• Observe student responses during the introductory activity of graphing and solving a basic system of inequalities.
• Observation of student graphs and layering of tracing paper in Part 2 of the activity allows you to make sure students understand that the solution region is the area where the graphs of the inequalities overlap each other.
• By observing the review responses (verbal and written) throughout the lesson, you can identify and assist individual students who might need an additional learning opportunity or additional practice with a particular skill.
• Student performance on the Lesson 3 Exit Ticket, handed out at the end of the class (M-A1-4-3_Lesson 3 Exit Ticket.docx and M-A1-4-3_Lesson 3 Exit Ticket KEY.docx), gives you a chance to see where students need further assistance with identifying constraints, graphing, or relating the information in the graph to the situation. On the exit ticket, students attempt to graph a system of inequalities and hand in the result.

To begin the lesson, provide students with the following situation, written on the board or an overhead projector. The problem should be visible to students throughout the first portion of the lesson. [IS.4 - Struggling Learners] Tell students, “A hat maker makes two different kinds of hats. He makes berets, which take 40 minutes per beret to make, and he can make top hats, which take 60 minutes per hat to make. He has a total of 6 hours in which to make hats.”

Before writing any inequalities to represent the situation, ask students what the constraints are; in other words, what is limiting how many or how few hats the hat maker can make. Students should respond that the time is a limiting factor―he only has 360 minutes of time to work in.

Ask students to write an inequality to represent the time constraint, using x to represent the number of berets the hat maker makes, and y to represent the number of top hats the hat maker makes. Ask a student to come to the board and write the inequality. (It should be 40x + 60y ≤ 360.)

Ask students to focus on the minimum number of each type of hat that the hat maker can make:

• “Is there a minimum number of berets he can make?” (yes)
• “Is there a minimum number of top hats he can make?” (yes)

Ask students to write an inequality for each of these constraints. Students should write x ≥ 0 and y ≥ 0.

Point out that there are really three constraints in this problem: time, the direction of the inequality (<, >, or +), and the fact that you can never make a negative number of any kind of hat. [IS.5 - Struggling Learners] Also, point out that there is a single inequality for each constraint; three constraints, three inequalities: “When we have multiple inequalities (or equations) that all correspond to the same problem or situation, we call it a system of inequalities (or system of equations). The good news about finding the solutions to systems of inequalities is that it isn’t much more difficult than finding the solutions to each inequality individually.”

Part 1

Draw a Cartesian plane on the board. [IS.6 - All Students] Briefly review the location and standard identification of the four quadrants. Ask students, “In which quadrant or quadrants are the solutions to the inequality x ≥ 0?”

Remind students that this particular inequality does not “care” what y is, as y can be any value, from -1,000,000 to to 7000. The only concern with this inequality is that the x-value is not negative. “Where are the points on the Cartesian plane that have a positive x-value or an x-value equal to zero?”

After students identify quadrants I and IV as well as the y-axis, lightly shade or crosshatch the appropriate portion of the Cartesian plane, using colored chalk if possible. Make sure to emphasize the y-axis as being included in the solution and being a solid line, either by tracing over it with white chalk or by tracing over it with colored chalk. Then, repeat the same kind of questioning for the inequality y ≥ 0. Once students have identified quadrants I and II as well as the x-axis, proceed with the following:

“Going back to our problem about the hat maker, we know that he cannot make a negative number of either kind of hat. Where do all the points that represent non-negative numbers of both x and y lie?” (quadrant I and the positive x- and y-axes)

Make sure to continually emphasize the inclusion of the appropriate axes and not just the appropriate quadrants on the graph; students often forget about the axes when focusing too much on the four quadrants. Graph and shade the appropriate part of the Cartesian plane, using a different color of chalk or a different direction/type of crosshatch pattern to help differentiate it from the first set of shading.

“So all the points that we’ve identified―do they represent all the possible solutions to our hat maker problem?” Students should recognize that they do not because we have not yet taken into account the time constraint. If students do not initially recognize this, ask students if the hat maker can make 100 berets and 200 top hats. Students should, at this point, be reminded of the time constraint: “So far, we’ve eliminated a large portion of the Cartesian plane, just by using the two inequalities that represent the impossibility of making a negative number of either kind of hat. But now, we have to eliminate the portion of the Cartesian plane that contains all the values that represent combinations of hats that take too long to make.”

Have students put the inequality that represents the time constraint (40x + 60y ≤ 360) into slope-intercept form to make it easier to graph. Students can check their work with their neighbor. (In slope-intercept form, the inequality is .)

Have a student come to the board and graph the inequality. Remind the student to pay attention to whether the inequality should be represented by a solid line or a dashed line. Also, as the student is graphing the line, ask the class how to decide which side of the line to shade. Students should respond to use a test point like (0, 0), which is a solution of the linear inequality, and so the side of the plane containing (0, 0) should be shaded. Provide colored chalk to the student or have him/her make a different crosshatch pattern to represent the solution to the inequality. Ask, “Do all the points below  represent solutions to the system of inequalities?” (no) [IS.7 - Struggling Learners]

Guide students toward the realization that only the points which satisfy all three inequalities in the system are valid combinations for the system. “How can we identify which part of the Cartesian plane contains solutions for all three inequalities?” (Identify the region that has all three colors of shading or all three styles of crosshatch.)

Outline the appropriate region to make it clear where the solution lies.

Ask, “How did the process used to arrive at this solution differ from just graphing a single inequality?”

Point out the feasible region. Students should recognize that they really just wrote and graphed each inequality on its own, and then identified the overlapping region. The process of identifying the overlapping region is the only new step; otherwise, it is just repeating the steps they already know to graph linear inequalities.

Before continuing, emphasize that finding solutions to systems of inequalities is just a repetition of previously-learned skills. Also, emphasize the importance of keeping work neat, since graphs that have three, four, or more inequalities can become messy. Encourage students to use colored pencils and also do their shading very lightly so that they can tell where different colors overlap.

Another approach that students can use is to draw an arrow on the graph of each line to show which side of the line to shade, then to shade the region all the arrows points to. This has the advantage of keeping the picture neater, but the disadvantage that it is difficult to picture where the shading goes when you have many inequalities to be graphed at once.

Tell students, “Let’s look at another example. Suppose we have the inequalities . What would the solution look like?”

Part 2

Have students work in pairs. Give each pair five pieces of tracing paper with Cartesian graphs on them. Write the following system on the board:

y > 0.5x – 1

y > –2x – 2

y ≤ –x + 5

y ≤ 2x + 2

Each pair should divide the four inequalities so each person in the pair gets two of them. Each student should then graph each of his/her two inequalities on separate pieces of tracing paper. Each student should shade lightly, but dark enough so that the shading and the line can be clearly seen through the tracing paper. (It will have to be able to be seen through four pieces of tracing paper by the end of the activity.)

Note: If regular paper is used instead of tracing paper, have each student graph both of his/her inequalities on the same sheet rather than two separate sheets to make the last step of drawing the solution easier.

While students graph their inequalities, walk around and check individual work. Remind students as they work to pay particular attention to the style of line (dashed or solid) and also to the direction of their shading. Remind students to use a test point (in this activity, (0, 0) is a suitable test point for all four inequalities) to determine which way to shade.

After each pair has graphed the four inequalities that make up the system, they should trade with their partner, who should verify that the inequalities are graphed correctly.

For the last part of the tracing activity, students should take all four pieces of tracing paper and lay them all on top of each other. (Holding the sheets against a window may help in terms of viewing the four graphs through the layers of tracing paper.) Then, have each pair take a fifth sheet of tracing paper with a Cartesian plane printed on it, and place it on top of their stack of tracing paper. They should be able to identify the region that is part of the solution of all four inequalities.

Have students pick out the four points that define the solution region. (Those four points are (–1, 0), (0, –2), (4, 1), and (1, 4).) Have students plot those points on their fifth piece of tracing paper and then connect them with the appropriate style of line (dashed or solid). Then, have students shade in the quadrilateral that represents the solution.

Have each pair of students compare their solution with other pairs. The solutions can be compared easily by placing the tracing paper with the solution quadrilateral atop one another and making sure they overlap correctly.

Review

Have students continue to work in their pairs from Part 2: “Imagine that you have a system of inequalities similar to the one you just worked on with your partner, but you are going to graph all the inequalities on a single Cartesian plane, rather than on separate ones using tracing paper. Think about the steps you have to go through, from the moment you are presented with the four inequalities. With your partner, write a series of clear, simple steps that someone can follow in order to find the solution to the system.”

Give students approximately 5 minutes to come up with a list of steps, and then have the class come back together. [IS.8 - Struggling Learners]

Ask one group for its first step, and ask other groups if they agree or disagree. If groups disagree, ask them what step(s) should come before the given first step. Continue this way until the class has arrived at a list of steps used to solve a system of inequalities. A possible series of steps could be:

1. Put each inequality into slope-intercept form.
2. Graph a single inequality by first graphing the associated line. Make sure to use solid lines (for ≤ or ≥) or dashed lines (for < or >) when you graph the line.
3. Choose a test point for the inequality and plug the coordinates of the test point in to the inequality. (Choose (0, 0) unless the line goes through the origin.)
4. If the test point makes the inequality true, shade on the side of the line that contains the test point; otherwise, shade the other side.
5. Repeat steps 2, 3, and 4 for each inequality in the system.
6. Identify the region that is part of the solution of each inequality in the system.

Have each pair copy the decided series of steps into their notes. Also, have each pair write their names on their original set of steps and turn them in. Emphasize that the pairs will not be scored or graded on their original set of steps. However, looking at the original steps that students came up with will help to anticipate potential points of confusion in both individual students and the class as a whole. Also, as the class leaves, hand students the Lesson 3 Exit Ticket (M-A1-4-3_Lesson 3 Exit Ticket.docx and M-A1-4-3_Lesson 3 Exit Ticket KEY.docx) to hand in for the next class.

Extension:

• Introduce students to linear optimization problems (for example, in which profit has to be maximized and is constrained by numerous inequalities). Introduce the corner-point principle (that maximum and minimum values are achieved at the corners of the solution region).