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Essential Question: How do we use matrices to solve problems? •How do we relate matrices to vertex-edge graphs? Standards: MM3A7: Students will understand and apply matrix representations of vertex-edge graphs. •MM3A7a: Use graphs to represent realistic situations. •MM3A7b: Use matrices to represent graphs, and solve problems that can be represented by graphs. Math 3 Activating Strategy Vertex-Edge Graph Adjacency Matrix A B A D A B B C E C D F G E F G C D E F G Math 3 Activating Strategy Vertex-Edge Graph A D B C E A B C A 0 1 1 B 1 0 1 C D F G Adjacency Matrix E F G D E F G AN OKEFENOKEE FOOD WEB Learning Task Recent weather conditions have caused a dramatic increase in the insect population of the Okefenokee Swamp area. The insects are annoying to people and animals and health officials are concerned there will be an increase in disease. Local authorities want to use an insecticide that would literally wipe out the entire insect population of the area. You, as an employee of the Environmental Protection Agency (EPA), must determine how detrimental this would be to the environment. Specifically, you are concerned about the effects on the food web of six animals known to populate the swamp. Consider the following directed vertex-edge graph (digraph) of a food web for the six animals and the insects that are causing the problem. Here each vertex represents an animal or insects. The direction of the edges indicates whether an animal preys on the linked animal. For example, raccoons eat fish. (Note: The food web shown is simplified. Initial producers of nutrients, plants, have not been included.) Adjacency matrices can be used in conjunction with digraphs. If we consider just the relationships between raccoons, fish, and frogs in the food web shown, an adjacency matrix would be R R FT F FT F 0 1 1 0 0 0 0 1 1 1. Construct the associated matrix F to represent this web. M B I M C B B R M F FT I C FT F R M I C FT F R 1. Construct the associated matrix F to represent this web. What does a row containing a single one indicate? What does a column of zeros indicate? 2. Which animals have the most direct sources of food? How can this be determined from the matrix? Find the number of direct food sources for each animal. 3. The insect column has the most ones. What does this suggest about the food web? More Practice with Vertex-Edge Graphs Worksheet To save paper, an office administrator is trying to deliver messages through oral communication between coworkers. The administrator tells one employee the message; the administrator then considers the message to be deliverable to every worker who has direct contact (talking directly to the designated employee) or second-hand contact (talking to someone who had direct contact with the designated employee). The vertex-edge graph below summarizes the direct contacts made between the six employees supervised by the administrator. Complete the adjacency matrix below to summarize the direct contacts made between the six employees. Ann Ann Bob Caf Dan Eve Fin Ann Bob Caf Bob Caf Dan Eve Fin Dan Eve Fin Complete the table below to show which employees will and will not get the message if the administrator tells the employee in the first column. Administrator tells: Who gets the message? Who doesn’t get the message? Ann Bob Caf Dan Eve Fin Ann Ann Bob Caf Dan Eve Fin Ann Bob Caf Bob Caf Dan Eve Fin Dan Eve Fin