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3.1: Basics of Geometry

Created by: CK-12

Points, Lines and Planes

I. Section Objectives

  • Understand the undefined terms point, line and plane.
  • Understand 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. Cross- curricular Study-Art

  • Use the painting on this website.
  • www.nationalgalleries.org/media_collection/6/GMA 1279.jpg
  • You can print it out or have students look at it on the computer.
  • One way to use this painting is to have students work in small groups to identify the points, lines, planes, segments and rays in the painting.
  • You could also use it as a spring board for the students to create their own black, white and gray piece of art that contains points, lines, planes, segments and rays.
  • You could even pair this up with the art teacher to really incorporate other disciplines into the mathematics classroom.

III. Technology Integration

  • One way to integrate technology is to use a drawing/painting program and have the students work to design their own artwork on the computer.
  • This could take several days, so be sure that you have access to the computers.
  • In the end, the students could present their material to the other students in the class and explain the geometric properties of their work.

IV. Notes on Assessment

  • When looking at student work, you want to include creativity in your assessment, but also look at the mathematics incorporated.
  • Did the student include all of the geometric elements?
  • Is the student able to describe the different elements to his/her peers?
  • Has the work been completed with care?
  • Offer feedback/correction as needed.

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. Cross- curricular Study- Surveying

  • Land surveyors measure distance all of the time.
  • They determine land boundaries, and property lines, as well as determine roads etc.
  • One possible activity is to have a land surveyor visit the class.
  • Have the surveyor bring in his/her tools also have them demonstrate the actual measuring to the students.
  • Maybe work on measuring a part of the school grounds.
  • Ask the presenter to be prepared to discuss the ways that geometry and measurement are featured in his/her job.
  • Have the students prepare questions and write a short report demonstrating what they have learned following the visit.

III. Technology Integration

  • Have the students research land surveying.
  • This can be done in class or as an at home assignment.
  • Completing this assignment will help the students to be prepared when the land surveyor comes to visit the class.
  • Ask students to keep track of the sites that they search and to jot down at least ten facts about land surveying as a career.
  • An extension into future careers could have the students research schooling and salary options for land surveyors.
  • Ask students to report on their findings.

IV. Notes on Assessment

  • One of the biggest points to assess in this activity is the student questions.
  • Ask the students to write down their questions and the answers to them.
  • This will help you to confirm that the students gave thought to their questions and were alert to the answers.

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. Cross- curricular- Astronomy

  • Begin this lesson by asking students to observe the night sky.
  • Tell them to make note of the different constellations that they observe.
  • Also ask the students to find these constellations on line or in a book and to bring the hard

copy of a picture of the constellation to class.

  • When students do this, you can begin a whole discussion about the angles in the constellations and the distance between stars.
  • This can lead right into the technology integration.

III. Technology Integration

  • www.geocities.com/angolano/Astronomy/PIinSky.html
  • Use the website for an in depth student of measurement in the sky.
  • This website looks at many different facets of geometry, measurement and astronomy.
  • Although some topics have not been discussed yet, have the student read it all anyway.
  • Then focus on the last section where students can see how to measure angles in the sky using their hands.
  • Then send them back out at night to rediscover the initial constellations using their hands to measure distances.

IV. Notes on Assessment

  • Assess student work in three sections.
  • First, discuss the initial constellations with the students.
  • What did the students discover?
  • Were they able to make connections between angles and the constellations?
  • Then move on to the astronomy site.
  • Finally, assess student observations once they had an understanding of how to measure using their hands.
  • What did this do to their initial observations?
  • Were the students able to expand on the initial sightings?

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. Cross- curricular-Architecture

  • Use the image from Wikipedia on A Frame Homes in this lesson. This is Figure01.04.01
  • www.en.wikipedia.org/wiki/File:Aframe2.jpg
  • You want the students to use the design of the A frame home to prove the theorems in this lesson.
  • Students are going to use the image to demonstrate the following:
  • 1. Congruent line segments
  • 2. Bisecting line segments
  • 3. Midpoints of line segments
  • 4. Congruent angles
  • 5. Bisecting angles
  • 6. Angle bisector postulate
  • Have students work in small groups, and then present their findings.
  • Another option is to have the students design their own A frame home.
  • When designing, the students will have to use the concepts in the chapter and apply them to the design of the home.

III. Technology Integration- Websearch

  • A great way to study the concept of bisection is through a websearch.
  • Have the students google “bisecting”
  • When they do this, many pages of images will pop up. For example, one image is of a fence bisecting two mountains. Another is an aerial view of a highway.
  • Ask students to select three different real- world images to work with.
  • You want the students to draw connections between the concepts in the text and the images that they have selected.
  • Ask the students to investigate how the terms bisect, congruent and midpoint applies to each image.
  • For example, the student might see that in the highway picture the roads are bisecting by other roads. In the mountain picture, the fence crosses the midpoint between the two mountains bisecting the distance between the two.
  • This activity will cause students to use higher level thinking skills. The connections may not be obvious.

IV. Notes on Assessment

  • Look at student work.
  • Are the students able to apply how each concept applies to the A frame home design?
  • Look at student designs- is the A frame home congruent?
  • Are the angles congruent?
  • You could choose to do some or all of the suggestions in this lesson, you are looking to see that the students understand the concepts and can apply them in real life situations. They could be doing this in a diagram, a presentation or a written explanation.
  • Assess student work accordingly.

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

Cross- curricular- Map of NYC

  • Use the image of a street map of Manhattan. This is Figure01.05.01.
  • www.aaccessmaps.com/show/map/us/ny/manhattan
  • Print a copy of this image for students to work with during the activity.
  • This map has many different examples of complementary and supplementary angles. As well as vertical angles.
  • Have the students work in pairs with a highlighter, colored pencils or markers.
  • The students are going to identify examples of each of the types of angles in the map.
  • Remind students to look at Broadway and at the way Broadway intersects the other streets.
  • Ask the students to make a list of the intersections on paper and how each angle fits the description.
  • You could also have the students enlarge the map (or do this ahead of time), and then use a protractor to measure the angles.
  • Ask students to share their work in small groups.

III. Technology Integration

  • Search maps from different cities.
  • You could use the map of the city that you live in.
  • You could ask the students to identify a city of their choosing.
  • Then complete the same exploration with a protractor on this map.

IV. Notes on Assessment

  • Check student maps.
  • Collect the student maps and their notes.
  • Have the students identified the angle pairs correctly.
  • If measurement was completed, is it accurate?
  • Provide students with feedback/comments on their work.

Classifying Triangles

I. Section Objectives

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

II. Cross- curricular-Triangle Creations

  • This is a fun way to explore triangles.
  • To prepare, you will need an assortment of one or more of the following items: gumdrops, marshmallows, toothpicks, tinkertoys, kynex
  • Be sure that the students understand the different types of triangles and have an example of each type.
  • Then have them create an example of each of the following triangles using the materials provided.
  • The students will have a GREAT time with this. It is very hands- on and since so much work has been done on the different triangle types, this will make the lesson fun.
  • Students need to label each type and example and be able to explain how and why it is that type of triangle.
  • Allow time for the students to present their creations.

III. Technology Integration- Geometric paintings

  • Use the following website for an investigation in geometric paintings using triangles. This is Figure 01.06.01
  • www.4.bp.blogspot.com/_wLt09kFTsi4/RdsN0RvBmgI/AAAAAAAAAAk/Oo6kiVl78AE/s400/BlackBeetle.jpg
  • Ask the students to work in small groups and identify the different types of triangles in this painting.
  • Then have the students create their own painting using the different types of triangles.
  • They should create a key and color code to show each type of triangle.
  • An alternative to this would be to have the students search and find triangles in other geometric paintings.
  • They can then use this one as a springboard for their own design.

IV. Notes on Assessment

  • Create a specific set of directions for the student art piece.
  • How many triangles do the students need to have in their design?
  • How many of each type?
  • Consider creativity.
  • Provide students with comments/feedback on their work.

Classifying Polygons

I. Section Objectives

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

II. Cross- curricular-Architecture

  • You will need a computer or a way to show this video during class.
  • You can go to www.futureschannel.com and have the students watch the short video on polygons in architecture.
  • Use this video as a discussion prompt.
  • Ask the students to identify how architecture is shaped by the use of polygons and how it depends on polygons.
  • Ask the students to brainstorm the many different types of polygons in the classroom.
  • Extension- ask the students to go home- take one room and write down all of the different types of polygons that they can find in that room.
  • Allow time for students to share their work.

III. Technology Integration

  • Have the students complete a websearch on different types of lens.
  • They can use Wikipedia for this.
  • Students are going to explore the concepts of concave and convex as it applies to lenses.
  • Ask the students to make a list of the different characteristics of concave polygons.
  • Then ask students to make a list of the characteristics of convex polygons.
  • Then have the student select five different lenses.
  • They need to create a description/explanation of how each lens is either concave, convex or neither.
  • Allow time for the students to share their work when finished.

IV. Notes on Assessment

  • Student work is assessed through discussion.
  • Be sure that all students have an opportunity to share.
  • You want to encourage this class lesson to include a lively engaging discussion.

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. Cross-curricular- Putting It Together

  • In this chapter, students have been working with all of the basics of geometry.
  • Now they are going to be combining these ideas to create their own word problem.
  • Students can use magazine pictures, clip art or other pictures in their problem.
  • Have the students work in pairs or small groups to design a word problem that uses some or all of the following concepts: lines, angles, triangles, polygons.
  • Students should demonstrate an answer key where students use the problem solving techniques from the chapter to solve the problem.
  • This may seem to be too broad an activity.
  • If so, you can give the students a topic to write their problems about such as sports.
  • Then all of the problems that the students write will have to do with the topic of sports.
  • Collect all student problems and answers when finished.

III. Technology Integration

  • Use the following website and have the students watch the video and geometry and dance.
  • www.thefutureschannel.com/dockets/realworld/dancing/
  • Use this as a discussion starting point.
  • Then you could show students a short clip from The Nutcracker (you will need to prep this ahead of time).
  • Ask the students to use the concepts from the first movie clip and apply it to the dancing in the Nutcracker.
  • This can be a fun way of showing how math is in places where you least expect it.

IV. Notes on Assessment

  • Collect student problems and answers.
  • Check all work for accuracy.
  • Provide students with feedback/correction.
  • You can use these problems as quiz questions or extra credit work.

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