Today was one of those frustrating days. It was evident that students hadn't looked at any of the problems over the weekend. In addition, the class just wouldn't settle down to think about or discuss problems.
I wanted to start by looking at the results for work on the second page of Finding the Value of a Relationship. This proved futile as students hadn't made their lists for problem 4 and had simply plugged in random degree measurements for problem 3.
We revisited that you needed to use the information given to help you answer the question. I drew and labeled the triangle and asked the class what trig ratios could be formed from the given information. Finally, someone mentioned that the tan(α) could be calculated using the ratio 12/8. I then told students to use their trig tables to find the angle that produced a tangent ratio of 1.5. The angle is approximately 56.5o. I then asked, knowing this, what the measure of the remaining angle would be. The non-response was overwhelming; you could hear crickets chirping, it was that quiet. Finally, a student questioned that shouldn't the sum of the angles sum to 180o? At least about half the class recognized that they should use Pythagorean theorem to calculate the third side length.
No one had bothered to make a list for the fourth problem, so I decided to move on. I might re-visit this a bit later but I didn't want to take the class time. I had hoped to use the lists to refine student thinking about trig ratios. However, since their thinking this morning was rough, at best, it just didn't seem like a good use of time.
Moving on to angles of elevation and depression, I had students read the description of the hiking situation for problem 5 and then asked them to draw and label a representation for the problem. Students were totally confused by the description. I had to demonstrate sight lines and the angles of elevation and depression.
I told students to hold one arm straight out in front of them. This was their sight line. I then told students to use their other arm and raise it up. The angle formed between the sight line and the raised arm was the angle of elevation. Similarly, if they lowered their raised arm, the would form an angle of depression. More students had an idea of what the two angles meant.
After a few minutes, I had four different students copy their drawings and labels on the board. I asked students to look over the drawings and comment on the similarities and differences they were seeing. Instead, I had a student ask which drawing was right. I pointed out that that was not the question I asked. I wanted to know what similarities and differences they were seeing among the four drawings.
We had a brief discussion, as there were some distinct differences and few similarities in the drawings. We then revisited the meaning of angles of elevation and angles of depression. At this point students saw that one of the drawings represented the situation better than the others.
We used this drawing to discuss the claims made in problem 5. First, I asked students about the two lines of sight drawn. A couple of students realized that the lines of sight should be parallel. I labeled the two lines as parallel. One of the interior angles was labeled as 23.5o and the alternate interior angle was labeled as 66.5o. I asked the class what type of angle pair did they form. A few students recognized they were alternate interior angles. I asked what the angle relationship was between the angles. Some other students said they would be congruent. I then asked why the angles were not the same angle. Again, there was silence. I decided to let students ponder this situation some more. I am hoping that they will come back with the conviction that the angles, indeed, should be congruent.
I asked students to complete sketches for the next three problems. There was about 10 minutes left in class and I thought we might be able to look at some sketches before class was over. Unfortunately, this activity went as poorly as the rest. Students were drawing random triangles, were not labeling what they had drawn, and, generally, made no discernible effort to represent the situation at hand.
I talked to several students and then the entire class about using their sketches to model the described situation. I left these sketches as homework. We'll take a look at the sketches and discuss the situations before revisiting the question about the relationship between angles of elevation and angles of depression.
Monday, February 29, 2016
Friday, February 26, 2016
Slope, tangent, and introducing inverse trig functions
I read through the exit slips I had students produce last class. On each slip, I made comments or wrote questions to push student thinking further. At the start of class, I asked students to write on the board their solutions to the last three problems (15-17) on the Relationships with Meaning packet.
At this point I checked to make sure students had calculated the slopes in problems 12-14. There was some disagreement on the value of the slope in problem 14. Everyone had that the slope would be negative but values were slightly different: -1/2, -3/2, -5/4. I used this as an opportunity to discuss how work could be checked. Other than students saying to ask others, they didn't have any way to verify their work. I pointed out that they could check slope using multiple points. This allows students to verify their own calculations rather than relying on others.
We then turned our attention to problems 15-17. There were no questions on problem 15, the work on problem 16 had failed to square the value √116. A few students immediately recognized the issue and corrected the error. Problem 17 was done correctly. Overall, students seemed fairly comfortable with the work.
At this point, I passed back the exit slips to each student. I then labeled as angle A the angle opposite the "height" leg of each triangle that was drawn for problems 15-17. I asked students to calculate the tan(A) for each triangle. I then asked students to look at the calculated slope, the tan(A) and what they had wrote the previous class about connections between slope and trigonometric functions. I had the class capture their thinking and connections in their notes.
I next moved into Finding the Value of a Relationship investigation. I started by showing students how to use the trig table to find an angle when they are given the value of the trig ratio. I want students to work through inverse trig functions using a trig table first so they can better connect the back and forth nature of angles to trig ratios and back to angles. Once this connection is solidified, I will show them how to use a calculator to get a more precise result.
Students started working through the first 6 problems. Questions arose about having to find all of the sides lengths. I had to clarify that they only needed for the first problem to identify the trig ratios they could calculate for the given values.
I asked students to do their best to complete the first 6 problems for homework. We'll start next class be looking at how well they did on these problems.
At this point I checked to make sure students had calculated the slopes in problems 12-14. There was some disagreement on the value of the slope in problem 14. Everyone had that the slope would be negative but values were slightly different: -1/2, -3/2, -5/4. I used this as an opportunity to discuss how work could be checked. Other than students saying to ask others, they didn't have any way to verify their work. I pointed out that they could check slope using multiple points. This allows students to verify their own calculations rather than relying on others.
We then turned our attention to problems 15-17. There were no questions on problem 15, the work on problem 16 had failed to square the value √116. A few students immediately recognized the issue and corrected the error. Problem 17 was done correctly. Overall, students seemed fairly comfortable with the work.
At this point, I passed back the exit slips to each student. I then labeled as angle A the angle opposite the "height" leg of each triangle that was drawn for problems 15-17. I asked students to calculate the tan(A) for each triangle. I then asked students to look at the calculated slope, the tan(A) and what they had wrote the previous class about connections between slope and trigonometric functions. I had the class capture their thinking and connections in their notes.
I next moved into Finding the Value of a Relationship investigation. I started by showing students how to use the trig table to find an angle when they are given the value of the trig ratio. I want students to work through inverse trig functions using a trig table first so they can better connect the back and forth nature of angles to trig ratios and back to angles. Once this connection is solidified, I will show them how to use a calculator to get a more precise result.
Students started working through the first 6 problems. Questions arose about having to find all of the sides lengths. I had to clarify that they only needed for the first problem to identify the trig ratios they could calculate for the given values.
I asked students to do their best to complete the first 6 problems for homework. We'll start next class be looking at how well they did on these problems.
Wednesday, February 24, 2016
Developing a connection between trigonometric ratios and slope
Today, we continued working through the final problems on the Relationships with Meaning packet. I started by asking a student to share their work on problem 6, the first homework problem from last class. I projected the work on the board and asked students to comment on the work. Typically I want students to say whether or not they agree or disagree, to ask questions, or to comment on things they notice.
For the most part, students were not commenting. I finally asked students to look at the diagram that was sketched. Finally, a couple of students recognized that one of the leg lengths was longer than the hypotenuse. Several others jumped in about this not being possible. We had a good discussion about making calculations and then using other ways to check for the reasonableness of the calculation. While it is sometimes painful to display incorrect work, it also exposed a mistake that many students had made and how they could use this to learn and do better next time.
With that correction, I had students re-work their answers to this problem. They also took advantage to fix their results for the next problem. The results were again displayed and this time, the student presenting realized that she had reversed the sine and cosine ratios. She jumped back up and corrected her mistake.
The final problem in this set involved the equation sin(B) = 1 / √2. Invariably, students freeze when facing values like this. I pointed out that √2 is a value written with two squiggly lines, but so is 37. We're just more used to seeing a value like 37 rather than √2. With that, students tackled the final problem. Some got stuck on how to use the Pythagorean theorem when one of the values was √2. They started to realize that √2 x √2 = √4. They also realized that √4 = 2. With this they could now tackle the trigonometric ratios.
As a final piece, I had students look at the sin(B) = 1 / √2 result. I again reinforced that the PSAT and ACT would not provide this as a possible answer and that they needed to simplify the expression so that there was no radical in the denominator. Many students recognized that multiplying the numerator and denominator by √2 would achieve the desired result. We now had
sin(B) = 1 / √2 = √2 / 2.
I had students proceed to the next three statements and determine whether or not these were true or false. With the aid of the trig table, students were able to correctly determine which statements were true.
We then moved to the final page of the packet. The first set of problems asked students to draw slope defining triangles. I was concerned about the wording and asked students what a slope defining triangle would look like. A few students suggested that the triangle should be a right triangle. I felt students could proceed ahead with the work and asked them to draw their triangles and calculate their slopes.
The first issue I saw as I walked around the room was that students wanted to use the entire line length drawn. This meant that they had triangle vertices that did not correspond to grid line intersections and their slopes were not correct. The next issue was that some students still did not know how to calculate slope. The third issue I ran into was that students had trouble identifying whether the slope was positive or negative.
I spent a good deal of time walking around and helping correct misunderstandings and confusion. The one-on-one work seemed to help and students started to show a better understanding of why slopes were positive or negative and how the slopes were calculated.
The final three problems involved finding missing side lengths and two of the problems involved square roots. I was pleased when many of the students tackled these problems and were actually doing them correctly. Some students still had a little issue with the hypotenuse length of √116. With a little help they realized that squaring √116 produces a value of 116. The other positive in the class's work with these three problems was that students were resisting the urge to grab a calculator to turn the square root into an approximate decimal value.
I concluded the class by asking them to think about their slope calculations and to make connections of their calculations with sine, cosine, and tangent. I want to see how many students can identify the connection at this point.
We'll continue next class with working on the next investigation packet.
For the most part, students were not commenting. I finally asked students to look at the diagram that was sketched. Finally, a couple of students recognized that one of the leg lengths was longer than the hypotenuse. Several others jumped in about this not being possible. We had a good discussion about making calculations and then using other ways to check for the reasonableness of the calculation. While it is sometimes painful to display incorrect work, it also exposed a mistake that many students had made and how they could use this to learn and do better next time.
With that correction, I had students re-work their answers to this problem. They also took advantage to fix their results for the next problem. The results were again displayed and this time, the student presenting realized that she had reversed the sine and cosine ratios. She jumped back up and corrected her mistake.
The final problem in this set involved the equation sin(B) = 1 / √2. Invariably, students freeze when facing values like this. I pointed out that √2 is a value written with two squiggly lines, but so is 37. We're just more used to seeing a value like 37 rather than √2. With that, students tackled the final problem. Some got stuck on how to use the Pythagorean theorem when one of the values was √2. They started to realize that √2 x √2 = √4. They also realized that √4 = 2. With this they could now tackle the trigonometric ratios.
As a final piece, I had students look at the sin(B) = 1 / √2 result. I again reinforced that the PSAT and ACT would not provide this as a possible answer and that they needed to simplify the expression so that there was no radical in the denominator. Many students recognized that multiplying the numerator and denominator by √2 would achieve the desired result. We now had
sin(B) = 1 / √2 = √2 / 2.
I had students proceed to the next three statements and determine whether or not these were true or false. With the aid of the trig table, students were able to correctly determine which statements were true.
We then moved to the final page of the packet. The first set of problems asked students to draw slope defining triangles. I was concerned about the wording and asked students what a slope defining triangle would look like. A few students suggested that the triangle should be a right triangle. I felt students could proceed ahead with the work and asked them to draw their triangles and calculate their slopes.
The first issue I saw as I walked around the room was that students wanted to use the entire line length drawn. This meant that they had triangle vertices that did not correspond to grid line intersections and their slopes were not correct. The next issue was that some students still did not know how to calculate slope. The third issue I ran into was that students had trouble identifying whether the slope was positive or negative.
I spent a good deal of time walking around and helping correct misunderstandings and confusion. The one-on-one work seemed to help and students started to show a better understanding of why slopes were positive or negative and how the slopes were calculated.
The final three problems involved finding missing side lengths and two of the problems involved square roots. I was pleased when many of the students tackled these problems and were actually doing them correctly. Some students still had a little issue with the hypotenuse length of √116. With a little help they realized that squaring √116 produces a value of 116. The other positive in the class's work with these three problems was that students were resisting the urge to grab a calculator to turn the square root into an approximate decimal value.
I concluded the class by asking them to think about their slope calculations and to make connections of their calculations with sine, cosine, and tangent. I want to see how many students can identify the connection at this point.
We'll continue next class with working on the next investigation packet.
Tuesday, February 23, 2016
Connecting trigonometric ratios
Today was not overly productive as students had struggled with determining the side lengths and trig ratios for a triangle that had sin(30o) = 1/2. I drew a triangle on the board and labeled one of the angles as the 30o angle. I asked students what possible side lengths could be given the sin(30o) = 1/2. One student volunteered that the opposite side length could be 5 units and the hypotenuse could be 10 units. I wrote these values down.
I then asked what the length of the second leg had to be. Some students jumped at using the Pythagorean theorem while others just sat there. I asked how they could find the missing side length and the students started to realize what they needed to do.
When I asked the class what the value was, one student replied the square root of 75. Others confirmed this. Many students wanted to use their calculator to get an approximate value. I told students this would be a good opportunity to practice simplifying radicals. I had to remind some to use factor trees and we worked through getting a value of 5sqrt(3).
The next struggle came with students calculating the remaining trigonometric ratios. They finally came out of their morning stupor and calculated the values for the cosine and tangent of 30o. I then had to remind them that they needed to calculate all the trigonometric ratios. They then realized they also needed to calculate the sine, cosine, and tangent of 60o. We then checked answers. The results provided more opportunity to simplify radical expressions. We looked at how to re-express 1/sqrt(3), for example.
At this point, I pointed out that they should be able to determine all side lengths and trigonometric ratios from just knowing a single trigonometric ratio. I wrote down that the tan(A) = 3/4 and asked students to determine all the side lengths and trigonometric ratios. This exercise actually went better. There were still a couple of students that were totally confused but the rest of the class appeared to be understanding what was needed and how to proceed.
I then asked students if there were any issues with solving the four equations/proportions. The only problem that students had questions about was the second problem that involved a proportion:
2 / 3 = x / 21. I wrote this on the board and asked students what they attempted or what they could do. One student commented that 3 x 7 = 21. I wrote this expression out for the denominator and noted that we could not just multiply the denominator by 7. Students then saw that the numerator would be 2 x 7 and that x = 14. I also noted they could eliminate the fraction by multiplying by 21 (again noting the multiplicative property of equality). This yielded the result that 21 ( 2 / 3 ) = x or 14 = x.
With this, we were at the end of class. I asked students to complete problems 6-8 on the fourth page of the Relationships with Meaning packet. These are a continuation of the work we had been doing in class. Problem 5 was actually the follow-up problem we had worked on. I told students that it shouldn't take long and that they should not spend more than 30 minutes working through the problems. I am hopeful that tonight's homework will go better than the previous night's assignment.
I then asked what the length of the second leg had to be. Some students jumped at using the Pythagorean theorem while others just sat there. I asked how they could find the missing side length and the students started to realize what they needed to do.
When I asked the class what the value was, one student replied the square root of 75. Others confirmed this. Many students wanted to use their calculator to get an approximate value. I told students this would be a good opportunity to practice simplifying radicals. I had to remind some to use factor trees and we worked through getting a value of 5sqrt(3).
The next struggle came with students calculating the remaining trigonometric ratios. They finally came out of their morning stupor and calculated the values for the cosine and tangent of 30o. I then had to remind them that they needed to calculate all the trigonometric ratios. They then realized they also needed to calculate the sine, cosine, and tangent of 60o. We then checked answers. The results provided more opportunity to simplify radical expressions. We looked at how to re-express 1/sqrt(3), for example.
At this point, I pointed out that they should be able to determine all side lengths and trigonometric ratios from just knowing a single trigonometric ratio. I wrote down that the tan(A) = 3/4 and asked students to determine all the side lengths and trigonometric ratios. This exercise actually went better. There were still a couple of students that were totally confused but the rest of the class appeared to be understanding what was needed and how to proceed.
I then asked students if there were any issues with solving the four equations/proportions. The only problem that students had questions about was the second problem that involved a proportion:
2 / 3 = x / 21. I wrote this on the board and asked students what they attempted or what they could do. One student commented that 3 x 7 = 21. I wrote this expression out for the denominator and noted that we could not just multiply the denominator by 7. Students then saw that the numerator would be 2 x 7 and that x = 14. I also noted they could eliminate the fraction by multiplying by 21 (again noting the multiplicative property of equality). This yielded the result that 21 ( 2 / 3 ) = x or 14 = x.
With this, we were at the end of class. I asked students to complete problems 6-8 on the fourth page of the Relationships with Meaning packet. These are a continuation of the work we had been doing in class. Problem 5 was actually the follow-up problem we had worked on. I told students that it shouldn't take long and that they should not spend more than 30 minutes working through the problems. I am hopeful that tonight's homework will go better than the previous night's assignment.
Monday, February 22, 2016
Conjectures about sines, cosines, and tangents
Today's focus was on working through the conjectures made on the second page of the Relationships with Meaning packet. Through my observations of and discussions with student, I knew there were some misconceptions out there. I wanted the class to uncover the truths of the given conjectures for themselves, as best they could.
After giving students a couple of minutes to share what they had concluded over night, I wrote the numbers 5-13 on the board. I then went around the room and asked different students whether they thought the conjecture for that number was true or not. I didn't ask for any explanations or make any comment as to the correctness of the conclusion. I find students are more willing to share if there isn't any immediate judgement being placed on their work. Oftentimes, even students who normally don't want to say anything in class are willing to share a simple answer.
After writing either a T or F next to each number, I opened the floor up for discussion. Almost immediately, students wanted to discuss the problems. We worked our way through the conjectures. There were some good explanations about why a statement should be true or not. As we progressed, there were a few statements that baffled students. In particular, conjectures 7, 11, 12, and 13 showed students were not in agreement or were totally incorrect in their thinking.
I had anticipated from yesterday's work that some of the conjectures may pose a problem. I had copied a trig table for each student's use and passed these out. I briefly explained how to use the trig table and then asked students to try out the conjectures in question.
I did have to walk around and explain to some students how to make use of the table in more detail. I also had to watch out for students misinterpreting conjectures 11 and 13. I found that this assistance and using the trig table made the squaring of sine values much more understandable.
Students readily could see that the sin(A) = cos(90o - A) and that sin2(A) + cos2(A) = 1. There was still a bit of confusion on the meaning of sin2(A) = sin(A2), but I was able to provide concrete explanation using the trig table.
We were running out of time, but I asked students to verify conjectures 11 and 13 if they hadn't already done so. I also asked students to complete the next problem about calculating all the trigonometric ratios when a triangle has sin(30o) = 1/2. I also asked students to practice solving the four equations and proportions that followed.
After giving students a couple of minutes to share what they had concluded over night, I wrote the numbers 5-13 on the board. I then went around the room and asked different students whether they thought the conjecture for that number was true or not. I didn't ask for any explanations or make any comment as to the correctness of the conclusion. I find students are more willing to share if there isn't any immediate judgement being placed on their work. Oftentimes, even students who normally don't want to say anything in class are willing to share a simple answer.
After writing either a T or F next to each number, I opened the floor up for discussion. Almost immediately, students wanted to discuss the problems. We worked our way through the conjectures. There were some good explanations about why a statement should be true or not. As we progressed, there were a few statements that baffled students. In particular, conjectures 7, 11, 12, and 13 showed students were not in agreement or were totally incorrect in their thinking.
I had anticipated from yesterday's work that some of the conjectures may pose a problem. I had copied a trig table for each student's use and passed these out. I briefly explained how to use the trig table and then asked students to try out the conjectures in question.
I did have to walk around and explain to some students how to make use of the table in more detail. I also had to watch out for students misinterpreting conjectures 11 and 13. I found that this assistance and using the trig table made the squaring of sine values much more understandable.
Students readily could see that the sin(A) = cos(90o - A) and that sin2(A) + cos2(A) = 1. There was still a bit of confusion on the meaning of sin2(A) = sin(A2), but I was able to provide concrete explanation using the trig table.
We were running out of time, but I asked students to verify conjectures 11 and 13 if they hadn't already done so. I also asked students to complete the next problem about calculating all the trigonometric ratios when a triangle has sin(30o) = 1/2. I also asked students to practice solving the four equations and proportions that followed.
Friday, February 19, 2016
Continuing work with sine, cosine, and tangent
Today we continued working with sines, cosines, and tangents. The class was working through parts 1 and 2 of Relationships with Meaning. I was pleased to see that students had very few questions and were realizing how opposite and adjacent sides were flipping based on the angle being used. I know that this has been an issue in past years but doesn't seem to be an issue this year. Other colleagues have expressed that students continue to struggle with this idea. I have to believe that the way the trig ratios were introduced and worked with before naming and using them helped.
As students moved into part 2, students started having questions. The problems that asked about relationships for sin(90o - A) confused students. I told students to substitute in side length values to help them. I demonstrated this with approximate side length values for a 30o, 60o, 90o triangle. At this point students were able to continue on and make sense of the relationships being described.
Because we were also doing class registration work for next year, we weren't able to get further. Students will be completing whatever pieces they hadn't finished yet over the weekend. We'll discuss their thinking next class before working on new problems.
As students moved into part 2, students started having questions. The problems that asked about relationships for sin(90o - A) confused students. I told students to substitute in side length values to help them. I demonstrated this with approximate side length values for a 30o, 60o, 90o triangle. At this point students were able to continue on and make sense of the relationships being described.
Because we were also doing class registration work for next year, we weren't able to get further. Students will be completing whatever pieces they hadn't finished yet over the weekend. We'll discuss their thinking next class before working on new problems.
Wednesday, February 17, 2016
Introducing Sine, Cosine, and Tangent
Today's class started off a bit weird as a student got sick in class. We were working out in the hallway before finally switching to another classroom.
I started students by comparing their work from the night before. Most students had completed the work successfully and were getting comfortable with the concept of opposite and adjacent sides. I had students continue to the third page of work in Are Relationships Predictable? While there were some questions, students were getting more comfortable with using the Pythagorean theorem to find the length of either a missing leg or the missing hypotenuse length. They also were comfortable calculating the side ratios for the given angle.
The next page started naming the ratios. I told students that the ratios that they had been calculating actually had names. I went through the names and wrote the corresponding side ratios underneath the name. I also drew a right triangle on the board and used two different colors to indicate which sides were adjacent and opposite based upon the angle being used. Some students readily connected that they were still calculating the same ratios, we were just shortening the reference to the ratios by naming them.
The question came up about how to round or record an answer such as square root of 48. I told students that for standardized tests, such as the SAT or ACT, they would be expected to simplify the radical. We briefly revisited how to factor and simplify radicals. Most students said they didn't remember this, until I mentioned factor trees, at which point they all remembered simplifying radicals. I'll practice and revisit this periodically as the class gets closer to taking the PSAT in April.
One thing I was pleased with was that some students started to see the relationship that the sine of one angle was the same as the cosine of the other angle. We didn't discuss this but it is a noticing that can be built upon and used to discuss why this relationship exists.
Students continued to work through these problems. The remaining problems were assigned as homework for those that didn't finish the work in class.
The geometry team had a meeting this morning to discuss where we were and where we were going. One part of the discussion centered around covering the Pythagorean theorem. Now, my class has been using the Pythagorean theorem, albeit, with some assistance. But, this is a topic they should be familiar with and one that I believe can be used with some assistance rather than re-teaching the Pythagorean theorem. A colleague said they were spending a couple of days going through the Pythagorean theorem and its converse.
I know that there is a constant complaint that students never remember what they learned the prior year. This is a huge issue of frustration. Yet, teachers don't hold students accountable for their prior years of learning. Instead, they turn around and re-teach topics. It is no wonder that students don't retain information because what they have learned is that the "important" topics will be re-taught to them and they don't have to worry about retaining any learning.
I don't believe this situation will change until we teachers learn to hold students accountable for their learning and put the burden of retaining information on students shoulders. Re-teaching topics over and over again simply promotes the very learning behaviors that cause so much frustration. Stop re-teaching and hold students accountable for their prior learning. You may be surprised at how well students respond and rally to the challenge.
I started students by comparing their work from the night before. Most students had completed the work successfully and were getting comfortable with the concept of opposite and adjacent sides. I had students continue to the third page of work in Are Relationships Predictable? While there were some questions, students were getting more comfortable with using the Pythagorean theorem to find the length of either a missing leg or the missing hypotenuse length. They also were comfortable calculating the side ratios for the given angle.
The next page started naming the ratios. I told students that the ratios that they had been calculating actually had names. I went through the names and wrote the corresponding side ratios underneath the name. I also drew a right triangle on the board and used two different colors to indicate which sides were adjacent and opposite based upon the angle being used. Some students readily connected that they were still calculating the same ratios, we were just shortening the reference to the ratios by naming them.
The question came up about how to round or record an answer such as square root of 48. I told students that for standardized tests, such as the SAT or ACT, they would be expected to simplify the radical. We briefly revisited how to factor and simplify radicals. Most students said they didn't remember this, until I mentioned factor trees, at which point they all remembered simplifying radicals. I'll practice and revisit this periodically as the class gets closer to taking the PSAT in April.
One thing I was pleased with was that some students started to see the relationship that the sine of one angle was the same as the cosine of the other angle. We didn't discuss this but it is a noticing that can be built upon and used to discuss why this relationship exists.
Students continued to work through these problems. The remaining problems were assigned as homework for those that didn't finish the work in class.
The geometry team had a meeting this morning to discuss where we were and where we were going. One part of the discussion centered around covering the Pythagorean theorem. Now, my class has been using the Pythagorean theorem, albeit, with some assistance. But, this is a topic they should be familiar with and one that I believe can be used with some assistance rather than re-teaching the Pythagorean theorem. A colleague said they were spending a couple of days going through the Pythagorean theorem and its converse.
I know that there is a constant complaint that students never remember what they learned the prior year. This is a huge issue of frustration. Yet, teachers don't hold students accountable for their prior years of learning. Instead, they turn around and re-teach topics. It is no wonder that students don't retain information because what they have learned is that the "important" topics will be re-taught to them and they don't have to worry about retaining any learning.
I don't believe this situation will change until we teachers learn to hold students accountable for their learning and put the burden of retaining information on students shoulders. Re-teaching topics over and over again simply promotes the very learning behaviors that cause so much frustration. Stop re-teaching and hold students accountable for their prior learning. You may be surprised at how well students respond and rally to the challenge.
Tuesday, February 16, 2016
Starting Right Triangle Trigonometry
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.
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.
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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