Exploring Marion Walter's Theorem Class: 10th Grade Geometry Unit: Post-Area Unit (Culminative of preceding units as well)

How This Unit Fits: Using the general skills of geometric proof and inductive/deductive reasoning, students will use the ideas of classical construction and trisection/congruence and relate it to the area of a triangle and the area of a hexagon.
Not only is this a good practice in the essential themes regarding proof and reasoning, but students will have to call upon various units in geometry so far this year (proof, constructions, area, triangle relationships, etc.).

Time Frame: Varies (Ideally, no more than 4 days)

Objective: SWBAT create a rigorous proof of Marion Walter's Theorem by utiliziing online geometry software and mentoring.

Curriculum Connection:
From the Maryland Geometry VSC
Goal 2: The student will demonstrate the ability to solve mathematical and real-world problems using measurement and geometric models and will justify solutions and explain processes used.
Expectation 2.1: The student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate.
Indicator 2.1.1 The student will analyze the properties of geometric figures.
Indicator 2.1.2 The student will identify and/or verify properties of geometric figures using the coordinate plane and concepts from algebra. Indicator 2.1.4: The student will construct and/or draw and/or validate properties of geometric figures using appropriate tools and technology. Expectation 2.2: The student will apply geometric properties and relationships to solve problems using tools and technology when appropriate. Indicator 2.2.3: The student will use inductive or deductive reasoning.

Essential Question: How can we pull together everything in the geometry toolkit in order to prove a "new" relationship?

Lesson Outline NOTE: Before this lesson starts, the teacher should have already contacted Making Mathematics and obtained at least one mentor for the class (through email). Also, teachers must PARTICIPATE in the research as well (complete the problem along side the students; you need to trust your mentor and your groups' math reasoning skills). - In order to both fully engage each student / make sure they are ready to do the rigor that is necessary for mathematical research, students will work in groups to complete the warm up problems. Whenever a group is finished, they are to post their solutions on the Edmodo site. Ideally, math is not about just "getting the right answer." Rather, we just try to use all of our tools and try to understand the solution / relationship ourselves. If a group finishes one question before yours, make sure you first only peek at the beginnings of the posted solutions for hints. If you eventually have to look at another student's solution, make sure you still post a solution in your "own math / words."
- As a class, we will review over these problems. The class will decide which group's solutions we will use to review (whichever ones where deemed most clear / helpful).
- Students will use either Geometer's Sketchpad or Cabri (Dynamic geometry software) and use Making Mathematics' visual representations of the figure's involved in Marion Walter's Theorem. Students will be asked to make one Blogger post answering part one of the research project (on Making Mathematics' website):

In the Figure 1, there is a small hexagon in the middle of the triangle. Will there always be a hexagon, no matter what the original triangle looks like?

- Next, students will read each other's blogs on blogger and discuss their findings on Edmodo.
- If any student becomes stuck, they will be encouraged to email our mentor (email will be posted). Try to use the hints on the website first and use the mentor as a last resort (he will respond almost instantly during the school day).
- Students who finish early will be asked to explore Geometer's Sketchpad in the special cases of 30-60-90 and 45-45-90 triangles.

- After groups finish up with the first part of the proof, students should be given open "think / work" time on the website in order to start the heart of the proof by exploring this question:

Can you relate the area of the hexagon to the area of the original triangle? Can you prove the relationship? You can create a figure like Figure 2 with geometry software.

-This will undoubtebly be the hardest part. Students should use help in this order:
1. Hints
2. Ask each other on Edmodo
3. Mentor's Email

-Periodically (if and when this takes students at least 2 days) the class will reconvene their findings in a group discusison on Edmodo.
- Making Mathematics provides teacher hints that can push classes that are very stuck with the theorem. If the class becomes overly frustrated or lost, at the teacher's discretion, the proper information from the teacher site should be disseminated.
- As groups start to finish their proofs, they will be asked to post their final results in Blogger.
- As a conclusion, the class will have an open discussion on Edmodo as they peer-review each other's proofs before they submit them to Making Mathematics in order to receive quick feedback from the group of mathematicians that work for Making Mathematics.

## Second Lesson Plan

Exploring Marion Walter's TheoremClass:10th Grade GeometryUnit:Post-Area Unit (Culminative of preceding units as well)How This Unit Fits:Using the general skills of geometric proof and inductive/deductive reasoning, students will use the ideas of classical construction and trisection/congruence and relate it to the area of a triangle and the area of a hexagon.Not only is this a good practice in the essential themes regarding proof and reasoning, but students will have to call upon various units in geometry so far this year (proof, constructions, area, triangle relationships, etc.).

Time Frame:Varies (Ideally, no more than 4 days)Objective:SWBAT create a rigorous proof of Marion Walter's Theorem by utiliziing online geometry software and mentoring.Curriculum Connection:From the Maryland Geometry VSC

Goal 2: The student will demonstrate the ability to solve mathematical and real-world problems using measurement and geometric models and will justify solutions and explain processes used.

Expectation 2.1: The student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate.

Indicator 2.1.1 The student will analyze the properties of geometric figures.

Indicator 2.1.2 The student will identify and/or verify properties of geometric figures using the coordinate plane and concepts from algebra.

Indicator 2.1.4: The student will construct and/or draw and/or validate properties of geometric figures using appropriate tools and technology.

Expectation 2.2: The student will apply geometric properties and relationships to solve problems using tools and technology when appropriate.

Indicator 2.2.3: The student will use inductive or deductive reasoning.

Essential Question:How can we pull together everything in the geometry toolkit in order to prove a "new" relationship?Multiple Intelligences Addressed:Logical Mathematical, Spatial, InterpersonalAssignment / Informational Resources:Warm Up Exercises, Research Project, Hints, Extension Problems, Teacher HintsMath Tools:Geometer's Sketchpad for Project, Cabri for Project, TI-92 Software for ProjectSocial Tech Resources:Edmodo, Blogger, GmailLocation:Computer LabLesson OutlineNOTE: Before this lesson starts, the teacher should have already contacted Making Mathematics and obtained at least one mentor for the class (through email). Also, teachers must PARTICIPATE in the research as well (complete the problem along side the students; you need to trust your mentor and your groups' math reasoning skills).-In order to both fully engage each student / make sure they are ready to do the rigor that is necessary for mathematical research, students will work in groups to complete the warm up problems. Whenever a group is finished, they are to post their solutions on the Edmodo site. Ideally, math is not about just "getting the right answer." Rather, we just try to use all of our tools and try to understand the solution / relationship ourselves. If a group finishes one question before yours, make sure you first only peek at the beginnings of the posted solutions for hints. If you eventually have to look at another student's solution, make sure you still post a solution in your "own math / words."- As a class, we will review over these problems. The class will decide which group's solutions we will use to review (whichever ones where deemed most clear / helpful).

- Students will use either Geometer's Sketchpad or Cabri (Dynamic geometry software) and use Making Mathematics' visual representations of the figure's involved in Marion Walter's Theorem. Students will be asked to make one Blogger post answering part one of the research project (on Making Mathematics' website):

In the Figure 1, there is a small hexagon in the middle of the triangle. Will there always be a hexagon, no matter what the original triangle looks like?

- Next, students will read each other's blogs on blogger and discuss their findings on Edmodo.

- If any student becomes stuck, they will be encouraged to email our mentor (email will be posted). Try to use the hints on the website first and use the mentor as a last resort (he will respond almost instantly during the school day).

- Students who finish early will be asked to explore Geometer's Sketchpad in the special cases of 30-60-90 and 45-45-90 triangles.

- After groups finish up with the first part of the proof, students should be given open "think / work" time on the website in order to start the heart of the proof by exploring this question:

Can you relate the area of the hexagon to the area of the original triangle? Can you prove the relationship? You can create a figure like Figure 2 with geometry software.

-This will undoubtebly be the hardest part. Students should use help in this order:

1. Hints

2. Ask each other on Edmodo

3. Mentor's Email

-Periodically (if and when this takes students at least 2 days) the class will reconvene their findings in a group discusison on Edmodo.

- Making Mathematics provides teacher hints that can push classes that are very stuck with the theorem. If the class becomes overly frustrated or lost, at the teacher's discretion, the proper information from the teacher site should be disseminated.

- As groups start to finish their proofs, they will be asked to post their final results in Blogger.

- As a conclusion, the class will have an open discussion on Edmodo as they peer-review each other's proofs before they submit them to Making Mathematics in order to receive quick feedback from the group of mathematicians that work for Making Mathematics.