Points, Lines and Their Properties
Lesson Plan

Developed byDavid McAdams
Last update28 Dec 2006
Grade Level9
Digital version
CopyrightUnpublished copyright work © 2005-2006, David McAdams, Orem Utah. This document may be duplicated for non-commercial educational use only.
Contact the author at DEMcAdams@usa.net.
  1. Lesson Model

    Direct Instruction Model

  2. Goal(s)/Standards
  3. Specific Objectives

    At the end of the lesson the student will be able to:

  4. Materials/Preparation
  5. Prerequisite Vocabulary
  6. Prerequisite Methodology

    Pedagogical Technique - Brainstorming

  7. Instructional Procedure
    1. Gathering Activity (optional).

      Have the students write down in their journals everything they can think of that has to do with a line or lines.

    2. Review
      1. Activate prior knowledge of the meaning of the word ‘line’ by having the students brainstorm answers to the question, "What is a line?" Have a scribe record all suggestions. Allow the students to propose their own ideas without comment.
      2. Help the students explore their definitions of a line by asking them to think of things that meet the various definitions, but are not lines. For example, if the definition is ‘something straight’ a flat desktop is straight, but is not a line.
      3. Help the student further explore their definitions of a line by asking them to think of things that are a line that are not included in the definition. Complete this item even if their definition is complete. This process will be used and expanded upon later in the unit.
    3. State the problem and objective
      1. State that mathematical definitions often differ from dictionary definitions. For advanced classes, explore why mathematical definitions differ (mathematical definitions are more exact to allow proofs using the thing being defined to use the properties of that thing).
      2. State and write on the board that, at the end of the lesson, the student will be able to give a mathematically correct definition of point and line, and be able to identify properties of a point and a line or lines.
    4. Introduce various definitions of a line and explore the strengths and weaknesses of each.
      1. Ask the class to create one or more definitions of a line using the ideas from the brainstorming.
      2. Add a few other definitions to the list on the board such as:
        • A long thin continuous mark1.
        • A path traced by a moving point5.
        • A continuous extent of length4.
        • A line is breadthless length3.
      3. Model note taking by making a table on the board as follows:
        Definition 1
        Depends on Strengths Weaknesses

        Have the students make similar notes in their workbook.

      4. Discuss each definition and its strengths and weaknesses.
    5. Formally define a line.
      1. State that in geometry, the definition of a line is axiomatic, meaning that it is taken as valid without proof. The attributes of a line are generally accepted to be:
        • Infinitely thin (having no width)
        • Infinitely long (having no starting or stopping point)
        • Straight (shortest distance between two points)
          Straight is taken to mean the shortest distance between two points. For advanced students, introduce the concept that the shortest distance between two points may not be what we commonly call straight. In hyperbolic geometry, it is in fact not what we call straight.
        • Continuous (having no empty spots or holes)
      2. Put various line segments, rays, and lines on the board and ask the class if any of these meet the definition of a line. Note that none of them do, as it is impossible to draw an infinitely thin, infinitely long, absolutely straight line. Introduce the concept of a representation: We assume the mark on the board or paper represents a line, even though it does not have the exact attributes of a line.
      3. Introduce rays and line segments as subsets of a line. Model note taking by writing the representation of each of these figures and the name.
    6. Points
      1. Have the students quickly brainstorm the attributes of a point, given the attributes of a line.
      2. Reinforce the attributes of a line: Infinitely thin and wide (has no width or breadth)
      3. Make connection between points and lines (a line is made up of an infinite number of points).
    7. Two lines
      1. State that two lines in the same plane may be parallel, perpendicular, or skew. Draw an example of each combination on the board or overhead. Form the students into groups and have each group write a mathematical description of parallel, perpendicular, and skew lines. After five minutes, combine each set of two groups into a larger group to combine description. After three minutes, have a representative of each group write their description on the board. Discuss each description and come up with a definitive description for the class.
    8. Formal Assessment - Quiz
      1. This quiz may be taken the same day to assist the students in moving short term memories to long term, or may be taken another day.
  8. Differentiation for Diverse Student Needs
  9. Assessments
  10. Other Resources
    1. Eather, Jenny, A Math Dictionary for Kids, http://www.teachers.ash.org.au/jeather/maths/dictionary.html
    2. Emints National Center http://www.emints.org/ethemes/resources/S00001219.shtml
    3. Euclid, Elements, Translated by Joyce, D.E., http://babbage.clarku.edu/~djoyce/java/elements/toc.html
    4. Quia Geometric Terms http://www.quia.com/jg/805list.html
    5. Random House Dictionary, Aug 1990
    6. WordIQ.com http://www.wordiq.com/definition/Line_(mathematics)
  11. Materials Masters

    Download Quiz
    Download Mathematical Definitions Rubric