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Lesson Plans

 

Perpendicular and Angle Bisectors

 

1.  Objective:  Students will use properties of perpendicular bisectors and angle bisectors to construct concurrent lines.  They should discover the concurrency theorems. 

 

State Objectives:  COS – 5, 9 AHSGE – VII.2, VII.4

 

Materials:  laptop computer, sketchpad, paper, pencil, textbook

 

Lesson Opener:  I will ask students to name some terms they remember from chapter.  I will make a concept map using the airliner and ask them how the words relate to real life situations.  I will include examples of each term on the board. 

 

Procedures:

Perpendicular Bisectors

1.       Students will work with one partner.  They will choose a partner. 

2.      Student will log onto sketchpad.

3.      Students will construct a segment.

4.      I will demonstrate a way to find the midpoint of a segment and how to construct a perpendicular line.

5.      Students will construct a triangle. 

6.      I will ask students to construct perpendicular bisectors.

7.      Students should already know the lines intersect at a point called the circumcenter.    They will label the point. 

8.      I will ask students to prove that the concurrent lines are perpendicular bisectors by measure the sides and angles.

9.      I will ask them to find another relationship that always exists with the circumcenter.

10.  They should discover that the circumcenter is equidistant from the vertices of the triangle by constructing those segments and measuring.  They may prove this theorem by constructing a circle with the circumcenter as the center.  They should prove that the theorem holds true with an acute, obtuse and equilateral triangle.  They will prove this by moving one of the vertices of the triangle. 

Angle Bisectors

1.      Students will construct an angle.

2.      I will demonstrate how to construct an angle bisector.

3.       Students will construct a triangle. 

4.      They will construct angle bisectors and construct the point of intersection.  They should label the point of concurrency as the incenter. 

5.      I will ask students to find a way to prove the lines are angle bisectors.  They should measure the angles.

6.      I will ask them to find any other special relationships.

7.      They should discover the theorem that angle bisectors are equidistant from the sides.  They may do this by constructing a circle inside the triangle.   

 

Closure:  We will discuss what the students have learned from this activity. 

 

Homework: P. 275-276 1-19

 

Evaluation:  Students will have a rubric to check off the steps as they are working on the assignment.  Students will check their homework. 

 

 

 

Medians and Altitudes

 

2.  Objective:  Students will use the properties of medians and altitudes to construct concurrent lines. 

 

State Objectives:  COS – 12  AHSGE – IV.2, VII.4

 

Lesson Opener:  We will review the terms median and altitude.  I will ask students to compare the properties of each. 

 

Procedures:

Medians

1.       Students will construct the midpoint of each side of the triangle. 

2.      They will construct the medians.  They should discover the medians intersect in one point.  They will label this point centroid. 

3.      I will ask them to measure the distance from each vertex to the median. 

4.      Students will develop a formula to find this distance.  They should use the measures to prove the concurrency theorem of medians. 

5.      Students will drag one of the vertices of the triangle to prove this theorem works with different kinds of triangles. 

Altitudes

1.       Students will construct a triangle.

2.      They will construct the altitudes of each triangle. 

3.      Students should discover that the altitudes are concurrent.  They will label this point the orthrocenter. 

4.      Students will move one point of the triangle to prove that the lines will still be concurrent regardless of the type of triangle. 

 

Closure:  We will discuss what the students have learned from this activity.  Students will write a reflection about the use of sketchpad to review concurrent lines.

 

Homework:  P. 282-283 1-23

 

Evaluation:  Students will have a rubric to check off the steps as they are working on the assignment.  I will check homework. 

 

 

 

Mid-segments

 

3.  Objective:  Students will construct mid-segments of a triangle and discover the mid-segment theorem. 

 

State Objectives:  COS – 12  AHSGE – IV.1, IV.2

Lesson Opener:  I will give students two numbers and ask them to find a number exactly between the two numbers.  I will ask them to explain the process of getting that number.  They should discover the midpoint formula. 

Procedures: 

  1.  Students will construct a triangle on a coordinate plane in sketchpad. 
  2. They will construct the midpoint of each side of a triangle.  They may do this by using the construct function in sketch pad or they can find the midpoint using the formula and then construct the points in the appropriate place. 
  3. Students will construct a segment from one midpoint to the other.  They will identify this segment as mid-segment. 
  4. Students will use this construction to discover the mid-segment theorem. 
  5. They should measure the length of the mid-segment and the side opposite the mid-segment and figure out the relationship. 
  6. They should also find a way to prove the mid-segment is parallel to the side opposite.  They should find the slope to prove this. 
  7. After they come up with some conjectures, they should construct the other mid-segments and see if the same conjectures hold true. 
  8. They should move one of the vertices of a triangle to see what happens with the mid-segments when the triangle changes form. 
  9. I will ask students to find the perimeter of the big triangle and the smaller triangle made with the mid-segments.  I will ask them to find the relationship. 

 

 

Closure:  We will discuss what the students have learned from this activity.  Students will write the theorem in their notebook. 

 

Homework:  P. 290-291 12-29

 

Evaluation:  Students will have a rubric to check off the steps as they are working on the assignment.  I will check homework. 

 

 

 

 

 

Polygons

 

 

4.  Objective:  Students will identify, name, and describe polygons using sketch pad.  Students will understand the sum of the measures of the interior angles of a quadrilateral. 

State Objectives:  COS – 3, 4  AHSGE – II.1, VII.4

Lesson Opener:  I will explain to the students that we are beginning a new chapter in which the big idea is polygons.  We will have a discussion about everything they already know about a polygon.  I will make list on the board.  I will ask the class to think about some ways to organize the data on the board in a way that we can add to it as we go. 

Procedures: 

  1.  Students will construct a parallelogram. 
  2. They will have to develop a method of construction for parallel lines. 
  3. They should be able to drag one vertex and the figure stays a parallelogram. 
  4. Students should draw in the diagonals and find a relationship between their lengths. 
  5. Students should look at the webpage http://www.math.com/tables/geometry/polygons.htm
  6. They should read the page and try to find any terms that they can identity on their quadrilateral. 
  7. I would like for them to add one or more sides and use other definitions.
  8. When students finish they should have used and constructed terms like concave, convex, pentagon, ect.
  9. Next students will develop a strategy calculate the sum of the interior angles of a polygon. 
  10. I will allow each group share their idea.  I will encourage each group to try to think of something that has not been said. 

 

Closure:  I will ask students if they think parallelograms are important in construction.  I will show them some pictures where they can identify the parallelograms.  We will discuss this. 

Homework:  P. 325-326 12-30

 

Evaluation:  I will check homework. 

 

 

 

 

Parallelograms

 

5.  Objective:  Students will discover and use the properties of parallelograms. 

State Objectives:  COS – 3, 4, 5  AHSGE – II.1, VII.4

Lesson Opener:  I will ask students to make a list of things they already know about a parallelogram.  I will show students some pictures of objects they use parallelograms, such as a bridge.  I will ask them to make sure the things they listed are true by looking at the picture. 

Procedures: 

  1.  Students will construct a parallelogram using geometers sketch pad. 
  2. They will use the drag test to make sure the construction is correct. 
  3. They will construct the diagonals.
  4. They will measure all of the sides and angles of the parallelogram using the measure function of sketch pad. 
  5. They will make a list of the anything they think will always be true.  For example, they should discover that the opposite sides and angles are congruent.  They should also discover that consecutive angles are supplementary and the diagonals bisect each other. 
  6. They will be able to prove these theorems by changing the size of the parallelogram to see which measures stay the same. 

 

Closure:  I will review the theorems that the students have discovered and ask the students to write them down in their notebook so they can review for homework. 

Homework:  P. 334 20-37

 

Evaluation:  I will check homework. 

 

 

 

 

 

Graphing Inequalities

 

6.  Objective:  Students will graph inequalities using sketchpad and a developed with sketchpad called Algebra in Motion. 

State Objectives:  COS 7, 8  AHSGE -

Lesson Opener:  I will show students an example of a system of linear inequalities graphed on a coordinate plane. I will ask students if they work out.  I will ask them how they know if they are overdoing it or not working hard enough.   This will be an example of a system of inequalities in two variables to help establish a person’s target zone for a workout. 

Materials:  graph paper, laptop

Procedures: 

  1.  Students will log onto laptop and look at the following website for example.   

http://www.mhhe.com/math/devmath/aleks/wt-ia/student/olc/graphics/author_ed/chp3sec23.htm

  1. I will ask students to look at the graph and see if they can figure out what the lines and variables mean. 
  2. I will give students a system of inequalities to graph using a sheet of graph paper.  They will work with a partner to do this.  They will plug in at least 4 points to find the solution.  They should check each other’s work. 
  3. Students will use the sketch pad program Algebra in Motion to graph the system of inequalities. 
  4. They will open sketch pad and then Algebra in Motion. 
  5. On Algebra in Motion, they will open inequalities.  They will plug in the inequalities in the appropriate place.  They will select graph.  The program will graph the inequalities. 
  6. The program has a point that can be moved around to different places.  It will state if that point is the solution or not.  They should check the same point they includes on the graph paper. 
  7. I will give students 3 other inequalities to graph using the program.  I will ask them to show me the solution. 
  8. A written explanation of the graph is required. Why did you use a solid or dotted line? Explain the shading chosen. Name the chosen test points and if they worked or not. Work for test points must be in the explanation.
  9. I will give students an example out of the book that will help each students find their target heart range.  (P. 390 example 5) They will graph this with a written explanation. 

 

Closure:  I will ask student to write a short refection of the assignment.  I ask if anyone will share their reflection.  I will explain that we will use the same steps they have learned to do linear programming. 

Homework:  P. 390 23-28

Evaluation:  I will use the following check sheet to grade the assignment. 

 

 

 

 

Algebraic Connections – Scoring guidelines        (1 = lowest, 5 = highest)

 

Graphing systems of linear inequalities

1.            Completion and correctness of all seven problems

 

                                1              2              3              4              5

2.            Graph on chart paper

 

                                Correctness

 

                                1              2              3              4              5

 

                                Neatness

 

                                1              2              3              4              5

 

                                Creativity

 

                                1              2              3              4              5

3.            Presentation

 

                                Calculator proficiency

 

                                1              2              3              4              5

 

                                Clear explanation

 

                                1              2              3              4              5

4.            Written statement

 

                                Clear explanation

 

                                1              2              3              4              5

 

                                All 4 test points worked correctly

 

                                1              2              3              4              5

 

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