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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 5 and find the length of \overline{BA},\overline{AD} and \overline{DC}.

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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