Lesson: The Next Dimension

Contributed by: Techtronics Program, Pratt School of Engineering, Duke University

The Andromeda Galaxy
How do we describe a location in space?
copyright
Copyright © NASA/JPL-Caltech, source: http://www.nasa.gov/mission_pages/galex/pia15416.html#.U-LTlPldW6M (The Andromeda Galaxy)

Summary

The purpose of this lesson is to teach students about the three-dimensional Cartesian coordinate system. Students also gain perspective on the size of our galaxy (the Milky Way) and the distance of a nearby spiral galaxy, the Andromeda galaxy (shown on the right) using a 3D model. 3D graphing is an important tool used by structural engineers to describe locations in space to fellow engineers.
This engineering curriculum meets Next Generation Science Standards (NGSS).

Engineering Connection

Engineers use a coordinate system whenever they create engineering drawings of something, and the Cartesian coordinate system described in this lesson is used most often.

Pre-Req Knowledge

Some experience with the two-dimensional Cartesian coordinate system is helpful, but is not required.

Learning Objectives

At the end of the lesson, students shoud:

  • Be able to find a point in space given the X, Y and Z coordinates.
  • Have a sense for the dimensions of the Milky Way galaxy, where the Sun is in within it, and the distance to a nearby galaxy.
  • Be able to give the X, Y and Z coordinates, given a point in space relative to a specified coordinate system and origin.

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Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

  • Analyze and interpret data to determine scale properties of objects in the solar system. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). (Grade 5) Details... View more aligned curriculum... Do you agree with this alignment?
  • Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
  • Information and communication systems allow information to be transferred from human to human, human to machine, and machine to human. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
  • The use of symbols, measurements, and drawings promotes a clear communication by providing a common language to express ideas. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). (Grade 5) Details... View more aligned curriculum... Do you agree with this alignment?
  • Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
Suggest an alignment not listed above

Introduction/Motivation

Cartesian coordinates. A line drawing shows three lines from one center point, identified as x, y and z.
The 3D coordinate system.

Use a flat surface such as a piece of paper, or a table top. Put your finger at a point on the surface, and ask students how they would describe the location of that point. If they have had experience graphing on the XY plane, they will probably figure out that they can give the coordinates of that point relative to a particular corner (over 5 inches, up 9 inches). If they do not get this on their own, ask leading questions to help them. For example, "Well how far away is the point from the side of the paper?" Once students grasp this, move your finger so that it is above the surface you are using. Then ask them again how they would describe the location of that point. They may say something like "above the paper." Ask them to be more specific. The goal is to get them to give a description such as "5 inches over from the left side, 9 inches up, and 6 inches above." Ask them to give you the three coordinates necessary to describe the location of the points.

Next, spend a short time discussing the terms in the vocabulary section.

It is helpful to describe some concrete examples of how the 3D coordinate system is used in everyday life. For example, when you describe the location of an office within a building, you are essentially using coordinates: "Go up 3 floors, do down the hall past four doors and turn right, it's the second room on your left." City blocks are another example of the use of a coordinate system. Directions from one house to another might read, "Go three blocks, then take a right and go four blocks." This is an example of a two-dimensional coordinate system.

Coordinate systems can also help us visualize where we are within our galaxy, as well as how far away a "nearby" galaxy is in space. Have students complete the Galactic Perspectives Worksheet. Students should be placed in teams of 2-3 and each team should be given a ruler.

Lesson Background and Concepts for Teachers

The purpose of this lesson is to familiarize students with 2-D and 3-D graphing and to use these tools to develop a sense of the spatial scales relevant to our galaxy including its spatial dimensions and relative distances between objects within the Milky Way and a neighboring galaxy. Have students complete the Galactic Perspectives student worksheet to achieve this goal.

Students can also learn the basics of graphing in three dimensions. Conduct the A Place in Space associated activity to accomplish this. Its worksheet guides students to first review finding points on the 2D (XY) plane, and then moves on to finding and describing points in a 3D space (X, Y ,Z). In order to get the most out of the activity, groups need their own set of axes. Instructions for how to build these as well as supplies needed are described in the activity's Procedure section.

Vocabulary/Definitions

axis: In math, a line used as a reference to describe the location of a point. For example, in the Cartesian coordinate system, an axis is a line marked zero at a certain point. An object's location can then be described by measuring how far away (on the line) it is from this origin, and in what direction. In many ways, an axis is like a number line that goes on forever in both directions (positive and negative). An axis is one dimensional.

Cartesian coordinate system: The rectangular coordinate system developed by mathematician Descartes. It consists of 2 or 3 axes (X, Y and Z) all at right angles to each other, and all intersecting at a specified origin.

dimension: A measure of spatial extent. What we see around us is a three-dimensional world, because objects have three dimensions (length, width and height). A line is in one dimension, an area (such as a rectangle drawn on a piece of paper) is in two dimensions, and a box has three dimensions.

Galactic Center: The center of the Milky Way galaxy

graph: A visual representation of a mathematical function or set of numbers. In the previous definition for ordered pair, if your desk surface could be thought of as a graph, and you graphed the point (2,3), this means you represented those numbers visually (or graphically).

ordered pair: A way to describe the location of a point within a plane, relative to a specified reference point. For example, if the front left corner of your desk is the origin, and you want to find a point given the ordered pair (2, 3) and you know that the unit you are using is inches, you would start at that front left corner of your desk, move two inches to the right, and then 3 inches toward the back of the desk, and you would be at that point.

origin: The specified reference point (0,0,0) in most coordinate systems.

plane: The set of all points between two intersecting lines. A plane is two dimensional, so it is a flat surface. A flat table top, for example, can be thought of as a plane.

Associated Activities

  • A Place in Space - Students practice finding points in space and then describing the locations of given points in space.

Lesson Closure

Coordinate systems can be used for more than mapping objects in space. A coordinate system is used by engineers in all designs. The coordinate system is used to specify dimensions for products. When an engineer designs a part, s/he specifies where each point on the part is located using a Computer Aided Design (CAD) program. Often, parts can be manufactured by sending a CAD drawing file to a machine that is designed to interpret the file and create the part.

  • We have learned how to locate a point given an origin, and the X, Y and Z coordinates.
  • We can also describe the location of a point by providing this information.
  • We have applied this information to construct a 3-D model of our galaxy and a neighboring galaxy, and have gained some insight into galactic distances and related spatial scales.

Attachments

Assessment

  • If students were able to satisfactorily complete the Galactic Perspectives Worksheet with little or no help from teachers or peers then they have demonstrated the ability to use 2-D and 3-D models to explore the spatial scales and dimensions of the Milky Way galaxy and the relative distance to the nearest spiral galaxy (the Andromeda galaxy).
  • If students were able to satisfactorily complete the A Place in Space activity worksheet with little or no help from teachers or peers then they have demonstrated a sound understanding of the basics of plotting coordinates in three dimensions.
  • Students should be able to locate a point in space, given its coordinates and an origin.
  • Students should be able to describe the location of a given point in space relative to some origin using coordinates.
  • Should have an enhanced perspective on the dimensions of our galaxy and appreciate the relative distance to other nearby galaxies

Lesson Extension Activities

A practical way to apply what students have learned in this lesson/activity is to have them design basic structures such as bridges or towers using a computer aided design (CAD) program.

Students could also calculate the scaled-distances to much farther away galaxies, and discover that these galaxies would be quite far away even using the scale model that was applied to the Milky Way.

Contributors

Ben Burnham

Copyright

© 2013 by Regents of the University of Colorado; original © 2004 Duke University

Supporting Program

Techtronics Program, Pratt School of Engineering, Duke University

Acknowledgements

This content was developed by the MUSIC (Math Understanding through Science Integrated with Curriculum) Program in the Pratt School of Engineering at Duke University under National Science Foundation GK-12 grant no. DGE 0338262. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Last modified: August 22, 2017

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