Ask, "When I say graph,
what do you think of?" [Expected answers include coordinate graphs, bar
graphs, pie graphs, data collections, grids, slope-intercept.] Show the
overhead. Tell students that in graph theory, a graph is a collection
of points, called vertices, and the edges that connect them. Note that
edges do not have to be straight; the only requirement is that they
have vertices as their endpoints. Mention that multiple edges can
connect the same two vertices and an edge can connect a vertex to
Discuss how a graph on paper can be a model of a real-world
situation. For example, the vertices of a graph might represent houses,
and the edges might represent streets. Ask students for other
situations that might be modeled by graphs. [Airports and routes, cell
phone signals and towers, assignment of class sections to rooms.]
Pass out the Graph Theory
activity sheet. Explain that graphs can be used to find the most
efficient way to carry out a task. For example, a road striper (the
machine that paints lines down the middle of a road) or a street
cleaner might want to travel along each street in a town exactly once.
By drawing a graph that includes streets as edges and intersections as
vertices, the driver can use a graph to determine the optimal path for
the job. A path that starts at a vertex and traces each edge exactly
once is called an Euler path, named after the mathematician Leonhard Euler (pronounced oil‑er). An Euler path that starts and ends at the same vertex is called an Euler circuit.
Have students trace paths on the graphs and decide if each graph has
an Euler path or circuit. Then, students should draw their own paths.
Encourage students to look for patterns and try to draw a conclusion
about what graph features make it possible to trace an Euler path or
circuit. If students are not ready to draw a conclusion, they will be
able to return to the activity sheet later in the lesson.
Distribute dot paper to each student. Have students draw an Euler
circuit on the dot paper, using the dots as vertices. Distribute sewing
needles and thread to each student. Explain that they are going to use
stitches to trace their circuits. Help students thread their needles,
if necessary. Then, tape the loose end of the thread to the back of the
paper near the starting point. Have students push the needle through
the starting point and then make a stitch by pushing the needle through
at the next vertex and repeating until all edges have been traced,
either on the front or the back of the paper. Students can use their
stitching to verify that they have not traced any edges more than once.
Below are some examples of circuits that have been stitched.
Have students look at the front and the back of the dot paper on
which they stitched their designs. They should note that the design is
different on the front and the back. Blackwork embroidery creates the
same image on the front and back sides of the canvas. Remind students
that their canvas is dot paper. Show students Blackwork 1
overhead. Explain that they are going to trace a path that, when
stitched, creates the same image on the front and the back. You will
track whether the stitches are on the front or the back of the paper by
recording it in the appropriate column. Note that stitches alternate
between front and back, depending on where the needle begins. The
beginning and ending vertex for each stitch is recorded in parentheses.
The first letter is where the stitch begins, and the second letter is
where it ends. When the stitches are listed in order, the ending vertex
of one stitch should be the same as the beginning vertex of the next
stitch. While analyzing the stitches listed, trace them on the graph
above, using different colors for front and back stitches. In the end,
each edge will have be been traced in two colors, indicating that the
image would be the same on the front and the back of the canvas.
Students may also want to draw this pattern on their own dot paper and
stitch the pattern to see better how it works.
Show the Blackwork 2
overhead. Explain that some designs are much more complicated and
require many more stitches. In these designs, the stitches are
numbered. The T indicates that the stitch is made on the first trip
around the pattern, and a B indicates that it is made on the return
trip. As you trace the stitches, again use two colors to indicate the
stitches on the front or the back. Have students help you problem solve
about where to begin each stitch so that the back image will look
identical to the front image. Again, when you are finished, each edge
should be traced in two colors.
Show the Blackwork 3
overhead. Again, the numbered stitches can be traced in two colors,
with color designating whether the stitch is on the front or the back
of the canvas. The T stitches are made between the starting point and
the turn around, and the B stitches are made after the turn around and
on the way to the finish. Ask students why they think this design is
called "Journey and Side Trips." Discuss the efficiency of the
stitching pattern; there are very few stitches required between the
turn around and the finish.
Since we actually stitch on both sides of the canvas, every edge on
the front will have a matching edge on the back. Explain that the
degree of a vertex is the number of edges that connect to it. Ask, "Why
will every vertex have an even degree?" [Each time an edge passes
through the front, there is also one on the back. The total degree will
always be a multiple of two.] Have students return to the Graph Theory
Answer Sheet and complete the last two questions.