I received an email notification yesterday morning about a new post on Lisa Bejarano's blog. It's an excellent post that talks about Lisa's internal thought process when deciding how to lead a lesson on simplifying complex fractions.
A couple of hours later, I see Dan Meyer tweet he commented on Lisa's post on his blog. I enjoyed Dan's analysis for a couple of reasons. He offers the thought that it is impossible to practice the process that Lisa takes with her lesson plan. I tend to agree with that opinion. Then, I what I really enjoyed was the nugget he left behind in his "BTW" section.
He links an article called "The Positive Effects of Blogging on Teachers". I hadn't seen this article before and it really hit home with me. Many of the items discussed relate directly to me and my professional growth and journey.
Thank you Lisa and Dan for helping me along my way!
Saturday, November 25, 2017
Wednesday, November 15, 2017
Desmos Transformation Golf + What My Assessment Looked Like
My geometry classes just recently completed our unit on transformations. I was super excited to be teaching transformations this year because a few weeks ago Desmos released one of their coolest activities to date: Transformation Golf: Rigid Motion. If you haven't checked out this activity, stop reading this post and go take a peek.
I wrestled with figuring out the best time to do this activity with students. Should I launch the unit with it? Do I do the activity after the unit as a performance task? When does this best fit?
I decided to do the activity as a review for the test. My intent was to help students solidify their understanding as well as to allow students to see that there is more than one composition of transformations that will yield the same result.
So I had students complete the activity in class. I used quite a bit of teacher pacing and paused students often to discuss some of their thinking during the activity. It's such a fun day to lead. Students found a lot of pleasure finding their own ways to complete the tasks. Take a look at the various ways students did challenge #8.
To wrap up the lesson, I had students complete an exit ticket to summarize their thoughts about the activity. Here are a few quotes from students:
"It was fun and I liked how we got to make different things different ways."
"It made me think in different ways than I normally would."
"I like how it made me think outside of the box and creatively."
"I liked that it was a little bit of a challenge."
As part of the unit test, I wanted to give students some problems that were similar to the Transformation Golf activity. I wanted students to have the opportunity to be creative and get the correct solution in more than one way. At the same time, I wanted the problems to be a bit challenging and I wanted to assess the students' understanding of transformations on the coordinate plane.
So I created six different problems that consisted of a pre-image figure and its image on a coordinate plane. (A link to the test problems is here.) Students were required to provide the list of steps needed to map the pre-image onto the image. The level of precision expected was this: for translations, I needed the translation rule. For reflections, I needed the equation of the line of reflection. For rotations, I needed the center and degree of rotation. Counterclockwise rotations were the default; students could rotate clockwise if they desired, as long as they noted the direction.
I was really stressing out about grading these problems because I knew there were many correct answers. I wouldn't be able to have one answer as a key; I would need to check each problem with a fine-toothed comb. With over 80 geometry students, I was worried about how long this task would take me.
Here is a sample of some the student responses. All of these solutions are for the same problem.
As it turns out, I found great joy in grading these problems. Yes, it took a bit of time...more time than it would have had I given my students a multiple choice assessment. But to see the creativity, thinking, and effort that students demonstrated was well worth my time.
I won't lie and say that all students did awesome work on this assessment. A common error was not being specific about the location of the center of rotation. [Often times the students intended the center to be the origin, but didn't specify. These errors led to a good conversation about precision.]
A few students who struggled mentioned that this assessment was tougher than the Desmos activity for two reasons. First, checking their work on the assessment was a bit tougher than checking on Desmos. The Desmos activity provides immediate feedback when a student performs a transformation. Second, the transformations on Desmos did not require use of coordinates, equations of lines, etc. I have a handful of students who still struggle with writing the equation of a line. They are able to draw / sketch the line of reflection when given a pre-image & image, but they are not able to write the equation of that line very well. These students were able to complete the Desmos activity without too much problem but struggled to complete this assessment correctly.
So, team at Desmos, here is my request. I LOVE the Transformation Golf activity. It made teaching transformations incredible enjoyable this year. I would love to see a "Transformation Golf: Round 2" activity that includes the x- and y-axes on the grid and that requires students to provide translation rules, equations of lines of reflections, and coordinates of centers of rotation in order to perform the transformations. My thought would be to have students start with the existing activity in order to learn some of the general transformation tools, and then send to the "Round 2" activity that ramps up the precision. Thanks in advance! ;-)
I wrestled with figuring out the best time to do this activity with students. Should I launch the unit with it? Do I do the activity after the unit as a performance task? When does this best fit?
I decided to do the activity as a review for the test. My intent was to help students solidify their understanding as well as to allow students to see that there is more than one composition of transformations that will yield the same result.
So I had students complete the activity in class. I used quite a bit of teacher pacing and paused students often to discuss some of their thinking during the activity. It's such a fun day to lead. Students found a lot of pleasure finding their own ways to complete the tasks. Take a look at the various ways students did challenge #8.
To wrap up the lesson, I had students complete an exit ticket to summarize their thoughts about the activity. Here are a few quotes from students:
"It was fun and I liked how we got to make different things different ways."
"It made me think in different ways than I normally would."
"I like how it made me think outside of the box and creatively."
"I liked that it was a little bit of a challenge."
As part of the unit test, I wanted to give students some problems that were similar to the Transformation Golf activity. I wanted students to have the opportunity to be creative and get the correct solution in more than one way. At the same time, I wanted the problems to be a bit challenging and I wanted to assess the students' understanding of transformations on the coordinate plane.
So I created six different problems that consisted of a pre-image figure and its image on a coordinate plane. (A link to the test problems is here.) Students were required to provide the list of steps needed to map the pre-image onto the image. The level of precision expected was this: for translations, I needed the translation rule. For reflections, I needed the equation of the line of reflection. For rotations, I needed the center and degree of rotation. Counterclockwise rotations were the default; students could rotate clockwise if they desired, as long as they noted the direction.
I was really stressing out about grading these problems because I knew there were many correct answers. I wouldn't be able to have one answer as a key; I would need to check each problem with a fine-toothed comb. With over 80 geometry students, I was worried about how long this task would take me.
Here is a sample of some the student responses. All of these solutions are for the same problem.
As it turns out, I found great joy in grading these problems. Yes, it took a bit of time...more time than it would have had I given my students a multiple choice assessment. But to see the creativity, thinking, and effort that students demonstrated was well worth my time.
I won't lie and say that all students did awesome work on this assessment. A common error was not being specific about the location of the center of rotation. [Often times the students intended the center to be the origin, but didn't specify. These errors led to a good conversation about precision.]
A few students who struggled mentioned that this assessment was tougher than the Desmos activity for two reasons. First, checking their work on the assessment was a bit tougher than checking on Desmos. The Desmos activity provides immediate feedback when a student performs a transformation. Second, the transformations on Desmos did not require use of coordinates, equations of lines, etc. I have a handful of students who still struggle with writing the equation of a line. They are able to draw / sketch the line of reflection when given a pre-image & image, but they are not able to write the equation of that line very well. These students were able to complete the Desmos activity without too much problem but struggled to complete this assessment correctly.
So, team at Desmos, here is my request. I LOVE the Transformation Golf activity. It made teaching transformations incredible enjoyable this year. I would love to see a "Transformation Golf: Round 2" activity that includes the x- and y-axes on the grid and that requires students to provide translation rules, equations of lines of reflections, and coordinates of centers of rotation in order to perform the transformations. My thought would be to have students start with the existing activity in order to learn some of the general transformation tools, and then send to the "Round 2" activity that ramps up the precision. Thanks in advance! ;-)
Saturday, November 11, 2017
"When Will I Ever Use This?"
In Geometry class last week, I shared a TED ED video with students titled "Pixar: The Math Behind the Movies". In the video, Pixar Research Lead Tony DeRose talks to a room full of students about some of the mathematics happening behind the scenes at Pixar.
One piece of the mathematics Tony talks about is something Pixar created in 1997 called "subdivision". Without giving away too much of the video, under the surface "subdividing" uses a bit of coordinate geometry and the concept of midpoints. On the surface, "subdividing" helps Pixar smooth the edges of their digital characters and makes the characters look a lot more life-like.
What I found interesting that is that this concept of "subdividing" was invented until 1997. I graduated HS in 1998, which means my high school geometry instruction dates back to somewhere between 1995-97. If I would have asked my high school math teacher at the time "When will I ever need to find the midpoint of a line segment?", he would not have been able to mention the concept of subdivision as an application for finding midpoints.
Likewise, it's safe to say that in five years, by the time my students are halfway done their undergraduate degrees, there will be math being used in the world that hasn't yet been invented. Whether it be an advanced statistical metric used to inform sports teams, some fancy new device that makes a iPhone X seem like an antique, or a parallel to Pixar's "subdivision", new math is being discovered and applied each year.
As math teachers, it's not a bad idea to have a list of occupations and examples that highlight some of the usefulness and application of mathematics. However, math teachers should also help students realize that we don't fully know how certain mathematical topics will be used in the future.
The video is under 8 minutes long; I invite you to watch it. It's really quite good.
One piece of the mathematics Tony talks about is something Pixar created in 1997 called "subdivision". Without giving away too much of the video, under the surface "subdividing" uses a bit of coordinate geometry and the concept of midpoints. On the surface, "subdividing" helps Pixar smooth the edges of their digital characters and makes the characters look a lot more life-like.
What I found interesting that is that this concept of "subdividing" was invented until 1997. I graduated HS in 1998, which means my high school geometry instruction dates back to somewhere between 1995-97. If I would have asked my high school math teacher at the time "When will I ever need to find the midpoint of a line segment?", he would not have been able to mention the concept of subdivision as an application for finding midpoints.
Likewise, it's safe to say that in five years, by the time my students are halfway done their undergraduate degrees, there will be math being used in the world that hasn't yet been invented. Whether it be an advanced statistical metric used to inform sports teams, some fancy new device that makes a iPhone X seem like an antique, or a parallel to Pixar's "subdivision", new math is being discovered and applied each year.
As math teachers, it's not a bad idea to have a list of occupations and examples that highlight some of the usefulness and application of mathematics. However, math teachers should also help students realize that we don't fully know how certain mathematical topics will be used in the future.
The video is under 8 minutes long; I invite you to watch it. It's really quite good.
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