Today I introduced rigid transformations. We started by discussing transformations and students were able to identify translations, reflections, rotations, and dilations (although they only described them and didn't name them).
I had mini-whiteboards with axes on them and I started the students off by plotting some points. This helped me see who might be struggling with coordinates and helped students refresh their memory on coordinates and plotting points.
Satisfied that students could correctly plot points, I proceeded to introduce reflections. I decided to use NCTM's Navigating through Geometry sequence for transformations. The first activity deals with reflections over the y-axis.
The beginning prompts dealt with drawing and tracing to create a reflection. Students began to have issues when they were asked to drawn a segment between two points they had selected. The notation seemed to baffle them, which was disheartening given how much time we had spent on notation and understanding what structure and information is given in prompts.
After moving beyond this hurdle, students were to identify how the two points (pre-image and image) were related to the line of reflection. Students readily recognized that the two points were the same distance from the reflection line, lying on opposite sides of the line.
The next piece became a problem. Students were to draw a triangle and then draw its reflection based upon what they had just discussed. Students wanted to fold the paper and trace out again or seemed totally lost. I realized that using the whiteboards and grids could help. I moved students to using these and they were able to proceed without much incident.
We discussed what was happening with the coordinates and students could more readily see that the pre-image point (x , y) became the image point ( -x, y) when reflecting over the y-axis (line x = 0).
At this point I deviated from the activity slightly by focusing on reflections over vertical lines. I asked students to pick a different vertical line, such as x = 2, and reflect their pre-image over this line. I told students to focus on what is happening to the x coordinate now. How does reflecting over a different line change things.
The class struggled mightily on this task. Many wanted to give up and several asked me to just tell them what the result was. I finally told the class that I expected them to make reasonable conjectures based upon what they were seeing. I told them if it takes the class two weeks to work through this issue then it will take two weeks.
Their homework is to come up with some conjectures. I'll see what they bring to class.
My hope is that we can understand the mathematics of reflections over vertical lines and the resulting expressions that describe these reflections. It should then be fairly easy to conjecture about the results we would see reflecting over horizontal lines and verifying the results.
My goal is for students to tackle lines that are not horizontal or vertical. I'll start with y = x and y = -x and then have them tackle a line such as y = x + 2. From there, I hope to look at more general lines such as y = 2x - 3.
After these explorations, I'll revert back to using more of the Navigating through Geometry activities.
Wednesday, September 30, 2015
Friday, September 25, 2015
Introducing algebraic proofs
Today I introduced algebraic proofs as a lead in to geometric proofs.
I started by having the expression, "Oh yeah, prove it!" on the board. I then asked students what it means to prove something. The discussion brought out using evidence to demonstrate a theory or statement was correct.
I wanted to have students consider rules and properties they work with when calculating or working with expressions. The idea was to pull out some of the properties that we would use as building blocks for algebraic proofs. This turned out to be a bit tougher than I expected.
I did provide an example:
For any two real numbers a and b, if a equals b then b = a. The meaning is shown in writing by "If a = b, then b = a." (This is the symmetric property of equality.)
It may work better to start with something even more simple such as the reflexive property, a = a.
Students slowly started putting things on the board, such as the additive equality, a x 0 = 0, PEMDAS (an acronym for order of operations), and a couple more. I then pointed out how these are accepted properties and rules. I used order of operations as a way to explain that the accepted order guarantees that any two people making a calculation from an expression will reach the same result.
I next put up the following nine properties:
1) addition property of equality
2) subtraction property of equality
3) multiplication property of equality
4) division property of equality
5) distributive property
6) substitution property
7) reflexive property
8) symmetric property
9) transitive property
I provided an example for the first property of writing it symbolically: if a = b then ac = bc for any other value c.
I asked students to write expressions for the other eight properties. I told them they could use their cell phone to search for assistance.
The students completed most of the other properties. These properties are the building blocks that we use for algebraic proofs.
The first piece I worked with was from an algebraic proof worksheet that a colleague found online. I talked through the first proof, asking the class at each step what allowed us to write the statement. I then wrote in the appropriate property. I told the class to use Q.E.D. to show that their proof was concluded.
Students were then turned loose on the next two proofs. I checked with students to verify they completed the second proof correctly. The remaining proofs were assigned as homework, although some students were able to finish these before leaving class.
I started by having the expression, "Oh yeah, prove it!" on the board. I then asked students what it means to prove something. The discussion brought out using evidence to demonstrate a theory or statement was correct.
I wanted to have students consider rules and properties they work with when calculating or working with expressions. The idea was to pull out some of the properties that we would use as building blocks for algebraic proofs. This turned out to be a bit tougher than I expected.
I did provide an example:
For any two real numbers a and b, if a equals b then b = a. The meaning is shown in writing by "If a = b, then b = a." (This is the symmetric property of equality.)
It may work better to start with something even more simple such as the reflexive property, a = a.
Students slowly started putting things on the board, such as the additive equality, a x 0 = 0, PEMDAS (an acronym for order of operations), and a couple more. I then pointed out how these are accepted properties and rules. I used order of operations as a way to explain that the accepted order guarantees that any two people making a calculation from an expression will reach the same result.
I next put up the following nine properties:
1) addition property of equality
2) subtraction property of equality
3) multiplication property of equality
4) division property of equality
5) distributive property
6) substitution property
7) reflexive property
8) symmetric property
9) transitive property
I provided an example for the first property of writing it symbolically: if a = b then ac = bc for any other value c.
I asked students to write expressions for the other eight properties. I told them they could use their cell phone to search for assistance.
The students completed most of the other properties. These properties are the building blocks that we use for algebraic proofs.
The first piece I worked with was from an algebraic proof worksheet that a colleague found online. I talked through the first proof, asking the class at each step what allowed us to write the statement. I then wrote in the appropriate property. I told the class to use Q.E.D. to show that their proof was concluded.
Students were then turned loose on the next two proofs. I checked with students to verify they completed the second proof correctly. The remaining proofs were assigned as homework, although some students were able to finish these before leaving class.
Saturday, September 19, 2015
Geometric constructions continued
The second day of constructions started with students practicing duplicating an angle. This embedded constructing segments of the same length, so it was good practice for both constructions introduced on the first day.
Some students were still struggling a bit with the congruent angle construction. As I worked with these students, I could see the light bulb going off as to what they were doing and why they were doing it.
I then presented the next challenge. I drew a line segment on the board and labeled it as segment AB. I then placed a point C on the board such that C was not on the segment. I told students their task was to construct a line parallel to segment AB that passed through point C.
I let students play around with this for about 5 minutes. Most were stymied. I told them to think about what they knew about parallel lines and angle relationships. This got a few students moving. I told students to look in their notes to see what the relationships were. Some said they had no notes on this topic. As I looked through their notes, I pointed out where this information was recorded.
More and more students started to realize they needed to make use of corresponding angles, yet were reluctant to draw a new line (the transversal) to help. I pushed them to think about how they would duplicate the angle without adding any new lines. Most came to realize that they indeed had to draw a transversal.
With the transversal drawn, many students still struggled with how to proceed. I encouraged them to think about the process they used for duplicating angles. Tentatively, students started to create the congruent angle they needed. I noticed many students wanted to fall back on using a protractor or making use of markings on the compass as to the width of the compass opening. I had to reinforce that the compass was the measuring device and that any markings on the device was irrelevant.
As I walked around, I noticed one girl had done something entirely different. I asked her to explain her method to me. She said she had used the compass to measure the separation between points A and C. She then went to point B and drew an arc the same distance. Next she measured the distance between points A and B. Placing the compass on point C, see drew an arc that intersected with the first arc she had drawn and labeled the intersection point D.
She said that point D was the same distance away from B as C was from A. She drew the segment CD and said this was parallel because C and D were the same distance away from the segment and so every point in between will also be the same distance. She had, in effect, constructed a parallelogram. I congratulated her on her thinking. She said it just made more sense to her based on one of the definitions of parallel lines that we had examined.
I had a student present the construction of the parallel line by using corresponding angles. We discussed this construction and students asked questions for clarification. I could tell that many still felt shaky on this construction. I then had the one girl share her parallel points construction. The reaction from the class was this was so much simpler and easier to understand. I had to agree.
I spent the last few minutes of the class going over a couple of problems from a quiz I gave last week. I was disappointed in students not thinking about the information given to them. I discussed two problems in particular. One, the problem stated a ray was an angle bisector and students had to justify the equation they set up to solve the problem. The second involved midpoints; many students wrote on the quiz that they didn't remember the midpoint formula.
This disturbed me because I hadn't asked them to remember formulas but, rather, to use reasoning and common sense to find solutions. These problems are perfect examples of why students do not perform well on state and national assessments. They have been hammered with memorizing formulas that have no meaning to them versus understanding the facts of the situation and using some basic knowledge to construct a solution. Hopefully the class will become more comfortable with this approach as we move along.
The geometry team has a common assessment scheduled for next week that covers the first unit. I will spend the two class days before working with students to get better at identifying the relationships they are seeing. Their algebra skills seem adequate to solve the equations that result in the problems, it's the recognition of the relationships that appears to be the problem.
I was to introduce algebraic proofs as part of this unit. There is no way that I can do this justice prior to the assessment, so I'll hold off assessing this piece until later.
Some students were still struggling a bit with the congruent angle construction. As I worked with these students, I could see the light bulb going off as to what they were doing and why they were doing it.
I then presented the next challenge. I drew a line segment on the board and labeled it as segment AB. I then placed a point C on the board such that C was not on the segment. I told students their task was to construct a line parallel to segment AB that passed through point C.
I let students play around with this for about 5 minutes. Most were stymied. I told them to think about what they knew about parallel lines and angle relationships. This got a few students moving. I told students to look in their notes to see what the relationships were. Some said they had no notes on this topic. As I looked through their notes, I pointed out where this information was recorded.
More and more students started to realize they needed to make use of corresponding angles, yet were reluctant to draw a new line (the transversal) to help. I pushed them to think about how they would duplicate the angle without adding any new lines. Most came to realize that they indeed had to draw a transversal.
With the transversal drawn, many students still struggled with how to proceed. I encouraged them to think about the process they used for duplicating angles. Tentatively, students started to create the congruent angle they needed. I noticed many students wanted to fall back on using a protractor or making use of markings on the compass as to the width of the compass opening. I had to reinforce that the compass was the measuring device and that any markings on the device was irrelevant.
As I walked around, I noticed one girl had done something entirely different. I asked her to explain her method to me. She said she had used the compass to measure the separation between points A and C. She then went to point B and drew an arc the same distance. Next she measured the distance between points A and B. Placing the compass on point C, see drew an arc that intersected with the first arc she had drawn and labeled the intersection point D.
She said that point D was the same distance away from B as C was from A. She drew the segment CD and said this was parallel because C and D were the same distance away from the segment and so every point in between will also be the same distance. She had, in effect, constructed a parallelogram. I congratulated her on her thinking. She said it just made more sense to her based on one of the definitions of parallel lines that we had examined.
I had a student present the construction of the parallel line by using corresponding angles. We discussed this construction and students asked questions for clarification. I could tell that many still felt shaky on this construction. I then had the one girl share her parallel points construction. The reaction from the class was this was so much simpler and easier to understand. I had to agree.
I spent the last few minutes of the class going over a couple of problems from a quiz I gave last week. I was disappointed in students not thinking about the information given to them. I discussed two problems in particular. One, the problem stated a ray was an angle bisector and students had to justify the equation they set up to solve the problem. The second involved midpoints; many students wrote on the quiz that they didn't remember the midpoint formula.
This disturbed me because I hadn't asked them to remember formulas but, rather, to use reasoning and common sense to find solutions. These problems are perfect examples of why students do not perform well on state and national assessments. They have been hammered with memorizing formulas that have no meaning to them versus understanding the facts of the situation and using some basic knowledge to construct a solution. Hopefully the class will become more comfortable with this approach as we move along.
The geometry team has a common assessment scheduled for next week that covers the first unit. I will spend the two class days before working with students to get better at identifying the relationships they are seeing. Their algebra skills seem adequate to solve the equations that result in the problems, it's the recognition of the relationships that appears to be the problem.
I was to introduce algebraic proofs as part of this unit. There is no way that I can do this justice prior to the assessment, so I'll hold off assessing this piece until later.
Thursday, September 17, 2015
Introducing Geometric Constructions
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.
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.
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.
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