Students will comprehend the relationship between equations and the slope-intercept form, as well as how to solve and graph two-variable equations and inequalities. They will: 
- represent real-world situations using two-variable equations and inequalities. 
- find solutions for two-variable equations and inequalities. 
- write an equation in slope-intercept form. 
- solve a two-variable equation for one variable. 
- solve a two-variable inequality for one variable. 
- graph two-variable linear equations and inequalities. 

- How can mathematics help us communicate more effectively? 
- How are relationships represented mathematically? 
- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems? 
- How may data be arranged and represented to reveal the relationship between quantities? 
 

- Coefficient: The coefficient of a term in an expression is the number that is multiplied by one or more variables or powers of variables in the term. 
- Function: A relation in which every input value has a unique output value. 
- Linear Equation: An equation of the form Ax + By = C, where A ≠ 0 and B ≠ 0. The graph of a linear equation is a straight line. 
- Rate of Change: The ratio of the change in the output value and change in the input value of a function. 
- Simultaneous Linear Equations: Two or more linear equations that share a set of variables and either exactly one solution, no solutions, or infinitely many solutions; often called a “system of equations.” 
- Slope: The measure of the steepness of a line; the ratio of the change in the y-values to the x-values of the linear function; rate of change. 
- y-intercept: The y-coordinate of the point where a line intersects the y-axis; the value of the function when x = 0.

- Memory Cards (M-8-2-3_Memory Cards) 
- Entrance Ticket (M-8-2-3_Entrance Ticket) 
- Graphing Inequalities (M-8-2-3_Graphing Inequalities and KEY) 
- Systems of Inequalities (M-8-2-3_Systems Extension and KEY) 
- envelopes

- The Entrance Ticket requires students to generate three equations and three inequalities utilizing appropriate variables and costs to indicate a combination of items equal to, greater than, or less than a certain amount. 
- Graphing Inequalities can be used to examine students' ability to express slope-intercept form, solve equations and inequalities as ordered pairs, and graph equations and inequalities. 
 

Active Engagement, Modeling, Explicit Instruction 
W: Students learn how to create equations and inequalities that represent real-world outcomes. They investigate methods for understanding and representing more complicated real-world situations, as well as solving related challenges. 
H: To complete the Entrance Ticket, students must develop their own version of the costs of grocery items, what quantities their selections represent, and how those variables are to be used. Grocery items and prices are familiar and accessible concepts. 
E: In Activity 1, students make use of the costs of combinations of items to obtain a sense of how variability of cost and quantity constrain or expand purchasing option in accordance with specified limits. 
R: Graphing Inequalities requires students to summarize step-by-step instructions for graphing linear inequalities and solving a two-variable inequality in a real-world scenario. To do so, students must develop an increased level of generalization. 
E: Graphing Inequalities summarizes key concepts and techniques that are important in reaching the objectives of the lesson. 
T: Students improve their comprehension of variables, equations, and inequalities by using thorough detailed examples and real-world applications. Student creation of sample equations and inequalities based on grocery lists and menus encourages individual learning, whereas class discussions provide an opportunity for revision and clarification. The end of Graphing Inequalities includes another component targeted to students' various learning styles. Visual learners benefit from the step-by-step equations and inequalities, which are accompanied by graphs. 
O: The lesson is designed in a way that takes students from writing equations/inequalities that represent real-world situations to solving such equations/inequalities. Students will learn the slope-intercept form of an equation and then end with graphing two-variable equations/inequalities. Students are encouraged to analyze real-world problems and gain a conceptual understanding related to the ideas of variables, equations, and inequalities. Students can relate the slope-intercept form of an equation to solutions for that equation. The lesson demonstrates the importance of leaving an equation in slope-intercept form and identifying potential solutions (actual values) for the equation. Students also receive a quick view of how these equations and inequalities look like when graphed. An interpretation of such graphs can be presented in a later unit. 

Give each student an Entrance Ticket (M-8-2-3_Entrance Ticket). Students will represent selections by writing three equations and three inequalities. Allow plenty of time for completion. Discuss the definition of variables; have students define variable and explain the role of a variable in an equation/inequality. Instruct students to explain the correctness of their equations/inequalities.


Activity 1: Menu

Display the following example of a menu:

"How do we figure out how much one sandwich costs if we know two sandwiches cost $7.30? Let's use s to represent the cost of one sandwich. We might write the equation below and solve for s:

2s = 7.30

s = 3.65

One sandwich costs $3.65.”


"What is the cost of one carton of milk, given that four cartons of milk minus the cost of one tossed salad is $4.25? Let m stand for the cost of one carton of milk and t represent the cost of a tossed salad. We might write the equation below and solve for m:

4m – t = 4.25                m indicates 1 carton of milk, while t represents tossed salad.

4m – 3.55 = 4.25          Substitute the value for t given in the menu.

4m = 4.25 + 3.55         Add 3.55 to each side.

4m = 7.80                     Perform the operation.

m = 1.95                       Divide both sides by 4.

"One carton of milk costs $1.95."


"Sometimes we don't know how much either item costs, but are given the quantities of each and the total. For example, we may know that the cost of 5 rolls and 3 sandwiches is $15.00. We would write the equation as 5r + 3s = 15." 

"What solution or ordered pair (r, s) would make this equation true? In other words, what values substituted for r and s would make the equation true?" 

Encourage students to discuss the possibilities. Here are several possibilities: 

r = 1.05; s = 3.25 or (1.05, 3.25) 

"When we substitute back into the equation, we have 

5(1.05) + 3(3.25) = 15 

15 = 15 

"Thus, the costs for a roll and sandwich could be $1.05 and $3.25, respectively."


"If we determine a value for one variable, we may solve the equation for the other variable. For example, if we assumed that r = 1.05, we can replace for r and solve for s. 

5r + 3s = 15 

5(1.05) + 3s = 15            Substitute the value of r. 

5.25 + 3s = 15                 Perform the operation. 

3s = 15 - 5.25                  Subtract 5.25 from both sides. 

3s = 9.75                          Perform the operation. 

s = 3.25                            Divide each side by 3." 

This method can be used to solve an equation with two variables. Lead students through the process of identifying three further solutions and discussing the concept of infinitely many solutions. 

Some other solutions include:

"Let’s substitute $2.25 for s and solve for r. 

5r + 3s = 15 

5r + 3(2.25) = 15             Substitute the value of s. 

5r + 6.75 = 15                  Perform the operation. 

5r = 15 - 6.75                   Subtract 6.75 from each side. 

5r = 8.25                           Perform the operation. 

r = 1.65                             Divide each side by 5. 

Thus . . . 

r = 1.65; s = 2.25              The solution is (1.65, 2.25).”



"Substitute $1.35 for r and solve for s. 

5r + 3s = 15 

5(1.35) + 3s = 15               Substitute the value of r. 

6.75 + 3s = 15                    Perform the operation. 

3s = 15 - 6.75                     Subtract 6.75 from each side. 

3s = 8.25                             Perform the operation. 

s = 2.75                               Divide each side by 3. 

Thus . . . 

r = 1.35; s = 2.75                The solution is (1.35, 2.75).


"Substitute $3.85 for s and solve for r. 

5r + 3s = 15 

5r + 3(3.85) = 15                Substitute the value of s. 

5r + 11.55 = 15                   Perform the operation. 

5r = 15 - 11.55                    Subtract 11.55 from each side. 

5r = 3.45                              Perform the operation. 

r = 0.69                                Divide each side by 5. 

Thus . . . 

r = 0.69; s = 3.85                 The solution is (0.69,3.86)." 


"We can also examine solutions to an inequality using the menu." 

"Suppose we know that four bottles of milk and three sandwiches cost less than or equal to $15.50. How might we write this scenario using an inequality? How could we find possible solutions?”

"We can write the inequality as…"

4m + 3s ≤ 15.50

"To find possible answers, we will follow the same procedure as before. Let's substitute a value for one of the variables and then solve for the other. 

"We will try a cost of $3.50 for a sandwich and substitute this value for s:"

4m + 3s ≤ 15.50
4m + 3(3.50) ≤ 15.50
4m + 10.50 ≤ 15.50
4m ≤ 15.50 – 10.50
4m ≤ 5
m ≤ 1.25

"A possible solution is (1.25, 3.50). In other words, a carton of milk must cost $1.25 or less, and a sandwich must cost $3.50 or less, for the total to be less than or equal to $15.50. Solving this type of problem occurs in everyday life when people decide whether or not they have enough money to buy items from a menu."

Have students come up with three more solutions to this inequality. Then have them write two more inequalities from the menu and discover four solutions. Hold a class discussion about equations, inequalities, variables, values substituted, and how they relate to one another.


Rewriting Equations and Inequalities in Slope-Intercept Form

"We have looked at several equations involving different letter variables. Let's look at some equations with two specific variables, x and y, and investigate the slope-intercept form of an equation. The slope-intercept form of an equation is described as a linear equation in the form of y = mx + b, where m represents the slope (constant rate of change) and b represents the y-intercept, or starting point." (Note: This brief definition assumes that students are already familiar with slope-intercept form and the terms "slope" and "y-intercept." If this is not the case, more discussion should take place here.) 

"Consider the equation: 4x + y = 15. Is this equation expressed in slope-intercept form?" (No.) 

"This equation is not expressed in slope-intercept form. A slope-intercept form of an equation is solved for y. In other words, y is isolated. Why might we want to transform this equation into slope-intercept form?" (Slope-intercept equations provide us with valuable information, such as the slope and y-intercept of the graph or problem.)

"To find the slope and y-intercept of the previous equation, we'll transform it into slope-intercept form. To do so, we need to solve the equation for y. We can do this by removing 4x from both sides. This process provides us: 

y = -4x + 15 slope-intercept form. 

"With the equation in slope-intercept form, we can simply calculate the slope and y-intercept. What is the slope?" (−4) "What is the y-intercept?" (15) 



“Let's create another equation in slope-intercept form. 

5x + 3y = 15 

3y = -5x + 15.                     Subtract 5x from both sides.

y = -\(5 \over 3\)x + 5                       Divide each side by 3.”

"In some cases, we have linear inequalities in two variables, x and y. In the same way that transforming such equations into slope-intercept form is beneficial for us, we may want to solve these types of inequalities for y. Consider the inequality: -2x + y < 12. 

"What could we do to rearrange this inequality so that y is isolated?" (Add 2x to both sides.) 

"By adding 2x to both sides, we obtain y < 2x + 12." 

"Let's look at another one:

-3x + 18y ≥ 6 

18y ≥ 3x + 6                Add 3x to both sides. 

y ≥ 3/18x + 6/18        Divide both sides by 18. 

y ≥ 1/6x + 1/3            Reduce all fractions."


Activity 2: Memory Game

Before the class, cut out the Memory Cards (M-8-2-3_Memory Cards) and place them in an envelope. Prepare enough envelopes for students to play in pairs or small groups, and distribute as needed. Instruct students to turn each card face down and mix it up so that the positions of specific cards are unclear. Students should take turns turning over two cards at a time. If the flipped cards reveal equivalent equations or inequalities, the student wins this pair. If the cards do not match, the student turns them back over, and the game continues with the next student in the group. The game ends when all of the cards are matched. The student with the most pairs wins.


Graphing Linear Equations and Inequalities in Two Variables

"Now we'll graph some linear equations and inequalities in two variables. In this unit's first lesson, you used a number line to graph linear equations and inequalities in one variable. In two variables, we graph equations and inequalities on a coordinate plane." 

Consider the equation: y - 5x = -10. Earlier, we covered the advantages of rewriting equations in slope-intercept form. Doing this also makes these equations much easier to graph." 

"Let's first solve for y:

y - 5x = -10                   Solve for y if the equation is not already stated in slope-intercept form.

y = 5x - 10                     Add 5x to each side.”

"Now that the equation is in slope-intercept form, how can we graph the corresponding line on a coordinate plane?" (First plot the y-intercept at (0, −10). Then count 'up five, over one' from point to point. (Note: Here, it is assumed that students have already graphed lines. If not, more discussion will be required.)

Demonstrate graph of line for students.

"Remember that the graph of a line shows all the sets of ordered pairs (x, y) that satisfy the equation." 



"Now, let's look at what happens to the similar inequality, y - 5x < -10. Use the same operations as tools in the same way to solve the inequality.

y - 5x < -10                Solve for y.

y < 5x – 10                 Add 5x to each side.”
______________________________________________________

y - 5x ≤ -10                Solve for y.

y ≤ 5x – 10                 Add 5x to each side.”
______________________________________________________

y - 5x > -10                Solve for y.

y > 5x – 10                 Add 5x to each side.”
______________________________________________________

y - 5x ≥ -10                Solve for y.

y ≥ 5x – 10                 Add 5x to each side.”
______________________________________________________

"Like the graph of a linear equation, the graph of a linear inequality needs to show all the sets of ordered pairs (x, y) that satisfy the inequality." 

Instruct students to list three ordered pairs that satisfy the inequality. Once students have completed this, ask them to write their ordered pairs on the board. Once all of the example solutions have been written on the board, plot the relevant points on a large coordinate grid for students to see.

"What do you see in the graph of this inequality? Do all of the ordered pairs line up in the graph of an equation?" (No. An inequality has significantly more solutions than an equation.) 

"As you can see, the graph of a linear inequality includes more than simply a line. Instead, when we graph a linear inequality, we are actually graphing a boundary line and a half plane of ordered pairs. The boundary line represents the line that would be graphed if the inequality were actually an equation (or what happens when we replace the inequality symbol with an equal sign). The half plane is the half of the coordinate plane that contains the solutions to the inequality. This half will be either above or below the boundary line. We shade the half plane containing the solutions to our inequality. Let's use our inequality y ≥ 5x – 10 as an example." 

"First, we graph the boundary line." 

Graph the line y = 5x - 10 on a coordinate plane so that students may see.

"The boundary line might be solid, dotted, or dashed. This is determined by whether the ordered pairs of the boundary line are included in the solution set for the inequality. What are some methods for determining whether the ordered pairs on the boundary line are part of the solution set to the inequality?" (Students may suggest testing an ordered pair from the boundary line by replacing the x and y values into the inequality. They should notice that in this situation, the boundary line IS included in the graph. Lead them to learn that one simple way to recognize this is that the inequality sign used is a greater than OR EQUAL TO symbol.) 

"Because our inequality use a greater than or equal to symbol, the boundary line is included in the graph and should remain solid. Now we must choose the suitable half plane to shade. This might be either 'above' or 'below' the line. How do we decide which half plane to shade?" (Students may consider testing an ordered pair from each half plane to determine which has x and y values that satisfy the inequality. Encourage students to learn that if a test point correctly satisfies the inequality, the half plane containing this test point is included in the inequality graph and should be shaded.) 

Shade the half plane on the coordinate grid so that students can see it.

"To summarize, to graph a linear inequality in two variables, we follow these three steps:

1. Graph it as an equation, and choose a dotted line for < or > and a solid line for ≤ or ≥. 
2. Select a test point (ordered pair) and substitute into the original inequality. 
3. If the point is a solution, shade the half-plane containing it. If the point isn't a solution, shade the other half plane."

Discuss the following more examples with the class:

Two-Step Linear Inequalities in Two Variables

"Let's look at some additional examples:

2y + 6x ≤ -6

2y ≤ -6x – 6                    Isolate the y term by subtracting 6x from each side.

y ≤ -3x – 3                      Divide each term on both sides of the equation by 2.”


“To graph the inequality, we must first replace it with an equal sign and represent it as an equation. We notice that the inequality is ≤, indicating that our line will be solid. We'll use the test point (0, 0) to figure out which half-plane to shade. Notice that any test point would work. The point (0, 0) produces a false statement, thus we will shade the half-plane that does not contain the point (0, 0)."

"Our next example is special since it contains a fraction for the slope:

-7x + 14y > 28               Isolate the y term by adding 7x to each side.

14y > 7x + 28                Divide each term on both sides of the equation by 14.”

“To graph the inequality, we must first replace it with an equal sign and represent it as an equation. We see that the inequality is >, thus our line will be dotted. We will use the same test point (0, 0) to decide which half plane to shade. The point (0, 0) results in a false statement, thus we shall shade the half plane that does not contain the point (0, 0).”

Activity 3: Graphing Linear Inequalities

Distribute the Graphing Inequalities (M-8-2-3_Graphing Inequalities and KEY). Instruct students to complete it. Walk around the class as students try to answer questions and ensure that all students are on the right track.

Extension:

Briefly discuss how to graph a system of linear inequalities. This entails plotting two inequalities on the same coordinate plane. The solution of the system is found where the solution regions for each particular inequality overlap. Give students the Systems Extension worksheets (M-8-2-3_Systems Extension and KEY). This worksheet includes four examples of graphing a linear system of inequalities in two variables. It is a natural follow-up to this lesson and can also serve as a review.