Lines, Rays, Line Segments, and Planes

Abstract

This lesson is designed to introduce students to lines, rays, line segments and planes.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to lines, rays, line segments and planes
  • have learned the differences between lines, rays, line segments and planes
  • have practiced graphing lines, rays, line segments and planes

Standards Addressed

Grade 3

  • Geometry

    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 4

  • Geometry

    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 5

  • Geometry

    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 9

  • Geometry

    • The student demonstrates understanding of position and direction when solving problems (including real-world situations).

Grade 10

  • Geometry

    • The student demonstrates understanding of position and direction when solving problems (including real-world situations).

Fourth Grade

  • Geometry

    • Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Grades 9-12

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Geometry

  • Data Analysis and Probability

    • Competency Goal 3: The learner will transform geometric figures in the coordinate plane algebraically.

  • Geometry and Measurement

    • Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Integrated Mathematics III

  • Geometry and Measurement

    • Competency Goal 2: The learner will use properties of geometric figures to solve problems.

3rd Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of the connection between the identification of basic attributes and the classification of two-dimensional shapes.

6th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine
    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine unit rates; and the use of scale to determine distance.

4th Grade

  • Geometry

    • Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and moveme
    • Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and movement within the first quadrant of a coordinate system.

3rd Grade

  • Geometry

    • 3.18 The student will analyze two-dimensional (plane) and three-dimensional (solid) geometric figures (circle, square, rectangle, triangle, cube, rectangular solid [prism], square pyramid, sphere, cone, and cylinder) and identify relevant properties, incl

4th Grade

  • Geomety

    • 4.14

5th Grade

  • Measurement

    • 5.11a The student will choose an appropriate measuring device and unit of measure to solve problems involving measurement of length — part of an inch (1/2, 1/4, and 1/8), inches, feet, yards, miles, millimeters, centimeters, meters, and kilometers

Student Prerequisites

  • Arithmetic: Students must be able to:
    • draw and understand the coordinate plane
    • graph a line
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag.
    • use a browser for experimenting with the activities.

Teacher Preparation

  • Access to a browser (optional for this lesson)
  • Pencil and paper
  • Ruler
  • Copies of supplemental materials for the activities:

Key Terms

coordinate plane

A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin

infinity

Greater than any fixed counting number, or extending forever. No matter how large a number one thinks of, infinity is larger than it. Infinity has no limits

line

A continuous extent of length containing two or more points

line segment

A piece of a line with endpoints at both ends

ray

A straight line that begins at a point and continues outward in one direction

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Begin drawing a number line on the board, and ask the students if they know what you are making.
    • Draw a vertical line through the zero point of your number line, as if you were going to turn it into a coordinate plane.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we will be learning about lines, rays and the coordinate plane.

  3. Teacher Input

    The discussion presents an outline to use for explaining to students the key words of the lesson.

    • Draw several lines, rays, line segments and planes that have characteristics in common. Write a function on the board, and show students how to graph it. Then make a ray and line segment that correspond to the given line.
    • Proceed by graphing a second line on the same sheet of graph paper. Again, draw a corresponding ray and line segment. Now that you have two lines on the page you can construct the corresponding plane.
    • After modeling how to draw similar lines, rays, line segments and planes, discuss with students how the figures are similar to each other.
    • Next, repeat the exercise, this time drawing non-corresponding lines, rays, line segments and planes.
    • After modeling how to draw non-similar lines, rays, line segments and planes, discuss how the figures are different from each other.

  4. Guided Practice

    Pass out 4 sheets of graph paper to each student (or use 2 double-sided sheets).

    Try another function, this time letting the students direct your moves as you graph it and draw some similar, and then non-similar, lines. Ask the students to follow you by doing the exercise with you on their graph paper.

    • After students graph the function you give them, instruct them to make a ray and line segment that correspond to the given line.
    • Proceed by giving them a second line to graph on the same sheet of graph paper. Then have them construct the corresponding ray and line segment. Now that you have two lines on the page you can construct the corresponding plane.
    • After they practice drawing similar lines, rays, line segments and planes, have them discuss how the figures they drew on their papers are similar to each other.
    • Next, have them repeat the exercise, this time drawing non-corresponding lines, rays, line segments and planes.
    • After they practice drawing non-similar lines, rays, line segments and planes, have them discuss how the figures they drew on their papers are different from each other.

  5. Independent Practice

    • Allow the students to work on their own or in groups of 2 - 4. Give them several other functions to work with to complete the exercise above, this time working independently or in groups. Monitor the room for questions and to be sure that the students are focused on the assignment.

  6. Closure

    You may wish to bring the class back together and have each group show their results. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outline

This lesson can be rearranged in several ways if there is only one available computer:

  • As this is an introductory lesson, students do not need to use computers. However, if there is time, groups of students could read the discussion before beginning the line drawing activity. This would give more time during one class period for students to complete the hands-on part of the lesson.

Suggested Follow-Up

After this lesson, students will have an intuitive understanding of functions and will have seen many examples of linear functions. The next lesson, More Complicated Functions , introduces students to more general linear functions.