Instructional notes - What's my path?

The first worksheet: What's my path?

Is to explore continuous paths.

Continuous path is drawn without retracing any part of it. Application - Draw the entire graph without lifting up your pencil or going over an edge twice.

It introduces nine graphs that have a path that can be traced with a continuous path without retracing any part of it (Eulerian).

Sample explanations using graph properties.

• Egg, star, spiral, and heart. They are one continuous path without nodes.
• Five-point star drawn inside a circle actually creates five vertices that each have exactly 4 lines touching them (2 from the star points and 2 from the circle segments). Because every single node has an even number of lines, it can absolutely be drawn in one continuous stroke.
• Others have one or two three node (odd) intersections.
• Stars in circles and spiral in pentagon have four node intersections.
• Egg, star, star in circle, and heart start and stop at the same place. All the rest start and stop at different places.

 

Instructional notes - Do I have a continuous path?

The worksheet: Do I have a continuous path?

Is to explore paths that are continuous and not continuous.

• Continuous - a path drawn without lifting a pencil and without retracing any part of it.

Identify properties used to describe graphs. And begin to understand and explain how nodes (how many lines touch a point) determine if a graph is, continuous or not.

Answer key

These three graphs are traceable, they have exactly zero or two nodes with an odd number of lines. If it has more than two odd nodes, it is impossible to trace in one go!

• The Classic Envelope House: A square with an X inside and a triangle roof. It has exactly two odd nodes (can you find them?) and the rest have four). You can trace it completely, but you must start at one odd node corner and finish at the other odd node. Different start and end points.
• The Infinity Loop (Figure-8): Two loops joined at a single central node. The central node has 4 lines touching it, and the outer loops are continuous. Because every single node has an even number of lines (4, 2, and 2), it contains a Euler Circuit. You can start anywhere and you will always finish exactly where you start.
• The Double-Triangle Hourglass: Two triangles stacked vertically, meeting at a single center vertex. The center vertex has 4 lines connecting to it, while the outer four corners each have exactly 2 lines. Since all vertices are even, it is perfectly traceable in one continuous loop. Start and stop at the same node.

These three graphs are not traceable in one continuous stroke without retracing a previous traveled line.

The Tic-Tac-Toe Grid: Four intersecting lines forming a central square and open outer edges. The four internal intersections each have 4 lines, but because of the outer 8 loose end points you cannot draw a full tic-tac-toe board in one continuous stroke without retracing a previous traveled line.

Circle with a plus symbol inside. There are four nodes with 3 lines intersecting. More than two odd intersections seems to make it impossible.

The four triangles is a graph similar to The Seven Bridges of Königsberg. Three of the nodes have 3 lines touching them, and one node has 5 lines (the central island). Because it has four odd nodes (3, 3, 3, and 5), it is mathematically impossible to trace without retracing a line or lifting your pencil.

Instructional notes - Exploration of graphs

The next set of worksheets are to explore graphs and systematically identify graph properties that describe their paths as continuous or not.

• Exploration 1
• Exploration 2
• Exploration 3 - Let's analyze networks!
• Exploration 4 -

Exploration 1 answer key

The box with the x in it. It cannot be traced without retracing a path. The two top corners each have 3 lines touching them (2 for the square edges, 1 for the diagonal). The two bottom corners also each have 3 lines touching them (1 side wall, 2 diagonals crossing). Because all four corners are odd nodes (degree 3), you will always get stuck and leave at least one line blank.

Exploration 2

Yes, No, Yes

Yes Yes No

Exploration 3

Complete the chart with new graphs or use any of the previous that may interest the learners.

Encourage noticing and sharing properties and any patterns or relationships discovered.

Exploration 4

Use ideas from the table and pick some graphs with common properties that might indicate a relationships or pattern.

 

Possible procedure with leading questions.

Use the data in the table to see how many ways the graphs that could be traced were traced. Identify them, write them below, and list their properties.

1. What did you notice for those that can be traced continuously?
2. What properties are common?
3. What conclusions can be made?

Use the data in the table for the graphs that could be traced. Identify them, write them below, and list their properties.

1. What did you notice for those that could not be traced continuously?
2. What properties are common?
3. What conclusions can be made?

Summary

1. How can we determine if a network will have a path or circuit?

Consider networks/graphs that can be traced. Did you discover there are two types?

• Ones that can be completely traced back to the starting position in a single continuous line without lifting your pencil or retracing any edge.
• Ones that start and stop in different places.

1. List properties for these two types.
2. Start & stop same place.
3. What properties are common for all of these?
4. Start and stop different places.
5. What properties are common for all of these?

 

Summary:

1. What did you discover about graphs with Zero odd vertices? The network can be traced and you will end up exactly where you started. This is called an Eulerian Circuit.
2. What did you discover about graphs with exactly two odd vertices? The network can be successfully traced, but you must start at one odd vertex and end your tracing at the other.
3. What did you discover about graphs with more than two odd vertices? The network cannot be traced in a single continuous line.