• Observe student responses throughout the activities, from suggesting possible solutions to more conceptual questions. This will allow you to guide students in the direction they need to go to solve the problem, as well as to assist them with the concepts.
• Observe student performance on the Inequality Worksheet as well as assisting students with the steps for solving linear inequalities.
• Student performance on the Lesson 2 Exit Ticket (M-A1-4-2_Lesson 2 Exit Ticket.docx and M-A1-4-2_Lesson 2 Exit Ticket KEY.docx) will show you how well students understand how to set up inequalities, as well as if students understand how to graph the inequalities. This allows you to assist students who might need additional practice, and see if a review of concepts is needed before proceeding to the next lesson.

To begin the lesson, provide the following situation to students: “Suppose that an athlete trains by running and jogging. When running, the athlete burns 8 calories per minute; when jogging, the athlete burns 5 calories per minute. The athlete wants to burn 240 total calories in her workout. The question is: What are all the possible combinations of running and jogging that will result in the athlete meeting her goal of burning 240 calories?”

Guide students through setting up the linear equation that represents the solutions to the problem. Students should identify the two variables (time running, x, and time jogging, y) as well as the constraint (burning 240 total calories). Ask students, [IS.4 - Struggling Learners]

“What equation can we use to represent the solutions to this situation?”

(8x + 5y = 240)

Once students have correctly identified the linear equation, proceed by examining the situation in more detail (and altering it to better represent a real-world situation).

• “Is it likely, if you are an athlete working out, that you want to burn exactly 240 calories?” (no)
• “What is more likely to be your goal—less than 240 calories, exactly 240 calories, or more than 240 calories?” (more than 240 calories)
• “If we want to show that the athlete wants to burn more than 240 calories, we can’t use an equal sign or an equation―instead, we want to use an inequality symbol.”

Erase the equal sign in the equation and replace it with an inequality sign so it reads

8x + 5y ≥ 240.

Tell students, “In many situations, an inequality represents the reality of the situation better than an equal sign.”

Activity 1

Tell the class, “We’ll continue to look at inequality signs with a few problems. In our problem about the athlete, how would our inequality change if, instead of burning at least 240 calories, she wanted to burn more than 240 calories?” Students should respond that it would change the inequality sign to be strictly greater than. Point out to students that the line underneath the inequality sign is really like part of an equal sign―it denotes equality, in addition to the inequality. Discuss: “How would our inequality sign change if our athlete’s goal was to actually burn 240 calories or less—a strange goal.” (The inequality sign would change to ≤.)

Repeat this question with the goal being burning strictly less than 240 calories.

Give each student a copy of Inequality Worksheet (M-A1-4-2_Inequality Worksheet and KEY.doc). Have students work in pairs, but each student should complete his/her own worksheet. Tell students that they will work on completing the entire worksheet in steps as a class. The first step is to write the inequality that represents each situation.

Encourage students to start by thinking of each problem as being like a linear equation. Students should identify the variables and also identify the constraint (time, etc.) and use those to write the equation. [IS.5 - Struggling Learners] Then, students should identify the portion of the problem that makes it an inequality and exchange the equal sign in their equation with the appropriate inequality sign, making sure to pay particular attention to whether the inequality sign is strictly less than (or greater than) or whether it is less than or equal to (or greater than or equal to).

Writing the inequalities to represent each situation is the only part of the worksheet that students should complete up to this point. After each pair has completed this part of the worksheet, have them compare with another pair who has completed the worksheet and make sure they have the same inequalities; if not, the foursomes should work together to resolve any problems.

Activity 2

Sketch the first quadrant of a Cartesian plane on the board. [IS.6 - Struggling Learners] Tell students, “Now, let’s continue to examine our problem about the athlete; we’ll come back and finish the next portion of the worksheet after we look at our sample problem. What if our athlete wants to meet her goal exactly? In other words, what if we consider all the solutions to the equation that we started with? What does our graph look like if we just consider the equation?”

Have students work in pairs with the equation 8x + 5y = 240 to determine what the graph looks like. Remind them that they can plot points or put the equation into slope-intercept form to best determine what it looks like. In slope-intercept form, the equation is:

“When graphing linear inequalities, the first step is to graph the line―think about it as if it were an equation.”

Graph the line on the Cartesian plane. Ask students, “Remember that she wants to burn at least 240 calories, so it’s okay if she burns exactly 240 calories. This means that all the solutions that make up our line are acceptable to the runner. Because the line represents some of the solutions to our inequality, we’re going to make the line solid.”

“What if her goal was to burn more than 240 calories; what if burning 240 calories exactly was not acceptable for our athlete? Would the solutions that make up the line actually be solutions?” (no)

“And how would that change our inequality, if the number of calories she wants to burn has to be strictly greater than 240?” (The inequality sign would change and just be a greater-than, rather than a greater-than-or-equal-to sign.)

“In that case, we would still start by graphing the line, but we would make it dashed instead of solid.”

Here, students should be reminded of graphing linear inequalities like x > 5 (or x ≥ 5) and their decision regarding whether to make an open circle or a closed circle, based on the inequality sign. Point out to students the analogue between making a closed circle and making a solid line (as well as making an open circle and making a dashed line).

Instruct students to return to their pairs and the Inequality Worksheets. Explain, “Now, with each of the three problems, you are going to complete the next step for graphing inequalities: graphing the line that represents the situation as if it were an equation. You’ve already written the equations, now you just have to graph them.” Remind students that they can plot points or convert each equation to slope-intercept form in order to graph it. [IS.7 - Struggling Learners]

“At this point, it makes a difference if the line is dashed or solid.” Students should note that the important part is whether the inequality is a “or equal to” inequality sign or not, and that this determines whether the line should be solid or dashed.

After each pair has completed this part of the worksheet, have them compare with another pair who has completed the worksheet and make sure they have the same lines; if not, the two pairs should work together to resolve misunderstandings or conflicts.

Activity 3

To begin this activity, tell students: “So far, we’ve just dealt with graphing the solutions as if it were an equation, rather than an inequality. Now, we’re going to tackle the inequality part. What are some possible amounts of time that the athlete can run and jog in order to meet her goal of burning at least 240 calories? Remember, we’ve already covered all the possible ways she can burn exactly 240 calories.”

As students provide possible amounts of time, plot them on the x-y coordinate (Cartesian) plane. Remind students that the athlete can burn as many calories as possible. For this example, ignore the facts that she may get tired, run out of time, etc. Encourage students to find “extreme” solutions (i.e., running for 1000 minutes) as well as “borderline” solutions―ones in which the athlete barely meets her goal.

When students have provided a dozen solutions or more, ask students what all the solutions have in common when plotted on the graph. They should note that they are all above the line. Ask, “Are there any points above the line that represent times for which our athlete would not meet her goal?” If students think there are, note than any point above the line has a point on the line that is directly “below” it, and that point represents burning exactly 240 calories. The point above the line represents exercising more, and thus has to represent burning more than 240 calories.

Continue, “Are there any points below the line that represent times for which our athlete would meet her goal?” Use the same reasoning to help students see that any point below the line is equivalent to less exercise than a point directly above it on the line, and thus must represent burning less than 240 calories.

“So our complete solution is really all the points on the line, where she burns exactly 240 calories, and every point above the line. To show this, since we can’t actually draw in dots for every single point, we just shade above the line to show that all of it represents solutions to our problem.”

Make sure to point out to students that, in graphing the solution for this problem, they must set aside some real-world constraints. For example, the athlete cannot run or jog for a negative amount of time, nor can the athlete run or jog for 1,000 minutes. Tell students that learning to graph linear inequalities without worrying about additional constraints is the first step in analyzing more detailed, realistic situations. The class will look at situations with multiple real-world constraints in the future.

Shade the appropriate area on the graph. Tell students, “This is the second step to graph linear inequalities―shading the appropriate side of the graph. Notice that the line that we graphed divides the entire x-y coordinate plane into two halves―one half represents the solutions to the problem, and the other half represents all the points that are not solutions. The trick is to determine which half is which. How did we know which half to shade in this problem?” [IS.8 - All Students]

Students should recall that they found a variety of different solutions to the problem―points which satisfied the inequality―and then shaded the part of the x-y coordinate plane that contained those points:

• “If all the solutions are always together on one half of the x-y coordinate plane, do we need to find a dozen different solutions to determine which side of the plane to shade?” (no)
• “How many solutions do we actually have to find in order to know which side to shade?” (one)
• “So we can pick any point we want―as long as it isn’t on the line, since we want to determine which side of the line to shade―and see if it is a solution. Since we can pick any point, would it make sense to pick a point like, say, x = 8.25 and y = 3.14?” (no)
• “Why not?” (Students should recognize that picking a “complicated” point makes the work much more difficult.)
• “What would be an easy point to pick and use for a test?” [(0, 0)]

Guide students toward selecting (0, 0) to answer the question. Have students, in pairs, substitute (0, 0) in for x and y in the first problem on the Inequality Worksheet. If students get stuck, work through the first problem on the board. Start with the inequality: All students should have 5x + 12y ≤ 240:

Then: 5(0) + 12(0) ≤ 240;

0 + 0 ≤ 240

0 ≤ 240

• “Is this a true statement? Is 0 less than or equal to 240?” (yes)
• “So, is (0, 0) a solution to our inequality―does it make our inequality true?” (yes)

Explain, “Since (0, 0) is a solution, we know all the solutions to our inequality for problem #1 lie on the same side of the line as (0, 0). So now, to finish problem #1, shade the side of the line that contains (0, 0).”

Once students have completed the shading, ask what they would have done differently if they plugged in (0, 0) and it did not make the inequality true. Remind students that the line they graphed divides the plane into two regions, one that has all the solutions and one that has all points that are not solutions. If (0, 0) is not a solution, then all the points on that side of the line must not be solutions. What does that mean about the other side of the line? (The other side is the solution.)

Have students work in pairs to complete the last two problems. After each pair has completed the worksheet, have them compare with another pair who has also completed the worksheet, and make sure they have the same inequalities and graphs; if not, the two pairs should work together to resolve any misunderstandings.

On the back of their worksheets, have students list the steps necessary to graph a linear equation. The steps should be:

1. Graph the associated line.
2. Pick and substitute a test point.
3. Shade the appropriate side of the line.

Students can turn in their completed Inequality Worksheet (with the steps on the back) before leaving class. In addition, hand out the Lesson 2 Exit Ticket (M-A1-4-2_Lesson 2 Exit Ticket.docx and M-A1-4-2_Lesson 2 Exit Ticket KEY.docx) for students to work on for next class.

Extension:

• Students begin to alter their solutions based on the real-world constraints of the problems (for example, that the athlete cannot run or jog a negative amount of time). Students should be encouraged not only to alter their solutions to represent the “real-world” solutions, but also to develop inequalities that represent these constraints (i.e., x ≥ 0, y ≥ 0).
• Students should be given linear inequalities to graph that go through (0, 0), helping them to remember that, while (0, 0) is almost always the most suitable test point, sometimes it is not a viable option.