Two Variable Inequalities

To begin the lesson, ask students to group up with other classmates whose birthday is in the same month as theirs. The goal is to have groups of twos or threes, so if any group has more students than that, ask the oldest student(s) of the group to enter another group. This is just a random and fun way of grouping.

This lesson is carried out with the use of Sketchpad.  Other similar software can be used as well. 

Project the following situation on the board. ResourceInequality Launch Problem.docx

Each student in the class lives in a house on the same block. Those in your group are your next door neighbors. The neighborhood committee has ruled that every house on the block build a rectangular garden in their front yard. Each home will receive 50 ft of fencing to surround their garden, and cannot use more than this amount. You can use less than 50ft and return the unused fencing.

Ask each group to place their values on a T-chart.

Example:

Most of the time, students will use up the entire 50 feet. Adel above did not. If you see that too many students are using up all 50 feet, remind and motivate some students to use less than 50 ft.

A “sometimes” error:

Some students may think about the area of the garden as being  50ft² and give dimensions like 5 and 10. Make sure that these students understand that area is not important here and that choosing 5 and 10, for example, means that they are using 30 feet of fence, the perimeter of the garden.

New Info/Application

Once every student (home) in the neighborhood knows the dimensions that their garden will have, project a four coordinate plane on the whiteboard. Sketchpad is good to use here because it allows you to plot points, graph a line through these, shade regions, and suggests other points on the line. The premade template is in the Resource Section.

See video: Inequality Sketchpad Narrative

If you have access to Geometer's Sketchpad, you can download the .GSP file by clicking here to open the preview and clicking the Download button: Sketchpad Inequality Template.gsp

First, ask those students who used the entire 50 ft of fence to provide the dimensions of their rectangular garden to you and plot these on the graph. (Part A of video)

(If time is available, I ask the students to go to the computer and enter their dimensions themselves, just to have the students move a bit and plot their own points plotted on the board. There’s a little excitement in that.)

They will see that all their points lie on the same line. Ask students to find the equation of the line in slope intercept form and then in standard form.

Once they have it, write it on the board, and then show students how to obtain it with the program (In Part A of video).

Then ask all the other students to provide their dimensions. I remind the class that these are the neighbors that did not use all the given fencing. (Part B of video) They will see when plotting, that these points lie off the line below it, but within the triangle formed by the axes and the line.

Questions to ask:

• Are these points solutions to the linear equation? (no)
• Which inequality symbol must we put in place of the = sign, to make all the students’ points, solutions to the inequality? (Students should say ≤, yet some students may say <. If so, ask someone to test a point that lies on the line, so they could see for themselves that the correct symbol is  ≤.

Rewrite the equation on the board as the inequality x + y  ≤ 25

Group work :  Part C of video

Now I plot each of the following points on the plane:

(0, 16), (25, 0), (-10, 5), (10, -5), (5, 14), (-5, -10), (20, 15), (35, -5)

Each group should:

1. Analyze each point and indicate whether they are solutions to the inequality.

2. If so, are they feasible solutions for the real world problem involved. Explain.

(students should state that points like (0, 16) and (25, 0) are solutions to the inequality, yet not reasonable for the task at hand because no dimension can measure 0 feet. By the same token the points with negative coordinates like (10, -5) are not reasonable solutions because no measure can be negative. The point (20, 15) is not a solution to the inequality. Tthese dimensions signify more than the given 50 feet of fencing.)

Tell students that the graph of a two variable inequality requires the shading of the half plane region where all its solutions lie.  Demonstrate this with sketchpad.

3. State where do the “reasonable” or feasible solutions to our real world problem lie?

(Students should say that the reasonable solutions lie in the triangular region confined by the two axes and the line not including the points on either of the axes. Shade this region using a different color using the program)

4. Ask students to provide, by observing the graph, other possible garden dimensions that were not used by anyone.

5. Ask the groups to write and graph the inequality that represents a situation similar to the garden problem but this time every student must build a rectangular garden using more than 40 feet of fencing. The graphing can be done on graph paper, one per group. Once they are finished, randomly call on a group to come up and present their work using sketchpad on the board for all to see. Some dimensions may not be practical, but they must be accepted. 

Students should have concluded with the graph below: 2nd Inequality

CLOSURE

To close the lesson, I hand out the resource: closing_inequalities

In this resource they are given a straight forward inequality instead of a real world problem. The steps for graphing it are provided and each student is to put the steps in the correct order. Although students should do these individually, I allow discussion.

Finally, I call on students to give me the steps one by one. As each step is given, I perform the step on the board immediately after and motivate them to do the same on their graph.  To be flexible, the step that asks to determine whether the line should be solid or dashed can be placed before or after finding the shaded region.

What to say as the steps are being performed.

• Ask students which points they think make good “test” points to decide the region to shade. The answer should be points clearly on one side or the other. The origin can be a quick easy one. Points too close to the boundary make bad test points.
• Tell students not to make the mistake of testing points into the equation used to graph the boundary but rather into the inequality. This is actually a bad mistake because it may indicate a weak understanding of linear equations and their solutions.
• Ask students how can they quickly determine whether the line, or boundary, will be solid or dashed
• To make sure that students know that the solutions are infinite, ask students if a point like (1,500, 120) is or isn’t a solution.
• Tell students that they should avoid choosing the half plane to shade by simply shading the side they think is above or to the right when the inequality sign is > or the left side when the sign is <. This may lead to errors. They should test a point.

HOMEWORK: HOMEWORK_inequalities.docx