DAy 3 activity:
Nearpod Lesson: http://np1.nearpod.com/sharePresentation.php?code=6bce8313983cd785dc9e49f0f4863c7e-0
List of Materials/Resources Required by Student:
-Computer or tablet to participate in lesson (my school provides laptops to students)
-Wi-fi or internet access
-Pencil & Paper for notes
-Access to Desmos, Weebly, Geogebra applet, Youtube, and Nearpod.
-Smart phone, digital camera, or scanner for homework (Students can arrive in the morning and use my equipment to post homework and watch videos)
Activity Rational: I am opening with a review of applying the Pythagorean theorem in this unit, as we will use it in our exploration. The unit begins with a short discovery task that has students applying knowledge of the Cartesian plane and Pythagorean theorem to derive an expression for a circle with Geogebra. From there, students will use rational about line segment length or transformations to use desmos to understand circle transformation (and oval if time permits) in the Cartesian plane. I am using discovery to help students arrive at formulas themselves. This activity also aligns directly to CCGPS standards in the unit and helps to use mathematics to model a situation.
Learning Objective: Students can find the equation for a circle in the Cartesian plane given a center and radius using knowledge of right triangle side length properties; Students can find the equation of circle given a polynomial equation using the process of solving for vertex form or completing the square (video in homework assignment)
Mathematical Practices developed in the unit:
-The ability to model a situation and write an expression to depict that model
-Using knowledge of right triangles and the Cartesian plane to develop an expression for a circle.
-Using quadratic expression conversions (standard form to vertex form) to aid in the analysis of circle expressions. Learning how to write and interpret circle expressions in different formats.
-Finding key features (radius and center point) of a circle in the Cartesian Plane.
Connection between technology and the learning objective:
-Geogebra app: Used to model a situation and find the expression for a circle. Students will be able to plan and get hands on with the model to help interpretation. They will see through movement that a circle is developed with the trace features.
-Desmos: Students will explore translations and transformations of circles through Desmos sliders and describe how those transformations will occur in their own words. They can relate translations of the center point to the right triangle and change of length values based upon the distance formula as well. Technology here is providing nice manipulatives for discovery.
-Nearpod presentation: Allows the teacher to collect student responses and assessment during class as well as keep the class on task/pace. The teacher can use the technology to help in the classroom teaching while aiding students who may fall behind. The teacher can also use student responses to facilitate classroom discussion.
-Youtube video: Used to teach and provide a flipped classroom for the next lesson segment. This will provide a resource to students outside of class as well as provide better time management for the teacher during the class. I think it also adds some entertainment to setting up a task and keeping the class on pace.
-Weebly: Used to collect homework results and provide homework to students. This allows students to submit questions and the teacher to quickly understand if students are ready to move on.
Teacher Guidelines:
The teacher should use the following nearpod slideshow:
http://np1.nearpod.com/sharePresentation.php?code=6bce8313983cd785dc9e49f0f4863c7e-0
Some editing may be needed for collecting the homework assignment on the last slide. The teacher should be able to follow along (answer details below) and facilitate discussion based on student response and assessment throughout the lesson.
List of Materials/Resources Required by Teacher:
-Computer or tablet to participate in lesson (my school provides laptops to students)
-Additional laptops or devices for students to use if they do not have them
-A smart board or projector to display results in front of the class as well as on personal devices (as you may need to pull up desmos or write something out further)
-Access to Youtube videos (test if your school blocks youtube)
-Wi-fi or internet access
-Pencil & Paper to provide to students
-Access to Desmos, Weebly, Geogebra applet, Youtube, and Nearpod on your machine (directions and troubleshooting can be found directly at each app site)
-Smart phone, digital camera, or scanner for students to hand in homework if you are collecting through weebly
-Teacher answers and guidelines below
-Printouts of activity questions if you do not want to use the Nearpod lesson.
Anticipated Student Actions/Potential Difficulties/Misconceptions:
Warm-up: Students are able to correctly find the solution (and write it in simplified form) using the Pythagorean theorem. I anticipate most students will get an unsimplified form with little to no issue. Some students might confuse sides and arrive at incorrect answers or perform the math incorrectly.
Task 1: Students will most likely have no issue with the first few questions. I anticipate most will see the 90 degree angle, but may need an explanation as to why they can prove that. Some students might need help in writing the expression for a circle or finding segment lengths in that activity. Teacher should focus on questioning and student discussion to clear items up.
Task 2: Students should be able to see the impact of the sliders for h and k on a circle and speak to it after Desmos discovery. They may need help building connections between the triangle distance formula and translations from previous units, so it is worth discussing the various explanations students will have for why it works. I hope you can get to the discovery of stretching your circle horizontally and vertically as well to develop ovals.
Homework: I do not anticipate students will have issues with the homework if they paid attention to today's lesson. I think they will be confused and need help tomorrow on the flipped classroom video, as it might be difficult for them to see the completing the square method. Soliciting questions prior to class via a form or email will help the teacher to prepare for tomorrow's lesson better.
Students may have issues with the technology or loads on the computer. Try to establish a system to be notified of these issues. If need be, have a few laptops with the apps loaded on them so students can continue working and class time is not wasted. Students can also work in groups and share computers if too many issues arise.
It is important to point out that the circle expression is not actually a function and describe why. You can discuss how to make it a function which is a very nice connection to make now as these students will have a radical functions unit down the road. It's never too early to talk about extraneous solutions! Also, you might want to connect circles and lines with potential solution sets to connect back to the original portions of the unit. You can also start to bring up the cut of a conic section that a circle is.
I do not anticipate a lot of misconception as the unit is pretty guided by the teacher. Some students might get confused as to why the formula subtracts h and k, which should be discussed. Also, some students might accidentally subract the x term from the y term here.
If you can stretch this activity, discuss inequalities with these functions as well. Do they actually make sense? How could they make sense?
Questions to pose to students:
I have outlined some good questions in my youtube homework video for the next lesson, questions for students have been incorporated into the presentation and outlined in the teacher guide below. You could ask what other real world applications could these expressions help with.
Sequencing of student responses:
-I would suggest if there are strong incorrect responses that the teacher collects in the Nearpod assessments to use those as a starting point for conversation first
-From there I would chose the answers that best fit the solutions outlined below
-If other formats or questions arise I would bring these up for discussion as well and discuss why they result in the same thing (recalling the same conversations in the quadratics unit)
-For the question about describing the impacts of (h, k) I would sequence the following:
1. A response on the translations that occur as they are applied to the (x,y) coordinates of the function
2. A response using the right triangle first discussed and how the distance coordinates have become (x-h, y-k) now
3. A discussion about how the center of the circle is at (h,k) so it is simply a transformation of the center while keeping the same radius
Assessments:
-Informal assessments are incorporated into the nearpod and have been outlined in the teacher guide below. You can also see who is in the application and following along as you go.
-A formal assessment and homework activity has been outlined and provided in the Nearpod as well as in the teacher guidelines below.
List of Materials/Resources Required by Student:
-Computer or tablet to participate in lesson (my school provides laptops to students)
-Wi-fi or internet access
-Pencil & Paper for notes
-Access to Desmos, Weebly, Geogebra applet, Youtube, and Nearpod.
-Smart phone, digital camera, or scanner for homework (Students can arrive in the morning and use my equipment to post homework and watch videos)
Activity Rational: I am opening with a review of applying the Pythagorean theorem in this unit, as we will use it in our exploration. The unit begins with a short discovery task that has students applying knowledge of the Cartesian plane and Pythagorean theorem to derive an expression for a circle with Geogebra. From there, students will use rational about line segment length or transformations to use desmos to understand circle transformation (and oval if time permits) in the Cartesian plane. I am using discovery to help students arrive at formulas themselves. This activity also aligns directly to CCGPS standards in the unit and helps to use mathematics to model a situation.
Learning Objective: Students can find the equation for a circle in the Cartesian plane given a center and radius using knowledge of right triangle side length properties; Students can find the equation of circle given a polynomial equation using the process of solving for vertex form or completing the square (video in homework assignment)
Mathematical Practices developed in the unit:
-The ability to model a situation and write an expression to depict that model
-Using knowledge of right triangles and the Cartesian plane to develop an expression for a circle.
-Using quadratic expression conversions (standard form to vertex form) to aid in the analysis of circle expressions. Learning how to write and interpret circle expressions in different formats.
-Finding key features (radius and center point) of a circle in the Cartesian Plane.
Connection between technology and the learning objective:
-Geogebra app: Used to model a situation and find the expression for a circle. Students will be able to plan and get hands on with the model to help interpretation. They will see through movement that a circle is developed with the trace features.
-Desmos: Students will explore translations and transformations of circles through Desmos sliders and describe how those transformations will occur in their own words. They can relate translations of the center point to the right triangle and change of length values based upon the distance formula as well. Technology here is providing nice manipulatives for discovery.
-Nearpod presentation: Allows the teacher to collect student responses and assessment during class as well as keep the class on task/pace. The teacher can use the technology to help in the classroom teaching while aiding students who may fall behind. The teacher can also use student responses to facilitate classroom discussion.
-Youtube video: Used to teach and provide a flipped classroom for the next lesson segment. This will provide a resource to students outside of class as well as provide better time management for the teacher during the class. I think it also adds some entertainment to setting up a task and keeping the class on pace.
-Weebly: Used to collect homework results and provide homework to students. This allows students to submit questions and the teacher to quickly understand if students are ready to move on.
Teacher Guidelines:
The teacher should use the following nearpod slideshow:
http://np1.nearpod.com/sharePresentation.php?code=6bce8313983cd785dc9e49f0f4863c7e-0
Some editing may be needed for collecting the homework assignment on the last slide. The teacher should be able to follow along (answer details below) and facilitate discussion based on student response and assessment throughout the lesson.
List of Materials/Resources Required by Teacher:
-Computer or tablet to participate in lesson (my school provides laptops to students)
-Additional laptops or devices for students to use if they do not have them
-A smart board or projector to display results in front of the class as well as on personal devices (as you may need to pull up desmos or write something out further)
-Access to Youtube videos (test if your school blocks youtube)
-Wi-fi or internet access
-Pencil & Paper to provide to students
-Access to Desmos, Weebly, Geogebra applet, Youtube, and Nearpod on your machine (directions and troubleshooting can be found directly at each app site)
-Smart phone, digital camera, or scanner for students to hand in homework if you are collecting through weebly
-Teacher answers and guidelines below
-Printouts of activity questions if you do not want to use the Nearpod lesson.
Anticipated Student Actions/Potential Difficulties/Misconceptions:
Warm-up: Students are able to correctly find the solution (and write it in simplified form) using the Pythagorean theorem. I anticipate most students will get an unsimplified form with little to no issue. Some students might confuse sides and arrive at incorrect answers or perform the math incorrectly.
Task 1: Students will most likely have no issue with the first few questions. I anticipate most will see the 90 degree angle, but may need an explanation as to why they can prove that. Some students might need help in writing the expression for a circle or finding segment lengths in that activity. Teacher should focus on questioning and student discussion to clear items up.
Task 2: Students should be able to see the impact of the sliders for h and k on a circle and speak to it after Desmos discovery. They may need help building connections between the triangle distance formula and translations from previous units, so it is worth discussing the various explanations students will have for why it works. I hope you can get to the discovery of stretching your circle horizontally and vertically as well to develop ovals.
Homework: I do not anticipate students will have issues with the homework if they paid attention to today's lesson. I think they will be confused and need help tomorrow on the flipped classroom video, as it might be difficult for them to see the completing the square method. Soliciting questions prior to class via a form or email will help the teacher to prepare for tomorrow's lesson better.
Students may have issues with the technology or loads on the computer. Try to establish a system to be notified of these issues. If need be, have a few laptops with the apps loaded on them so students can continue working and class time is not wasted. Students can also work in groups and share computers if too many issues arise.
It is important to point out that the circle expression is not actually a function and describe why. You can discuss how to make it a function which is a very nice connection to make now as these students will have a radical functions unit down the road. It's never too early to talk about extraneous solutions! Also, you might want to connect circles and lines with potential solution sets to connect back to the original portions of the unit. You can also start to bring up the cut of a conic section that a circle is.
I do not anticipate a lot of misconception as the unit is pretty guided by the teacher. Some students might get confused as to why the formula subtracts h and k, which should be discussed. Also, some students might accidentally subract the x term from the y term here.
If you can stretch this activity, discuss inequalities with these functions as well. Do they actually make sense? How could they make sense?
Questions to pose to students:
I have outlined some good questions in my youtube homework video for the next lesson, questions for students have been incorporated into the presentation and outlined in the teacher guide below. You could ask what other real world applications could these expressions help with.
Sequencing of student responses:
-I would suggest if there are strong incorrect responses that the teacher collects in the Nearpod assessments to use those as a starting point for conversation first
-From there I would chose the answers that best fit the solutions outlined below
-If other formats or questions arise I would bring these up for discussion as well and discuss why they result in the same thing (recalling the same conversations in the quadratics unit)
-For the question about describing the impacts of (h, k) I would sequence the following:
1. A response on the translations that occur as they are applied to the (x,y) coordinates of the function
2. A response using the right triangle first discussed and how the distance coordinates have become (x-h, y-k) now
3. A discussion about how the center of the circle is at (h,k) so it is simply a transformation of the center while keeping the same radius
Assessments:
-Informal assessments are incorporated into the nearpod and have been outlined in the teacher guide below. You can also see who is in the application and following along as you go.
-A formal assessment and homework activity has been outlined and provided in the Nearpod as well as in the teacher guidelines below.
| day3teacherfile.docx |