Today we started the unit on right triangle trigonometry. I began by asking students what trigonometry meant to them. Students identified the prefix tri and said it had something to do with three. I broke the word down tri | gonometry and wrote 3 under the tri. Some students connected gon with polygons and said it had to do with a figure or sides. I separated the word further tri | gon | ometry and wrote sides under the gon. Another student said the metry had to do with measure. I wrote measurement under ometry. I then said that we were using triangles as a way to measure things and would expand on how similar triangles can be used.
With that, we started the investigation using Are Relationships Predictable? package. Of course, this investigation required students to draw a 30o, 60o, 90o triangle, measure its sides, and then calculate ratios. As usual, some students still struggled with using a protractor to draw the appropriate angles and others had difficulty measuring sides lengths.
Once students had triangles drawn, I had a lot of questions, not unexpectedly, about which side was an opposite side and which was an adjacent side. We went through these using an example triangle I drew on the board. Many students were okay at this point, but I found a few that had inverted the ratio, calculating hypotenuse / adjacent rather than adjacent / hypotenuse, for example. I also encountered a couple of students that were either mixing angles and side measurements together or working exclusively with angle measurements as they tried to calculate their ratios.
We did reach a point as a class where students could see that the ratios, within given measurement error, were the same, regardless of the size of the triangle drawn. We then proceeded to the next piece where two triangles, each with a missing side length, were given. Students needed to calculate the missing side length and then calculate out given side ratios.
Students seemed baffled about how to find the missing side length. I reminded the class that these were right triangles, which prompted several to use the Pythagorean theorem to find the missing distance. As other students caught onto this idea, I walked around to check their work. The given side lengths were for one leg and the hypotenuse. What I saw was students calculate the missing side length as if it were the hypotenuse. As a result, they calculated a leg length that was longer than the hypotenuse. None of the students recognized this error.
I then had to remind the class that the hypotenuse was the longest side in a right triangle. If they calculated the missing side length and it was longer than the hypotenuse they did something wrong.
One student had calculated the missing side length as 4 when the hypotenuse was 6 and the other leg was 3. I asked her to recheck her calculations. At this point, class was ending. I asked the class to complete this second page. I also pointed out that the side length they calculated should be greater than 4 and less than 6.
I'll see how they did with this piece and proceed on with some work with right triangles and naming the ratios that the class has been calculating.
Tuesday, February 16, 2016
Wednesday, February 10, 2016
Transitioning to right triangle trig
The next unit focuses on right triangle trigonometry. We'll start in on this unit next week. I will be using investigations from the Mathematics Vision Project, starting with Are Relationships Predictable? This material builds directly from the work on similarity, so there is some nice continuity that can be drawn upon.
Tuesday, February 9, 2016
Floodlight Shadows conclusion
Today we wrapped up work on the floodlight shadows. It was interesting to see that students could intuitively express what they believed would happen with the shadows as the person walked away or toward a floodlight but they could not solve the problem. There continued to be guesses about lengths without supporting evidence. A few students knew they would need similar triangles and proportional relationships but didn't know how to proceed.
Even after discussing the situation and sketching out lines and triangles, students still struggled to use mathematical structure to model the situation. There is a distinct lack of understanding of how using math models can help to simplify, to describe, and to explain situations. So, while students made conjectures about the results, they had no way of supporting or confirming their conjectures. As I told the class, I could guess the shadow will be 9 feet long or 4 feet long, but without supporting evidence, it is simply a guess.
We then looked at the three sample solutions. Students discussed these and answered questions about the methods. I asked students to complete their work and self-evaluation forms and collected their work. I'll go through the responses and provide some additional feedback and comments.
Next class will be an assessment of their learning over quadrilaterals and similarity. I have one problem on the test that will draw on the modeling of shadow lengths. I'll be interested to see how students fare on this particular question.
Our next unit moves into right triangle trigonometry. This should be a natural progression from the similarity and triangle work that we have been using.
Even after discussing the situation and sketching out lines and triangles, students still struggled to use mathematical structure to model the situation. There is a distinct lack of understanding of how using math models can help to simplify, to describe, and to explain situations. So, while students made conjectures about the results, they had no way of supporting or confirming their conjectures. As I told the class, I could guess the shadow will be 9 feet long or 4 feet long, but without supporting evidence, it is simply a guess.
We then looked at the three sample solutions. Students discussed these and answered questions about the methods. I asked students to complete their work and self-evaluation forms and collected their work. I'll go through the responses and provide some additional feedback and comments.
Next class will be an assessment of their learning over quadrilaterals and similarity. I have one problem on the test that will draw on the modeling of shadow lengths. I'll be interested to see how students fare on this particular question.
Our next unit moves into right triangle trigonometry. This should be a natural progression from the similarity and triangle work that we have been using.
Monday, February 8, 2016
Floodlight Shadows
Today, the focus was on modeling a situation with geometric structures. The class started with going through questions on the multiple choice practice problems. The main issues on these questions were those involving given coordinates. For whatever reason, students don't think to sketch out the points. Once they do, answering the question becomes much simpler.
For the modeling problem, I used the Mathematical Assessment Project formative assessment problem Floodlight Shadows. Because of some time constraints, I modified the lesson structure somewhat.
I started by giving students the floodlight shadow task. I told them to work on their own for 10 minutes. As I monitored their progress, I ended up providing some extra time for their individual work. Many students could draw a representation but had difficulty overlaying mathematical structures that would help them. They also were coming up with answers for which they had no justification. I pushed them to justify why their response would be correct.
After, I had students work in groups. I provided a blank sheet of legal-sized paper to record their results.Some groups continued to struggle while others readily started drawing triangles to overlay on their drawings. For those struggling groups, I asked them what structures could be overlaid to help them model the situation. At this point many started to realize the triangular structures that were present.
The groups that originally started using triangles to model the situation also started to use similarity and scale factors to derive answers.
I asked the class to complete work tonight. Tomorrow we'll look at the sample responses provided in the lesson package and share out some of the class' thinking on solutions.
For the modeling problem, I used the Mathematical Assessment Project formative assessment problem Floodlight Shadows. Because of some time constraints, I modified the lesson structure somewhat.
I started by giving students the floodlight shadow task. I told them to work on their own for 10 minutes. As I monitored their progress, I ended up providing some extra time for their individual work. Many students could draw a representation but had difficulty overlaying mathematical structures that would help them. They also were coming up with answers for which they had no justification. I pushed them to justify why their response would be correct.
After, I had students work in groups. I provided a blank sheet of legal-sized paper to record their results.Some groups continued to struggle while others readily started drawing triangles to overlay on their drawings. For those struggling groups, I asked them what structures could be overlaid to help them model the situation. At this point many started to realize the triangular structures that were present.
The groups that originally started using triangles to model the situation also started to use similarity and scale factors to derive answers.
I asked the class to complete work tonight. Tomorrow we'll look at the sample responses provided in the lesson package and share out some of the class' thinking on solutions.
Friday, February 5, 2016
Discussion on similar triangles and other figures
Today was focused on students sharing and discussing responses to the questions they completed on the Similar Triangles and Other Figures investigation packet.
I first asked several students which of the five conjectures they thought were true. I wrote the conjecture numbers on the board. After getting responses from five of them, I had various students explain why they either thought the conjecture was true or not. Students were providing good counter-examples to disprove some conjectures and had solid reasoning to explain why other conjectures would be true.
As we proceeded through the questions, I called on different students. Overall, the responses were mathematically sound. We had a nice discussion around the AAA and AA theorems. We went through the proportion solutions and the only questions that came up were on the last two: one involving an expression in the denominator and the other involving square roots. I worked through these two problems and explained my thinking and results as I completed the problems.
For the last problem, the issue always comes up about simplifying radicals and results. For my part, I typically don't care the form in which the final answer is given; I care more about the thinking that went into finding the answer. However, I realize that standardized tests, especially multiple choice tests, typically simplify a response.
With this idea in mind, I explained to my class that while I would not mark them off for not simplifying, they should still practice since, as sophomores, they will be taking the PSAT test this Spring and the SAT test next year. I went through my thinking process for simplifying and what I was trying to accomplish. Student comments as I worked through the process showed they were making connections and understanding the hows and whys.
It turns out I had made copies of a selection of SAT style multiple choice questions covering quadrilaterals and similarity. I passed these out and asked students to work through them. I'll go over any problems they struggled with over the weekend and then work through and indirect measurement investigation in preparation for our unit test next week.
I first asked several students which of the five conjectures they thought were true. I wrote the conjecture numbers on the board. After getting responses from five of them, I had various students explain why they either thought the conjecture was true or not. Students were providing good counter-examples to disprove some conjectures and had solid reasoning to explain why other conjectures would be true.
As we proceeded through the questions, I called on different students. Overall, the responses were mathematically sound. We had a nice discussion around the AAA and AA theorems. We went through the proportion solutions and the only questions that came up were on the last two: one involving an expression in the denominator and the other involving square roots. I worked through these two problems and explained my thinking and results as I completed the problems.
For the last problem, the issue always comes up about simplifying radicals and results. For my part, I typically don't care the form in which the final answer is given; I care more about the thinking that went into finding the answer. However, I realize that standardized tests, especially multiple choice tests, typically simplify a response.
With this idea in mind, I explained to my class that while I would not mark them off for not simplifying, they should still practice since, as sophomores, they will be taking the PSAT test this Spring and the SAT test next year. I went through my thinking process for simplifying and what I was trying to accomplish. Student comments as I worked through the process showed they were making connections and understanding the hows and whys.
It turns out I had made copies of a selection of SAT style multiple choice questions covering quadrilaterals and similarity. I passed these out and asked students to work through them. I'll go over any problems they struggled with over the weekend and then work through and indirect measurement investigation in preparation for our unit test next week.
Wednesday, February 3, 2016
Similar Triangles and Other Figures investigation
Yesterday we had a snow day, so no school but lots of snow shoveling. Today we finished going through the last few problems on triangle dilation sheet and then moved to the next investigation.
The Similar Triangles and Other Figures investigation provides a formal definition for similarity and delves into AAA and the AA corollary for triangles. I asked students to answer the first seven questions on their own. I wanted them to get their thinking down before talking with others. My hope is that they would start to identify conflicting thoughts and start to resolve them.
The class worked through these, albeit slowly. Some students would call me over and discuss their conflicting thoughts. As they discussed their ideas, they started to realize which direction to take. As students started to discuss responses and ideas with each other, I heard many good discussions about examples and counter-examples.
I let students continue working through the questions and following problems. I decided to delay discussing responses until next class. I asked the class to finish the questions as best they can.
I am hopeful that misconceptions about similarity will be resolved for everyone by the end of the class. My intent is to provide some practice problems similar to what will be given on the unit test. I will then conclude the unit with an investigation that looks at using similarity to estimate shadow lengths and heights of objects.
The Similar Triangles and Other Figures investigation provides a formal definition for similarity and delves into AAA and the AA corollary for triangles. I asked students to answer the first seven questions on their own. I wanted them to get their thinking down before talking with others. My hope is that they would start to identify conflicting thoughts and start to resolve them.
The class worked through these, albeit slowly. Some students would call me over and discuss their conflicting thoughts. As they discussed their ideas, they started to realize which direction to take. As students started to discuss responses and ideas with each other, I heard many good discussions about examples and counter-examples.
I let students continue working through the questions and following problems. I decided to delay discussing responses until next class. I asked the class to finish the questions as best they can.
I am hopeful that misconceptions about similarity will be resolved for everyone by the end of the class. My intent is to provide some practice problems similar to what will be given on the unit test. I will then conclude the unit with an investigation that looks at using similarity to estimate shadow lengths and heights of objects.
Monday, February 1, 2016
Still working with triangle dilations
Well, today did not progress as far as I hoped. Students still had quite a few of the tasks from the Triangle Dilations packet. I told students to continue to work on these problems. I hoped that they could work on these for 15-20 minutes, we could then discuss responses, and then move into the next investigation set. Instead, the class worked on the problems for almost the entire period. I took time at the end of the period to have a class discussion covering the first four pages. Next class we'll wrap up the discussion and move into the next investigation tasks.
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