4.1: Basics of Geometry
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. Multiple Intelligences
This section is designed to assist educators in differentiating instruction with the multiple intelligences in mind.
 Visual Learners one way to assist visual learners with this lesson is to use the actual objects mentioned in the lesson. Where there is a map or a globe mentioned, use an actual map and a globe. This will also assist students with special needs in making a connection with the material.
 Kinesthetic Learners allow move time so that students can walk around the classroom identifying points, lines and planes in their surroundings. Request that students make a list of the things that they find.
 Interpersonal Learners have students work in pairs or small groups to discuss their findings from the “walk around” activity. This engages students who need to talk about their work to gain a better understanding of a concept.
III. Special Needs/Modifications
This section is designed to assist with any modifications or to assist students who face learning challenges.
 Be sure that all of the vocabulary words are written on a board or overhead as they are presented and discussed. Request that students copy this information into a notebook. Reading the terms, hearing them discussed, seeing them written again and writing the words themselves assists students in retaining information.
 Write each postulate on the board as it is discussed.
 Example 7 Expand this for all learners.
The goal here is to assist students in grasping and learning each term/postulate and its definition. The more students interact with each term and concept, the more they will remember what has been taught.
 Draw an example of each vocabulary word. Example, draw three collinear points.
 Draw an example that illustrates each postulate.
 Allow students to have an interpersonal connection by discussing their drawings with a peer.
IV. Alternative Assessment
There are many ways to assess student understanding during a lesson. This section provides a few ideas for that.
 Walk around and observe students as they work. Are students on task? Are they working diligently? Is the conversation appropriate to what is being taught?
 Use peers to assess each other. With the activity in Example 7, have the students assess the accuracy of each other’s work and correct any inconsistencies. If time allows, you could even have a presentation part where students share their findings.
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. Multiple Intelligences
 Activity with Example 3 This activity will address the following intelligences visual, interpersonal, kinesthetic, logical mathematical.
The students work in pairs. One member in the pair draws a line segment that he or she has measured to find the distance. The other member also draws a line segment that he/she has measured. Then they switch drawings. Student A must figure out the length of student B’s line segment, and student B must figure out the length of student A’s line segment.
 Extension with postulates. In this lesson, there are two key postulates. One is the Ruler Postulate and the other is the Segment Addition Postulate. The students then take their work and determine which line segments are examples of the postulates. They can even exchange with another group to accomplish this.
This extension includes visual, interpersonal, kinesthetic, logical mathematical intelligences.
 Discussion of activity by having the students share their examples, answers and reasoning with the entire class or in small groups, the intrapersonal intelligence is included as students share their personal insights into their work.
 Activity with Example 5 and 6 hand out grid paper. Ask students to draw a coordinate plane and provide given distances on the board/overhead. Then allow the student time to draw a line segment with this distance. Then provide a time for sharing/feedback from the exercise.
III. Special Needs/Modifications
 Begin class with a brief review of previously learned material. This can be done with words on the board/overhead or as a class discussion.
 Review what distance means and what it means to estimate.
 Write all vocabulary on the board as it is brought up in the lesson. Request that students take the time to copy this information in their notebooks.
 Ruler Postulate defined on board.
 Segment Addition Postulate defined on board.
 Be sure that students are given plenty of time to think through their work and be sure that all students have finished examples before going over the answers. Sometimes, special needs students require more time to complete tasks and will stop working if the answers to a particular question are given before they have finished.
IV. Alternative Assessment
 Observe students as they work in groups. Notice which students need assistance or seem lost. Make a note of who each student is and set aside a time to check in with each of these students.
 Create an observation checklist of things to watch for when students are completing exercises in a group.
 Pay close attention to student thinking during discussions before and after an activity.
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. Multiple Intelligences
 Activity with identifying rays and angles.
Have students work in small groups. Assign one group rays and the other group angles. Using rulers, the students need to design a series of either rays or angles. You can use index cards for this activity. Then have the groups switch cards. The angle group needs to name all of the rays that the other group has drawn. The ray group needs to name all of the angles that the angle group has drawn. Then the groups exchange answers and check each other’s work. This involves discussion and peer tutoring as well.
Addresses the following intelligences

 Linguistic students discuss their answers and thinking
 Logical mathematical students draw their angles and rays
 Spatial visual students draw angles and rays
 Interpersonal students share their thinking in a group
 Intrapersonal students explain their answers in a group
 Activity with Protractors
Provide students with drawings of several different angles. You can use the angles that were drawn in the previous activity. Have students measure their angles using protractors. Then have the students all share in a class discussion.
III. Special Needs/Modifications
 Begin each lesson with a review of previously learned vocabulary words and information. This helps students to recall what they have learned in a previous lesson. It also decreases the number of confused students once an assignment has been given.
 Write all vocabulary on the board or overhead. Request that students write these terms in their notebooks.
 Vocabulary for this lesson
 Ray include symbol notation and an example
 Angle include symbol notation and a diagram with sides and vertex labeled.
 Right angle include drawing
 Perpendicular include symbol
 Acute angle include drawing
 Obtuse angle include drawing
 Straight angle include drawing
 Protractor Postulate
 Angle Addition Postulate
IV. Alternative Assessment
 Use an observation checklist to observe students as they work.
 Pay attention to the questions asked during the lesson.
 Make a note of students who are having difficulty. Consider flexible grouping to assist these students in 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. Multiple Intelligences
 We can differentiate this lesson by organizing the content into a table. This is done as part of a class discussion. It is not done ahead of time and then presented. Creating the chart is meant to be interactive. Since this lesson works with line segments and angles, we can use these as the two columns of our table. Here is a sample of a table and how to organize it for the students.
Line segment  Angles 

Congruent (show example)  Congruent (show example) 
Segment midpoint  Show vertex and sides 
Show symbols  Show symbols 
Segment midpoint postulate  Angle bisector postulate 
 Be sure to explain each concept and how they are different and similar depending on whether you are working with line segments or angles.
 This helps the students to see the connections between the concepts.
II. Multiple Intelligences: Linguistic, logical mathematical, spatial visual, interpersonal, intrapersonal
III. Special Needs/Modifications
 Review previously learned information. One way to do this is with students working in pairs to quiz each other.
 Write all vocabulary on the board/overhead. Request that students copy this information in their notebooks.
 Vocabulary
 Congruent with symbol
 Segment
 Midpoint
 Segment Midpoint Postulate
 Segment bisector
 Angle bisector postulate
IV. Alternative Assessment
 When creating the table, be sure to include all students in the discussion.
 Refer students back to the information in the lesson to assist with adding in the information.
 Make a note of which students have a strong grasp of the material. Be sure to pair those students up with students that seem to be having difficulty when working on in class assignments.
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. Multiple Intelligences
 In this lesson, students are going to work on understanding the relationship between pairs of angles. One way to assist students in doing this is to create a chart that compares and contrasts the different relationships.
 Go through all of the material in the lesson first. Be sure that the students have a basic understanding of the terms and concepts in this lesson. You want to use the activity to expand student knowledge and understanding.
 To do this, students are going to work in small groups. Review what that compare means to look at the similarities between things, and that contrast means to look at the differences between things.
 Hand out chart paper and markers to each group.
 Request that students compare and contrast supplementary angles, complementary angles, linear pairs and vertical angles. Ask them to include drawings to justify what they are comparing and contrasting.
 Then allow time for the students to share their chart work with the rest of the class.
 II. Multiple Intelligences linguistic, logical mathematical, bodily kinesthetic, spatial visual, interpersonal, intrapersonal
III. Special Needs/Modifications
 Sometimes, special needs students will have difficulty remembering how to do previously learned skills. Here are some prerequisite skills to review prior to beginning this lesson.
 Solving one step equations
 Solving multi step equations
 Combining like terms to solve an equation
 Write all vocabulary words on the board/overhead. Request that students copy this information in their notebooks. Students will need this information to complete the activity.
 Vocabulary
 Adjacent
 Congruent
 Complementary angles
 Supplementary angles
 Linear pairs
 Line pair postulate
 Vertical angles
 Vertical angles theorem
IV. Alternative Assessment
 This is a great lesson to use an observation checklist. Make the checklist prior to teaching the lesson. Then use it while groups complete and present their charts. It will provide you with clear things to look and listen for when teaching this lesson.
Classifying Triangles
I. Section Objectives
 Define triangles
 Classify triangles as acute, right, obtuse, equiangular
 Classify triangles as scalene, isosceles, or equilateral
II. Multiple Intelligences
 This activity is a drawing activity that involves students creating a design and then classifying the triangles within the design.
 The students are given plain paper and a ruler. They are told to create a page of triangles created by intersecting lines.
 Once they have finished, ask them to create a key and to color (use crayons or colored pencils) to color in the different triangles found in the design.
 Students can be asked to finish this design for homework.
 Multiple Intelligences logical mathematical and spatial visual
III. Special Needs/Modifications
 Provide students with a diagram of a triangle with the vertices labeled, the sides labeled and the angles labeled. Be sure that students understand where to find the interior angle. They will need this to classify the triangles.
 Write all vocabulary words on the board/overhead. Request that students copy these notes into their notebooks.
 Vocabulary for classifying by angles
 Right
 Obtuse
 Acute
 Equiangular
 Vocabulary for classifying by side lengths
 Scalene
 Isosceles
 Equilateral
IV. Alternative Assessment
 Use observation to assess students as they work. Most students will need assistance creating a key to show how their design has been colored. You may want to provide an example of this and then see how the students do following directions.
 You can pair students up to work together too. This may help students who are having a difficult time with the activity.
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. Multiple Intelligences
 This activity has students begin working alone and then they work in a small group. The purpose is to assist students with developing a deeper understanding of concave and convex figures.
 Students begin alone. They can choose to draw either four concave polygons or four convex polygons. They don’t tell anyone else what they have chosen.
 When finished, the students join a small group. Then they exchange papers and they must show, by drawing lines, whether the figures they have been given or concave or convex.
 Students need to justify their answers.
 Peers correct each other’s work.
 Multiple Intelligences linguistic, logical mathematical, spatial visual, interpersonal, intrapersonal
III. Special Needs/Modifications
 Polygon drawing with sides and vertices labeled on the board.
 Write all vocabulary words on the board. Request that students copy these words in their notebooks.
 Complete the distance formula examples slowly on the board/overhead. Be sure that the students are following along.
 Add another example using the distance formula. Use exercise and find the length of and .
IV. Alternative Assessment
 Collect all student work when the groups have finished. Review their work and see how the students have justified whether their figure was concave or convex. This will show you a lot about how students were thinking as they worked on the assignment.
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. Multiple Intelligences
 The great thing about this lesson is that each way of solving a problem can be identified with a specific intelligence. This lesson can assist each student in understanding how he/she works best.
 Begin by presenting all of the information in the lesson. Request that students take notes too.
 Then go through a brief discussion on multiple intelligences. Ask the students to try to identify how they learn best. Have them write this down on a piece of paper. You can refer back to this discussion throughout your teaching and help students to further define the ways that each of them learns best.
 Once students have identified how they learn best, reorganize the class according to learning styles.
 Then ask each group (you may need to subdivide if groups are large) to solve the exercises at the end of the section according to how the group learns.
 Students will easily leap into this, but if not help them with an example or two.
III. Special Needs/Modifications
 Review concept the Pythagorean Theorem it is mentioned in the lesson, but not reviewed.
 Write the steps to simplifying a problem on the board. Review what it means to “simplify” something.
 What is this problem asking for?
 What do I need to know to find the answer?
IV. Alternative Assessment
 Make notes about the groups of students when organized according to how they learn best.
 This can be very valuable when assisting students in learning.
 For example, a visual learner could better understand a concept presented verbally by drawing a picture. When you are aware of which category each student falls in, you can better address his/her needs when teaching.
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Feb 22, 2012Last Modified:
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