All Hands on Deck

Note: The purpose of this lesson is to look at linear relationships through algebraic, graphical, tabular and analytical representations.

	Background on Kedging When a sailing vessel, like USS Constitution, encounters a period of no wind, the Ship can still move forward by rowing 2 small, kedging anchors in the Ship's boats, out in front of the Ship and dropping them to the ocean floor. When the crew on board pulls in the anchors, the Ship is naturally dragged forward in the water until it is above the kedging anchors (before the anchors are pulled out of the water). Although this is slow, back-breaking work, the process can be repeated until the wind returns. In 1812, Constitution was able to escape a squadron of British ships using this technique.

Objectives
After completing this lesson, students will be able to:

	1.	Interpret, sketch, and analyze graphs from situations.
	2.	Evaluate linear functions.
	3.	Model functions using rules, tables, and graphs.
	4.	Write a linear function rule given a table or a real-world situation.
	5.	Use inductive reasoning in continuing number patterns.

Massachusetts Math Curriculum Frameworks

	•	AI.N.4 Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers. (10.N.4)
	•	AI.P.1 Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g. Fibonacci Numbers), linear, quadratic, and exponential functional relationships. (10.P.1)
	•	AI.P.3 Demonstrate an understanding of relations and functions. Identify the domain, range, dependent, and independent variables of functions.
	•	AI.P.4 Translate between different representations of functions and relations: graphs, equations, point sets, and tabular.
	•	AI.P.5 Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x and y intercepts from its graph or from a linear equation that represents a line. Find a linear equation describing a line from a graph or a geometric description significance of a positive, negative, zero, or undefined slope. (10.P.2)
	•	AI.P.11 Solve everyday problems that can be modeled using linear, reciprocal, quadratic, or exponential functions. Apply appropriate tabular, graphical, or symbolic methods to the solution. Include compound interest, and direct and inverse variation problems. Use technology when appropriate. (10.P.7)

Materials

	Worksheets (pdf):
	•	Writing Linear Equations: Communicating and Connecting What We Know About Kedging
	•	Writing Linear Equations: Communicating and Connecting What We Know About the Water Supply

Procedure

Define kedging (using the explanation above) for your students.

The equations that will result from these problems will be in slope-intercept form. Commonly referred to as y equals mx plus b form (y = mx + b) where m represents the slope and b represents the initial value or y-intercept. The slope, or rate of change, is always the number in front of the independent variable, which in this case is the x. In the following problems, n represents the independent variable.

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The water problem is a decreasing graph with an initial value. The student should be able to predict the time when the sailors would run out of water, which could lead into a discussion of the daily diet and also of ways in which food and water were replenished.

Equation for the water problem: G = 45,000 – 440n

Try to make a connection between the negative slope and the graphical connotation.

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The kedging problem is an increasing graph without an initial value. The students could use the data to estimate the distance traveled per number of kedges and also estimate the number of kedges it took to outrun the British fleet.

Equation for the kedging problem: D = 650n

Try to make a connection between the y-intercept and the graphical connotation. Since the initial value equals zero, the graph crosses the origin of the graph. Another name for a linear relationship that has an initial value of zero is direct relation because the dependent variable, D, can be solved for by simply multiplying by 650 by the number of days, n.

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