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Collinear and Coplanar Points

Run the Collinear and Coplanar Points MicroSim Fullscreen

<iframe src="main.html" width="100%" height="500px" scrolling="no"></iframe>

Description

This diagram illustrates the fundamental geometric concepts of collinear and coplanar points through four contrasting panels. Students can visually distinguish between points that share the same line (collinear) or plane (coplanar) and those that do not.

Panel 1 (Top Left) shows collinear points A, B, C, and D all positioned on a single blue line, demonstrating the definition of collinearity.

Panel 2 (Top Right) presents non-collinear points P, Q, and R arranged in a triangular formation, with dashed lines connecting them to emphasize that no single line passes through all three points.

Panel 3 (Bottom Left) displays coplanar points W, X, Y, Z, and V all lying on the same plane, represented by a semi-transparent purple parallelogram with a grid pattern.

Panel 4 (Bottom Right) illustrates non-coplanar points with M, N, and O on a plane and point P elevated above it, with a dashed line showing the perpendicular distance from P to the plane.

You can include this MicroSim on your website using the following iframe:

<iframe src="https://dmccreary.github.io/geometry-course/sims/collinear-coplanar-points/main.html" width="100%" height="600px" scrolling="no"></iframe>

Learning Objectives

Students will be able to:

  • Analyze point arrangements to determine whether sets of points are collinear or coplanar (Bloom's Taxonomy: Analyzing)
  • Distinguish between examples and non-examples of collinear points
  • Identify coplanar points in 2D and 3D representations
  • Understand that three or more points lying on the same line are always collinear
  • Recognize that four or more points may or may not be coplanar

Lesson Plan

Introduction (5 minutes)

Begin by reviewing the definitions: - Collinear points: Points that lie on the same line - Coplanar points: Points that lie on the same plane

Guided Exploration (10 minutes)

  1. Examine Panel 1: Ask students to identify what makes points A, B, C, and D collinear. Emphasize that you can draw a single straight line through all four points.

  2. Contrast with Panel 2: Have students explain why P, Q, and R are NOT collinear. Discuss how connecting these points forms a triangle rather than a single line.

  3. Study Panel 3: Introduce the concept of a plane as a flat surface that extends infinitely. Point out how all five points (W, X, Y, Z, V) rest on the same plane.

  4. Analyze Panel 4: Use the 3D perspective to show how point P is elevated above the plane containing M, N, and O. The dashed line helps visualize the perpendicular distance.

Practice Activities (10 minutes)

  • Have students identify collinear and coplanar point sets in the classroom (corners of desks, ceiling tiles, etc.)
  • Ask students to draw their own examples and non-examples of collinear and coplanar points
  • Challenge students: "Can three points ever be non-coplanar?" (Answer: No, any three points define a plane)

Assessment Questions

  1. If four points are collinear, are they also coplanar? (Yes - a line lies in infinitely many planes)
  2. Can two points be non-collinear? (No - any two points define exactly one line)
  3. What is the minimum number of points needed to define a plane? (Three non-collinear points)

Extension

For advanced students, introduce the concept that: - Any two points are always collinear (a line can be drawn through them) - Any three points are always coplanar (a plane can be drawn through them) - Four or more points may or may not be coplanar

Standards Alignment

  • CCSS.MATH.CONTENT.HSG.CO.A.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Metadata

  • Grade Level: 9-12 (High School Geometry)
  • Subject Area: Mathematics - Geometry
  • Topic: Foundations of Geometry - Points, Lines, and Planes
  • Duration: 15-20 minutes
  • Prerequisites: Understanding of basic geometric terms (point, line, plane)

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