After devoting a class to just practicing work with angle relationships and midpoints, I needed to move on to geometric constructions. I wanted to take a more inquiry-based approach to this topic.
My first challenge was thinking about what exposure students may have had to compasses. I know that I have seen a lot of movies with sailing ships and the chart scenes normally included the use of a compass. I decided to look up a clips that showed the use of a compass in ship navigation.
To start class, I held up a compass and asked students if they had ever seen this before. When I asked where, a couple of students said it was something pirates used. Perfect! I asked the class what pirates would use the compass for. Most shrugged their shoulders. A few replied that they could draw circles with them.
I then showed the first clip I found. It's short and has no sound but shows someone using a compass with a chart. I asked the class what the person was doing with the compass. The class responded it appeared the person was using it to measure distance. I re-emphasized the idea of using the compass to measure distance.
It was now time to show the second clip. This clip shows how the compass is used to measure distance and make markings with arcs.
I then drew a line segment on the board and labeled it as segment AB. I marked a third point C on the board. I told the class the challenge was to make an exact copy of AB so that AB = CD by using the compass as a distance measuring device and a ruler solely to draw straight lines.
I then let students struggle through the challenge. Some students were done quickly. Checking their work, I asked how they copied the line. These students said they used the ruler. I told them that wasn't allowed. The only device they had for measuring distance was the compass. I told them to think about what they saw on the video.
Slowly students started to get the idea. I did have to walk around a lot and talk through how the compass could be used to measure distance with quite a few groups. After everyone had the general idea, I asked them to draw another segment and then make a copy that was congruent. This time students seemed to get what they needed to do.
I next drew an angle on the board and labeled the vertex A. I drew a second point B and told them the challenge was to make an exact copy of angle A so that the measures of both angles were the same. Again, they were to use only the compass and straight edge.
Students worked on this for 15 plus minutes. I walked around and checked on their work. Many students had drawn two angles with both angles having side segments that were the same length. I asked how they knew the angles were the same measure. They were stumped by this. Others had measured the separation of the rays but hadn't considered that they weren't measuring the width of the angles from equivalent points.
After about 15 minutes, several students were honing in on some productive ideas. One student in particular said he though he had it. He went through his process and it was exactly what I would have shown if I were giving step-by-step instructions. I had him share his method with his group before letting him share it with the class.
As a class, we discussed why this process duplicated the angle and related it back to the video and distance measurement done in the clip. I asked students to try using this to duplicate their angle.
By this time, almost the entire class was comfortable with duplicating the length of a line segment. They weren't as comfortable with the ideas of using the different lengths to ensure the angles were congruent.
Next class, we'll work on duplicating another angle and then I'll turn them loose on trying to construct a parallel line through a point not on the given line.
Overall, I was pleased with the outcome of this class. Students gained a better understanding of how to use a compass to measure distance, how to use arcs as markings, and how to construct congruent line segments. They also were exposed to how to put these ideas together to construct congruent angles.
Thursday, September 17, 2015
Friday, September 11, 2015
Making connections in geometry
The last few classes have been focused on simple explorations of angle relationships, such as linear pairs, vertical angles, and angles formed by transversals cutting across parallel lines. I continue to reference the parallel parking scenario to motivate where and how these angles come about.
I've had to work in algebraic expressions as values to help prepare the class for questions they may see on common assessments. I have been hard pressed to come up with scenarios that would naturally generate these expressions; it's something I'll need to work on.
For one set of practice problems, students indicated they were getting stuck on some problems. Rather than working through specific situations, I asked students to step back and focus on the angle relationships they are seeing and how they relate to each other. I told them not to worry about the values they were given for different angles. I drew two intersection lines and asked them to tell me the different relationships they saw in the four angles. I then gave a couple of expressions for two angles and asked them how this fit into the situation and could be used. The students that said they were stuck said this helped. I told them to keep working on the problems and we'll discuss them next class.
My purpose is to have students focus on the pieces and how they can be put together to answer questions. By breaking problems down into the components they know and understand, I believe students can piece things back together and become better problem solvers.
Today's class wrapped up with a question tied back to the parallel parking situation. I told them we are turning 45o in relation to the original position of the parking car. As the car backs up at this angle it forms an angle with the curb going away from the car. I asked what was the size of the angle.
It was interesting that students concluded the angle would be 135o. I asked them how they knew this. Many stumbled around with rather unconvincing arguments. I asked them to focus on the angle relationships they were seeing. Some students started to recognize that the 45o angle was a corresponding angle to the angle paired with our angle of interest and that these two angles formed a linear pair. Perfect. Students were using geometry but didn't realize why the angle had to be 135o. I'm hopeful I can push them further into asking themselves why things work.
I've had to work in algebraic expressions as values to help prepare the class for questions they may see on common assessments. I have been hard pressed to come up with scenarios that would naturally generate these expressions; it's something I'll need to work on.
For one set of practice problems, students indicated they were getting stuck on some problems. Rather than working through specific situations, I asked students to step back and focus on the angle relationships they are seeing and how they relate to each other. I told them not to worry about the values they were given for different angles. I drew two intersection lines and asked them to tell me the different relationships they saw in the four angles. I then gave a couple of expressions for two angles and asked them how this fit into the situation and could be used. The students that said they were stuck said this helped. I told them to keep working on the problems and we'll discuss them next class.
My purpose is to have students focus on the pieces and how they can be put together to answer questions. By breaking problems down into the components they know and understand, I believe students can piece things back together and become better problem solvers.
Today's class wrapped up with a question tied back to the parallel parking situation. I told them we are turning 45o in relation to the original position of the parking car. As the car backs up at this angle it forms an angle with the curb going away from the car. I asked what was the size of the angle.
It was interesting that students concluded the angle would be 135o. I asked them how they knew this. Many stumbled around with rather unconvincing arguments. I asked them to focus on the angle relationships they were seeing. Some students started to recognize that the 45o angle was a corresponding angle to the angle paired with our angle of interest and that these two angles formed a linear pair. Perfect. Students were using geometry but didn't realize why the angle had to be 135o. I'm hopeful I can push them further into asking themselves why things work.
Saturday, September 5, 2015
What does bad notation buy you?
As I was teaching my geometry class on Friday, I went into the need for good notation and labeling as ways to understand and begin to problem solve. Labeling items helps to identify different characteristics and enables our brains to start to begin absorbing pertinent facts. Hopefully, the brain makes connections to similar situations or relates one idea to another to begin the problem solving process.
Notation is an often overlooked weapon in the problem solving arsenal. We teach lots of notation to be memorized but too often overlook how important good notation is to solving problems and making math useful.
I have a couple of examples that I can think of where notation actually made mathematics more useful and expanded its role in the world. The first being the use of Hindu-Arabic numerals and the use of 0 in writing numbers.
The Romans had an expansive empire that lasted for centuries but did little in the advancement of mathematics. Yes, they were wonderful engineers and ruthless conquerors but they did little new in the way of mathematics. One reason that has been hypothesized is that Roman numerals hinder mathematics.
Let's try this. Don't convert these values, try to use them as a Roman would. What is the answer to the following addition problem?
Notation is an often overlooked weapon in the problem solving arsenal. We teach lots of notation to be memorized but too often overlook how important good notation is to solving problems and making math useful.
I have a couple of examples that I can think of where notation actually made mathematics more useful and expanded its role in the world. The first being the use of Hindu-Arabic numerals and the use of 0 in writing numbers.
The Romans had an expansive empire that lasted for centuries but did little in the advancement of mathematics. Yes, they were wonderful engineers and ruthless conquerors but they did little new in the way of mathematics. One reason that has been hypothesized is that Roman numerals hinder mathematics.
Let's try this. Don't convert these values, try to use them as a Roman would. What is the answer to the following addition problem?
XCIV + LVI = ?
It's not the easiest problem to work out, is it?
Now, what about this problem?
94 + 66 = ?
There is quite a difference in working through these problems. One yields and answer of CLX, the other answer of 160. Oh, wait, they are the same value but the connection between the problem and result are not as clear when adding with Roman numerals.
The use and spread of Hindu-Arabic numerals led to a rapid expansion of mathematics in the western world.
Flash ahead to the 1600's and early 1700's in Europe. The calculus wars raged during this period and while most of Europe adopted Leibniz's notation, England stuck with Newton's. The weaker notation that Newton developed slowed mathematical progress in England while mathematics across Europe flourished, expanding mathematics influence across many fields.
The next time you work with notation or have students work and understand notation, think about whether the notation is flexible and useful or cumbersome.
Mathematics should help students to easily model situations and to make their lives easier not more difficult. Good notation can help in this effort.
Wednesday, September 2, 2015
Labeling and notation in geometry
In my last post, I concluded with Day 7. The next two days started with a focus on labeling and notation. I used an expanded model of the parallel parking situation.
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.
Sunday, August 30, 2015
Start of Inquiry-based Geometry
The first two weeks of my school year are now finished. It's been quite busy as my school implemented a new study-hall policy for all freshman that involved team-teaching for the first two weeks. As a result, I haven't had a lot of extra time (I still have five different preps, four of which a singletons).
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:
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:
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!
Friday, May 29, 2015
Geometry - an inquiry-based approach aligned to common core for next year?
I'll be teaching geometry for the first time in five years. During the past five years I have made a concerted effort to taking an inquiry-based approach in my classes. I've had a relatively easy time implementing many of my ideas because my statistics and discrete math classes are singleton preps.
I'll be on a team of eight for geometry. I will be pushing the group to take an inquiry-based approach. I am hopeful that I can at least get the group to move into that direction.
I will document my progress during the school year as I develop or modify lesson plans and unit flow for an inquiry-based approach to geometry. I intend on using this blog to capture what is working and what is not working.
If you have used an inquiry-based approach in high school geometry, I'd love to hear from you.
If you are interested in taking an inquiry-based approach in high school geometry, comment back and let's get a discussion going.
Now that the school year is over, I'll be off to Kansas City for the AP Statistics reading.
Enjoy your summer; I'll definitely be back in the fall, if not sooner.
I'll be on a team of eight for geometry. I will be pushing the group to take an inquiry-based approach. I am hopeful that I can at least get the group to move into that direction.
I will document my progress during the school year as I develop or modify lesson plans and unit flow for an inquiry-based approach to geometry. I intend on using this blog to capture what is working and what is not working.
If you have used an inquiry-based approach in high school geometry, I'd love to hear from you.
If you are interested in taking an inquiry-based approach in high school geometry, comment back and let's get a discussion going.
Now that the school year is over, I'll be off to Kansas City for the AP Statistics reading.
Enjoy your summer; I'll definitely be back in the fall, if not sooner.
Graph coloring lessons for high school discrete math
I incorporated a graph coloring piece within my graphs and trees unit. For this I made use of Tom Zaremba et al's Map Coloring lessons. There are a few minor tweaks that are needed and a few extensions and modifications that I made.
Below are my notes regarding my modifications:
Chromatic color of graph should reference planar graphs since discussing maps. Will not work for non-planar graphs (see K5).
Fish tank graph should show edge BE not edge BD on the applications page.
For coloring algorithms, have students write out their algorithm. They should try their algorithm on the South America map. Students then swap their algorithm instructions and try to follow instructions. Discuss findings and those that appear to find a graphs chromatic color. Test these further.
Coloring algorithm needs to address that no two adjacent vertices are colored the same color.
Have students write out algorithm for edge coloring. Try this on South America and then exchange.
Discuss graph coloring and scheduling type problems.
Below are my notes regarding my modifications:
Chromatic color of graph should reference planar graphs since discussing maps. Will not work for non-planar graphs (see K5).
Fish tank graph should show edge BE not edge BD on the applications page.
For coloring algorithms, have students write out their algorithm. They should try their algorithm on the South America map. Students then swap their algorithm instructions and try to follow instructions. Discuss findings and those that appear to find a graphs chromatic color. Test these further.
Coloring algorithm needs to address that no two adjacent vertices are colored the same color.
Have students write out algorithm for edge coloring. Try this on South America and then exchange.
Discuss graph coloring and scheduling type problems.
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