The Euler Family Reunion:
Helping Students Visualize, Construct, Sketch, and
Understand Basic Geometric Concepts
Martha Barker, Candace Brame, William Flythe, and LeatonjaSallee
Discrete Mathematics for Teachers
MATH 539, Summer 2009
The National Council of Teachers of Mathematics (NCTM) recommended that discrete mathematics be
"an integral part of the school mathematics curriculum for all grade levels in their Principles and
Standards for School Mathematics in 2000. This new and rising branch of contemporary mathematics
which is widely used in business and industry emerged in the middle of the twentieth century with the
expansion of the computer revolution. This branch of mathematics, because it is useful, contemporary,
and pedagogically powerful, in addition to being a substantial and active field of mathematics, is a
strong link to technology and today's schoolchildren - tomorrow's technological workforce. In working
with discrete mathematics, students strengthen their skills in reasoning, problem solving,
communication, connections and representations in various settings. Students provided with
opportunities to explore discrete math principles find a smooth transition into future math curriculum
with a deeper understanding of many real-life situations as it motivates and interests students.
During the middle school years, students are introduced to concepts that are related to discrete
mathematics through geometry and probability and statistics. Students are required to identify and
understand topics and vocabulary dealing with vertices, points, lines, graphs as well as patterns. In
2009, the Virginia Standards of Learning (SOL) will initiate the revision of all math standards to introduce
students to concepts of algebraic and geometric thinking in earlier grade levels. Through this earlier
introduction, students begin to associate these basic skills through the real-world as well as through the
pages of a textbook. Recognizing the discrete math topics earlier and in the context of the real-world,
students will be able to associate with these ideas and how they affect one's daily life.
This unit has been prepared to provide opportunities for understanding and concept strengthening as
listed in the National Council of Teachers of Mathematics standards that were adopted in 2000. The unit
was created in accordance to mathematics and algebra standards of 2001 as well as introduced through
basic geometry standards and NCTM guidelines. The lessons are adaptable for various levels as well as
for advanced seventh-grade (as enrichment) or eighth and ninth grade classes (as remediation). Daily
lessons and activities have been developed to be effective with both high and low-achieving students, as
well as with high and low-ability students. These lessons have activities that may be used with
accommodations for those students with learning disabilities by using the material as games, activities,
and review material to arouse student interest. The pacing of these lessons is based on eighth and/or
ninth grade classroom in semi-rural to rural settings; where minorities range from small to large
percentages of approximately
20 65% of school communities and where the classroom population averages from
20 25 students provided by block scheduling in a 90-minute setting.
This unit, inclusive of discovery and teacher-lead lessons, manipulatives, and
high-interest visual content is supported by NCTM standards for middle school students.
EULER FAMILY REUNION
DAY 1: INTRODUCTION TO EULER CIRCUITS ("EULER FAMILY REUNION" TRIP ACTIVITY)
Students will be introduced to and recognize concrete situations using vertex-edge graphs. Discovery of
the properties of these graphs will include class and group discussions based on initial directions.
Students will begin by touring the town of their ancestors. This is a day of discovery about the paths
and circuits formed by intersecting roads and streets. This activity will help students focus on analysis of
basic characteristics then describing and communicating about their discoveries.
DAY 2: GRAPH BASICS & VOCABULARY ("EULER FAMILY GENEAOLOGY")
Students will identify key concepts from Day 1 and attach appropriate mathematical vocabulary.
Students will use a variety of activities to reinforce vertex-edge graphs and their understanding of how
the circuit family came-to-be.
DAY 3: EULER THEOREM ("CRASHING THE EULER FAMILY COOKOUT" ACTIVITY)
Students will analyze, recognize, and determine which graphs are Euler Circuits, and/or pathways based
on characteristics learned previously and extend this through explanation of what is NOT an Euler
Through visual study students will describe and apply properties of the "family" members' and how to
distinguish imposters from family members. Various activities will help reinforce their command of
introductory vocabulary and basic understanding. ASSESSMENT: VOCABULARY QUIZ
DAY 4: CONSTRUCTION OF EULER CIRCUITS
("PICTURE DAY AT THE EULER OIL COMPANY, INC." ACTIVITY)
Students will represent Euler's through a picture capturing of various illustrations and characteristics of
Euler paths, cycles and circuits. Note special features.
Students will construct various Euler forms and label any subgraphs. Interpretations of existing graphs
will also be included.
DAY 5: CONSTRUCTING AND INTERPRETING EULER CIRCUITS
("MEETING Al GeRithm AT THE POOL PARTY")
Students will show understand and describe all the paths and rules for constructing Euler circuit in a
connected graph through algorithms.
Practice in formulating and designing connected graph circuits will enhance student understanding.
DAY 6: EULERIZING BASICS ("WEDDING AT THE EULER FAMILY REUNION". REVISIT THE "COOKOUT"
ACTIVITY. HOW DO WE MARRY THE NON-EULERS INTO THE FAMILY?)
Students will differentiate cycles, circuits, paths, etc and through the modification process (eulerization)
create a Euler Circuit.
Eulerizations process is compared to the marriage process where a Non-Euler is transposed into a
EULER CIRCUIT FOLDABLE
Adapted from the Euler Circuit foldable created by Tom Dagley
THIS IS A FOLDABLE TEACHERS CAN DO WITH STUDENTS TO SUMMARIZE CONCEPTS IN THE EULER
CIRCUIT UNIT. IT CONTAINS FILL IN THE BLANK INSERTS TO HELP STUDENTS FOCUS ON THE CONCEPTS
TO BE LEARNED.
MATERIALS NEEDED: COLOR PAPER (COMPUTER PAPER OR CONSTRUCTION PAPER), FOLDABLE
INSERTS, GLUE STICKS/SCOTCH TAPE, SCISSORS, MARKERS/COLOR PENCILS/CRAYONS (OPTIONAL),
DIRECTIONS: HAVE STUDENTS CUT OUT EACH OF THE INSERTS BELOW. GIVE STUDENTS A PIECE OF
COLOR PAPER. HOLDING THE PAPER IN PORTRAIT ORIENTATION, HAVE STUDENTS MEASURE ABOUT 1
INCH FROM THE LEFT EDGE OF THE COLOR PAPER AND DRAW A VERTICAL LINE DOWN THE ENTIRE
PAGE. INSTRUCT STUDENTS TO FOLD THE RIGHT EDGE OF THE PAPER UP TO THE LINE THEY JUST DREW,
CREASING THE PAPER ON THE RIGHT SIDE.
HAVE STUDENTS MAKE 5 DIFFERENT TABS (APPROXIMATELY EQUAL IN SIZE) ON THE FLAP OF THE
FOLDABLE. LABEL THE TABS AS FOLLOWS: 1) GRAPH BASICS, 2) VOCABULARY, 3)EULER'S THEOREM, 4)
EULER'S CIRCUIT, AND 5) EULERIZING BASICS.
USING SCISSORS, HAVE STUDENTS CUT SLITS IN THE FLAP OF THE FOLDABLE SO IT LOOKS LIKE THERE
ARE 5 DOORS THAT OPEN TO THE LEFT OF THE FOLDABLE. HAVE STUDENTS PASTE IN THE INSERTS
BELOW AS YOU COVER THE CONCEPT IN CLASS, FILLING IN THE BLANKS AS YOU GO. ONCE FINISHED,
YOU MAY HAVE STUDENTS HOLE-PUNCH THE LEFT SIDE OF THEIR FOLDABLES SO THEY MAY BE STORED
IN THEIR BINDERS.
*YOU MAY CHOOSE TO GIVE STUDENTS EXAMPLES AND NON-EXAMPLES OF ACTUAL EULER CIRCUITS
ON THE BACK OF THE FOLDABLE.
Graphs consist of vertices (nodes which represent objects) and edges
(lines that represent relations between objects).
Every edge has at least 2 vertices.
Every vertex, x, has neighbors--vertices that share an edge.
Degrees: the number of edges connected to a vertex.
Walk: a connectedsequence of vertices and edges: V-E-V-E
Path: a walk without repeated vertices
Circuit: a closed walk that starts and ends at the same vertex.
Connected Graph: A graph in which a pathexists between any two
Non-connected Graph: A graph that is notconnected to any other graph
by a path.
An Euler Circuit covers all the edges exactlyone time.
A graph has an Euler Circuit iff ("if and only if"):
1. The Graph is connected.
2. All vertices have even degrees.
Finding an Euler Circuit is EASY!
Is the graph connected? Do all of the vertices have even degrees?
If you answered yes to both, we can keep going!
Start from any vertex and find a path. If all the edges are not
covered, remove the edges from the path found.
Start from another vertex and find another path. If all the edges
are not covered again, remove the edges from the path found
and repeat until all the edges are covered.
Combine the paths at common vertices.
MAKE an EULER from a circuit: (Eulerize a connected graph with odd
Identify all vertices with odd degrees (there must be an even
number of vertices with odd degrees).
Pair up odd degrees with nearest neighbor by drawing an
edge between the two odd degrees.
All vertices will be even, and an Euler circuit can be found.
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