Author: Dennis Clark
469 Mahoney Road
Oliver Springs, TN 37840
Telephone 865 435-9054
email dn.clark@comcast.net
Grade Level 9-12
Class Duration 3 hours
Prerequisites Algebra 1
Geometry 1
Basic familiarity with the binary number system.
Knowledge of the XOR or EOR (exclusive Or) binary operation.
Short Overview This lesson plan outlines the necessary and optional steps to
teach students how to draw N-space cubes (hypercubes) using both
an intuitive visual and a mathematical approach. It introduces
them to a research problem to find maximal length paths
(using a few special rules) in hypercubes. They will use
geometry, topology, algebra and the binary arithmetic exclusive
OR (XOR or EOR) for an activity that they are almost guaranteed
never to have studied. Students are shown that not all problems
have known answers.
Prep Time 3-6 hours
Because this lesson teaches an unfamiliar subject, the teacher
should prepare well in advance of the class. The author will
gladly answer questions by email or telephone.
Objectives Learn to draw 4-space cubes by hand with an intuitive, visual
approach.
Learn to draw 4-space cubes using a binary, mathematical approach.
Use the binary XOR exclusive OR operation to create adjacent
corners of N-cubes also known as N-space cubes or hypercubes.
Discover 3, 4, and 5-space maximal length paths (snakes) by hand.
Name maximal length snakes using binary node names.
Name maximal length snakes using an edge (line) numbering method.
Expand (t+2)**N to see the coefficients and discover their
significance to N-cubes.
Discover n-cube's relation to Pascal's Triangle.
Learn rudimentary definition of topology and how it applies to
hypercubes.
Discover why the problem of finding snakes explodes
computationally at 7-space.
Define the Computer as a tool that extends the reach of your mind.
Realize that the answers to problems are not always known.
That is what is fun about research!
Steps Overview Read "How to Draw N-space Cubes" found online.
Read "Chasing Snakes in N-space Cubes" also found online at
http://howtoadapt.com/ncubes
Read this lesson plan and decide on any optional exercises to use.
Present the lesson in one or more sessions.
Steps Optional Employ student(s) to create 3-D n-cubes by drilling holes in
wooden balls and testing (pre-assembling) n-cubes. The holes
in the wooden balls should be drilled ahead of time so that
students can assemble the N-cubes in class.
Advertise "Coming Soon: N-space Snakes!" using posters and handing
out Snake "business cards".
Vocabulary N-cube
N-space cube
hypercube
Pascal's Triangle
Snake
Twisted Circle Cube
Untwisted Circle Cube
Pascal's Triangle Cube
Geometry
Topology
XOR (EOR) Exclusive OR
Binary number system
Assessment Students will be able to:
Discover, map, and verify a 5-space snake on an 5-cube template
visually (without math). Enhances spatial relations.
Demonstrate the exclusive OR (XOR) binary operation.
Create 5-space cubes using the XOR binary operation, rules about
adjacent corners of hypercubes and the naming of hypercube
vertices with binary numbers.
Demonstrate that the coefficients of (t+2)**N represent the
numbers of points, lines, squares, cubes, and other sub
N-space cubes contained in a given N-space cube.
Define and find maximal length paths (snakes) in 5-space
hypercubes. They will be able to verify that a snake is both
maximal length and valid.
Name snakes using binary-named vertices that occur in a snake on
a hypercube.
Name snakes using the "edges" or lines that constitute a snake in
a hypercube.
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Lesson Plan