Lattice Land: Polygon Exploration

Christina Pei
Mathematics
1-2 Class Periods (45 minutes each)
Geometry
v1

Overview

In this Lattice Land unit, students will be exploring the lattice and lattice polygons--an array of dots on a plane, such that there is one dot at each coordinate (x,y), where x and y are integers. In other words, {(x,y)|x,y∈Z}. This lesson can be the first in a unit using Lattice Land, a stand-alone exploration of geometric entities, or it may follow the series on Lattice Land Triangles or Lattice Land Squares. Students are asked to identify and define geometric entities: points, line segments, triangles, etc. There are some familiar geometric objects that cannot be drawn in Lattice Land, for example the circle. Students should explore some of these impossible Lattice shapes and discuss theories for why they are impossible. These discussions should also help cement what does and does not define a polygon. Students should also feel free to get creative. They can draw diagrams of their choosing, and may opt to explore in depth topics in symmetry, patterns, or optical illusions, and the like.

Standards

Computational Thinking in STEM
  • Modeling and Simulation Practices
    • Using Computational Models to Find and Test Solutions
  • Computational Problem Solving Practices
    • Troubleshooting and Debugging
  • Systems Thinking Practices
    • Communicating Information about a System
    • Investigating a Complex System as a Whole
    • Understanding the Relationships within a System

Activities

  • 1. The model: Lattice Land - Explore
  • 2.
  • 3. Share Your Work

Student Directions and Resources

1. The model: Lattice Land - Explore


  1. Open the file “Lattice Land – Explore”
  2. Click the “setup” button
    • You should see a black area with an array of yellow dots. This is called a “lattice.”
  3. Click “go”
  4. Select an “action”:
    • “draw-segment”
      • Click on any yellow dot in the black area. Next click on any other dot.
      • You will draw a line segment between the two dots.
    • “draw-polygon”
      • Click on any dot. Next click on any other dot. Then another, and another.
      • Continue linking dots.
      • When you click back on your first dot, you will have completed your polygon.
      • You can “check-area” for the last polygon you drew.
  5. You may clear all segments (“clear-segments”) or all polygons (“clear-polygons”), or reset everything (“setup”),
  6. See figure below for an example of segments and polygons:

                    

2.


Play around. Explore what kind of shapes you can make, and what you are not able to make in Lattice Land. Be creative. Ask questions. Share with friend.

You can also customize your drawing:

  1. Change the size of your world using the “width” and “height” sliders. Click “setup” to apply any changes.
  2. Notice that the width and height refer to the number of units (distance) and not the number of dots.
  3. At any time you may change the color of your segments. Just click the “segment-color” button and choose from the color palette.

 


Question 2.1

Play around. What kind of shapes you can make Lattice Land? List the types of shapes you can make and DEFINE each type of shape.

(For example: I can draw right triangles. A right triangle is a triangle that has one angle which measures 90 degrees.)



Question 2.2

What is a polygon? Be as specific and detailed as possible in your definition of polygon.

Note: It is not enough to say "a closed shape" because a circle is also a closed shape but not a polygon



Question 2.3

Can you draw a lattice polygon (that is, a polygon in Lattice Land)? Why or why not?



Question 2.4

What kind of shapes are not polygons? Draw at least 2 examples of shapes you can draw in Lattice Land that are not polygons.

Note: Draw your sketch in the sketchpad below


Question 2.5

Describe what kind of shapes are not polygons. How can you refine your definition of polygon so that it excludes such shapes?



Question 2.6

Define what is a circle. Can you draw a lattice circle (that is, a circle in Lattice Land)? Why or why not?

3. Share Your Work


Discuss your work as a group and/or as a class.


Question 3.1

Record your final definitions. Define any terminology that others defined and you may have missed in your definition. Did your definitions change? Did you discover anything new about Lattice Land?



Question 3.2

Did you notice anything else of interest that you'd like to share?