Lesson Plan Edf Ca2

Teacher: James Westberry Subject/Class: Geometry – A1 Unit Name: Relationships in Triangles Date: 4-10-2009 Lesson Title: Bisectors, Medians, and Altitudes Sunshine State Standard: Students define and construct altitudes, medians, and bisectors, and triangles congruent to given triangles. (MA.912.G.4) Benchmark: Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter. (MA.912.G.4.2) Learning Objective: Given instruction on triangles, students will be able to define, calculate, apply, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenters, incenters, centroids, and circumcenters with 80% accuracy. Introduction: The class is asked how it would be best to balance a paper triangle (also known as a paper football) of any different angle measures on the end of a pencil tip, and if there is some special way to determine a center of gravity for the paper triangle. The teacher asks the students to make a paper triangle and encourages those who know how to make one to help the students who don’t know how to make one. These will be used at the end of class if time permits as an extra credit activity. The teacher then tells the class that triangles have several “centers” within them that are formed by using bisectors, medians, and altitudes of triangles. The teacher tells the students that there are different ways to manipulate the angles and sides of a triangle to gather more information about the triangle. The teacher mentions that there are four different kinds of centers formed by manipulating the triangle four different ways. Knowing these centers is crucial for determining where the different centers of gravity for a triangle are. Students will be using rulers and protractors to accurately construct bisectors, medians, and altitudes. The teacher reminds the students they know that there are different kinds of triangles that exist, and that triangles generally have the same properties, no matter what kind of angle measures are inside the triangle, unless it is specifically stated otherwise (such as only being able to explicitly use the Pythagorean Theorem for right triangles). Students will also be learning how to use their graphing calculators to help find points of concurrency, and will also be going to the computer lab to learn more about the properties of triangles later in the week, using a program with excellent animation and content. Presentation of Content: 1. Scaffolding: Students will be given a chart to copy down on the board for the different

terms that will be used and how they relate to one another. Initially, the teacher will use this chart as a reference guide until it becomes mundane or unnecessary to use it. 2. Conspicuous Strategies: The teacher will give the first example of how to manipulate a triangle by showing how to construct the perpendicular bisectors of a triangle and how to find the point of concurrency with these three lines. The teacher will talk aloud throughout the whole process and explain when it is appropriate to use this method and when to use it for similar methods (that is, with angle bisectors, medians, and altitudes). 3. Examples and Models: Examples will include what bisectors, medians, and altitudes are like in its most basic form (instead of a triangle, just a line segment or an angle). The teacher will also show why this can only be used for triangles, and not for other shapes (such as squares or hexagons) in finding out information about the polygon. 4. Checking Student Understanding: The teacher will check for understanding by drawing pictures that illustrate the types of bisectors, medians, and altitudes used, and pictures that do not illustrate the desired constructions, and by calling on students at random for each concept learned. 5. Reteaching and Extension Activities: The teacher will give a short worksheet to be completed independently or with the help of a buddy. The teacher will offer to all of the students a choice to make a creative chart that organizes all of the information learned during the lesson after completing a worksheet on the lesson. The teacher will give the answers after about 10 minutes has passed, and students can either watch the teacher work out each solution on the board or they can do their extra credit creative assignment (which is not required). The paper triangles made at the beginning of the class may also be used as extra credit if the student can demonstrate how to balance the paper triangle on the tip of a pencil, and how they arrived to their conclusion using the concepts they learned in today’s lesson. Practice and Feedback: 1. Guided Practice: The teacher goes through example problems on the board as he goes through the lesson. Students are to work on problems and take notes independently. The teacher observes individual students for signs of progress and will supply additional help, if needed, before moving onto the next concept. Students are to especially refer to and write down the chart that explains each concept (most of which are listed in the objective). 2. Independent Practice: Using an independent study worksheet, students learn to identify, use, and construct bisectors, medians, and altitudes within various triangles to determine the four different points of concurrency within the triangle. The teacher will circulate among the students to provide help, if needed. Students, at this time, may work one-on-one with each other for help. The

teacher then gives the answers for the problems, and then goes step-by-step through each problem. 3. Judicious Review: The teacher reviews these concepts in at least one additional lesson on the unit of relationships within triangles. He surveys the content of the unit for appropriate times to review these concepts. Lesson Summary: 1. Review Objective: The teacher writes the objective on the board and reads it

together with the entire class. He asks the students if they are now able to identify, use, and construct bisectors (angle and perpendicular), medians, and altitudes, along with all the points of concurrency (circumcenter, incenter, centroid, and orthocenter) that they make. The teacher explains that they will know if they have mastered the material after they take an assessment. He reminds the students of the lecture, example problems, and the worksheet that they completed that helped them learn the material. He tells them that they have learned a lot about triangles already and how they have many special properties. He tells the students that their next focus will be on inequalities and triangles. Assessment: 1. Procedures: Give students a quiz with 5 problems that assess the four

different types of constructions to a triangle and the four points of concurrency they make. Use different numbers (such as whole numbers, fractions, decimals, square roots) based on student mathematical levels. Give students verbal and written directions to construct the bisectors, medians, and altitudes and to identify the circumcenter, incenter, centroid, and orthocenter. 2. Judge Performance: Students are expected to identify all but two of the concepts given and be able to construct at least two of the four constructions. Review will take place after the assessment, with special attention to the problems with the most trouble. Conduct an error analysis for students who do not master the objective to guide reteaching activities. Accommodations: Students may need to be given extra time to complete the quiz. Some students may also need the quiz to be read to them. Some students may need an isolated environment to complete the quiz. Materials:

Rulers, Protractors, Paper (for writing and to make paper triangles), Writing Utensils.

Quiz 5.1 – Bisectors, Medians, and Altitudes Name:_______________________________ Date:________ Class:________ Directions: Fill in the blank with the correct answer and turn in at the end of class.

1. When three or more lines intersect, they are said to be ___________________. 2. A line that passes through a line segment so that the line segment is divided into two equal parts, and the intersection of the two lines form four right angles, is called a(n) ______________________________. 3. A line that divides an angle into two equal angles is a(n) ____________________. 4. A line whose endpoints are the vertex of a triangle and the midpoint of the opposite side is called a(n) __________________. 5. A line segment from the vertex of a triangle to the opposite side of the vertex, and that is perpendicular to the opposite side, is called a(n) ____________________. 6. Name all four of the points of concurrency that result from the four different types of line segments that we can draw inside of a triangle. a. ______________________ b. ______________________ c. ______________________ d. ______________________ For one extra credit point on the next test, write next to each point of concurrency the type of line segment that is used to form it. You must match ALL of the points of concurrency with the correct line segments to get the extra credit.

Answer Key: 1. 2. 3. 4. 5.

Concurrent Perpendicular Bisector Angle Bisector Median Altitude 6. (any order) a. circumcenter (perpendicualr bisector), b. incenter(angle bisector) , c. centroid (median), d. orthocenter (altitude)