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5.1: Basics of Geometry

Created by: CK-12

Points, Lines and Planes

I. Section Objectives

  • Understand the undefined terms point, line and plane.
  • Understand the defined terms, including space, segment, and ray.
  • Identify and apply basic postulates of points, lines and planes.
  • Draw and label terms in a diagram.

II. Problem Solving Activity- Global Architecture

  • Objective: The objective of this activity is to have students recognize the postulates connected with points, lines and planes in real life architecture.
  • Here are the postulates.
    • Line Postulate- There is exactly one line through any two points.
    • Plane Postulate- There is exactly one plane that contains any three non- collinear points.
    • Postulate- A line connecting points in a plane also lines with the plane.
    • Postulate- the intersection of two distinct lines will be a single point.
    • Postulate- the intersection of two planes is a line.
  • You can either print the pictures, use computer displays or slides.
  • The students need to use the following postulates and identify examples of each postulate as displayed in the picture.
  • Students should be encouraged to use mathematical language as they describe and write about each example of a postulate.
  • Figure01.01.01- The Guggenheim Museum in Spain http://en.wikipedia.org/wiki/Guggenheim_Museum_Bilbao
  • Figure 01.01.02- Eiffel Tower http://en.wikipedia.org/wiki/File:Tour_Eiffel_Wikimedia_Commons.jpg
  • Figure01.01.03- St. Basil’s Cathedral http://en.wikipedia.org/wiki/Saint_Basil%27s_Cathedral

III. Meeting Objectives

  • Students found examples of points, lines and planes in each architectural figure.
  • Students identified and applied the basic postulates associated with points, lines and planes.

IV. Notes on Assessment

  • There are several different illustrations of each postulate in the architecture provided. This is an activity where students are not only expected to be able to locate an example of each, but also where they need to use mathematical language to write about how each postulate is shown in the architecture.
  • An example of this is in the picture of the Guggenheim. You can see that two of the planes intersect in exactly one line. The sun is even shining directly on the line.

Segments and Distance

I. Section Objectives

  • Measure distances using different tools.
  • Understand and apply the ruler postulate to measurement.
  • Understand and apply the segment addition postulate to measurement.
  • Use endpoints to identify distances on a coordinate grid.

II. Problem Solving Activity- Town Design

  • Students are assigned the task of using measurement to design their own town.
  • Students will need rulers, pencils, colored pencils and chart paper.
  • Each town needs to have the following buildings in it: a post office, a police station, a bank, a park, a school and some houses. The students can expand this list if they choose.
  • Each town map needs a scale to determine the distances from one building to another building. This scale could be expanded to include standard and metric measurement.
  • Each students needs to develop a key that shows the measurements from one building to another.
  • Distances must be actual distances measured with rulers and matched to scale. This could be expanded to include both standard and metric measurement.
  • After the design has been completed, the students need to write a series of directions and distances for someone to travel around their town. The distances should be clear enough for any other student to follow.
  • Students pair up with a peer to check and make sure that student directions/measurements are accurate.
  • Finally, allow students time to share their designs in small groups or with the entire class.

III. Meeting Objectives

  • Students were required to measure distance using different tools.
  • Street measurement is a real life example of the ruler postulate.

IV. Notes on Assessment

  • Observe students as they work on this assignment.
  • You can create a rubric where each piece of the assignment is worth points.
  • For example, the scale is worth 4 points.
  • Then student grading is calculated out of the total possible points.
  • Check final work for accuracy.
  • Are the directions/measurements clear and accurate?
  • Does the town contain all of the essential buildings?
  • Provide student feedback and grading according to a rubric.

Rays and Angles

I. Section Objectives

  • Understand and identify rays.
  • Understand and classify angles.
  • Understand and apply the protractor postulate.
  • Understand and apply the angle addition postulate.

II. Problem Solving Activity- Angle Hunt

  • Use Figure01.03.01 in this activity.
  • Students are going to use the figure to apply each of the section objectives.
  • Students will need rulers, colored pencils or markers and protractors.
  • First, students take the drawing and find ten different angles.
  • They need to use letters and label each of the following angles.
  • Next, they make a list of each of the ten angles and classify each.
  • Then, they apply the protractor postulate to measure each of the ten angles.
  • Finally, they apply the angle addition postulate and create four different combinations of angles to calculate total measures.
  • When finished, pair up students and have them check each other’s work.
  • Each student needs to provide their peer partner with verbal and written feedback.
  • Then allow students time to share feedback in small groups.

III. Meeting Objectives

  • Students demonstrated understanding angles.
  • Students demonstrated classifying angles.
  • Students applied the protractor postulate.
  • Students applied the angle addition postulate.

IV. Notes on Assessment

  • Create a rubric to grade each students work.
  • Were ten angles labeled and identified?
  • Are the measurements of each angle accurate?
  • Did students successfully use the angle addition postulate?
  • Provide students with feedback and grading on their work.

Segments and Angles

I. Section Objectives

  • Understand and identify congruent line segments.
  • Identify the midpoint of line segments.
  • Identify the bisector of a line segment.
  • Understand and identify congruent angles.
  • Understand and apply the Angle Bisector Postulate.

II. Problem Solving Activity-Revolving Door Design

  • Use the two diagrams from this Wikipedia site. These are Figure01.04.01 and Figure01.04.02
  • http://en.wikipedia.org/wiki/Revolving_door
  • Students are going to be assigned the task of designing their own revolving door.
  • Point out that the diagram of the revolving door has four wings to it.
  • The students are going to be assigned the task of designing a revolving door with at least six wings in it.
  • Students will need rulers, protractors, pencils, and paper.
  • They can choose to add more wings, but the revolving door needs to have at least six in it.
  • Here are the specifics of the assignment:
  • Design a revolving door with at least six wings.
  • Each angle must be congruent.
  • Label each angle measure using a protractor.
  • Identify line segments that are bisected.
  • Identify the midpoint of each line segment.
  • Label each part of the revolving door and demonstrate congruency.

III. Meeting Objectives

  • Students will demonstrate an understanding of line segments, angles, congruency and bisecting angles in this lesson.
  • Students will also demonstrate measuring angles and identifying angles.

IV. Notes on Assessment

  • Assess student work by thinking about each of the following points.
  • Were the students successful in executing a design that matches the specifics of the assignment?
  • Are the angles of the wings congruent?
  • Are the angle measures labeled?
  • Is it clear that students understand the concepts discussed in the lesson?

Angle Pairs

I. Section Objectives

  • Understand and identify complementary angles.
  • Understand and identify supplementary angles.
  • Understand and utilize the Linear Pair Postulate.
  • Understand and identify vertical angles.

II. Problem Solving Activity- Visualize It

  • Students are going to go on a search for different types of angle pairs. This can be done in the classroom, but it would be best to expand it to the entire school or outside.
  • If possible, allow the use of digital cameras.
  • If this is not possible, students can draw sketches of the places where they locate each type of angle pairs.
  • Students can photograph or draw each.
  • They will need rulers, pencils, chart paper, clip boards.
  • Students need to locate three examples of each.
  • They first find three examples of complementary angles.
  • Three examples of supplementary angles.
  • Three examples of vertical angles.
  • Students must write a description of each example and explain why it is a complementary angle pair, supplementary angle pair or vertical angle pair.
  • Print student pictures and create a display of student work.

III. Meeting Objectives

  • Students will identify and write about complementary angles. This demonstrates understanding.
  • Students will identify and write about supplementary angles. This demonstrates understanding.
  • Students will identify and write about vertical angles. This demonstrates understanding.

IV. Notes on Assessment

  • Have students work in groups to assess each other’s work.
  • Request that students read each description of the angle pair to be sure that it describes each angle pair in mathematical terms.
  • You want to see that students are using measurements such as 90° for complementary angles, and that they are demonstrating that vertical angles are congruent.

Classifying Triangles

I. Section Objectives

  • Define triangles.
  • Classify triangles as acute, right, obtuse or equiangular.
  • Classify triangles as scalene, isosceles or equilateral.

II. Problem Solving Activity- Bicycle Design

  • Begin by showing students the following short movie clip from this website.
  • http://www.thefutureschannel.com/dockets/hands-on_math/bicycle_design/
  • Allow time for a student discussion about the short video.
  • Ask students what they observed about bicycle design from the video.
  • Write these notes on the board.
  • Allow time for questions and then give students the assignment.
  • Students are going to create their own bicycle design.
  • They need to use at least two different types of triangles in the design.
  • Students will need paper, rulers, pencils and colored pencils.
  • Students can design their own bicycle.
  • When finished, they need to label each type of triangle used and label it according to side length of and angles.
  • Then students may decorate their design.
  • Allow time for students to share their work.

III. Meeting Objectives

  • Students will define triangles by using them in their bicycle design.
  • Students will classify the triangles used according to side length.
  • Students will classify the triangles used according to angle measure.

IV. Notes on Assessment

  • Study each student design.
  • Does the design have at least two different types of triangles in it?
  • Are the triangles labeled according to side length?
  • Are the triangles labeled according to angle measure?
  • Provide students with feedback/corrections.

Classifying Polygons

I. Section Objectives

  • Define polygons.
  • Understand the difference between convex and concave polygons.
  • Classify polygons by number of sides.
  • Use the distance formula to find side lengths on a coordinate grid.

II. Problem Solving Activity- Polygon Sort

  • This activity requires students to sort polygons in three different ways.
  • 1. According to whether or not it is a polygon
  • 2. Convex or concave
  • 3. According to the number of sides
  • To prepare this activity, you will need to create or copy different polygons. You want an assortment of polygons and non- polygons, convex polygons, concave polygons and regular polygons (i.e. quadrilaterals, hexagons, etc.). Then you can place these all around the room.
  • When students begin the activity, they need to hunt for a specific number of figures. You could have each student find three different ones to work with. Then they can choose one for each exercise.
  • Then you can do a sorting exercise.
  • For example, “All of the polygons sit down. All of the non- polygons stand up.”
  • Then you can ask for a few examples from each group to explain why they are or are not a polygon.
  • Next, you can do another sort. Concave figures to the front of the room. Convex to the back of the room.
  • Same thing- ask for students to demonstrate why the figure is concave or convex.

III. Meeting Objectives

  • Students will be required to define polygons.
  • Students will demonstrate an understanding between concave and convex polygons.
  • Students will classify polygons according to the number of sides.

IV. Notes on Assessment

  • Assess student understanding by checking each “sorting exercise”
  • Also ask different student for feedback about why they “sorted” their polygons the way that they did.
  • Allow time for feedback and student questions.

Problem Solving in Geometry

I. Section Objectives

  • Read and understand given problem situations.
  • Use multiple representations to restate problem situations.
  • Identify problem- solving plans.
  • Solve real- world problems using planning strategies.

II. Problem Solving Activity- Camping Fire Expansion

  • Use the diagram on page 65 of the text. This will be Figure01.08.01.
  • Here is an expansion on the earlier problem.
  • Students can work in groups on this problem.
  • The fire has begun to spread. It had spread to a tent that is fifty feet north of her tent. It has also spread to additional tent that is twenty- five miles south of the river and fifty feet south of the original tent. Two other campers have begun helping with the fire problem. How can all three minimize their distances? What is the shortest distance any one of them can run to put out the fire?

III. Meeting Objectives

  • Be sure that the students have completed the work in problem 7 of the text before tackling this problem.
  • If they have, then this problem should be a natural extension of the original one.
  • Encourage students to follow the problem solving steps. There are two parts to this problem. Be sure that the students identify each part.
  • Then have students draw a diagram to show the original tent and the two new tents as well. Students can label them A, B and C.
  • Finally, using a scale, have students measure the distances.
  • Who has the shortest distance to run?
  • Ask students to show their work and to justify their thinking.

IV. Notes on Assessment

  • Walk around as students work on this problem.
  • Offer assistance when necessary and remind the students of the problem solving steps.
  • When finished, allow students an opportunity to share their work.
  • If there are different answers, ask groups to justify their answer.

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