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    Artifact2 allen (3) Artifact2 allen (3) Document Transcript

    • Thomas Allen Dr. Adu-Gyamfi 11/19/13 TRIANGLE CENTERS Statement of Mathematical Investigation: Students will explore the concepts of the various centers of a triangle and apply them to a real world application and justify their answers with educated responses. A gardener is trying to figure out where to put a sprinkler within his yard will hit his 3 islands of plants that form a triangle. He is trying to decide which of the four centers of a triangle will maximize his sprinklers reach. Below is rough sketch of his yard, use this picture and the given dimensions as well as your knowledge of the FOUR triangle centers to decide if where he should place it. Choose one of the following as the solution to the gardener’s problem: Incenter, Circumcenter, Orthocenter, and Centroid. NOTE: use the centers of the circles to approximate the placement of the sprinkler, also the sprinkler has a spraying radius of 11feet and the dimensions are in feet NOT cm. D m DF = 15.86 cm F DE = 10.44 cm m EF = 20.71 cm E
    • INCENTER D m DF = 15.86 cm F DE = 10.44 cm G m EF = 20.71 cm E I created the angle bisectors for each interior angles of the given triangle DEF. I then marked the intersection of the lines that bisected each angle which became the Incenter labeled G. D m DF = 15.86 cm m DG = 4.44 cm G F m GF = 13.51 cm m EF = 20.71 cm DE = 10.44 cm m GE = 8.38 cm E I created segments from each vertex of the triangle to the Incenter point G of the triangle EDF. I then color coordinated each of the measures of the segments for organization.
    • Question 1: Can all three plants be watered from the one sprinkler by using the Incenter? How can you decide whether this particular triangle center is sufficient for the gardener’s problem by using just the properties of this particular center? ANSWER No this center will only be able to reach plants E and D. By definition the Incenter is the circle within the triangle which touches each side of the triangle making it impossible for the sprinkler to reach plant F. CIRCUMCENTER D m DF = 15.86 cm F m EF = 20.71 cm DE = 10.44 cm G E I started by finding the midpoints of each segment of the triangle and then creating the perpendicular bisectors of the midpoint in relation to the segment to which that point lied on. I then marked the intersection of the perpendicular bisectors with point G which then became the Circumcenter of triangle EDF.
    • D m DF = 15.86 cm F m GD = 10.58 cm m EF = 20.71 cm DE = 10.44 cm m FG = 10.58 cm G m EG = 10.58 cm E Here I created the segments from the vertices of the triangle to the Circumcenter G and color coordinated their respective measures with the segments themselves. Question 2: Can all three plants be watered from the one sprinkler by using the Circumcenter? How can you decide whether this particular triangle center is sufficient for the gardener’s problem by using just the properties of this particular center? ANSWER Yes this center can be used because it touches all of plants E, D or F. Making it the most effective choice of placements for the sprinkler. The Circumcenter is the point that lies equidistant from each vertex. In this case with the given spraying radius of the sprinkler of 12 feet this center proves to be highly effective.
    • CENTROID D m DF = 15.86 cm F DE = 10.44 cm G m EF = 20.71 cm E I first created the midpoints of the segments of the triangle EDF and then created segments from these midpoints to the opposing vertices of the midpoint. I then created the intersection point of the three segments which became point G which is the Centroid of triangle EDF. D m DF = 15.86 cm m DG = 5.70 cm F m FG = 11.79 cm DE = 10.44 cm G m EG = 9.57 cm m EF = 20.71 cm E
    • Here I created segments from the vertices of triangle EDF to the Centroid G and measured the distance from this center to the vertices. I color coordinated each segment and its respective length for organizational purposes. Question 3: Can all three plants be watered from the one sprinkler by using the Centroid? How can you decide whether this particular triangle center is sufficient for the gardener’s problem by using just the properties of this particular center? ANSWER: This choice of center is better than using the Incenter however it is not as effective as the Circumcenter option for the sprinkler. The centroid is used for calculating the center of gravity of a triangle which in this case is of no relevance to the gardener’s situation. It does however accomplish the task of watering plants E and D. G D m DF = 15.86 cm F DE = 10.44 cm m EF = 20.71 cm E Here I created the perpendicular lines to each vertex and the opposing side of the triangle. This created the orthocenter of the triangle which is labeled G in the picture above.
    • G m GD = 4.35 cm m GF = 18.40 cm D m GE = 14.00 cm m DF = 15.86 cm F DE = 10.44 cm m EF = 20.71 cm E Here I created segments using each vertex and the Orthocenter G to represent the distance between the vertices and the Orthocenter. I color coordinated each segment and its respective length for organizational purposes. Question 3: Can all three plants be watered from the one sprinkler by using the Orthocenter? How can you decide whether this particular triangle center is sufficient for the gardener’s problem by using just the properties of this particular center? ANSWER: No this option is the least effective for the placement of the sprinkler. It is only capable of being able to reach plant D. Using the properties of an orthocenter and its relationship with triangles one would understand that the triangle created by the gardener’s plants forms and obtuse triangle. This means that the orthocenter will outside of the triangle and in this case specifically it lies far from being considered an option. Also, the sprinkler lies outside of the gardener’s lawn which isn’t going to make the neighbors very happy.
    • Content. Describe:COMMON CORE STANDARDS CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★ Describe:Standards of mathematical Practice CCSS.Math.Practice.MP1- Make sense of problems and persevere in solving them. CCSS.Math.Practice.MP3- Construct viable arguments and critique the reasoning of others. CCSS.Math.Practice.MP4- Model with mathematics. Pedagogy. Pedagogy includes both what the teacher does and what the student does. It includes where, what, and how learning takes place. It is about what works best for a particular content with the needs of the learner. 1. Describe instructional strategy (method) appropriate for the content, the learning environment, and students. This is what the teacher will plan and implement. Teacher will introduce the problem Teacher will allow the students to gather in groups of 2 to explore the topic. Teacher will circulate the room and ask open ended probing questions to facilitate the conceptual thinking and understanding of the topic. 2. Describe what learner will be able to do, say, write, calculate, or solve as the learning objective. This is what the student does. Students will be able to describe, explain, analyze, apply and evaluate real world situations. Students will be able to defend their solutions with sound deductive reasoning. Students will be able to create their own problems and solve them with proficiency and accuracy. 3. Describe how creative thinking--or, critical thinking, --or innovative problem solving is reflected in the content. The content of the lesson implores the students to solve one problem in more than one way to see which solution will be the most efficient depending on their results. Technology. 1. Describethe technology Students- GSP is computer software that helps users model mathematics and create mobile illustrates and depictions of mathematical problems geometrically to help the user conceptualize the topic of interest. Within this lesson, GSP will be used to create a model of each triangle center with the calculations noted below each illustration. It will help the students create a visual representations while still employing them to defend their representations by requiring them to organize and evaluate each strategy using the procedural mathematics. 2. Describe how the technology enhances the lesson, transforms content, and/or supports pedagogy. The technology enhances the lesson by engaging students to create accurate visual representations of the problem as they work through the appropriate calculations.
    • The technology helps transform the content by allowing students to quickly create representation of the problem expediting conceptual understanding through modeling. GSP supplements pedagogy by allowing the teacher to be able to discern student’s thoughts and conceptions of the topic by their visual representations more quickly than if they were to read through written mathematical procedural steps. It also allows students to manipulate the end product to continue further investigation of the topic or concept. 3. Describe how the technology affects student’s thinking processes. Due to the technology helping the students organize and expedite the modeling process this gives the students much more time to analyze what is going on and recognize relationships. Reflect—how did the lesson activity fit the content? How did the technology enhance both the content and the lesson activity? The lesson’s activity fit the lesson very well and allows the teacher to be able to quickly evaluate student’s conceptual understanding more efficiently. It also allowed the student continue their education on the topic by permitting them to manipulate their models for questions that they or the teacher may come up with. The technology enhanced the content by engaging the students and leading them to create accurate depictions of the concept/s, thereby increasing their understanding and retention of the subject matter.
    • Lesson Plan Template MATE 4001 (2013) Title: Triangles Subject Area: Math II Grade Level: 9th to 10th Concept/Topic to teach: The four different centers of a triangle and their relevance to real world applications. Learning Objectives: Students will be able to describe, explain, analyze, apply and evaluate real world situations and defend their solutions with sound deductive reasoning with the aid of GSP. Essential Question What question should students be able to answer as a result of completing this lesson? Students will be able to answer the question of which center will prove to be the most appropriate for the problem posed? Standards addressed: Common Core State Mathematics Standards: CCSS.Math.Content.HSG-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★ Common Core State Mathematical Practice Standards:  CCSS.Math.Practice.MP1- Make sense of problems and persevere in solving them.  CCSS.Math.Practice.MP3- Construct viable arguments and critique the reasoning of others.  CCSS.Math.Practice.MP4- Model with mathematics. Technology Standards: Copy and paste from NCDPI  HS.TT.1.1 - Use appropriate technology tools and other resources to access information (multi-database search engines, online primary resources, virtual interviews with concept expert).  HS.TT.1.2 – Use appropriate technology tools and other resources to organize information ( e.g. online note-taking tools, collaborative wikis).
    • Required Materials: Students: paper/pencil for personal notes and computers with GSP software for modeling and computations. Notes to the reader: Students already have a minor understanding of the various centers of a triangle the purpose of this lesson is for them to explore and utilize what they know and then draw conclusions about the relationship between this center and the triangle. Time: Assume 90minutes Time Teacher Actions Student Engagement Before Review previous concepts and introducing the problem of the Gardener. Students will be reviewing over their previous lessons notes and listening to further instruction from the teacher about upcoming class activity. Circulating the room and asking open ended probing questions while monitoring student’s progress and checking their understanding and reasoning through their visual representations. Students will be in self assigned groups of two and working on solving the problem. Once completed students will come up as a group and present their findings and defend their solutions to the class and the instructor. Reviewing the concepts learned and introducing briefly the next lesson’s topic. Making corrections to their work if needed and paying attention to what is expected in the future. 20 min During 60 min After 10 min Reflection By writing a lesson plan that incorporated GSP into the class it made the subject more interesting and engaging. It allowed me to see the benefits of using such software, such benefits include: quicker and more effective evaluation of student’s grasp of the concepts, allowing me to circulate the room and ask different probing questions to each group. By allowing students to be in groups this alone helped me circulate through the room quicker. For the students this lesson would help them see the problem in more than just letters and numbers but as well as a selfcreated representation of each concept. The program adds to their conceptual understanding by allowing them to manipulate each scenario and resolve questions that the instructor has asked as well as individual curiosity.