I taught a lesson on using the distance formula, one of the geometry standards for math I. The Common Core standard that this lesson was designed to address was: G-GPE-4- "Use Coordinate sot prove simple geometric theorems algebraically." The distance formula is an important tool to do this.
What I learned
The core of the lesson centers around using the distance formula to find the distance between points on a coordinate plane. This is going to be part of my student's "toolbox" when it comes to proving triangle congruence. What I found as I taught this is that students level of engagement was lower when they were not looking at visuals. They still prefer using visual diagrams, and resist the rigor of problems that don't lend themselves to algebraic tools instead of graphical ones.
This is a central challenge of coordinate geometry. Formalizing geometric figures into coordinate points on a plane is very relevant to today's world, since any application of geometry--in construction, industrial design, computer graphics--is going to use computers and arithmetic/algebraic tools, not physical tools like rulers and protractors. However, there is so much computation involved that it's easy for students to get bogged down in the computation, and make little mistakes. Geometry software does exist for taking some of the tedium out of computing distances by hand, but teaching students how to use geometry software would take a lot of time, too. On balance, even though the computation does get tedious, I think it's best to keep students attending to precision even if they don't always enjoy it. I know with practice they will become more efficient and accurate.
I also learned that students, while they have had practice with the pythagorean theorem before, do not recall it with much consistency. They also struggle with the concept of square roots. For now, they only need to compute them, but in Math II they will need to understand the meaning of a root and the structure of expressions that contain roots.
Assessment results
I found that in the ticket-out-the-door assessment, 26 of the 36 students correctly put the numbers into the distance formula, and that 24 of those computed the distance correctly. This is probably too large a number of students who are still making an error in switching around x and y-values.
While I did give a brief demonstration of how to check for reasonable answers through identifying the quadrant that each point was in, I might demonstrate this earlier as a useful tool for students who will improve their accuracy with spacial or visual reasoning.
What I would change
My site instructional coach, after I had a conversation about how the lesson went, helped me come up with a couple of ways to keep students engaged and better focus their interest.
First, in my initial demonstration during the "ongoing assessment" portion of my lesson, I showed students a pair of triangles that was congruent but that was hard to accurately determine what rigid motions would carry the first onto the second. Students had trouble accepting that there was only one way to do this, because another series of rigid motions looked so close to accurate that I had to convince them that it wasn't. My coach suggested I make the triangle such that it would be clearer.
Secondly, my practice problems all included triangles or quadrilaterals for students to compare distances. My coach noted that I only need to provide them line segments (two points) to really give them practice with the concept.
Finally, he suggested, as a way of making the class more engaging and interactive, that I do a partner activity where one students creates three line segments, where two of them are congruent, and then trade with a partner have have the partner determine it.
What I enjoyed
I really enjoyed the beginning of the lesson where students were able to correct their thinking about how to find a distance between two points, and how I could see that they started to consider the geometric complexity of working in two variables.
In general, I enjoy teaching and get excited when I see students practicing skills that I know will help them master the standards and move on in their education.
What I learned about planning
I taught this standard last year, but this lesson seemed to go much better because it was part of a unit that was more cohesively planned. I learned through this process how it's really helpful for student engagement if the series of topics in a unit build on one another, and that students are given the chance to think about how all the different tools fit together, instead of just learning a series of apparently random skills.
What I learned about teaching
I learned that as a teacher, I tend to underestimate the time students will need to process and practice a skill, and that I need to work more variety into the methods for learning so that students can maintain engagement, and not get discouraged or bogged down in long problems if the skill they are learning can be broken into smaller segments.
What I learned
The core of the lesson centers around using the distance formula to find the distance between points on a coordinate plane. This is going to be part of my student's "toolbox" when it comes to proving triangle congruence. What I found as I taught this is that students level of engagement was lower when they were not looking at visuals. They still prefer using visual diagrams, and resist the rigor of problems that don't lend themselves to algebraic tools instead of graphical ones.
This is a central challenge of coordinate geometry. Formalizing geometric figures into coordinate points on a plane is very relevant to today's world, since any application of geometry--in construction, industrial design, computer graphics--is going to use computers and arithmetic/algebraic tools, not physical tools like rulers and protractors. However, there is so much computation involved that it's easy for students to get bogged down in the computation, and make little mistakes. Geometry software does exist for taking some of the tedium out of computing distances by hand, but teaching students how to use geometry software would take a lot of time, too. On balance, even though the computation does get tedious, I think it's best to keep students attending to precision even if they don't always enjoy it. I know with practice they will become more efficient and accurate.
I also learned that students, while they have had practice with the pythagorean theorem before, do not recall it with much consistency. They also struggle with the concept of square roots. For now, they only need to compute them, but in Math II they will need to understand the meaning of a root and the structure of expressions that contain roots.
Assessment results
I found that in the ticket-out-the-door assessment, 26 of the 36 students correctly put the numbers into the distance formula, and that 24 of those computed the distance correctly. This is probably too large a number of students who are still making an error in switching around x and y-values.
While I did give a brief demonstration of how to check for reasonable answers through identifying the quadrant that each point was in, I might demonstrate this earlier as a useful tool for students who will improve their accuracy with spacial or visual reasoning.
What I would change
My site instructional coach, after I had a conversation about how the lesson went, helped me come up with a couple of ways to keep students engaged and better focus their interest.
First, in my initial demonstration during the "ongoing assessment" portion of my lesson, I showed students a pair of triangles that was congruent but that was hard to accurately determine what rigid motions would carry the first onto the second. Students had trouble accepting that there was only one way to do this, because another series of rigid motions looked so close to accurate that I had to convince them that it wasn't. My coach suggested I make the triangle such that it would be clearer.
Secondly, my practice problems all included triangles or quadrilaterals for students to compare distances. My coach noted that I only need to provide them line segments (two points) to really give them practice with the concept.
Finally, he suggested, as a way of making the class more engaging and interactive, that I do a partner activity where one students creates three line segments, where two of them are congruent, and then trade with a partner have have the partner determine it.
What I enjoyed
I really enjoyed the beginning of the lesson where students were able to correct their thinking about how to find a distance between two points, and how I could see that they started to consider the geometric complexity of working in two variables.
In general, I enjoy teaching and get excited when I see students practicing skills that I know will help them master the standards and move on in their education.
What I learned about planning
I taught this standard last year, but this lesson seemed to go much better because it was part of a unit that was more cohesively planned. I learned through this process how it's really helpful for student engagement if the series of topics in a unit build on one another, and that students are given the chance to think about how all the different tools fit together, instead of just learning a series of apparently random skills.
What I learned about teaching
I learned that as a teacher, I tend to underestimate the time students will need to process and practice a skill, and that I need to work more variety into the methods for learning so that students can maintain engagement, and not get discouraged or bogged down in long problems if the skill they are learning can be broken into smaller segments.
Comments
Post a Comment