Points, Lines and Their Properties
Lesson Plan

Direct Instruction Model

• From National Council of Teachers of Mathematics
  • Reasoning and Proof: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Recognize reasoning and proof as fundamental aspects of mathematics
    • Make and investigate mathematical conjectures
  • Communication: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
    • Use the language of mathematics to express mathematical ideas precisely.
  • Connections: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Recognize and use connections among mathematical ideas
    • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
    • Recognize and apply mathematics in contexts outside of mathematics
  • Representation: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
    • Create and use representations to organize, record, and communicate mathematical ideas
    • Use representations to model and interpret physical, social, and mathematical phenomena
• From Utah Educator Network:
  • Geometry Content
    • 3.1.2 - write good definitions for terms. Include symbolic representation where appropriate
    • 3.1.5 - Sketch, label and recognize each of the following: parallel, perpendicular and skew lines
  • Process Objective
    • Problem solving:
      • draw a diagram
      • clarify understanding
      • check for reasonableness
      • guess and check
      • identify counter examples
    • Reasoning and proof:
      • investigate mathematical conjectures
      • formulate counter examples
      • realize that observing a pattern does not constitute proof
    • Communication:
      • Class and group discussions using precise language
      • Journal
      • Express mathematical ideas coherently
    • Connections:
      • establish connections among mathematical expressions and physical models
      • use real-world applications
      • explore historical and multicultural contributions to math
    • Representation:
      • use a variety of visual representations and tools (protractor, compass, straight edge, manipulatives)
      • represent patterns verbally, numerically, geometrically and algebraically

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

• Write a meaningful definition of a point, a line, a ray, a line segment, an intersection, perpendicular lines, parallel lines, and skew lines.
• Identify strengths and weaknesses of mathematical definitions.
• Show an understanding of mathematical definitions by writing a definition.

• This lesson requires the use of a straight edge and drawing surface for the teacher. This can be a white board, an overhead transparency, or other.
• The author recommends a large version of the definition rubric to display in the classroom for reference.
• Each student will need a straight edge and a drawing surface such as paper or personal whiteboard to draw figures.
• Each student will need a worksheet and a definition rubric. Download quiz.

• Continuous
• Definition
• Distance
• Length
• Line
• Point
• Proof
• Space

Pedagogical Technique - Brainstorming

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.

• Blind students will need accommodations to understand, interpret, and draw geometric figures.
• Students with cognitive difficulties will need additional assistance bridging from the concrete to the abstract.
• Some students may want to orally describe lines and points rather than in writing.

• An informal assessment will be performed by listening to students' contributions to discussion.
• An informal assessment will be performed by reviewing the students' notes.
• A formal assessment will be performed with a quiz.

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)

Download Quiz
Download Mathematical Definitions Rubric