. I wasn't expecting much in the way of formal responses, but wanted to see what reasoning and geometry they might bring into the problem. This was their homework assignment.
DAY 3: We started class working through the parallel parking space problem that was assigned as homework. Keep in mind, through the entire process, I kept referring to the behaviors as students moved from working as individuals to pairs to groups.
For the group work, I asked students to consider what roles may be needed. We had a good discussion about the need for someone to capture the team work, someone to keep track of time, someone to ask questions and probe the results for weaknesses. One student said there should be a leader. I held off on this role until the end. At this point, I said that a team doesn't need a leader, it needs a facilitator to help ensure things flow smoothly and that progress is being made. I wanted to avoid any taking a leadership role as the groups should work together as a team, coalescing toward ideas on their merits as opposed to someone saying this is what they should do.
We then proceeded to a whole class discussion. A couple of groups referenced the length of the diagonal of the car and one group actually used rectangles to represent the car and said the space would have to be at least the length of the diagonal. We didn't quite finish the presentations, so the final group had to wait until the next class.
DAY 4: We started by looking at the final group's solution. It was obvious the student had googled a formal result and was presenting a fairly complex formula that involved a lot more factors than what had been discussed in class. I let it go other than to clarify which letters represented which aspects of the situation.
I then focused on the rectangle representation and the length of the diagonal. I connected this to the idea that we are simply using math to model a situation, focus on some key aspects, and try to draw some understanding of the situation.
With this in mind I asked students how much space would be needed if the parking car were 4 feet wide and 11 feet long? Again, I referenced the working behaviors and asked students to work on their own for a few minutes. Then I had them pair and then work in their groups. As I walked around the room several students said they were stuck. Seeing that the entire group was stuck I asked them what kind of figure was made by the diagonal and were there anything about the triangle formed they could use. For most, this was all they needed to connect back to the Pythagorean theorem.
Once the class was on board, I gave a couple more dimensions for them to practice. It was all a very natural flow with no question about why they needed to know the length of a diagonal.
For homework, I gave them three ways to define parallel lines and asked them to agree or disagree with each statement, explain why they made their choice, and which made the most sense to them.
DAY 5: This class went through discussion and presentations about parallel lines. There were some good discussions about the three statements. Most were confused by the parallel lines are perpendicular to a third line. The confusion came in the use of the third line. Students felt this line was unnecessary and could make the statement possibly false. I left this as an open-ended question. We'll be looking at parallel lines and transversals, at which time we can revisit this idea.
To help with notation and definitions, we worked through a basic geographic terms graphic organizer. It was designed as a jigsaw, where each student was to received one card. But given the length of the parallel line discussion, I elected to have students try to complete as much of the organizer as they could with their groups. I told them to write in pencil in case they needed to correct or erase any entries.
We then worked through placing the cards on the grid, with some discussion, and completed the organizer.
DAY 6: For the start of this day, the class used their completed graphic organizer to help them complete a practice worksheet. There were a couple of new terms on the practice sheet, such as co-linear, which we had to discuss.
After this, it was off to the computer lab to have the class sign up in Khan Academy. I had set up a geometry class and had all my students sign up for the class with me as a coach. The geometry team is trying to use Khan Academy geometry lessons for homework. I haven't done this yet, but did have the class practice a lesson on identifying rays, lines, segments, and the such. It was a natural follow-up to the worksheet they had completed earlier. It also allowed them to learn how to get hints and help as they worked through the lesson.
DAY 7: This turned into a very interesting class. I really didn't have something set to do but wanted to begin introducing the idea of a coordinate system.
I had three rectangles of different colors drawn in a Smart notebook file. I had drawn a line (arrows at both ends) to represent the curb of the street, and had a coordinate plane displayed below.
I re-iterated how we were using the rectangles to model the cars and the line to model the curb. I had a picture of a google driverless car on the slide as well. I asked the class how they thought instructions were given to the car. There was some discussion about using sensors. I asked the class if they thought instructions like, "hang a left at the next corner" were used. The class agreed that was probably not happening.
I described how google has been mapping the world and basically establishing a grid system that underlies their work. We can do the same thing with our model of a car parallel parking. The question I posed to the class was, "Where should the origin of our graph be placed?"
I asked students to consider this on their own. Where would they place the origin and what did they perceive to be the strengths and weaknesses of that placement? (I was referencing work behaviors again.) I then had students discuss their selections in pairs and then in groups of four with the goal that the group should come to a consensus as to where to place the origin.
I asked someone to place the origin where they thought was best. The first group placed the origin at the center of the parking car. I wrote this on the board with advantages and disadvantages written immediately below. I had the class discuss what they perceived as advantages and disadvantages of this placement. It was a good discussion.
The next group placed the origin halfway between the two parked cars and lying on the curb line. I clarified that the origin would be at the midpoint of the two parked cars and wrote this on the board. We then went through the advantages and disadvantages.
A third group said they would place the origin at the midpoint of the two parked cars and also at the midpoint of their width. Again, advantages and disadvantages were discussed.
A final origin was placed on the curb line, aligned with the front-end of the car parked to the back. We again discussed advantages and disadvantages.
The whole point of this discussion was for students to realize that it is personal choice that dictates where the origin should be placed. By critically evaluating each choice, students can decide for themselves what makes the most sense.
As I have already told the class on several occasions, math should make your life simpler, not more complicated. Use the math to isolate key aspects and help you better understand the situation.
I noted that several placements required finding midpoints. With the discussion out of the way, I had students practice finding the midpoints working from a practice worksheet. Again, students didn't question why they were practicing this skill, they just dove right in and tackled the task. It was natural to work on midpoints because they said they needed midpoints to place the origin of their graph.
Reflection on the first two weeks:
I am encouraged by how the class has come together the first two weeks. I was able to invoke rules that were easy for me to convey and easy for students to understand. I was able to practice work behaviors that will help students in this class and in their future.
The use of parallel parking as an anchor problem is working out well. It will be easy to work this into discussions of parallel lines and transversals, transformations, geometric constructions, area, much more. I only wish I had all of this mapped out better versus coming up with stuff more or less at the last minute.
One final point. We had back to school night and I laid out what I was doing with parallel parking as the anchor problem. Parents seemed very pleased with the idea that the geometry was being made relevant to their children's lives.
I will continue to post on activities and progress in geometry. Hopefully, I will do this more frequently than every other week.
Until the next post, hope your school year gets off to a great start!
Whoa! You need geometry to park your park.
I asked students to consider how much space you would need between two cars in order to parallel park. I wasn't expecting much in the way of formal responses, but wanted to see what reasoning and geometry they might bring into the problem. This was their homework assignment.
DAY 3: We started class working through the parallel parking space problem that was assigned as homework. Keep in mind, through the entire process, I kept referring to the behaviors as students moved from working as individuals to pairs to groups.
For the group work, I asked students to consider what roles may be needed. We had a good discussion about the need for someone to capture the team work, someone to keep track of time, someone to ask questions and probe the results for weaknesses. One student said there should be a leader. I held off on this role until the end. At this point, I said that a team doesn't need a leader, it needs a facilitator to help ensure things flow smoothly and that progress is being made. I wanted to avoid any taking a leadership role as the groups should work together as a team, coalescing toward ideas on their merits as opposed to someone saying this is what they should do.
We then proceeded to a whole class discussion. A couple of groups referenced the length of the diagonal of the car and one group actually used rectangles to represent the car and said the space would have to be at least the length of the diagonal. We didn't quite finish the presentations, so the final group had to wait until the next class.
DAY 4: We started by looking at the final group's solution. It was obvious the student had googled a formal result and was presenting a fairly complex formula that involved a lot more factors than what had been discussed in class. I let it go other than to clarify which letters represented which aspects of the situation.
I then focused on the rectangle representation and the length of the diagonal. I connected this to the idea that we are simply using math to model a situation, focus on some key aspects, and try to draw some understanding of the situation.
With this in mind I asked students how much space would be needed if the parking car were 4 feet wide and 11 feet long? Again, I referenced the working behaviors and asked students to work on their own for a few minutes. Then I had them pair and then work in their groups. As I walked around the room several students said they were stuck. Seeing that the entire group was stuck I asked them what kind of figure was made by the diagonal and were there anything about the triangle formed they could use. For most, this was all they needed to connect back to the Pythagorean theorem.
Once the class was on board, I gave a couple more dimensions for them to practice. It was all a very natural flow with no question about why they needed to know the length of a diagonal.
For homework, I gave them three ways to define parallel lines and asked them to agree or disagree with each statement, explain why they made their choice, and which made the most sense to them.
DAY 5: This class went through discussion and presentations about parallel lines. There were some good discussions about the three statements. Most were confused by the parallel lines are perpendicular to a third line. The confusion came in the use of the third line. Students felt this line was unnecessary and could make the statement possibly false. I left this as an open-ended question. We'll be looking at parallel lines and transversals, at which time we can revisit this idea.
To help with notation and definitions, we worked through a basic geographic terms graphic organizer. It was designed as a jigsaw, where each student was to received one card. But given the length of the parallel line discussion, I elected to have students try to complete as much of the organizer as they could with their groups. I told them to write in pencil in case they needed to correct or erase any entries.
We then worked through placing the cards on the grid, with some discussion, and completed the organizer.
DAY 6: For the start of this day, the class used their completed graphic organizer to help them complete a practice worksheet. There were a couple of new terms on the practice sheet, such as co-linear, which we had to discuss.
After this, it was off to the computer lab to have the class sign up in Khan Academy. I had set up a geometry class and had all my students sign up for the class with me as a coach. The geometry team is trying to use Khan Academy geometry lessons for homework. I haven't done this yet, but did have the class practice a lesson on identifying rays, lines, segments, and the such. It was a natural follow-up to the worksheet they had completed earlier. It also allowed them to learn how to get hints and help as they worked through the lesson.
DAY 7: This turned into a very interesting class. I really didn't have something set to do but wanted to begin introducing the idea of a coordinate system.
I had three rectangles of different colors drawn in a Smart notebook file. I had drawn a line (arrows at both ends) to represent the curb of the street, and had a coordinate plane displayed below.
I re-iterated how we were using the rectangles to model the cars and the line to model the curb. I had a picture of a google driverless car on the slide as well. I asked the class how they thought instructions were given to the car. There was some discussion about using sensors. I asked the class if they thought instructions like, "hang a left at the next corner" were used. The class agreed that was probably not happening.
I described how google has been mapping the world and basically establishing a grid system that underlies their work. We can do the same thing with our model of a car parallel parking. The question I posed to the class was, "Where should the origin of our graph be placed?"
I asked students to consider this on their own. Where would they place the origin and what did they perceive to be the strengths and weaknesses of that placement? (I was referencing work behaviors again.) I then had students discuss their selections in pairs and then in groups of four with the goal that the group should come to a consensus as to where to place the origin.
I asked someone to place the origin where they thought was best. The first group placed the origin at the center of the parking car. I wrote this on the board with advantages and disadvantages written immediately below. I had the class discuss what they perceived as advantages and disadvantages of this placement. It was a good discussion.
The next group placed the origin halfway between the two parked cars and lying on the curb line. I clarified that the origin would be at the midpoint of the two parked cars and wrote this on the board. We then went through the advantages and disadvantages.
A third group said they would place the origin at the midpoint of the two parked cars and also at the midpoint of their width. Again, advantages and disadvantages were discussed.
A final origin was placed on the curb line, aligned with the front-end of the car parked to the back. We again discussed advantages and disadvantages.
The whole point of this discussion was for students to realize that it is personal choice that dictates where the origin should be placed. By critically evaluating each choice, students can decide for themselves what makes the most sense.
As I have already told the class on several occasions, math should make your life simpler, not more complicated. Use the math to isolate key aspects and help you better understand the situation.
I noted that several placements required finding midpoints. With the discussion out of the way, I had students practice finding the midpoints working from a practice worksheet. Again, students didn't question why they were practicing this skill, they just dove right in and tackled the task. It was natural to work on midpoints because they said they needed midpoints to place the origin of their graph.
Reflection on the first two weeks:
I am encouraged by how the class has come together the first two weeks. I was able to invoke rules that were easy for me to convey and easy for students to understand. I was able to practice work behaviors that will help students in this class and in their future.
The use of parallel parking as an anchor problem is working out well. It will be easy to work this into discussions of parallel lines and transversals, transformations, geometric constructions, area, much more. I only wish I had all of this mapped out better versus coming up with stuff more or less at the last minute.
One final point. We had back to school night and I laid out what I was doing with parallel parking as the anchor problem. Parents seemed very pleased with the idea that the geometry was being made relevant to their children's lives.
I will continue to post on activities and progress in geometry. Hopefully, I will do this more frequently than every other week.
Until the next post, hope your school year gets off to a great start!
No comments:
Post a Comment