Today was a much better day. I guess the class just needed to settle back into the school routine.
I started off today with watching about 10 minutes of the movie October Sky. I'm using this as a follow-up to the ClassDojo videos on mindset. I asked the students to think about the ClassDojo videos and try to identify characters with fixed mindset versus growth mindset. The first 10 minutes establishes the setting and major characters. After viewing the clip I asked the class for observations and questions. We had a brief discussion about some of the behaviors they observed and about the historical setting of the film. My plan is to continue watching the film in 10 minute increments on our block days (we have eight more of these for the remainder of the semester).
While students would have liked to continue watching the video (anything but math, right?), I asked students if they had come up with estimates for the number of cookie boxes being held in the Nissan Rogue. No one had any answers, so I moved on to the next investigation. I intend to follow up with this problem later.
Next, I displayed the following statement: The surface area of a cube can never be less than its volume. I then asked whether or not this statement was true. I told students I would check class progress in 10 minutes and then started walking around and asking questions.
As I talked to students, they had some difficulty expressing their thoughts. Several times I heard that they knew what they were thinking but were having trouble articulating their thoughts. For these students, I asked if they could come up with an example that would help get their thoughts out. Others stated their belief about the truth of the statement. For these, I asked how they could convince the rest of the class about their belief. What evidence could they provide that would help them.
As time progress, students became even more involved in the task. I had small cubes that students could use to help them organize their thoughts. Others started drawing representations and examining side lengths and the resulting surface areas and volumes in a systematic manner. A couple of students honed in on the side length of 6 as a critical value to examine.
I let students work a bit longer than 10 minutes since I saw so much productive thinking and discussion taking place. I made note of who did what and how far they progressed and then pulled the class together for some share out. I asked groups who were still struggling with articulating their thinking to share a bit about what they were focused on. I progressed to students who had done some additional work and calculations share what they discovered. I then had students who had focused on the side length of 6 to share what they did and why they focused on that side length.
I then posed the question of why 6 was such a critical value. Students struggled some with expressing why this was a critical value. One student had actually written out an inequality x·x·6 > x·x·x. I asked this student to present this result and explain where it came from. In the presentation and resulting questions that arose, the class did a nice job trying to understand the relationship and importance of the 6 side length.
After this discussion, I wrote surface area above the left side and volume above the right side. I reminded students that they worked with similar inequalities in Algebra 1. Just as working with an equality, we could divide both sides with x·x and end up with 6 > x. This means that
surface area > volume as long as the side length is less than 6. When the side length equals 6 the two quantities are equal and when the side length is greater than 6 then the statement that surface area is greater than volume is no longer true.
We then pursued the next investigation; I displayed the following:
You are a burn victim in a hospital and need skin grafts for 70% of your body.
A doctor walks in, examines you and orders 162 cm2 of skin for you.
Should you think "Thank goodness!" or "Oh, no!"
I asked students to respond as to whether or not they should be concerned or not. This problem proved more difficult for students to grasp. Some tried to look up how much skin covered a human body. As I walked around the room, I began asking students how tall they were. I also asked them what their waist size was. Could they use those to help estimate their skin coverage. They still struggled with this. I took a sheet of paper and demonstrated how rolling the paper to form a cylinder made the sheet act like a wrapper. I then asked how big a wrapper would they need to cover themselves. This seemed to break the logjam and students started calculating a surface area.
The next issue came up with having the their personal measurements in in2 and having the given value in cm2. Students started to ask the relation between centimeters and inches. I provided the conversion that 1 in = 2.54 cm. There were still some points of confusion about conversions which I helped students work through. I used desktops as a way to explain that in2 represents a count of the number of grids that could be formed on the desktop. This seemed to help students differentiate between the measurement of one side of their desktop versus the grid cells covering their desktop.
As we came toward the end of the class, students started to see that 162 cm2 of skin was not nearly enough skin. They then started to wonder how much would be covered by 162 cm2? I explained that this would be equivalent to around a 10 inch by 3 inch patch.
With that, I passed out a sheet explaining how to read three dimensional math diagrams. These often show up on standardized tests and I wanted to help prepare them for their upcoming PSAT test.
Next class, I am away at our annual math competition. I will have students get some practice calculating surface areas and volumes while I am gone.
Wednesday, March 30, 2016
Tuesday, March 29, 2016
Girl scout cookies - opening of surface area and volume
Today was the first day back from Spring break. Perhaps it was too early to get things going like I planned, so while I believe the activity and investigation are solid, what transpired was not overly effective today.
I started class by showing Girl Scout Cookies as posted by Dan Meyer on 101 questions. For those who don't know Dan Meyer, he has created and presented many engaging math activities and is worth checking out.
So, I played the following video to start class.
I then asked the class what questions come to mind. I gave the class a little time to discuss with their partner and then did a share out. The first question to come out was, "Why are we watching this video?" I told the class the questions should focus on the situation being depicted.
The next question to come out was, "What does this have to do with math?" Not great, but at least it related. There were long periods of silence as students struggled to come up with questions. The following is a representation of questions that came up:
I started class by showing Girl Scout Cookies as posted by Dan Meyer on 101 questions. For those who don't know Dan Meyer, he has created and presented many engaging math activities and is worth checking out.
So, I played the following video to start class.
I then asked the class what questions come to mind. I gave the class a little time to discuss with their partner and then did a share out. The first question to come out was, "Why are we watching this video?" I told the class the questions should focus on the situation being depicted.
The next question to come out was, "What does this have to do with math?" Not great, but at least it related. There were long periods of silence as students struggled to come up with questions. The following is a representation of questions that came up:
- How many boxes does the cargo bay hold?
- Why does it have a Tennessee license plate?
- How would that many boxes affect the miles per gallon of the vehicle?
- Why didn't they fill the vehicle up entirely?
- How many more boxes could fit into the rest of the vehicle?
There actually weren't very many more questions than this, but I think you get the idea of how this was going.
I asked students what information they would need to answer the question about how many boxes would the cargo bay hold. One student said they would like to know the dimensions of the cargo bay. A second said they would like to know the dimensions of a cookie box. There was nothing else forthcoming.
I asked students how they could estimate the size of a cookie box. Again, no one had any idea. I asked how many students had ever sold or eaten girl scout cookies. Almost the entire class had, but still they had no idea about how large the box size would be.
I told students the could look up the dimensions of the cargo space. Students found that the cargo bay was 9.4 cubic feet. I asked what a cubic foot meant. No one had any idea. We discussed this a bit, but there was no connections being made.
The dimensions of the box are given in centimeters. I asked how centimeters and feet or inches were related. Students found online that 1 in = 2.54 centimeters and that 1 ft = 30.48 centimeters. I asked what they would do with this information.
One student suggested multiplying the 9.4 by the 30.48. The result is 286.512. I asked what this value meant and what units were associated with the value. Again, students had no idea.
All of the students are in a science class of one sort or another. I asked if they had ever seen unit conversions. None recognized nor remember converting units in science class. I explained that there are 30.48 cm / ft and that there are 9.4 ft3 of cargo space. Multiplying these two values together has the effect of 9.4 ft ft ft x 30.48 cm / ft. When doing this multiplication, one student recognized that we would end up with a ft2 and the value is 286.512 ft2 cm. The resulting units are still not useful at this point.
This fiasco took up the entire period and students still had not estimated how many boxes would fit in the cargo space. I told students to use whatever knowledge they had to come up with an estimate. I'll see what they come up with next class.
This fiasco took up the entire period and students still had not estimated how many boxes would fit in the cargo space. I told students to use whatever knowledge they had to come up with an estimate. I'll see what they come up with next class.
Thursday, March 17, 2016
Conclusion of right triangle trigonometry and using a concept map for assessment
Last class was the conclusion of right triangle trigonometry. Students worked on the final three application problems in the Solving Right Triangles Using Trigonometric Relationships packet. These proved challenging because of the vocabulary used in the descriptions. Students felt comfortable solving for the required values. For example, problem 33 discusses pitch and trusses, which are unfamiliar vocabulary for students. I provided a mini-lesson on house building so they could better understand what was being described and how the actual roof is laid on top of the trusses.
Common issues that came up in solutions were not converting all measurements to the same units, which caused students to mis-identify angle sizes. There were also some issues, based on the description, as to whether the description was referencing the horizontal distance or the distance along the hypotenuse. There were some good discussions as to the thinking about these situations.
As a lead in to the test next class, I asked students to construct a concept map with the central question of "What is trigonometry?" I often provide a vocabulary list and a list of linkage words to assist students. This time I just put the central question on the board and let students go.
[Update 2/23/2017: If you're interested in learning more about concept maps, check out 3 Ways Concept Maps Help You Learn.]
I did tell students that the concept map would be part of the unit assessment. I intend to make the test worth 40 points and then have the concept map be worth 10 points. I have used concept maps for review purposes, for assessments, and for previewing upcoming material. In this case, I want to see how students are making connections regarding right triangles, trigonometric ratios, and similarity.
Next class is the test and then its our one week Spring break. I won't have any posts until we return and begin looking at surface area and volume.
Common issues that came up in solutions were not converting all measurements to the same units, which caused students to mis-identify angle sizes. There were also some issues, based on the description, as to whether the description was referencing the horizontal distance or the distance along the hypotenuse. There were some good discussions as to the thinking about these situations.
As a lead in to the test next class, I asked students to construct a concept map with the central question of "What is trigonometry?" I often provide a vocabulary list and a list of linkage words to assist students. This time I just put the central question on the board and let students go.
[Update 2/23/2017: If you're interested in learning more about concept maps, check out 3 Ways Concept Maps Help You Learn.]
I did tell students that the concept map would be part of the unit assessment. I intend to make the test worth 40 points and then have the concept map be worth 10 points. I have used concept maps for review purposes, for assessments, and for previewing upcoming material. In this case, I want to see how students are making connections regarding right triangles, trigonometric ratios, and similarity.
Next class is the test and then its our one week Spring break. I won't have any posts until we return and begin looking at surface area and volume.
Tuesday, March 15, 2016
Wrapping up solving right triangles
Today we wrapped up work on solving right triangles. I gave time for students to finish working on problems 13-30 of the Solving Right Triangles Using Trigonometric Relationships packet. The students had some confusion about problems 19-30. I had to explain that they were trying to match up equivalent expressions and values. Once they understood the exercise they were able to proceed ahead with some success.
I had a students paper projected on the board in order to discuss responses to problems 13-18. There were some points of contention as to results but largely the class seemed successful in completing these problems.
A couple of questions about labeling diagrams came up. These related to which angle should be designated "A" and which "B." As the class discussed this, students started to realize that it didn't matter which was labeled which way, as long as the corresponding side lengths associated with the angle were also labeled properly. I was pleased to see that students were starting to think abstractly in the sense that the diagram orientation didn't matter, it was how the diagram was labeled that dictated its properties.
For the matching exercise of items 19-30, students struggled with this. Many students were matching things like sin(A), cos(A), tan(A) thinking these formed the group of all the basic trig ratios for angle A. There were other items that were left unmatched. Part of the problem lay with students' ability to work with expressions like 1 / (a/b). Their algebraic skills and reasoning are not where they need to be when working with values like these. I worked with several students individually to help with their thinking about these compound fractions.
As we went through the matches, my sense was that students understood why sin(B) = cos(A) but weren't as sure about why 1 / tan(A) = tan(B) or why sin(A) / cos(A) = tan(A).
We'll work through and discuss the last three problems in this packet and then cover any questions students may have. The trig test will be Friday; there will be 10 free-response problems. The following week is our spring break. When we return, we'll dive into surface area and volume.
I had a students paper projected on the board in order to discuss responses to problems 13-18. There were some points of contention as to results but largely the class seemed successful in completing these problems.
A couple of questions about labeling diagrams came up. These related to which angle should be designated "A" and which "B." As the class discussed this, students started to realize that it didn't matter which was labeled which way, as long as the corresponding side lengths associated with the angle were also labeled properly. I was pleased to see that students were starting to think abstractly in the sense that the diagram orientation didn't matter, it was how the diagram was labeled that dictated its properties.
For the matching exercise of items 19-30, students struggled with this. Many students were matching things like sin(A), cos(A), tan(A) thinking these formed the group of all the basic trig ratios for angle A. There were other items that were left unmatched. Part of the problem lay with students' ability to work with expressions like 1 / (a/b). Their algebraic skills and reasoning are not where they need to be when working with values like these. I worked with several students individually to help with their thinking about these compound fractions.
As we went through the matches, my sense was that students understood why sin(B) = cos(A) but weren't as sure about why 1 / tan(A) = tan(B) or why sin(A) / cos(A) = tan(A).
We'll work through and discuss the last three problems in this packet and then cover any questions students may have. The trig test will be Friday; there will be 10 free-response problems. The following week is our spring break. When we return, we'll dive into surface area and volume.
Monday, March 14, 2016
More solving right triangles
As we move toward a test on right triangle trigonometry later this week, the class is getting in its last couple of days of practice. We worked on problems 13-18 in the Solving Right Triangles Using Trigonometric Relationships packet. Students had completed the previous problems without many issues and were doing well on these problems, albeit they still worked slowly. I left whatever problems they hadn't completed as homework. We'll go through answers tomorrow. I'm anticipating that we may need some class time to complete this work. We'll finish with the last three application problems over the next two class periods and then have a test. There will be 10 free response problems of varying degrees of difficulty with several application problems and a few solving right triangle problems that represent the work the class has been completing.
Friday, March 11, 2016
Trig ratios or scale factors
Today's class started by going through the answers of problems 8-15 on the Solving Right Triangles Using Trigonometric Relationships packet. I let students provide their responses and then we discussed any questions or differences. This piece went smoothly with very few questions and differences. I did a thumbs-up, thumbs-down check with the class as to their comfort and ability to solve problems like these. While very few thumbs were straight up, most students' thumbs were pointed in a positive direction.
We then moved on to the next twelve problems. These involve similar figures and parallel lines and them move into solving right triangles. The first six problems proved challenging for students to think about. As I worked with students, answering their questions, I tried to have students focus on the aspects that they recognized. By and large, they recognized many characteristics of the problems. The main issue was that there were not enough markings on some of the figures to proceed without making some assumptions about parallelism or the presence of right angles. Once these issues were resolved, the work proceeded well.
As students were working through the first six problems, it was interesting to watch their approaches. When you assume that the figures presented are similar and that some of the triangles are right triangles, students could use trigonometric ratios to establish proportions for answering the questions. When I mentioned that the triangles were similar and to think about scale factors, some students started establishing proportional relationships from that perspective. Because we've been working with trigonometric ratios for a while now, many of the students preferred to make use of the trigonometric ratios as opposed to using scale factors. Either way, they would end up with the same proportional relationship. It was interesting to watch and see who made connections between the two.
Because of the questions that arose in problems 1-6, many students jumped to working on problems 7-12. They felt comfortable working with these and knew how to proceed.
I assigned the problems out of 1-12 that they hadn't finished as homework. We'll check work next class before proceeding ahead with the next set of problems.
We then moved on to the next twelve problems. These involve similar figures and parallel lines and them move into solving right triangles. The first six problems proved challenging for students to think about. As I worked with students, answering their questions, I tried to have students focus on the aspects that they recognized. By and large, they recognized many characteristics of the problems. The main issue was that there were not enough markings on some of the figures to proceed without making some assumptions about parallelism or the presence of right angles. Once these issues were resolved, the work proceeded well.
As students were working through the first six problems, it was interesting to watch their approaches. When you assume that the figures presented are similar and that some of the triangles are right triangles, students could use trigonometric ratios to establish proportions for answering the questions. When I mentioned that the triangles were similar and to think about scale factors, some students started establishing proportional relationships from that perspective. Because we've been working with trigonometric ratios for a while now, many of the students preferred to make use of the trigonometric ratios as opposed to using scale factors. Either way, they would end up with the same proportional relationship. It was interesting to watch and see who made connections between the two.
Because of the questions that arose in problems 1-6, many students jumped to working on problems 7-12. They felt comfortable working with these and knew how to proceed.
I assigned the problems out of 1-12 that they hadn't finished as homework. We'll check work next class before proceeding ahead with the next set of problems.
Thursday, March 10, 2016
Continued work on solving right triangles
Class time was spent today continuing to work on the Solving Right Triangle Using Trigonometric Ratios scenarios. While students made some progress at home, there was still a lot of uncertainty and confidence in what they were doing.
I gave the class some time to compare what they had done at home and finish finding solutions to the problems. As the class continued to work through the scenarios I could see that overall, they were gaining confidence in their work.
I asked students to put the numeric answers to problems 4-7 on the board. I then asked the class if they agreed with the answers or not. The only problem where there was a difference of opinion was for problem 4. I had the representation drawn on the board and everyone agreed that it was a correct representation. The differences came in which trigonometric ratio was most appropriate to use and in the algebraic process to solve the equation. When given the equation tan(35o) = .7002 = 100 / x and asked to solve for x, there are still too many students who want to multiply both sides of the equation by 100. I asked students what happens if we multiply both sides of this equation by 100? On the left side, the result is 70.02. On the right, students started to recognize that the value is 1002 / x or
10,000 / x. This caused some cognitive dissonance which got some students thinking about what value they would have to multiply by in order to move forward. They slowly began to realize what they needed to do was multiply by x and then divide by .7002.
I had students proceed on to the next set of problems. At this point, the class seemed comfortable with what they needed to do. As they got to problems 13-15, the only questions they were asking were clarification or to check that they were proceeding correctly. I left whatever they hadn't completed as homework. I also asked that they start work on the next six problem, which involve working with angle relationships of parallel lines. I reminded students about our work last semester so that they would remember that they have notes and resources to draw upon as the proceed ahead with these problems.
I gave the class some time to compare what they had done at home and finish finding solutions to the problems. As the class continued to work through the scenarios I could see that overall, they were gaining confidence in their work.
I asked students to put the numeric answers to problems 4-7 on the board. I then asked the class if they agreed with the answers or not. The only problem where there was a difference of opinion was for problem 4. I had the representation drawn on the board and everyone agreed that it was a correct representation. The differences came in which trigonometric ratio was most appropriate to use and in the algebraic process to solve the equation. When given the equation tan(35o) = .7002 = 100 / x and asked to solve for x, there are still too many students who want to multiply both sides of the equation by 100. I asked students what happens if we multiply both sides of this equation by 100? On the left side, the result is 70.02. On the right, students started to recognize that the value is 1002 / x or
10,000 / x. This caused some cognitive dissonance which got some students thinking about what value they would have to multiply by in order to move forward. They slowly began to realize what they needed to do was multiply by x and then divide by .7002.
I had students proceed on to the next set of problems. At this point, the class seemed comfortable with what they needed to do. As they got to problems 13-15, the only questions they were asking were clarification or to check that they were proceeding correctly. I left whatever they hadn't completed as homework. I also asked that they start work on the next six problem, which involve working with angle relationships of parallel lines. I reminded students about our work last semester so that they would remember that they have notes and resources to draw upon as the proceed ahead with these problems.
Tuesday, March 8, 2016
Mis-representations of solving right triangle scenarios
Well, today was an interesting class. I started by having two different students draw their representations for the first problem of the Solving Right Triangles packet. I had two different students do the same for the second problem. Once these were up on the board, I asked the class to comment on or to ask questions about what they were seeing.
I waited, but nothing was forth-coming. I asked students to focus on the problem 1 representations. I wondered aloud why the wall heights were different for the two drawings. Students were a bit confused about why the drawings were different. Finally, a student said they thought the side opposite the 65o angle would be smaller. We then had a discussion about which side should be longer. I asked students how they could determine this. There were a few suggestions but finally someone said that the tangent represents the slope of the side and that a larger angle would have a bigger slope and therefore would be taller. It was a nice, mathematical response that the class could agree with. We also discussed the labeling of opposite and adjacent sides as the two drawings had the labels flipped-flopped. This difference was quickly resolved by the class.
We then moved to the second problem's representations. In both cases, the drawing indicated that the hypotenuse was 15' long. A few students questioned the drawings and a student drew what they thought the scenario looked like, showing the 15' length as the horizontal leg. Many students agreed with this representation and we had a brief discussion about why this drawing worked.
I then had different students come to the board and provide their solutions to each problem. There were some questions about the labeling of opposite and adjacent sides, about which trigonometric ratios were most appropriate to use, and about whether the calculations were correct. We were able to resolve these issues without too much difficulty.
At this point I asked students to label their opposite and adjacent sides for the remaining problems and work through solutions. Students still struggled with some of the representations, especially the third problem. They also struggled with actually performing the algebraic calculations necessary to derive a solution.
For problem 3, the issue was which length represented the 6 mile stretch. The first reading might lead one to believe that the horizontal length is 6 miles. I'll admit that was what I initially thought. But upon reading the description again, the road segment is 6 miles, which is the hypotenuse in the representation. This was confusing for students.
I asked students to complete finding solutions for the remaining problems. We'll go through these solutions next class and then proceed with working on the next set of problems.
I waited, but nothing was forth-coming. I asked students to focus on the problem 1 representations. I wondered aloud why the wall heights were different for the two drawings. Students were a bit confused about why the drawings were different. Finally, a student said they thought the side opposite the 65o angle would be smaller. We then had a discussion about which side should be longer. I asked students how they could determine this. There were a few suggestions but finally someone said that the tangent represents the slope of the side and that a larger angle would have a bigger slope and therefore would be taller. It was a nice, mathematical response that the class could agree with. We also discussed the labeling of opposite and adjacent sides as the two drawings had the labels flipped-flopped. This difference was quickly resolved by the class.
We then moved to the second problem's representations. In both cases, the drawing indicated that the hypotenuse was 15' long. A few students questioned the drawings and a student drew what they thought the scenario looked like, showing the 15' length as the horizontal leg. Many students agreed with this representation and we had a brief discussion about why this drawing worked.
I then had different students come to the board and provide their solutions to each problem. There were some questions about the labeling of opposite and adjacent sides, about which trigonometric ratios were most appropriate to use, and about whether the calculations were correct. We were able to resolve these issues without too much difficulty.
At this point I asked students to label their opposite and adjacent sides for the remaining problems and work through solutions. Students still struggled with some of the representations, especially the third problem. They also struggled with actually performing the algebraic calculations necessary to derive a solution.
For problem 3, the issue was which length represented the 6 mile stretch. The first reading might lead one to believe that the horizontal length is 6 miles. I'll admit that was what I initially thought. But upon reading the description again, the road segment is 6 miles, which is the hypotenuse in the representation. This was confusing for students.
I asked students to complete finding solutions for the remaining problems. We'll go through these solutions next class and then proceed with working on the next set of problems.
Monday, March 7, 2016
Solving right triangle scenarios
As we move closer to wrapping up the right triangle trigonometry unit, the emphasis is on solving right triangles. I am using the Solving Right Triangles Using Trigonometric Ratios section from Mathematics Vision Project.
I started the class by going through the answers to the last three problems for the Finding the Value of a Relationship packet. Overall, students seemed to understand what the answers should be or, if they missed an answer, why they missed the answer.
Next, I passed out the next set of problems and emphasized that everyone should be able to draw a representation of the described situation. I let students work through the first couple of problems. Although the class tends to be slow at working through the relationships they can use to answer the question, they are getting better at understanding when they made an incorrect decision. For example, students find that they get leg lengths greater than the hypotenuse when they choose the wrong trigonometric ratio or apply the ratio incorrectly, perhaps by swapping the roles of opposite and adjacent side. They know they did something wrong but aren't always sure how to correct it.
Some students still struggle with the ideas or just aren't putting effort into learning how to use trigonometric ratios. For these students, I asked them to make the representations since they should be able to do at least that.
I let students work through the first few problems. For homework, I told the class to come next class with drawings for all seven problems. We'll discuss the first 2 problems and share out what people did. I am hopeful that this will give some needed direction for students who are still struggling with the material.
I started the class by going through the answers to the last three problems for the Finding the Value of a Relationship packet. Overall, students seemed to understand what the answers should be or, if they missed an answer, why they missed the answer.
Next, I passed out the next set of problems and emphasized that everyone should be able to draw a representation of the described situation. I let students work through the first couple of problems. Although the class tends to be slow at working through the relationships they can use to answer the question, they are getting better at understanding when they made an incorrect decision. For example, students find that they get leg lengths greater than the hypotenuse when they choose the wrong trigonometric ratio or apply the ratio incorrectly, perhaps by swapping the roles of opposite and adjacent side. They know they did something wrong but aren't always sure how to correct it.
Some students still struggle with the ideas or just aren't putting effort into learning how to use trigonometric ratios. For these students, I asked them to make the representations since they should be able to do at least that.
I let students work through the first few problems. For homework, I told the class to come next class with drawings for all seven problems. We'll discuss the first 2 problems and share out what people did. I am hopeful that this will give some needed direction for students who are still struggling with the material.
Friday, March 4, 2016
First right triangle trigonometry quiz
Just a brief update today. I gave a quiz today: 10 free response problems and 5 multiple choice questions. I told the class each free response problem was worth 1 point and that the multiple choice problems were practice preparation for the PSAT. These would be worth 1/2 point and could be applied toward point recovery on the free response questions (basically they are being treated as extra credit).
I was expecting the quiz to take almost the entire period. It did and then some. Only about 1/3 of the class completed the quiz during the period. I then assigned the quiz as a take-home quiz. For those completing the quiz during class, I won't cap the extra-credit at 10 points. I don't like to penalize students who may take more time to process through problems. This way students who actually did finish during class get some added benefit, so they don't feel like they are missing out on the benefit of finishing the quiz at home. I'll probably be a bit more forgiving when scoring work as well.
I'm hoping to wrap-up the right triangle trig work next week. We'll have a text covering all the trig material the following week.
I was expecting the quiz to take almost the entire period. It did and then some. Only about 1/3 of the class completed the quiz during the period. I then assigned the quiz as a take-home quiz. For those completing the quiz during class, I won't cap the extra-credit at 10 points. I don't like to penalize students who may take more time to process through problems. This way students who actually did finish during class get some added benefit, so they don't feel like they are missing out on the benefit of finishing the quiz at home. I'll probably be a bit more forgiving when scoring work as well.
I'm hoping to wrap-up the right triangle trig work next week. We'll have a text covering all the trig material the following week.
Thursday, March 3, 2016
Using ClassDojo growth mindset videos
Every Wednesday is a block day (90 minute class) for geometry. I decided to begin these classes showing videos about growth mindset. Helping instill growth mindset in students has been a school-wide initiative for the last couple of years.
ClassDojo developed five brief cartoons to communicate the ideas behind growth mindset. The first video in the series was released on January 19. A new chapter was then released each week after that. I held off starting the series immediately in case the published release schedule slipped.
The ClassDojo videos are highly entertaining and engaging. In addition, they provide valuable information about growth mindset, the power of perseverance and effort, and the benefit of learning from mistakes. My geometry students love these videos. They actually pay attention.
I have brief discussions after each video. I may ask students if they have any personal experiences that are similar to what occurred in the video or I may share a personal experience of my own. The thing that I notice is that the level of effort that students put into class and their attitude about working through problems has improved dramatically.
Sadly, we have now completed the entire ClassDojo video series. Some students suggested that we just start over again (they like the series that much). The final episode involves a failed rocket launch and the efforts to go back, to learn from the mistakes, and to try again. It reminded me of the film October Sky.
I asked my class how many had seen the movie. Only a couple of students had but agreed that the film and the video we just viewed had similarities. I am now thinking about watching October Sky in short segments and then discussing the growth and fixed mindsets exhibited during the film. I want to keep the class thinking about how facing and working through challenges can lead to personal growth and accomplishment.
If you can, check out the ClassDojo video series. They are well worth the time.
ClassDojo developed five brief cartoons to communicate the ideas behind growth mindset. The first video in the series was released on January 19. A new chapter was then released each week after that. I held off starting the series immediately in case the published release schedule slipped.
The ClassDojo videos are highly entertaining and engaging. In addition, they provide valuable information about growth mindset, the power of perseverance and effort, and the benefit of learning from mistakes. My geometry students love these videos. They actually pay attention.
I have brief discussions after each video. I may ask students if they have any personal experiences that are similar to what occurred in the video or I may share a personal experience of my own. The thing that I notice is that the level of effort that students put into class and their attitude about working through problems has improved dramatically.
Sadly, we have now completed the entire ClassDojo video series. Some students suggested that we just start over again (they like the series that much). The final episode involves a failed rocket launch and the efforts to go back, to learn from the mistakes, and to try again. It reminded me of the film October Sky.
I asked my class how many had seen the movie. Only a couple of students had but agreed that the film and the video we just viewed had similarities. I am now thinking about watching October Sky in short segments and then discussing the growth and fixed mindsets exhibited during the film. I want to keep the class thinking about how facing and working through challenges can lead to personal growth and accomplishment.
If you can, check out the ClassDojo video series. They are well worth the time.
Wednesday, March 2, 2016
Solving triangles using trigonometric ratios
Today's class didn't cover a lot of new territory, but it was a productive day.
We started at looking at the first two solving triangle problems (#4 and #5) in the Finding the Value of a Relationship packet. Approximately 20% of the class felt like they got it, another 30%-40% tried and but weren't confident about what they did, and the remainder didn't try. This was actually a better attempt/completion rate then I have had on the last few assignments.
I decided to spend time working through and checking work on these problems. I projected the two problems on the board and had students discuss what they attempted and what results they got. It was quite productive as students would ask questions or make comments about their results and why they felt a result was either correct or not. Covering these two problem in a discussion format like this with breaks in between for students to make calculations and look at their work took a while. Overall, students seemed to understand what they should be doing.
At this point, I used problem #4 to demonstrate using a calculator to find trig ratio values and inverse trig values. I did this mainly to help with preparation for the PSAT test that they will take next month. Some students were excited about using their calculator for the calculations, but many went right back to using the trig tables that they've been using.
I turned them loose on the next two problems. They class jumped in and tackled these problems. There were a lot of good discussions, I was called over and asked good clarifying questions or to check processes and results.
As I walked around, I helped students who were struggling or stuck. The basic issue with these students was where to begin. I would ask them which angle they wanted to work with. They would pick an angle and then I would ask them for the side with a given length, which side (opposite, adjacent, or hypotenuse) did that length represent. I would then ask students which trig ratios could they calculate for the given side. Students responded well to these prompts and could proceed from there. For the last problem, these students didn't require the prompts but asked clarifying questions and asked for feedback on their processes.
Students actually pushed themselves hard. It took a while for students to work through these problems because they were checking their work and then checking against others work. Even though we didn't get through many problems, I felt there was a large amount of progress in students comfort level with using trig ratios.
I assigned the last three problems in the packet as homework. This will provide a good indication of how well students are grasping solving triangles with trig ratios. We'll look at these problems and there will be a quiz on trig ratios next class.
We started at looking at the first two solving triangle problems (#4 and #5) in the Finding the Value of a Relationship packet. Approximately 20% of the class felt like they got it, another 30%-40% tried and but weren't confident about what they did, and the remainder didn't try. This was actually a better attempt/completion rate then I have had on the last few assignments.
I decided to spend time working through and checking work on these problems. I projected the two problems on the board and had students discuss what they attempted and what results they got. It was quite productive as students would ask questions or make comments about their results and why they felt a result was either correct or not. Covering these two problem in a discussion format like this with breaks in between for students to make calculations and look at their work took a while. Overall, students seemed to understand what they should be doing.
At this point, I used problem #4 to demonstrate using a calculator to find trig ratio values and inverse trig values. I did this mainly to help with preparation for the PSAT test that they will take next month. Some students were excited about using their calculator for the calculations, but many went right back to using the trig tables that they've been using.
I turned them loose on the next two problems. They class jumped in and tackled these problems. There were a lot of good discussions, I was called over and asked good clarifying questions or to check processes and results.
As I walked around, I helped students who were struggling or stuck. The basic issue with these students was where to begin. I would ask them which angle they wanted to work with. They would pick an angle and then I would ask them for the side with a given length, which side (opposite, adjacent, or hypotenuse) did that length represent. I would then ask students which trig ratios could they calculate for the given side. Students responded well to these prompts and could proceed from there. For the last problem, these students didn't require the prompts but asked clarifying questions and asked for feedback on their processes.
Students actually pushed themselves hard. It took a while for students to work through these problems because they were checking their work and then checking against others work. Even though we didn't get through many problems, I felt there was a large amount of progress in students comfort level with using trig ratios.
I assigned the last three problems in the packet as homework. This will provide a good indication of how well students are grasping solving triangles with trig ratios. We'll look at these problems and there will be a quiz on trig ratios next class.
Working with angles of elevation and depression
Today's class started by looking at the representations that students had created for the three scenarios given on page 3 of the Finding the Value of a Relationship packet. I had two representations drawn for each scenario using a different student for each drawing.
We started with the first scenario. Both drawings were good representations. I asked the class the similarities and differences they were seeing and if either drawing represented the described situation. Most students agreed that both drawings represented the scenario and that there were only minor differences (such as the orientation of the ladder leaning to the left or right) between the two.
Satisfied that the class felt this was a good representation, I asked the class how high up the wall the ladder reached. Students pondered this for a few moments. I then heard students saying that the height had to be less than 10 feet since that was the length of the hypotenuse. Others started using the Pythagorean theorem to find the third side length. Overall, I was pleased from the responses and discussions I was hearing.
From here, I asked what angle the ladder formed with the floor. This question baffled students for a bit. I asked the class if they could use any trig ratios to help identify the angle. Slowly, students started considering sines, cosines, and tangents. Students started calculating the ratio values but weren't sure what to do with these. I reminded them to use their trig tables. At this point many were able to determine the angle measurement created by the ladder and floor.
We then move onto the second scenario. The drawings were not as complete or correct as for the first scenario. We discussed what students were seeing. One drawing showed 3000 feet as the horizontal distance while the other showed the 3000 feet as the altitude. It took a while and some guidance to create a good representation.
I then asked the class who had flown on commercial airline flights and how high airplanes flew. A few students had flown and most of the class knew that an commercial airplane flies at an altitude of approximately 30,000 feet. I told the class that planes would have an initial descent and then go into their final descent. We assumed the final descent started at an altitude of 20,000 feet and that the angle of descent would be the 15o given in the scenario. I asked students how far away from the airport would the plane be when it started its final descent.
This posed a whole new issue for students. They were having trouble identifying the triangle they should work with and the relationships that they could use to answer the question. After much questioning and individualized help, I had the class moving in the right direction. Slowly, students started calculating their answers. At first, they questioned their calculations as their result was 74,641 feet. This translates to approximately 14 miles. I relayed how real flights feel like they are landing in the middle of nowhere at they make their approach since they start their final descent from so far away.
We next moved to the third scenario. Both representations were good and the class agreed. I then asked students to determine how far away the object was from the building base. Students fumbled around for this but several started to recognize this was basically the same situation as the landing airplane. For those not making the question, I asked them what connections they could make between the two situations.
We closed with looking at the next two problems (#4 and #5) in the packet. I assigned these for homework. We'll start by looking at how this work went at the start of next class.
We started with the first scenario. Both drawings were good representations. I asked the class the similarities and differences they were seeing and if either drawing represented the described situation. Most students agreed that both drawings represented the scenario and that there were only minor differences (such as the orientation of the ladder leaning to the left or right) between the two.
Satisfied that the class felt this was a good representation, I asked the class how high up the wall the ladder reached. Students pondered this for a few moments. I then heard students saying that the height had to be less than 10 feet since that was the length of the hypotenuse. Others started using the Pythagorean theorem to find the third side length. Overall, I was pleased from the responses and discussions I was hearing.
From here, I asked what angle the ladder formed with the floor. This question baffled students for a bit. I asked the class if they could use any trig ratios to help identify the angle. Slowly, students started considering sines, cosines, and tangents. Students started calculating the ratio values but weren't sure what to do with these. I reminded them to use their trig tables. At this point many were able to determine the angle measurement created by the ladder and floor.
We then move onto the second scenario. The drawings were not as complete or correct as for the first scenario. We discussed what students were seeing. One drawing showed 3000 feet as the horizontal distance while the other showed the 3000 feet as the altitude. It took a while and some guidance to create a good representation.
I then asked the class who had flown on commercial airline flights and how high airplanes flew. A few students had flown and most of the class knew that an commercial airplane flies at an altitude of approximately 30,000 feet. I told the class that planes would have an initial descent and then go into their final descent. We assumed the final descent started at an altitude of 20,000 feet and that the angle of descent would be the 15o given in the scenario. I asked students how far away from the airport would the plane be when it started its final descent.
This posed a whole new issue for students. They were having trouble identifying the triangle they should work with and the relationships that they could use to answer the question. After much questioning and individualized help, I had the class moving in the right direction. Slowly, students started calculating their answers. At first, they questioned their calculations as their result was 74,641 feet. This translates to approximately 14 miles. I relayed how real flights feel like they are landing in the middle of nowhere at they make their approach since they start their final descent from so far away.
We next moved to the third scenario. Both representations were good and the class agreed. I then asked students to determine how far away the object was from the building base. Students fumbled around for this but several started to recognize this was basically the same situation as the landing airplane. For those not making the question, I asked them what connections they could make between the two situations.
We closed with looking at the next two problems (#4 and #5) in the packet. I assigned these for homework. We'll start by looking at how this work went at the start of next class.
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