Students should work in pairs on this Launch Question. The question is deliberately similar to prior problems dealing with solving two variable equations. The exception is that now the entire $25 does not need to be spent. I have students try to read and understand the problem and attempt to determine the difference (MP7). I avoid being too helpful at this point. I let students try to work out why this question is different from the one they solved in a previous class (MP1).
Without giving any instruction, I will have one student from each pair come to the board to post one of their possible solutions. In my experience, many students will still choose points that are on the line. If I am lucky, one or two students will choose points that are below the line.
The opening of this section of the lesson depends on whether or not some students choose points below the spending limit on the first problem. If not, I will post one and then have students do a Think-Pair-Share around the viability of this proposed solution. When students are sharing their ideas, I will try to guide them towards the understanding that they are allowed to spend less than $25 and coming up with other options.
Next, I will have students work with a partner to determine a rule that could be used for this situation. I may also have them come up with at least three more points that satisfy their rule. I want them to experience the idea that it should be very easy for them to generate combinations because there are so many.
As usual, most students will come up with whole number values. I will remind them to think about solutions that are not whole numbers. Again, I will give them concrete prompts like, "What if you spend $10.15 and your friend spends $8.87. Is the sum of those two numbers still less than $25?" As students volunteer possible coordinates, I will add them to the graph. Students should also experience the futility of trying to mark every one. Once they begin to get the point, I want them to work together to to try to figure out a way to represent all of the solutions graphically (MP4). If necessary, I will guide them towards the idea of shading in all of the area that we want. But, I do want my students to own this one (see Collaboration: Building Communication and Student Ownership).
I have included a Computer Generated Graph to represent this solution. However, in the context of the problem we would only one the shaded area that lies in the first quadrant. I will display this graph for students and show them how this actually represents the inequality x+y<=25.
I will ask my students to work in pairs on the solving_two_variable_inequalities_investigation. My students will need some guidance with certain aspects, I have included points of emphasis below. It will be a judgment call as to which parts we discuss as an entire class. However, in my experience it is sometimes better to let students try to think through the material first as this will help build their thinking skills and perseverance (MP1).
Students should understand that a test point serves the purpose of determining which way to shade. We do this by asking a simple question, "Is the test point in the solution or not?" In other words, does the test point make the inequality true or false. If the test point makes the inequality true than other points on that side of the line will also make the inequality true.
Teaching Point: When a boundary line becomes very steep, sometimes it is difficult for students to discern up and down and they start thinking about left and right. I just show students to put their pencil on the line and draw a vertical line up (or down) and this will show them the side of the line that is above or below, respectively.
Points of Emphasis:
1) Question 1 will allow students to try a variety of test points on either side of the boundary line to see that some will make the equation true and others will make it false. While students will normally only use one or two test points, this will help them build understanding around the concept.
2) Question 1d gets at the idea that if you reverse the inequality symbol then the shading would change from one side of the line to the other. You can quickly demonstrate this to your students using a software program like Desmos.com or Geogebra.
3) Question 2 may need to be emphasized with the entire class. This is typically a concept that is glossed over in most algebra classrooms and replaced by the rule <,> means a dotted line. Choose some points that are on the boundary line and show students that they make the inequality false. Because of this we need a way to show that the line is there but none of the points on the line are solutions (answer: a dotted or dashed line).
Because students are choosing their own test point this is also a nice place to discuss the idea of choosing (0,0). I like using the origin as a test point because of the ease of calculation. I remind students that if the origin is on the boundary line they would have to choose a different test point.
4) Question 3 will allow students to practice what they have just learned. I may carefully go over (b) and (c) with the entire class. Graphing a single variable inequality in the coordinate plane may still be somewhat confusing for students. I explain it as follows: If x <-3 this just means that is long as the x-value of the point is less than -3 the inequality is true. The y-value can be any real number.
Today's Ticket Out the Door asks students to construct an inequality based on a description and graph all of the solutions to that inequality in the coordinate plane. I require all students to show their work on this so that I can see the flow of their thinking in determining the following:
(1) Where is the boundary line? Is it solid or dashed?
(2) What test point should I choose that is above or below the boundary line?
(3) What did the test point tell me? Is it a solution or not?
(4) Do other points in the shaded area satisfy this inequality?