22.1: Graph Features of Inequalities (5 minutes)
In this activity, students practice recognizing the slope and intercepts of inequalities.
For each inequality:
- What is the \(x\)-intercept of the graph of its boundary line?
- What is the \(y\)-intercept of the graph of its boundary line?
- Plot both intercepts, and then use a ruler to graph the boundary of the inequality.
\(2y \geq 4x - 8\)
\(2x + 3y < 12\)
The purpose of the discussion is to recall the similarities and differences for isolating a variable in an inequality and in an equation.
- "If each of these inequalities were equations, what would be different about the way you thought about the question?" (Most of the solution steps would be the same, but I wouldn't have to worry about reversing the direction of the inequality in any circumstances.)
22.2: Fruits and Running (20 minutes)
The purpose of this activity is to give students practice writing and graphing equations from situations. In the next activity and in the supported Algebra 1 lesson, students will write and graph inequalities Students then assess the meaning of points off of the line in the situation provided and interpret whether those points might be valid in the situation.
Write an equation that helps to answer the question about the situation. Then draw a graph that represents the equation.
- Jada goes to an orchard to pick plums and apricots to make jam. She picks 20 pounds of fruit altogether. If she picks \(a\) pounds of apricots, how many pounds of plums does she pick?
- Consider the point \((5,16)\). Is it possible for the weight of the fruit to be represented by that point in this situation? Explain your reasoning.
- In a video game, a character can run at a top speed of 30 miles per hour, but each additional pound that the character carries lowers the maximum running speed by 1 mile per hour. What is the maximum running speed of the character when they are carrying \(w\) pounds?
- Consider the point \((10,15)\). Is it possible for a character in this game to be represented by that point in this situation? Explain your reasoning.
The purpose of the discussion is to examine points on and off the lines and determine what they might mean in the situations.
- "What are the coordinates for a point on the line that represents the scooter situation? What do the coordinates mean in that situation?" (The point \((1,22)\) is on the line. This means that renting the scooter for 2 hours should cost $22.)
- "What are the coordinates for a point above the line that represents the scooter situation? What do the coordinates mean in that situation? Is it realistic?" (The point \((2,30)\) is above the line. This means paying $30 for 2 hours with a scooter. It only makes sense if the person is overpaying for some reason like if they broke part of the scooter and have to pay extra.)
- "The point \((5,30)\) is above the line that represents the maximum run speed for the videogame character. Describe a situation where this might be possible." (If the character has an item that ignores the carry weight or something that boosts their run speed, then maybe this is possible.)
22.3: Matching Graphs and Inequalities (15 minutes)
In this activity, students take turns with a partner matching graphs of inequalities to their symbolic and verbal descriptions. Students trade roles, explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).
Monitor for groups that use:
- whether the inequality is less than or greater than and whether the shading is to the upper left or lower right
- whether the equation is strictly less than or greater than, or less than or equal to, or greater than or equal to, and whether the graphed line is dashed or solid
- testing solutions in the situation, equation, and graph
- the boundary equation to determine the correct graphed line
Monitor for groups that struggle to make sense of the graphs involving a vertical or horizontal line.
Arrange students in groups of 2.
If necessary, demonstrate how to set up and and find matches. Choose a student to be your partner. Mix up the cards and place them face up. Point out that the cards contain either a graph or an inequality. Select cards with different styles and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree, for example, by explaining your mathematical thinking or asking clarifying questions. Give each group a set of cut-up cards for matching.
- Take turns with your partner to match graphs, inequalities, and constraints.
- For each match that you find, explain to your partner how you know it’s a match.
- For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
Ask previously identified students to share their strategies in this order:
whether the equation is strictly less than or greater than, or less than or equal to or greater than or equal to, and whether the graphed line is dashed or solid
whether the inequality is less than or greater than and whether the shading is to the upper left or lower right
testing solutions in the situation, equation, and graph
If no students used the testing solutions method, introduce it as a possibility to avoid getting confused in situations where there is a less than sign but the shading is above the line, e.g., for the graph on Card 2 which goes with the equation on Card 5.
Suggested questions to help students understand and connect strategies:
- "Why do the graphs with dotted lines go with the situations on Cards 9 and 10?" (Possible answer: Because if a kid is 4 feet tall, which would be on the line, they couldn’t go in the play area, so the line has to be dashed.)
- "What would you look for in the symbolic inequalities to see if the line should be dashed or solid? Why?" (The line should be solid only when the inequality is true when both sides are equal. A solid line means that points on the line make the inequality true.)
- "What would you have done if all the graphs had been dashed and all the symbols and situations had been strictly less than or greater than? What other strategies could you have used?" (Consider the slope and intercept of the boundary line and which side of the line is shaded.)
- "Do less than symbols always go with graphs that are shaded below the lines?" (Possible answer: No, because Card 2 and Card 5 go together, and the shading is above the line, but the symbol is a less than symbol.)
- "What are other ways you could write the equation on Card 5? Would any of those ways be easier to match to the graphs and situations? Why?" (Possible answer: \(y > x - 4\) is equivalent to \(4 > x - y\), and you can see that the \(y\) values above the line \(y = x - 4\) will be shaded because you can see that points with \(y\) values greater than the ones on the line are in the solution set.)
- "In which situations does it make sense to match cards with the less-than symbol to shading below the line, and in which situations might you need to test points to be sure?" (When the \(y\) is isolated on the left side of the inequality, using the symbol to shade above or below the line may make sense. It is always safe to test points to be sure.)