After investigating connections between rotations and reflections I started wondering about generalizations. I felt this would be a good way for students to practice working with rotations and gain some better insight into the difference between a procedure working in specialized situations or working in general.
I posed the question to the class whether the methods they used in the investigation (connecting line segments created from two different pairs of points between the pre-image and image) would always identify the center of rotation. I also asked if the two reflecting lines always provide a way to determine the angle of rotation and the center of rotation.
Tables investigated and discovered that the process of connecting two line segments as per the investigation did not work in general. This is a valuable lesson because too often students learn some procedure and start applying without regard to when it is appropriate to apply.
The second piece led to the conclusion that the center of rotation is at the point of intersection of the two reflecting lines and can be used to help determine the degree of rotation.
I then posed the reverse of this, given a pre-image and image figure, how do you determine the reflecting lines and from them identify the center of rotation and the degree of rotation.
I intend to use this as a springboard back to constructions. I am hoping that students make some connection to bisectors (perpendicular bisectors in particular).
Also, my intention is to use MS Paint to investigate dilations. I can use the coordinates of the figure and then change the size by percentage and look at the new coordinates. I think this will bring dilations to life and provide a more understandable setting as to what is going on and why.
Friday, October 16, 2015
Monday, October 12, 2015
Connecting reflections and translations - proving relationships
Today did not progress as I hoped. Students did not make any progress on why the three segments connecting pre-image to double-reflected points were all the same length. There is a distinct lack of initiative and strong desire to be told what to do that can be disheartening.
I decided to try work through the problem as a proof. Unfortunately, not thinking this through, I tackled all three segments at once, which resulted in extensive repetition and loss of the central issue. Doing this again, I would prove that one segment was equal to twice the distance between the parallel lines and then say that the exact same argument would apply to the other two segments.
As part of this proof, we needed a working definition of what it means to reflect a point. One student suggested a reflection was a flip of 180o over a given line. I drew an example that did not have the two flipped points equally distant from the reflection line. Students realized that the definition needed to include that the two points were the same distance from the reflection line. While the definition could be tightened up a bit, the critical aspect of equal distance apart was what was needed for the proof.
After plodding starting to plod through the proof, I could see that it was going to be too lengthy and confusing doing all three segments. I stopped along the way knowing that the pieces of thinking about what was given and how they relate had been communicated.
I then asked students to address how they knew if the three segments were parallel. Again, there was not much activity going on. Was it Monday doldrums or just a lack of interest? I told students to review angle relationships associated with parallel lines and transversals. At this point I had to ask the class about different angle relationships and their associated names.
The problem now made use of the reflection definition; a flip of 180o equated with the lines being perpendicular. Now, all the angle pairs are 90o and the converse relationships are in play.
The class ended with students tasked with determining how to position or to determine the location of the parallel lines given a specific reflection. As I walked around the room asking groups their thoughts before the end of class, I could see there was a lack of understanding about the concept of generalizing results. Yet something else that needs to be addressed.
I decided to try work through the problem as a proof. Unfortunately, not thinking this through, I tackled all three segments at once, which resulted in extensive repetition and loss of the central issue. Doing this again, I would prove that one segment was equal to twice the distance between the parallel lines and then say that the exact same argument would apply to the other two segments.
As part of this proof, we needed a working definition of what it means to reflect a point. One student suggested a reflection was a flip of 180o over a given line. I drew an example that did not have the two flipped points equally distant from the reflection line. Students realized that the definition needed to include that the two points were the same distance from the reflection line. While the definition could be tightened up a bit, the critical aspect of equal distance apart was what was needed for the proof.
After plodding starting to plod through the proof, I could see that it was going to be too lengthy and confusing doing all three segments. I stopped along the way knowing that the pieces of thinking about what was given and how they relate had been communicated.
I then asked students to address how they knew if the three segments were parallel. Again, there was not much activity going on. Was it Monday doldrums or just a lack of interest? I told students to review angle relationships associated with parallel lines and transversals. At this point I had to ask the class about different angle relationships and their associated names.
The problem now made use of the reflection definition; a flip of 180o equated with the lines being perpendicular. Now, all the angle pairs are 90o and the converse relationships are in play.
The class ended with students tasked with determining how to position or to determine the location of the parallel lines given a specific reflection. As I walked around the room asking groups their thoughts before the end of class, I could see there was a lack of understanding about the concept of generalizing results. Yet something else that needs to be addressed.
Friday, October 9, 2015
Making arguments in geometry and the need for vocabulary and notation
Today we continued looking at the connection between reflections and translations. Students struggled with explaining why the three segments connecting double-reflected images had to be parallel.
I had students talk things over in their groups. As I walked around asking what they were thinking, most groups expressed that they knew in their heads what they wanted to say but couldn't express it well. I told them to not worry about how rough or awkward the communication was, but to try to get their thoughts out.
The results were interesting. Most groups expressed conjectures or possible theorems about the relationship. For example, one group stated if the segments connecting the reflected points were parallel then the result would have to be a translation. Another group stated that if the segments were not parallel then the result could not be a translation. These indicated that the students were trying to think about the situation in a mathematical way.
One group said they were focused on the distance between the reflected figures and the given parallel lines. They weren't sure how to proceed with their thoughts but felt that the distances would be relevant to concluding the results were translations. Another student thought that the first reflection flipped the figure 180o. The second reflection over a parallel line would then flip the figure back and therefore it would be a translation. This was a good intuitive way of thinking about what was happening.
I used this as an opportunity to discuss maths vocabulary and notation. The class understood how they were struggling to communicate what they were thinking and seeing. This is why vocabulary and notation was developed. Sometimes it took centuries to create but the need to express and communicate thoughts drove the vocabulary and notation. Students were able to better appreciate the value of learning maths vocabulary and notation.
I then asked students to focus on the idea of distance. They've been working on this aspect of reflections and know that a reflected point is the same distance away from the line of reflection as the original point. I want to see if students can use this result to determine that all three line segments must be equal. I'll see what they come up with next class.
I had students talk things over in their groups. As I walked around asking what they were thinking, most groups expressed that they knew in their heads what they wanted to say but couldn't express it well. I told them to not worry about how rough or awkward the communication was, but to try to get their thoughts out.
The results were interesting. Most groups expressed conjectures or possible theorems about the relationship. For example, one group stated if the segments connecting the reflected points were parallel then the result would have to be a translation. Another group stated that if the segments were not parallel then the result could not be a translation. These indicated that the students were trying to think about the situation in a mathematical way.
One group said they were focused on the distance between the reflected figures and the given parallel lines. They weren't sure how to proceed with their thoughts but felt that the distances would be relevant to concluding the results were translations. Another student thought that the first reflection flipped the figure 180o. The second reflection over a parallel line would then flip the figure back and therefore it would be a translation. This was a good intuitive way of thinking about what was happening.
I used this as an opportunity to discuss maths vocabulary and notation. The class understood how they were struggling to communicate what they were thinking and seeing. This is why vocabulary and notation was developed. Sometimes it took centuries to create but the need to express and communicate thoughts drove the vocabulary and notation. Students were able to better appreciate the value of learning maths vocabulary and notation.
I then asked students to focus on the idea of distance. They've been working on this aspect of reflections and know that a reflected point is the same distance away from the line of reflection as the original point. I want to see if students can use this result to determine that all three line segments must be equal. I'll see what they come up with next class.
Thursday, October 8, 2015
Reflecting over the line y = 2x and connecting translations to reflections
Another short post. I had students try reflecting over y=2x. This was a struggle because students wanted the reflected points to behave nicely, i.e. reflect onto nice coordinates, which they weren't. I had to pass out tracing paper so that students could see that the image points they were drawing were not actually the reflection points. On the positive side, they stayed with the effort for over 30 minutes.
Students could see that the reflection points were not behaving well and that the nice symmetry of reflecting over a horizontal line, a vertical line, y=x, or y=-x was gone. Based on where they were, I determined that I needed to move on from this for now. Trying to investigate reflections over y=-2x or y=2x+3 would not be productive.
Since this was a block period, I did a quick brain break to re-energize the class and then moved on to translations. I had a triangle on a coordinate plane and asked what the translation of this figure by translating four points to the right and 3 points down would look like. I told students to guess at the meaning if they weren't sure what a translation was.
Students easily performed this task. We wrote out the notation (x, y) --> (x + 4, y - 3) as suggested by the class. I asked what the general expression would be for translating the x by a points and y by b points. One student suggested (x, y) --> ( x ± a, y ± b), which I hadn't expected. We tried a couple of more translations that I provided through expressions and then started to explore connections between translations and reflections.
I asked the class what connections they could think of between reflections and translations. No ideas were forth coming. I told the class we were going to explore this. The Slide Me Now activity from Navigating through Geometry provided the basis for this investigation. Students completed the first four questions during class. Their homework is to complete questions 5 & 6. I told the class to use what they know about parallel lines, transversals, and the angle relationships they learned to help them.
Next class I'll see what they come up with. I'm going to try to push this to begin looking a geometry proofs. I plan on completing the first two questions in the extension section of this investigation as well.
I'm thinking to revisit the reflection over y=2x and looking at why the segment connecting the pre-image and image points must be perpendicular to the line of reflection. This could lead into relationship of slope to perpendicular lines. I might also look at how to construct a segment bisector or perpendicular line as offshoots of this. I'm still debating which direction I want to take this piece.
Students could see that the reflection points were not behaving well and that the nice symmetry of reflecting over a horizontal line, a vertical line, y=x, or y=-x was gone. Based on where they were, I determined that I needed to move on from this for now. Trying to investigate reflections over y=-2x or y=2x+3 would not be productive.
Since this was a block period, I did a quick brain break to re-energize the class and then moved on to translations. I had a triangle on a coordinate plane and asked what the translation of this figure by translating four points to the right and 3 points down would look like. I told students to guess at the meaning if they weren't sure what a translation was.
Students easily performed this task. We wrote out the notation (x, y) --> (x + 4, y - 3) as suggested by the class. I asked what the general expression would be for translating the x by a points and y by b points. One student suggested (x, y) --> ( x ± a, y ± b), which I hadn't expected. We tried a couple of more translations that I provided through expressions and then started to explore connections between translations and reflections.
I asked the class what connections they could think of between reflections and translations. No ideas were forth coming. I told the class we were going to explore this. The Slide Me Now activity from Navigating through Geometry provided the basis for this investigation. Students completed the first four questions during class. Their homework is to complete questions 5 & 6. I told the class to use what they know about parallel lines, transversals, and the angle relationships they learned to help them.
Next class I'll see what they come up with. I'm going to try to push this to begin looking a geometry proofs. I plan on completing the first two questions in the extension section of this investigation as well.
I'm thinking to revisit the reflection over y=2x and looking at why the segment connecting the pre-image and image points must be perpendicular to the line of reflection. This could lead into relationship of slope to perpendicular lines. I might also look at how to construct a segment bisector or perpendicular line as offshoots of this. I'm still debating which direction I want to take this piece.
Tuesday, October 6, 2015
Reflecting over lines that are not vertical or horizontal
Things got back on track today. I had a brief discussion at the start of class. I explained that I expected them to struggle, that I expected them to discuss the problem, and that I expected them to ask questions of others or me. What I will not abide is intellectual laziness; the class sitting around waiting to be told exactly what steps to take. I related this to later in life when they are trained in a job. The training doesn't cover all situations. The expectation is that an employee will be able to apply their training to the situation at hand.
With that we explored reflections over the line y = x. I asked students to conjecture what would happen in this situation. I instructed students to close their eyes and picture the grid with the line y = x on it. I asked them to imagine what would happen as an object was reflected over this line. We discussed their thoughts, with many thinking about how the coordinates might go from positive to negative. One girl conjectured that this would be equivalent to reflecting over the x-axis and then over the y-axis. The students were ready to start exploring.
Some students did not know what the line y = x looked like and I had to assist them to get things started. The other thing that kept cropping up was that students would still end up reflecting as if it was over a vertical line or horizontal line. Once they got this down, they were able to see the result that the pre-image point (x, y) ends at the image point (y, x).
I asked the class to consider what if the line was y = -x. Students still felt that the x and y values would flip. A couple thought, in addition, that the coordinates would become negative. After exploring the results, the class saw that (x, y) --> (-y, -x).
We had time to start exploring further. I asked what would happen when reflecting over the line y=2x? Again, some students weren't sure how to graph the line. I reminded them how to create a table and then they were off exploring. The class ended at this point, so we'll pick up where we left off last time.
I really like this exploration as it is bringing back algebra review that students obviously need. Hopefully the energy level and effort will continue.
With that we explored reflections over the line y = x. I asked students to conjecture what would happen in this situation. I instructed students to close their eyes and picture the grid with the line y = x on it. I asked them to imagine what would happen as an object was reflected over this line. We discussed their thoughts, with many thinking about how the coordinates might go from positive to negative. One girl conjectured that this would be equivalent to reflecting over the x-axis and then over the y-axis. The students were ready to start exploring.
Some students did not know what the line y = x looked like and I had to assist them to get things started. The other thing that kept cropping up was that students would still end up reflecting as if it was over a vertical line or horizontal line. Once they got this down, they were able to see the result that the pre-image point (x, y) ends at the image point (y, x).
I asked the class to consider what if the line was y = -x. Students still felt that the x and y values would flip. A couple thought, in addition, that the coordinates would become negative. After exploring the results, the class saw that (x, y) --> (-y, -x).
We had time to start exploring further. I asked what would happen when reflecting over the line y=2x? Again, some students weren't sure how to graph the line. I reminded them how to create a table and then they were off exploring. The class ended at this point, so we'll pick up where we left off last time.
I really like this exploration as it is bringing back algebra review that students obviously need. Hopefully the energy level and effort will continue.
Friday, October 2, 2015
Reflecting over a vertical line
This will be a brief post. I continued with students trying to determine the relationship between reflected points and the line over which they points were reflected.
I encouraged students to write their results on the board (vertical line used, pre-image coordinates, image coordinates). After getting a half-dozen results on the board, I asked students to look for patterns and connections. Not much was happening, so I told students to get out of their seats and take a closer look for connections.
Students returned to their seats and some discussions began. Different students started interacting with other groups. Some students continued to try new values. I let the struggles continue and periodically asked what the connection was between the coordinates and the line used.
Finally, two girls went to the board on their own and were arguing/discussing the relationship. Their discussion centered on the idea of midpoints. After they had satisfied themselves, I asked the two to share their discussion with the rest of the class. There was a lot of discussion and questions about this idea, but gradually the class started to come together with the idea.
I checked their understanding by giving two x-coordinates and asking what line was reflected over. I then assigned them the task of coming up with an expression that could determine the image coordinate given the line of reflection and the pre-image point.
This entire exploration took 45 minutes, but they started to figure things out.
I encouraged students to write their results on the board (vertical line used, pre-image coordinates, image coordinates). After getting a half-dozen results on the board, I asked students to look for patterns and connections. Not much was happening, so I told students to get out of their seats and take a closer look for connections.
Students returned to their seats and some discussions began. Different students started interacting with other groups. Some students continued to try new values. I let the struggles continue and periodically asked what the connection was between the coordinates and the line used.
Finally, two girls went to the board on their own and were arguing/discussing the relationship. Their discussion centered on the idea of midpoints. After they had satisfied themselves, I asked the two to share their discussion with the rest of the class. There was a lot of discussion and questions about this idea, but gradually the class started to come together with the idea.
I checked their understanding by giving two x-coordinates and asking what line was reflected over. I then assigned them the task of coming up with an expression that could determine the image coordinate given the line of reflection and the pre-image point.
This entire exploration took 45 minutes, but they started to figure things out.
Wednesday, September 30, 2015
Introducing rigid transformations in geometry
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.
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.
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