Today we moved back to work with Triangle Dilations. This work starts with students constructing dilations. Students grabbed their compasses and started working right away. Having worked through constructing dilations last class really helped. There were no questions and students were able to work through the initial constructions.
Listing proportions were a bit more troublesome. I had to clarify what was being asked. I decided to have students focus on the relationship of segments, for example, AB = 2A''B'', etc. Students could understand relationships expressed this way and it tied into the idea of scale factor. I'll wait until we go through responses to get into how to take these relationships and re-express them as proportions. I also am waiting to discuss their findings on the proportion relationships until next class.
The review of angle relationships was interesting. Students recognized that we had worked with these relationships but quite a few struggled with naming one or the other of the relationship names. I thought it was good review to keep the vocabulary up front with the class.
The next section on creating the described dilation seemed to be going well with the class. As I walked around, there really wasn't any questions about this section and the drawings appeared to be correct. Most of the class was either working on this section or just completing this section when class ended.
I asked students to complete the remainder of the questions as homework. I also told them that they should use a ruler to complete any remaining constructions.
The class worked throughout the entire period. I was encouraged by the work I saw produced and the small amount of support that the students needed to complete the tasks.
Next class we'll discuss responses to the questions. I want to flesh out the side relationships that exist between the pre-image and image in a dilation. I also want to get students to think about the side relationships in terms of a proportional relation.
Friday, January 29, 2016
Wednesday, January 27, 2016
Drawing dilations using a compass
Today we looked at the ideas connected with drawing dilations and finding centers of dilations. Class started by checking on responses to the Photocopy Faux Pas questions. Students seemed comfortable with identifying scale factors and finding corresponding side lengths.
One issue that needed clarification was the idea that a scale factor is actually a numeric value. This came about on the first problem in which the scale factor used was 2. When asking the class the scale factor, students replied that the scale factor doubled. While this was the result of using a scale factor of 2, double was not the scale factor.
Another issue that presented itself was that students did not understand the idea of center of a dilation. The scenarios presented in the last set of questions baffled students. The common thinking was the center of dilation was between the pre-image and image figures. I didn't press the issue but knew that we needed more work to build the understanding of center of a dilation.
I also checked their understanding about the issue of lengthening the rubber band chain that was used in the last class. Some thought the image would get larger but there was little confidence in the responses.
The next investigation I was planning on using was Triangle Dilations. Since this investigation required students to construct dilations and know centers of dilation, I didn't want to tackle this investigation just yet. First, I wanted students to learn how to construct dilations using a compass and practice a bit more with scale factors and work with centers of dilation.
A colleague had used the Drawing Dilations exploration from the On Core Mathematics curriculum. This exploration explains how to use a compass to draw dilations and provides work with using dilations in a coordinate plane. The dilation constructions require the construction of segment bisectors, which is a nice tie-in to previous geometric construction work. By the end of the exploration and work, students have a much better understanding of the center of a dilation.
This work took the entire class. Students had some basic questions and some needed reminding about segment bisector construction. The most difficulty occurred with dilations in the coordinate plane. Students repeatedly wanted to add the scale factor rather than multiply the scale factor. I tried to relate the idea of a scale factor to a continuous expansion or contraction. This seemed to help.
By the end of class students were feeling very comfortable with the compass constructions and understood how to apply scale factors in the coordinate plane. More importantly, they also had a better understanding about the center of dilations and what would happen with using longer rubber band chains to draw dilations.
When I asked the class what would happen if we used a longer rubber band chain, the class confidently stated the image would become larger. When I asked the class where the center of dilation is for the photocopy scenario, students recognized that the upper left corner was the center of the dilation. With these ideas solidified, I will proceed to the Triangle Dilations investigation next class.
One issue that needed clarification was the idea that a scale factor is actually a numeric value. This came about on the first problem in which the scale factor used was 2. When asking the class the scale factor, students replied that the scale factor doubled. While this was the result of using a scale factor of 2, double was not the scale factor.
Another issue that presented itself was that students did not understand the idea of center of a dilation. The scenarios presented in the last set of questions baffled students. The common thinking was the center of dilation was between the pre-image and image figures. I didn't press the issue but knew that we needed more work to build the understanding of center of a dilation.
I also checked their understanding about the issue of lengthening the rubber band chain that was used in the last class. Some thought the image would get larger but there was little confidence in the responses.
The next investigation I was planning on using was Triangle Dilations. Since this investigation required students to construct dilations and know centers of dilation, I didn't want to tackle this investigation just yet. First, I wanted students to learn how to construct dilations using a compass and practice a bit more with scale factors and work with centers of dilation.
A colleague had used the Drawing Dilations exploration from the On Core Mathematics curriculum. This exploration explains how to use a compass to draw dilations and provides work with using dilations in a coordinate plane. The dilation constructions require the construction of segment bisectors, which is a nice tie-in to previous geometric construction work. By the end of the exploration and work, students have a much better understanding of the center of a dilation.
This work took the entire class. Students had some basic questions and some needed reminding about segment bisector construction. The most difficulty occurred with dilations in the coordinate plane. Students repeatedly wanted to add the scale factor rather than multiply the scale factor. I tried to relate the idea of a scale factor to a continuous expansion or contraction. This seemed to help.
By the end of class students were feeling very comfortable with the compass constructions and understood how to apply scale factors in the coordinate plane. More importantly, they also had a better understanding about the center of dilations and what would happen with using longer rubber band chains to draw dilations.
When I asked the class what would happen if we used a longer rubber band chain, the class confidently stated the image would become larger. When I asked the class where the center of dilation is for the photocopy scenario, students recognized that the upper left corner was the center of the dilation. With these ideas solidified, I will proceed to the Triangle Dilations investigation next class.
Tuesday, January 26, 2016
Introducing similarity - dilations and scale factors
Similarity was introduced today. I used the Photocopy Faux Pas investigation package. Here is a run-down of how the lesson unfolded.
First, I told students we were going to look at similarity. Since this was the first look, we would work through an investigation to better understand the concepts.
I held up a copy of the first two pages of the packet (copied 2-sided). I told students not to be concerned with the questions on the back side for now. I asked students to read through the text one time to help establish context. Basically, I wanted to have students read through the text like they would a story. I then told them that they needed to read through the story a second time, highlighting or annotating key words, math ideas and pertinent information.
I passed out a copy to each student and let them read through and annotate. I had to remind students that this was not a group activity at this point and to read through the text twice: one time for context and one time for detail. I walked around to check on student progress and to gauge their level of engagement with annotating the text. Overall, this piece progressed smoothly.
Next I told students to imagine they were the boss. I asked the class to think about why things went wrong and how they could be corrected. With this perspective in mind, I asked the class to complete four of the questions on the back-side of the page (skipping question 2 for now). I told students to answer to the best of their ability given their current understanding.
I had tables share their responses to the fifth question and then we did a class share out. The list included things like initial position, starting size, desired end size, size of paper, etc. Students were clearly thinking about relevant issues.
We then went back to question two. This step required use of a rubber band stretcher. A colleague found instructions for a Two Band Stretcher Activity. I handed out the first page instructions to each table group and then either page 3 or page 4 to every student. I passed out two rubber bands to each student and demonstrated how to tie the rubber bands together. After the second pass through the tying instructions there were only 3 students who didn't have their rubber bands tied. Table mates helped these individuals so everyone was set.
I explained the process of having the knot of the rubber bands trace along the figure. They were not to look at what their pencils were doing. At this point the class started tracing their figures. Some students complained that their rectangle drawing went off the paper. My response was, "Well, what happened in the photocopy situation?" Students realized the same issue arose.
I asked the class to consider what they were seeing and to go back through their responses to the other four questions. I wanted the class to consider whether any of their responses changed based upon the rubber band drawing results.
I then passed out pages 3 and 4 from the Photocopy Faux Pas investigation packet. I briefly connected dilations and scale factors to image manipulation in software packages. I discussed the meaning of scale factor and then had students work on determining scale factors and side lengths.
As the class drew to a close I asked students to think about where the center of dilation was for the photocopier. At this point, a student asked what impact having a longer rubber band would have on the image drawing. It was a good question that was asked right when class ended. I asked students to think about how they would respond to that question as well.
Tomorrow we'll address those questions, discussion their ideas on the location of centers of dilations described in the problems and then move into the next investigation piece.
A colleague said that her students were still confused about the idea of doubling an image size. Did this mean doubling side lengths of did it mean doubling area. It will be interesting to see how my class thinks about the idea of doubling as we move forward.
First, I told students we were going to look at similarity. Since this was the first look, we would work through an investigation to better understand the concepts.
I held up a copy of the first two pages of the packet (copied 2-sided). I told students not to be concerned with the questions on the back side for now. I asked students to read through the text one time to help establish context. Basically, I wanted to have students read through the text like they would a story. I then told them that they needed to read through the story a second time, highlighting or annotating key words, math ideas and pertinent information.
I passed out a copy to each student and let them read through and annotate. I had to remind students that this was not a group activity at this point and to read through the text twice: one time for context and one time for detail. I walked around to check on student progress and to gauge their level of engagement with annotating the text. Overall, this piece progressed smoothly.
Next I told students to imagine they were the boss. I asked the class to think about why things went wrong and how they could be corrected. With this perspective in mind, I asked the class to complete four of the questions on the back-side of the page (skipping question 2 for now). I told students to answer to the best of their ability given their current understanding.
I had tables share their responses to the fifth question and then we did a class share out. The list included things like initial position, starting size, desired end size, size of paper, etc. Students were clearly thinking about relevant issues.
We then went back to question two. This step required use of a rubber band stretcher. A colleague found instructions for a Two Band Stretcher Activity. I handed out the first page instructions to each table group and then either page 3 or page 4 to every student. I passed out two rubber bands to each student and demonstrated how to tie the rubber bands together. After the second pass through the tying instructions there were only 3 students who didn't have their rubber bands tied. Table mates helped these individuals so everyone was set.
I explained the process of having the knot of the rubber bands trace along the figure. They were not to look at what their pencils were doing. At this point the class started tracing their figures. Some students complained that their rectangle drawing went off the paper. My response was, "Well, what happened in the photocopy situation?" Students realized the same issue arose.
I asked the class to consider what they were seeing and to go back through their responses to the other four questions. I wanted the class to consider whether any of their responses changed based upon the rubber band drawing results.
I then passed out pages 3 and 4 from the Photocopy Faux Pas investigation packet. I briefly connected dilations and scale factors to image manipulation in software packages. I discussed the meaning of scale factor and then had students work on determining scale factors and side lengths.
As the class drew to a close I asked students to think about where the center of dilation was for the photocopier. At this point, a student asked what impact having a longer rubber band would have on the image drawing. It was a good question that was asked right when class ended. I asked students to think about how they would respond to that question as well.
Tomorrow we'll address those questions, discussion their ideas on the location of centers of dilations described in the problems and then move into the next investigation piece.
A colleague said that her students were still confused about the idea of doubling an image size. Did this mean doubling side lengths of did it mean doubling area. It will be interesting to see how my class thinks about the idea of doubling as we move forward.
Monday, January 25, 2016
A final look at quadrilaterals in the coordinate plane
The common unit assessment we're using includes a couple of coordinate proof problems. Since we hadn't worked in the coordinate plane for a week or so, I decided it would be helpful to revisit this work before moving to similarity.
A colleague had put together four problems that covered this material. Each problem gave coordinates for a different quadrilateral. In the first problem, they had to calculate the slopes of the sides to show opposite sides were parallel. In the second, students had to calculate the length of the sides to show that opposite sides were congruent. In the third, students had to show that diagonals bisected each other. In the fourth, they used protractors to show that opposite angles were congruent.
This work went fairly well. Some students stilled had questions about calculating side lengths or slopes. For the most part they seemed comfortable with the tasks. For a couple of students, they really just wanted a formula to use.
I pointed out that they should try to work with what they know. If a problem didn't have pieces they know, think about how they could break the problem down into pieces that they knew. For example, the distance formula is essentially the Pythagorean theorem used for a specific purpose. However, problems often don't provide the right triangle dimensions needed. Students need to ask themselves how they could create a right triangle to use the Pythagorean theorem, since they typically know this theorem well. The idea started to click for some of these students and they were able to proceed ahead on their own.
I did pass out a quiz and told students it was a take-home quiz. The quiz emphasized concepts as opposed to solving problems for x. I allow notes to be used on quizzes and tests, so I felt there wouldn't be much difference in results by having the quiz completed at home. This also saved me a class period for investigation and instruction versus assessment.
Tomorrow we start similarity. Today, another colleague worked through the first lesson I am using tomorrow. She was really pleased with the engagement and learning that took place. I'm excited to see how the lesson goes in my class.
A colleague had put together four problems that covered this material. Each problem gave coordinates for a different quadrilateral. In the first problem, they had to calculate the slopes of the sides to show opposite sides were parallel. In the second, students had to calculate the length of the sides to show that opposite sides were congruent. In the third, students had to show that diagonals bisected each other. In the fourth, they used protractors to show that opposite angles were congruent.
This work went fairly well. Some students stilled had questions about calculating side lengths or slopes. For the most part they seemed comfortable with the tasks. For a couple of students, they really just wanted a formula to use.
I pointed out that they should try to work with what they know. If a problem didn't have pieces they know, think about how they could break the problem down into pieces that they knew. For example, the distance formula is essentially the Pythagorean theorem used for a specific purpose. However, problems often don't provide the right triangle dimensions needed. Students need to ask themselves how they could create a right triangle to use the Pythagorean theorem, since they typically know this theorem well. The idea started to click for some of these students and they were able to proceed ahead on their own.
I did pass out a quiz and told students it was a take-home quiz. The quiz emphasized concepts as opposed to solving problems for x. I allow notes to be used on quizzes and tests, so I felt there wouldn't be much difference in results by having the quiz completed at home. This also saved me a class period for investigation and instruction versus assessment.
Tomorrow we start similarity. Today, another colleague worked through the first lesson I am using tomorrow. She was really pleased with the engagement and learning that took place. I'm excited to see how the lesson goes in my class.
Friday, January 22, 2016
Wrapping up work with quadrilaterals
Today's class concluded our work with kites and trapezoids. Students, generally, felt that the assigned problems were some of the easiest they had so far. There were still some points of confusion that needed to be addressed, especially with regard to problems involving kites. These centered around realizing that one pair of opposite angles had to be congruent and that one diagonal was the perpendicular bisector of the other diagonal.
To conclude the work with quadrilaterals and begin paving the way for work on similarity, I used pages 3 and 4 of the Parallelism Preserved investigation. Students seemed to breeze through identifying quadrilaterals and, when asked, were able to justify their responses with the specific characteristics they used.
The next three questions caused a bit of confusion, but students were able to make reasonable arguments as to when parallelism was preserved. The final problems reviewed triangle congruence theorems. As we move into exploring triangle similarity theorems, the review will be helpful.
We'll check responses next class and have an assessment. I'm still deciding whether to make this a partner quiz or a take-home quiz. I intend on using the Understanding similarity in terms of similarity transformations tasks to work through the unit on similarity. There will be a summative assessment over quadrilaterals and similarity once this unit is complete.
To conclude the work with quadrilaterals and begin paving the way for work on similarity, I used pages 3 and 4 of the Parallelism Preserved investigation. Students seemed to breeze through identifying quadrilaterals and, when asked, were able to justify their responses with the specific characteristics they used.
The next three questions caused a bit of confusion, but students were able to make reasonable arguments as to when parallelism was preserved. The final problems reviewed triangle congruence theorems. As we move into exploring triangle similarity theorems, the review will be helpful.
We'll check responses next class and have an assessment. I'm still deciding whether to make this a partner quiz or a take-home quiz. I intend on using the Understanding similarity in terms of similarity transformations tasks to work through the unit on similarity. There will be a summative assessment over quadrilaterals and similarity once this unit is complete.
Wednesday, January 20, 2016
Kites and trapezoids
The class wrapped up working with parallelograms today. The class spent about 20 minutes completing questions and discussing problems. We then went through responses. I displayed a page of responses from a randomly chosen student and then the class commented or questioned the results. There were some good discussions and some ideas were clarified and some misunderstandings identified.
We then dove into working with kites and trapezoids. I had three examples of each on the board. For the trapezoids, I was sure to include one example of an isosceles trapezoid. I then had students refer to the quadrilateral properties grid and asked them to consider the examples and try to complete the grid columns for kites and trapezoids.
I walked around and answered questions students had. I also asked students their reasoning and thinking about their entries. The main focus I was trying to have students consider the properties that parallelograms always have and what this would mean for kites and trapezoids. Students seemed to struggle with the connections but finally started to realize that if a parallelogram always had diagonals that bisected each other then kites and trapezoids could never have this property.
The properties that parallelograms sometimes have were a bit trickier. For example, all side lengths are congruent is always true for a rhombus; kites and trapezoids can never have all side lengths congruent. On the other hand, a rhombus always has diagonals that are perpendicular but this does not preclude a kite from also always having perpendicular diagonals.
I referenced traditional kites that students may have used as children to make connections to the diagonal structures that are present in a kite.
I passed out a sheet that contained definitions and theorems concerning kites and trapezoids. This sheet also provided a couple of worked examples and a few problems. Their homework is to complete the problems for next class, which will provide additional problem and work practice.
We then dove into working with kites and trapezoids. I had three examples of each on the board. For the trapezoids, I was sure to include one example of an isosceles trapezoid. I then had students refer to the quadrilateral properties grid and asked them to consider the examples and try to complete the grid columns for kites and trapezoids.
I walked around and answered questions students had. I also asked students their reasoning and thinking about their entries. The main focus I was trying to have students consider the properties that parallelograms always have and what this would mean for kites and trapezoids. Students seemed to struggle with the connections but finally started to realize that if a parallelogram always had diagonals that bisected each other then kites and trapezoids could never have this property.
The properties that parallelograms sometimes have were a bit trickier. For example, all side lengths are congruent is always true for a rhombus; kites and trapezoids can never have all side lengths congruent. On the other hand, a rhombus always has diagonals that are perpendicular but this does not preclude a kite from also always having perpendicular diagonals.
I referenced traditional kites that students may have used as children to make connections to the diagonal structures that are present in a kite.
I passed out a sheet that contained definitions and theorems concerning kites and trapezoids. This sheet also provided a couple of worked examples and a few problems. Their homework is to complete the problems for next class, which will provide additional problem and work practice.
Assessing work with parallelograms
At this point, parallelograms have been worked through from a number of different perspectives. I wanted to assess informally students' understanding of parallelogram properties and the relationships of different parallelograms. The Guess My Parallelogram packet fit my needs perfectly.
Students spent the entire class working through the problems. As I walked around the class, students were having good mathematical discussions about the questions. I did have to encourage some students to refer to the quadrilateral property grid for help. There were very few questions.
One thing I liked about the problems was that it included some review of using a compass and straight edge to construct segment and angle bisectors. As we move to dilations, we'll be using a compass and straight edge to construct the image of a dilation.
Students were still working on the questions at the end of class. I asked that they finish any non-construction questions for next class. I'll give the class about 15 minutes at the start to complete any questions and constructions and to discuss their answers with their table partners. We'll then go through responses as a class.
We'll be moving on to kites and trapezoids next.
Students spent the entire class working through the problems. As I walked around the class, students were having good mathematical discussions about the questions. I did have to encourage some students to refer to the quadrilateral property grid for help. There were very few questions.
One thing I liked about the problems was that it included some review of using a compass and straight edge to construct segment and angle bisectors. As we move to dilations, we'll be using a compass and straight edge to construct the image of a dilation.
Students were still working on the questions at the end of class. I asked that they finish any non-construction questions for next class. I'll give the class about 15 minutes at the start to complete any questions and constructions and to discuss their answers with their table partners. We'll then go through responses as a class.
We'll be moving on to kites and trapezoids next.
Subscribe to:
Posts (Atom)
