Lesson 22

In this lesson, students will be writing linear inequalities that represent constraints in situations and graphing the solution regions. To prepare for that work, students review writing and graphing an equation that represents a situation.

If needed, explain or show additional images of grass sod and flower beds to students who might be unfamiliar with these landscaping terms.

Invite students to share their equation and graph. Discuss with students:

• "In this situation, what does a point on the line mean?" (A combination of square feet of grass sod and square feet of flower beds that the homeowner could have if she spent her entire budget.)
• "What does the vertical intercept of the graph mean?" (The square feet of flower beds she could have if she installs no grass sod.)
• "What does the horizontal intercept of the graph tell us?" (The square feet of grass sod she could install if she installs no flower beds.)

This is the first of a series of activities in which students write an inequality to represent a constraint in a situation. Students graph a related equation, interpret the coordinate pairs of points on the graph and on either side of the graph, and test the pairs of values to see if they make the inequality true. They then use these observations to determine the solution region to the inequality.

Students also consider whether all the points in the solution region are necessarily meaningful or feasible in the situation. In doing so, they reason quantitatively and abstractly (MP2) and practice evaluating the reasonableness of their solutions in context (MP4).

Graphing technology should not be used in this activity and the other activities in the lesson.

Briefly explain to students who are unfamiliar with landscape materials what artificial turf and gravel are (or show additional pictures).

Consider arranging students in groups of 2 and giving them quiet time to work on the first set of questions, followed by time to share their thinking before moving on to the rest of the activity.

Select students to share their interpretations of the two points on either side of the line. Make sure students understand that one point represents a combination of gravel and artificial turf that meets the budget constraint and that the other point does not. The region in which each point belongs can be interpreted in the same way. 

Also make sure students understand why points on the boundary line are included in the solution set of \(3x + 15y \leq 3,\!000\).

To encourage students to evaluate the reasonableness of their solutions in terms of the situation being modeled, discuss questions such as:

• "Both \((100,50)\) and \((50,100)\) are in the solution region. Both points mean a total of 150 square feet of landscape materials. Does it make a difference which option is chosen?" (It does not make a difference in terms of the total area, but it does in terms of cost. It might also make a difference to the plants and to the overall appearance of the yard.)
• “All the points in the shaded region represent amounts of gravel and turf that are within the homeowner's budget. Are all these options equally good and desirable?" (Most likely not. For example, if the homeowner needs both materials, pairs such as \((0,100)\) or \((450,0)\) are probably not desirable because they mean buying only one material but not the other. The point \((0,0)\) is also in the solution region, but buying no materials is probably also not an option because it would not help the homeowner with her landscaping goals.) 

In this activity, students continue to work with inequalities in two variables in context. They write an inequality that represents the constraints in a situation, graph its solutions, and interpret points in the solution region.

Earlier, students saw that some points in the solution region might satisfy an inequality (mathematically) but might not work in the given situation because other considerations were at play. Here, they see another reason that some values that satisfy an inequality might be unfeasible in the situation—namely, that the pair of values must be whole numbers. When students think about the plausibility of these values in context, they engage in an aspect of mathematical modeling (MP4).

As students work, look for those who plot discrete points to represent the solution region and those who shade a part of the plane.

Arrange (or keep) students in groups of 2. Give students a few minutes of quiet time to work on the first three questions, and then time to discuss their responses before they continue with the last two questions. 

Some students may have trouble graphing the line that delineates the solution region from non-solution region because they are used to solving for the variable \(y\), but here the variables are \(b\) and \(n\). Ask these students to decide which variable to solve for based on the graph that has been set up. Ask them to notice which quantity is represented by the vertical axis.

If some students plotted discrete points and some shaded the region, choose one of each and display these for all to see. Ask students why each one might be an appropriate representation of the solutions to the inequality.

Highlight that the discrete points represent the situation more accurately, because it is impossible to sell a fraction of a bracelet or 2.75 necklaces. It is, however, tedious to plot a bunch of points to show the solution region. It is much easier to shade the entire region, but with the understanding that, in this situation, only whole-number values make sense as solutions.

This optional activity offers an additional opportunity for writing a linear inequality in two variables to represent constraints and for graphing and interpreting the solutions.

In previous activities, the solution region of an inequality \(Ax + By \leq C\) lies below a boundary line and the solution region of an inequality \(Ax + By \geq C\) lies above a boundary line. In this activity, students encounter an example in which the symbol \(\geq\) does not correspond to a solution region above the boundary line. (In the given situation, pairs of values that generate more revenue for the concert are points below the graph of \(Ax+By=C\).)

The work here reinforces the importance of reasoning about points on either side of a boundary line, rather than simply assuming that < or \(\leq\) means shading below the line and > or \(\geq\) means shading above the line. It allows students to practice reasoning quantitatively and abstractly (MP2).

Some students may find it challenging to graph the boundary line (\(25t -1,\!250c = 20,\!000\)) by identifying the intercepts. The horizontal intercept is fairly easy to find, but the graph intersects the vertical axis at a negative value (and a negative number of concerts does not make sense in this situation). Ask students to find at least one other point (besides the vertical intercept) that is a solution to the equation.

Students who rewrite the equation in slope-intercept form and find the slope to be 0.02 may also find it difficult to interpret. Ask them to try writing the slope as a fraction (\(\frac{2}{100}\)).

Invite students to share their inequality and the graph of the solution region.

Focus the discussion on the last two questions—on how students knew which combinations of tickets and concerts would enable the band to meet its goal and would raise more money. Highlight responses that involve testing pairs of values to see if they satisfy the inequality or to make comparisons.

If not mentioned in students' explanations, point out that even though the inequality has a \(\geq\) symbol, the solution region is below the boundary line, not above it. Stress the importance of not blindly shading a region based on the symbol in the inequality.