NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 15: Graphing Solutions to Inequalities
Students graph solutions to inequalities taking care to interpret the solutions in the context of the problem.
Rapid White Board Exchange (10 minutes): Inequalities
Students complete a rapid whiteboard exchange where they practice their knowledge of solving linear inequalities in the form and .
Discussion/Exercise 1 (10 minutes)
Two identical cars need to fit into a small garage. The opening is feet inches wide, and there must be at least feet inches of clearance between the cars and between the edges of the garage. How wide can the cars be?
Encourage students to begin by drawing a diagram to illustrate the problem. A sample diagram is as follows:
Have students try to find all of the widths that the cars could be. Challenge them to name one more width than the person next to them. While they name the widths, plot the widths on a number line at the front of the class to demonstrate the shading. Before plotting the widths, ask if the circle should be open or closed as a quick review of graphing inequalities. Ultimately, the graph should beMP.7
Describe how to find the width of each car.
· To find the width of each car, I subtract the minimum amount of space needed on either side of each car and in between the cars from the total length. Altogether, the amount of space needed was
or Then, I divided the result, , by since there were cars. The answer would be no more than or
Did you take an algebraic approach to finding the width of each car or an arithmetic approach? Explain.
· Answers will vary.
If arithmetic was used, ask, “If is the width of one car, write an inequality that can be used to find all possible values of .”
Why is an inequality used instead of an equation?
· Since the minimum amount of space between the cars and each side of the garage is at least feet inches, which equals , the space could be larger than feet inches. If so, then the width of the cars would be smaller. Since the width in between the cars and on the sides is not exactly feet
inches, and it could be more, then there are many possible car widths. An inequality will give all possible car widths.
If an algebraic approach was used initially, ask, “How is the work shown in solving the inequality similar to the arithmetic approach?”
· The steps to solving the inequality are the same as in an arithmetic approach. First, determine the total minimum amount of space needed by multiplying by . Then, subtract from the total of and divide by .
What happens if the width of each car is less than feet?
· The amount of space between the cars and on either side of the car and garage is more then feet
What happens if the width of each car is exactly feet?
· The amount of space between the cars and on either side of the car and garage is exactly feet
What happens if the width of each car is more than...