For the past two classes, the focus has been on working with triangles. Students are starting to improve on names and definitions but still have a long way to go. Although there were only two students who could name vertical angles before, after working through problems in which vertical angles are used, more students can identify the angles.
I have noticed a tendency for students to work through identifying congruent parts and then not to take a step back and look at what they had shown to be congruent parts. As a result, they finish finding congruent pieces and then just start guessing which congruence theorem applies. I keep telling students to take a step back and look at what they have now identified as congruent. This helps a little, but I am not confident in the transference. Unless students can do this on their own, they will be lost.
The problems the class has been working on cover a wide variety of topics in order to get them prepared for our upcoming test. I have been trying to push them to be able to write proofs on their own without any extra help. There seems to be slight progress on this front. Their issues are tied to what I just previously discussed; they get lost in the details and never take a step back to look at the whole picture.
Perhaps it will help if they are stopped periodically in their work and asked what they have demonstrated so far and what they still need to demonstrate? I need to check research to see if there is anything that will help on this front.
We are going to finish up working through these problems and then we'll take the test. Since the test doesn't cover mid-segments and related topics, I am skipping the topic in order to catch up with the rest of the geometry team. I was planning on using the fire hydrant and warehouse placement investigations as outlined in the NCTM teacher's blog. It's unfortunate I don't have the time right now to fit these in; I think they would be productive investigations.
Up next, quadrilaterals and coordinate geometry proofs. I'm starting to look around at what's available from an inquiry-based approach. I sure hope there are some good pieces I can use.
Thursday, November 19, 2015
Monday, November 16, 2015
Working with triangle congruence
Today, we started by reviewing what we had learned so far about triangle congruence. I listed out SSS, SAS, and SSA and indicated which worked and which didn't. I then said the focus so far had been on looking at triangle sides. What happens when you focus on angles instead.
I wrote out the three equivalent statements for angles: AAA, ASA, and AAS. I felt this laid out some symmetry to the situation that would help students better relate to why we would investigate these options. I gave students some time to try these.
I walked around to see how things were progressing. There was the usual help with how to use a protractor but I was also amazed at how many students would just guess rather than try drawing the figures need. I asked how they came to their conclusion and the said they just guessed. What was the basis for the guess? It just looks like it wouldn't or would work. How can you know without having any foundation for your guess?
I encourage students to actually draw out their figures and measure their angles and sides. If nothing else, maybe they'll get better at using a protractor. The classroom discussion that resulted was mixed since there were still a decent number of students who just thought about it to draw a conclusion. We finally did get out that ASA and AAS worked but that AAA did not work.
We then practiced working through some congruence problems and a congruence proof. The proof gave the statements and students were to justify the statements with a reason. There was still a lot of struggle with the reasons although I am seeing some improvement.
The last two steps of the proof involve using a vertical pair and one of the triangle congruence theorems. Students didn't know what the name for the vertical pair was and couldn't articulate that because the angles were a vertical pair they were congruent. It's really difficult to write or complete proofs if you don't know that vocabulary and the properties of things.
Since no one seemed to remember or have this in their notes, the homework for tonight is to find out what the name of angles are when they are formed by two intersecting lines. They are also to identify the properties of these angle.
I wrote out the three equivalent statements for angles: AAA, ASA, and AAS. I felt this laid out some symmetry to the situation that would help students better relate to why we would investigate these options. I gave students some time to try these.
I walked around to see how things were progressing. There was the usual help with how to use a protractor but I was also amazed at how many students would just guess rather than try drawing the figures need. I asked how they came to their conclusion and the said they just guessed. What was the basis for the guess? It just looks like it wouldn't or would work. How can you know without having any foundation for your guess?
I encourage students to actually draw out their figures and measure their angles and sides. If nothing else, maybe they'll get better at using a protractor. The classroom discussion that resulted was mixed since there were still a decent number of students who just thought about it to draw a conclusion. We finally did get out that ASA and AAS worked but that AAA did not work.
We then practiced working through some congruence problems and a congruence proof. The proof gave the statements and students were to justify the statements with a reason. There was still a lot of struggle with the reasons although I am seeing some improvement.
The last two steps of the proof involve using a vertical pair and one of the triangle congruence theorems. Students didn't know what the name for the vertical pair was and couldn't articulate that because the angles were a vertical pair they were congruent. It's really difficult to write or complete proofs if you don't know that vocabulary and the properties of things.
Since no one seemed to remember or have this in their notes, the homework for tonight is to find out what the name of angles are when they are formed by two intersecting lines. They are also to identify the properties of these angle.
Friday, November 13, 2015
Tweeting triangles and triangle congruence
Last class I had students draw a triangle. and asked them to consider what were the fewest parts of a triangle that could be communicated so that the triangle could be copied exactly. I set this up as follows:
Second, you have just created what you consider to be the most beautiful triangle ever created. You are so excited about your triangle that you want to share it with the world. You go to your Twitter account to send a tweet and realize that the small number of characters you can send may be a problem.
First, draw what you consider to be the most beautiful triangle. Measure and record all of the side lengths and angles for your triangle. Be as precise as you can.
Second, you have just created what you consider to be the most beautiful triangle ever created. You are so excited about your triangle that you want to share it with the world. You go to your Twitter account to send a tweet and realize that the small number of characters you can send may be a problem.
Undaunted you decide to see what the fewest number of sides and angles of your triangle that you can tweet and still have followers be able to re-draw your beautiful triangle exactly as you have drawn it. Your task is to determine the fewest number of sides and angles of your triangle that you can tweet. Think about this problem and decide what you would do.
I gave students a couple of minutes and then have students pair up and see if their method works. I walk around to see what they come up with. I had students share out their findings, which follow:
- give 2 angles and 2 sides
- an equilateral triangle with the side length
- right or isosceles triangle and two side lengths
- 2 congruent side lengths and 1 angle
Many students had elected to draw isosceles or right triangles. As I walked around, one group had concluded that you just need 3 pieces of information about the triangle, deciding that AAA and SSA would also work.
I decided to tackle this by having the one group state that just three parts of the triangle were needed. I wrote this on the board. I focused on the right or isosceles triangle and two side lengths first. I didn't want to tackle right triangles yet, so I asked students to focus on the isosceles triangles.
I asked the class to re-create an isosceles triangle I was thinking of, the congruent side lengths were 5". After pondering this, students realized that there wasn't enough information. Without the angle formed by the congruent sides, they could not replicate the triangle. So, we basically needed to know the two sides and the angle formed by those two sides.
Does this work for any triangle? I asked students to try this out. There was still some question about the process working, mostly due to students' poor use of protractors and rulers (although they are much better than they were). I asked them to all draw a triangle that had side lengths of 3" and 4" with the angle formed by these sides being 70o. When they compared their drawings they realized that they had drawn congruent triangles. I wrote under the three parts side-angle-side and the abbreviation SAS. At the end of the line I wrote works.
I asked if the order given for the two sides and angle mattered. What would happen if you were told that a side of 3" joined a side of 4" and at the other end of the 4" side there was a 70o angle? Students determined that this didn't work. I wrote side-side-angle SSA and then didn't work on the board.
I next used the equilateral triangle as a springboard. In this situation, you are essentially communicating the lengths of all three sides. Would the idea of communicating all three side lengths work for any triangle? I gave them triangle side lengths of 3", 4", and 6" and asked them to draw the triangle to see if they had the same triangle. After the typical struggles and my referencing back to what was going on when they copies angles using a compass and straight-edge, the class decided that providing three sides worked. I wrote side-side-side SSS and works after it.
We were running out of time so I asked the class to consider the analogous situations of ASA and AAS, which flipped the SAS and SSA relationships. I'll use next class to work through these and also bring to attention of the entire class the use of AAA.
I asked the class to re-create an isosceles triangle I was thinking of, the congruent side lengths were 5". After pondering this, students realized that there wasn't enough information. Without the angle formed by the congruent sides, they could not replicate the triangle. So, we basically needed to know the two sides and the angle formed by those two sides.
Does this work for any triangle? I asked students to try this out. There was still some question about the process working, mostly due to students' poor use of protractors and rulers (although they are much better than they were). I asked them to all draw a triangle that had side lengths of 3" and 4" with the angle formed by these sides being 70o. When they compared their drawings they realized that they had drawn congruent triangles. I wrote under the three parts side-angle-side and the abbreviation SAS. At the end of the line I wrote works.
I asked if the order given for the two sides and angle mattered. What would happen if you were told that a side of 3" joined a side of 4" and at the other end of the 4" side there was a 70o angle? Students determined that this didn't work. I wrote side-side-angle SSA and then didn't work on the board.
I next used the equilateral triangle as a springboard. In this situation, you are essentially communicating the lengths of all three sides. Would the idea of communicating all three side lengths work for any triangle? I gave them triangle side lengths of 3", 4", and 6" and asked them to draw the triangle to see if they had the same triangle. After the typical struggles and my referencing back to what was going on when they copies angles using a compass and straight-edge, the class decided that providing three sides worked. I wrote side-side-side SSS and works after it.
We were running out of time so I asked the class to consider the analogous situations of ASA and AAS, which flipped the SAS and SSA relationships. I'll use next class to work through these and also bring to attention of the entire class the use of AAA.
Thursday, November 12, 2015
Introducing congruent triangles
Last class was devoted to introducing and discussing congruent shapes. I started the introduction by asking students to draw a simple figure, perhaps using a straight edge of graph paper. I then asked the class to make an exact copy of the figure they drew.
With these in hand, I asked the class how they knew that the figures were exactly the same? This led to a discussion about characteristics that each figure needed. The class boiled it down to each figure having the same number of sides with matching side lengths and that all the angles had to match in measurement.
I then tied this back to rigid motions. With rigid motions we are creating a pre-image and image that are congruent. I referenced how we labeled both images and then proceeded to show different pre-image and image figures from rigid motions placed on a graph.
The task for the class was how to mark the figures to show which pieces were congruent. There was a bit of hesitancy at first, but a student came to the board and appropriately marked the congruent sides. Another student then marked the congruent angles. This happened with the second pair of figures as well.
The third pair of figures presented a slight wrinkle. As before, a student marked congruent sides and then congruent angles. The pair presented were isosceles triangles. As the rest of the class looked at the markings, a couple of hands shot up. One girl asked whether two sides should actually be marked congruent. Others in the class agreed, noting that the triangle was isosceles and should have two congruent sides. (They were using actual side length to determine the congruence.) Then another student said that the base angles should also be marked as congruent.
I was pleased with students recognizing the anomaly in the last pair. I was also pleased with how comfortable students were with marking congruent sides and congruent angles.
I mentioned that what they had identified was that corresponding parts of congruent triangles are congruent. This was the first mention of what is commonly abbreviated CPCTC. I didn't dwell on this, but wanted to point it out in case they ran across this terminology.
I gave students some practice problems, identifying corresponding parts or writing congruence statements. I used the practice sheet problems that showed congruence statements as the guide of how to write a congruence statement. They readily picked it up and seemed comfortable working through the problems.
I next started the thinking about how much information is needed to convey congruence between triangles. I used a tweeting triangle lesson that I have used over the years.
The tweeting triangles lesson outline is posted on my website. I'll have students try their ideas next class and we'll, hopefully, build all of the triangle congruence shortcuts.
With these in hand, I asked the class how they knew that the figures were exactly the same? This led to a discussion about characteristics that each figure needed. The class boiled it down to each figure having the same number of sides with matching side lengths and that all the angles had to match in measurement.
I then tied this back to rigid motions. With rigid motions we are creating a pre-image and image that are congruent. I referenced how we labeled both images and then proceeded to show different pre-image and image figures from rigid motions placed on a graph.
The task for the class was how to mark the figures to show which pieces were congruent. There was a bit of hesitancy at first, but a student came to the board and appropriately marked the congruent sides. Another student then marked the congruent angles. This happened with the second pair of figures as well.
The third pair of figures presented a slight wrinkle. As before, a student marked congruent sides and then congruent angles. The pair presented were isosceles triangles. As the rest of the class looked at the markings, a couple of hands shot up. One girl asked whether two sides should actually be marked congruent. Others in the class agreed, noting that the triangle was isosceles and should have two congruent sides. (They were using actual side length to determine the congruence.) Then another student said that the base angles should also be marked as congruent.
I was pleased with students recognizing the anomaly in the last pair. I was also pleased with how comfortable students were with marking congruent sides and congruent angles.
I mentioned that what they had identified was that corresponding parts of congruent triangles are congruent. This was the first mention of what is commonly abbreviated CPCTC. I didn't dwell on this, but wanted to point it out in case they ran across this terminology.
I gave students some practice problems, identifying corresponding parts or writing congruence statements. I used the practice sheet problems that showed congruence statements as the guide of how to write a congruence statement. They readily picked it up and seemed comfortable working through the problems.
I next started the thinking about how much information is needed to convey congruence between triangles. I used a tweeting triangle lesson that I have used over the years.
The tweeting triangles lesson outline is posted on my website. I'll have students try their ideas next class and we'll, hopefully, build all of the triangle congruence shortcuts.
Tuesday, November 10, 2015
Why do students struggle with geometric proofs?
We've spent a couple of days working through puzzle type problems that involve angle relationships, both triangular and angles formed by lines (intersecting, parallel, etc.). Students have been engaged in working through these though they still struggle to different degrees. There is a tendency to not see what is given, to not think about what is given actually means, and to not use what has been solved as a piece of the next solution. After working through challenging problems for a couple of days they are doing better with these.
The other piece that continues to challenge students are geometric proofs. Since we've been working on problems involving the exterior angle theorem of a triangle, I thought this would be a simple proof for students to tackle.
I presented the scenario, with diagram, and asked them to describe how they could prove the following theorem: The measure of an exterior angle to a triangle equals the sum of the measures of the two non-adjacent interior angles of the triangle.
Walking around and talking with different groups and then discussing as a class showed they were thinking about the proof in a proper way. Students described that the exterior angle and the adjacent interior angle formed a linear pair and were supplementary. They explained that the three interior angles summed to 180o. They said you could substitute and subtract to show the result. They could say all of this but they couldn't write it out.
As I walked around, I could see that students would leave steps out. How do you know that those angle measurements sum to 180o? Why do you say these two angle measurements equal the exterior angle's measurement?
Some of the struggle comes from a lack of understanding the vocabulary. You cannot write a proof if you don't know what a linear pair is and what properties it possesses. Part of it is that students do not have experience in writing step-by-step detailed instructions. Writing a proof is akin to writing a computer program. You cannot skip steps or the computer won't understand what you want done. Proofs are the same. You can't skip a step or the reader is left scratching their head as to how you got from step A to step B.
We're told that students should have a level of proficiency in writing proofs by this time in the semester. I don't think that is realistic. Without having the background of writing detailed step-by-step instructions, students do not have any basis upon which to draw.
It is necessary to build this basis of writing detailed instructions that can lead to proof. I will be experimenting with some ideas. For example, have students construct a figure with compass and straight edge and then write instructions so that another student can replicate the figure. We are moving into congruence next. so I'll be able to test this out to see if it helps.
The other piece that continues to challenge students are geometric proofs. Since we've been working on problems involving the exterior angle theorem of a triangle, I thought this would be a simple proof for students to tackle.
I presented the scenario, with diagram, and asked them to describe how they could prove the following theorem: The measure of an exterior angle to a triangle equals the sum of the measures of the two non-adjacent interior angles of the triangle.
Walking around and talking with different groups and then discussing as a class showed they were thinking about the proof in a proper way. Students described that the exterior angle and the adjacent interior angle formed a linear pair and were supplementary. They explained that the three interior angles summed to 180o. They said you could substitute and subtract to show the result. They could say all of this but they couldn't write it out.
As I walked around, I could see that students would leave steps out. How do you know that those angle measurements sum to 180o? Why do you say these two angle measurements equal the exterior angle's measurement?
Some of the struggle comes from a lack of understanding the vocabulary. You cannot write a proof if you don't know what a linear pair is and what properties it possesses. Part of it is that students do not have experience in writing step-by-step detailed instructions. Writing a proof is akin to writing a computer program. You cannot skip steps or the computer won't understand what you want done. Proofs are the same. You can't skip a step or the reader is left scratching their head as to how you got from step A to step B.
We're told that students should have a level of proficiency in writing proofs by this time in the semester. I don't think that is realistic. Without having the background of writing detailed step-by-step instructions, students do not have any basis upon which to draw.
It is necessary to build this basis of writing detailed instructions that can lead to proof. I will be experimenting with some ideas. For example, have students construct a figure with compass and straight edge and then write instructions so that another student can replicate the figure. We are moving into congruence next. so I'll be able to test this out to see if it helps.
Thursday, November 5, 2015
Triangle properties
I tried to use workstations and have students investigate different triangle properties: isosceles triangle base angle theorems, exterior angle theorem, and correspondence of angle length to side length. These investigations came from the Discovering Geometry book and included using compass, protractor, and patty paper.
The investigations went okay. I particularly liked the exterior angle theorem investigation and the correspondence of angle size to opposite side length. For whatever reason, students got bogged down with the two isosceles triangle base angle theorems (two congruent sides results in two congruent angles and the converse).
I was expecting these investigations to go quicker than they did. After collecting sheets to see what students captured, it appeared that many of them did not pick up on key properties. I have to hope that as we continue to work with these properties that they will make sense. I intend to keep referencing back to these investigations, which I hope will reinforce the importance of making sense of what is happening during an investigation.
I intend to have students work through the exterior angle theorem and then complete an angle puzzle that a colleague found online.
The investigations went okay. I particularly liked the exterior angle theorem investigation and the correspondence of angle size to opposite side length. For whatever reason, students got bogged down with the two isosceles triangle base angle theorems (two congruent sides results in two congruent angles and the converse).
I was expecting these investigations to go quicker than they did. After collecting sheets to see what students captured, it appeared that many of them did not pick up on key properties. I have to hope that as we continue to work with these properties that they will make sense. I intend to keep referencing back to these investigations, which I hope will reinforce the importance of making sense of what is happening during an investigation.
I intend to have students work through the exterior angle theorem and then complete an angle puzzle that a colleague found online.
Tuesday, November 3, 2015
Working with triangles
The next unit in geometry covers triangles. I started by using a graphic organizer to see what students already know and correct any misconceptions. The organizer lists names of triangles (scalene, isosceles, equilateral, obtuse, etc.) down one side and properties (no congruent angles, 2 congruent angles, 3 congruent sides, 1 right angle, etc.) across the top. I had students fill out what they could, discuss in groups and then discuss as a class. I asked the class to complete this in pencil as I knew there would be corrections.
This led to some good questions and discussions about whether certain properties could exist. We worked through these questions and misconceptions and then used the information to either name or create described triangles.
The next class started by working through a couple more problems using naming to identify triangles. This class turned into a long, drawn-out affair because students couldn't find the side lengths of a triangle given its coordinates. I was baffled by this. I finally drew a single line segment on the graph and asked how they would determine the length of the segment. Students readily said that you would create a right triangle and use the Pythagorean theorem. It became apparent that students did not recognize triangle sides as line segments. Once we got over this hurdle we were able to complete the remainder of the practice problems (with some struggles on computation). Unfortunately, this took the entire class.
Today we worked through proving the interior angle sum theorem for triangles. I started with three progressive problems: the first gave to values and then asked the measure of the third angle, the second gave one value and two expressions for angle measure and asked for the value of x, the third gave three expressions for angle measurement and asked for the value of x. I anticipated that moving from the first to the second would give some students pause, which it did. Once they realized that the same interior sum property held, they were able to move on.
I then asked students what properties or theorems could be used to help prove that the interior angle sum theorem was true. After briefly discussing in groups we did a share out. Many of the groups thought if we could somehow relate the interior angles to a straight line it would help. (Many students were thinking along the line of unfolding the triangle.) One group suggested using a protractor and compass. A couple of groups thought the triangle angles all had to be less than 180o and that perhaps that could be made use of.
I drew a line parallel to one side. I could relate this to their thinking about somehow relating the angles to a straight line. I instructed them to look at the angle relationships when a transversal crosses a pair of parallel lines. How are the angle relationships connected.
This again was a struggle for the class. I kept reminding students to draw things out and make connections. Finally, a couple of students drew out the triangle, extended the sides to lines and then drew a parallel line. They started to mark congruent angles and saw the alternate interior angles connected to angles in the triangle.
I wrote out the first few steps in the proof and asked students to try and finish a two-column proof based on this start. It will be interesting to see if any are able to complete the thinking.
This led to some good questions and discussions about whether certain properties could exist. We worked through these questions and misconceptions and then used the information to either name or create described triangles.
The next class started by working through a couple more problems using naming to identify triangles. This class turned into a long, drawn-out affair because students couldn't find the side lengths of a triangle given its coordinates. I was baffled by this. I finally drew a single line segment on the graph and asked how they would determine the length of the segment. Students readily said that you would create a right triangle and use the Pythagorean theorem. It became apparent that students did not recognize triangle sides as line segments. Once we got over this hurdle we were able to complete the remainder of the practice problems (with some struggles on computation). Unfortunately, this took the entire class.
Today we worked through proving the interior angle sum theorem for triangles. I started with three progressive problems: the first gave to values and then asked the measure of the third angle, the second gave one value and two expressions for angle measure and asked for the value of x, the third gave three expressions for angle measurement and asked for the value of x. I anticipated that moving from the first to the second would give some students pause, which it did. Once they realized that the same interior sum property held, they were able to move on.
I then asked students what properties or theorems could be used to help prove that the interior angle sum theorem was true. After briefly discussing in groups we did a share out. Many of the groups thought if we could somehow relate the interior angles to a straight line it would help. (Many students were thinking along the line of unfolding the triangle.) One group suggested using a protractor and compass. A couple of groups thought the triangle angles all had to be less than 180o and that perhaps that could be made use of.
I drew a line parallel to one side. I could relate this to their thinking about somehow relating the angles to a straight line. I instructed them to look at the angle relationships when a transversal crosses a pair of parallel lines. How are the angle relationships connected.
This again was a struggle for the class. I kept reminding students to draw things out and make connections. Finally, a couple of students drew out the triangle, extended the sides to lines and then drew a parallel line. They started to mark congruent angles and saw the alternate interior angles connected to angles in the triangle.
I wrote out the first few steps in the proof and asked students to try and finish a two-column proof based on this start. It will be interesting to see if any are able to complete the thinking.
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