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.
Friday, October 9, 2015
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.
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.
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