# Inequalities Lesson

Grade Level:  This lesson is intended for students in an algebra class.  Before beginning the lesson, students will need an understanding of how to graph inequalities.

Overview:  The purpose of this activity is for students to use TI-Nspire handhelds to explore the properties of graphs of inequalities, and to gain an appreciation for the real-life applications of the topic.  This lesson represents an effective use of technology because it helps students develop the skills needed to work with complex problems that can be solved using systems of inequalities.  When moving on to these more involved situations, the calculator takes care of tasks like graphing and shading the equations and finding the points of intersections; this leaves students with more time for critical thinking and higher-level analysis.

Resources:

Virginia State Standards of Learning (SOLs) Addressed:
Algebra 9:
The student will solve systems of two linear equations in two variables both algebraically and graphically and apply these techniques to solve practical problems. Graphing calculators will be used both as a primary tool for solution and to confirm an algebraic solution.
Geometry 2:  The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation.
AFDA 5:  The student will determine optimal values in problem situations by identifying constraints and using linear programming techniques.

Algebra
Geometry
Problem Solving
Connections

Preparation:
1. Review each of the three student activity worksheets from the Resources section above.  Each one discusses a different aspect of graphing inequalities.  Since the worksheets are independent of one another, you can choose any or all of them to be used in your class.  For this document, though, we will focus mainly on the second worksheet (linear programming).

2. For each of the activities you have chosen to give to your class, make copies of the student directions.  Note: each of the three activities contains the student directions, followed by the teacher notes, both in the same file.  You will probably want to ensure that your students don't have the teacher notes included in the papers you give to them.

3. Note that we are using the TI-Nspire instead of the TI-83/TI-84s listed in the worksheets.  Make sure to review the procedure for graphing inequalities on the Nspire, which are described further down this page.  These instructions will most helpful if you have chosen to focus on the second worksheet (linear programming).

Procedure:
1. Review with your students the procedure for graphing inequalities on the TI-Nspire, as the directions listed in the student worksheets pertain to graphing inequalities on the TI-83/TI-84.

2. Allow students to work in groups of 2 or 3.  For each of the activities you have chosen for your class, all you need to do is give your students sufficient time to follow the directions contained in the student worksheets you have passed out to them.  Make sure to walk around the classroom as your students work, assisting them and clarifying and questions or misconceptions.

3. Be aware of the following differences between the worksheet (which is written for the TI-83/TI-84) and this lesson plan (which is writte for the TI-Nspire):
• Page 49: the Nspire does not allow you to shade just the region which is the intersection of all the inequalities.  Every inequality must be shaded seperately, and the region of interest is where all of these seperate shadings overlap.
• Page 49: to find the corner points of this region of interest, on the Nspire you will need to find the intersection of the appropriate lines instead.

Graphing Inequalities on the TI-Nspire:
1. Inequalities can only be graphed on certain kinds of pages within your document.  Either begin a new document, or insert a new page to your current document, and choose the "Graphs and Geometry" category.

2. You will see a blank Graphs and Geometry page appear, with a prompt at the bottom of the screen asking you to input an equation for f1.

3. Delete the part of the prompt that says "f1(x)=" and replace it with an inequality.

4. Press the Enter key and your inequality will be graphed on the screen.

5. To add another inequality, press the Menu button, and under the Graph Type option, choose Function.

6. The same way you entered the first inequality, delete the part of the prompt which says "f1(x)=" and replace it with your own inequality.  To get a greater-than-or-equals or less-than-or-equals sign, press Ctrl + > or Ctrl + <.  Then press Enter to graph your new inequality.

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8. Continue graphing as many inequalities as desired, using the same steps.  For this demonstration, I graphed one more inequality.

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10. To find points of intersection, first open the menu by pressing the Menu button.  Then move down to Points & Lines, and choose Intersection Points.

11. Move your cursor so that it is hovering over the first inequality you want to find the intersection point for, and press the mouse click button (in the center of the pointer wheel) or press the Enter button.  Then move your cursor so it is hovering over the second inequality, and press Enter again.  The intersection point will automatically be labeled and displayed on the screen.

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13. The more inequalities you graph, the more difficult it becomes to tell which region of the screen is the darkest one (i.e. the region that fulfills all of the inequalities being graphed).  To help fix this, you can change the level of shading for each inequality.  Open the menu, and under the Actions category, choose Attributes.  Then move your cursor until it is hovering over the line you want to change properties for, and click the center mouse button or press Enter.

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15. You will see three small boxes pop up.  Using the up/down/left/right buttons on the mouse wheel, you can change the shading color and line thickness for your selected graph.

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17. You can view a list of all the inequalities you are graphing, as well as edit the equations for these inequalities.  To do this, move your cursor to the bottom left corner of the screen and click on the >> mark.  This opens a prompt at the bottom of the screen, but instead of entering anything, press the mouse wheel right and highlight the ^ mark.  Click again and the list of all your inequalities will be displayed.  You can press the up/down buttons on the mouse wheel to change which line you are highlighting, and you can edit the line you are currently highlighting like normal.

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