• Students’ performance in understanding the behavior of equations and functions is greatly affected by their skills in basic tasks such as combining like terms. Where individual students show weaknesses in this area, emphasize practice.
• Ask students to tell in their own words what the difference is between an equation and a function. Answers such as an equation is a condition and a function is a rule represent the kind of global and conceptual thinking that should drive the lesson.

The focus of this lesson is on teaching how to graph linear equations using technology. Students will use a graphing calculator to find the solution to linear equations. A conceptual understanding of solutions, as found through a graphical approach, is the main objective of this lesson.

While graphing linear equations, students encounter both linear equations in one variable and in two variables. It is important that students conceptually understand the differences in solutions and solution process for these two linear equation types.

Part 1: Graphing Linear Equations: Discovery

Start class with the following examples of linear equations in one variable written on the board:

“We might also encounter examples with no solution, such as the following” [IS.3 - Struggling Learners] (put these on the board as well):

“How can these equations have no solutions?” Explain, in the first example, combining like terms  leads to the contradiction 0 = 9. In the second example,  leads to another contradiction, 4 = 0. When a statement seeming to be an equation contains contradictory information, it means that the statement is not an equation.

“How can you discern graphically whether a linear equation has a solution? What will the graph look like? Before we get into a discussion of linear equations without solutions, let’s think about how solutions can be derived from graphs of linear equations in one variable and those in two variables.”

“First, let us understand that any letter can be used to represent a variable. If just one variable is present in the linear equation, then we say it is an equation in one variable. However, if two different variables are present, we say it is an equation in two variables.” [IS.4 - Struggling Learners]

“Let’s create a chart to summarize what we know about the previous linear examples. We will also use the chart to provide ideas related to how solutions appear graphically, what the solutions actually represent, and confirmation of our ideas with an algebraic approach. We will start the chart together. Then, you will work with a partner to hypothesize how to find the solutions to the different equations, describe what the solutions will look like and represent, and justify your ideas with an algebraic method or check.” Start the chart together with the class and fill in the second column. Distribute copies of the Linear Equations 1 handout (M-A1-3-2_Linear Equations 1 and KEY.docx).

Class Chart Example [IS.5 - Struggling Learners]

Answers to Class Chart, Version 1

Activity 1: Chart, Part A

With a partner, students complete the chart, focusing on developing a consensus on how to graphically find solutions to each linear equation type: equations in one variable and equations in two variables. Note: The table feature can be used to pinpoint values not easily discerned from the graph. Hopefully, students begin to see that x-intercepts and/or point of intersection can be used to find the solution to linear equations in one variable, whereas any point on the line can represent a solution to an equation in two variables. This distinction is important and often difficult for students to understand conceptually. If necessary, assist the class by explaining this concept, maybe by filling in one of the rows in the chart with the class to demonstrate.

Chart, Part B

Have students describe what solution means when discussing linear equations in two variables. “Discuss with your partner whether or not a solution to an equation in two variables means the same as a solution to an equation in one variable. (Hint: You may discuss the idea of placeholder versus exact solution.)” Note: At this early stage of the lesson, students may discover that equations in one variable have a precise solution, or one exact solution, that satisfies the equation. However, a linear equation in two variables uses a type of placeholder effect, where the values are “variable.” Any value can be used as the input value, thus creating a unique output value.

Hold a class discussion to review any concerns, questions, revelations, etc. Also, pose the following questions:

• “With this exploration, did you pause to think about the y-intercept and what that value represented?” (the point where the graph intersects the y-axis)
• “Was that the solution?” (yes, when x = 0)
• “Was it the only solution?” (no)
• “If so, to what type of equation was the y-intercept a solution? In other cases, what did it represent?” (the value for y when x = 0)

Explain about the y = screen window in graphing calculators: “The y = screen is the window in your graphing calculator that records which equation is to be graphed. This window typically holds a cursor following a ‘y =’ prompt.” Then ask, “Is it at all confusing that the equations written into the y= screen look exactly as if they are written in two variables, but have a specific solution, not many solutions, as evidenced by evaluation of one particular x-value? Then, we also input equations with x and y into the y= screen, intending for y to represent ‘y.’ How can we alleviate any ambiguity? [IS.6 - Struggling Learners] We will explore this idea more in Part 2 of this lesson. We will also discuss the two methods used to find solutions to linear equations in one variable.”

Part 2: Graphing Linear Equations: Practice

Tell students, “Let’s revisit the linear equations from before:

Using a graphical approach, we will determine the solution to each equation. From our previous activity, have you decided on a definition for solution?” Let the class discuss; then give the answer. The solution(s) is/are the numerical value or values that make(s) the equation true. In some cases, there is only one such value. In other cases, the values for the variables are indeed “variable” in that they change as the input value changes.

“We will use the chart you filled in and compare our responses across the board.” Have students help complete the class chart. Make the completed chart available to students in electronic form. Sample observations are recorded below. Use these to guide the discussion. Provide this information to students: The table feature can be used to pinpoint values not easily discerned from the graph.

Distribute copies of the Linear Equations 2 worksheet (M-A1-3-2_Linear Equations 2 and KEY.docx).

Answers to Class Chart, Version 2

Ask the class, “For linear equations in one variable, why is it that the intersection of the two pieces of the equation and the x-intercept of the entire equation give the same solution?” Give students time to ponder this important question.

“To answer this question, let’s take the following linear equation in one variable:

We know we can find the solution graphically by either finding the intersection of , or finding the x-intercept of the equation .”

“Any time that we have two expressions set equal to one another, the intersection of the two expressions is the solution. If we check with a table, an x-value of 2 gives 2 for the y-value of the equation (2 is the y-value for the second equation for all x-values; it has a slope of 0). However, if we transform the equation in order to set it equal to 0 (subtracting 2 from each side), we are saying the equation equals 0. Thus, the statement, or equation, is true whenever the y-value is 0. The y-value is 0 at the x-intercept. Thus, the x-intercept also provides the solution.”

“What does the y-intercept indicate in each equation? Let’s examine the y-intercept for each equation.” Let students explore and see what they find about the y-intercept. Example answers are below in the table. Distribute copies of the Two Variables handout (M-A1-3-2_Two Variables and KEY.docx).

Example Answers

“These ideas lead us into the discussion of difficulties associated with understanding the usage of the y= screen, as opposed to the actual equation, written in one or two variables. In other words, even with an equation written in one variable, we can transform the equation, setting it equal to 0, and entering it as y = ______. However, we know that it does not have a variable x-value. There is indeed one solution. How does this differ from an equation originally written in two variables and entered as is? Why can we enter an equation in one variable as y = ____?” (Allow time for discussion or questions.)

Remind students that when an equation has two variables, the equation is true for many different values of the two variables.

“Taking the equation

we will record the things we know so far:

1. 1.      We have an equation in one variable.
2. 2.      We can write it as .
3. 3.      We can enter it into the graphing calculator as .”

Ask, “Why is the x-intercept the only solution? Why not all points on the line?”

Explain, “The reason for this is that the solution occurs when y = 0, hence the y-intercept.”

“However, if we have the equation

we know a different set of things:

1. 1.      We have an equation in two variables.
2. 2.      We can write it in slope-intercept form as .
3. 3.      We enter it as is into the y= screen.”

Ask, “What is the solution? We are not transforming the equation to y = 0. Therefore, our solution does not have to be an x-value that results in a y-value of 0. However, and important to note, it can be a solution. There are infinitely many solutions, as shown as points on the line of the graph. Depending on what the equation is describing, it may or may not be significant that when y = 0, the solution is x = −7.”

“How do you discern the difference? You note the differences in the beginning, prior to entering the equation into the calculator. Look to see if you have two variables or one before beginning; this will tell you what is going on.”

Equations with No Solution

“Thus far we have examined equations with solutions. You will often encounter equations without solutions. Let’s look at the two examples from the beginning of the lesson:

When examining the equations, consider the following questions:

• What makes them different?
• How does the graph look?”

“If you enter each expression on the left and right sides of the equal sign into the first and second y= screens, the graphs will be parallel. However, if you set the equation equal to 0 and enter the one equation into one y= screen, you will get a line with a slope of 0, or a horizontal line through the y-intercept. This fact translates to the following: for all x-values, you get a y-value of ____.”(Ask students to fill in the blank, guiding them as necessary). “If you set the output value to 0, then this isn’t possible!”

Part 3: Applications of Previous Learning

Assign the class to do the following: “Using your understanding of graphing and solutions to equations in one variable and two variables, create one of the following:

• a PowerPoint presentation.
• a scholarly article.
• another presentation.” (This option will be reviewed and accepted by you.) [IS.7 - Struggling Learners]

Explain the assignment: “The purpose of the presentation is to illuminate linear equation examples and solutions in the real world. You must present at least five equations for three different topics (15 equations in all). For example, you may present equations modeling business outcomes, science, consumer spending, etc. The equations must be accompanied by a graph with highlighted solution(s). The solution should be clearly identified and connected to the context of the situation. The context may be presented in a variety of ways, including word-problem form, a graphic, a text description, and so on. For example, if you have the equation

representing the cost of 4 apples and 2 bananas, with A representing the cost of an apple and B representing the cost of a banana, you must state the possible variable costs for the given total cost.”

“Prepare a 5–10 minute presentation. Highlight a specific area and be prepared to discuss any concerns, problems along the way, surprises, etc. Each presentation will be placed as a resource on the class Web site.” [IS.8 - Struggling Learners]

To review the lesson, hold a class discussion related to any questions, difficulties, or epiphanies experienced during the lesson. The ending activities are higher-level and require much thought. Students may have several questions.

Also hold a class discussion on the ideas of solutions and graphing. Ask, “What new ideas did you learn? Do you view solutions differently? How? Does graphing make it easier or harder to view solutions? Was the table feature helpful?”

Extension:

• Divide students into groups of three to four. Ask the groups to make a brief presentation (5 minutes), connecting and comparing the following: x-intercepts,
  y-intercepts, solutions, linear equations in one variable, linear equations in two variables, graphs, and tables.
• Ask students to answer the following question: If the x- and y-axes represent horizontal and vertical quantities similar to the length and width of a rectangle, what kind of geometric objects could represent one more dimension and how could you describe it in terms of x and y? For example, in a three-dimensional x-y coordinate (Cartesian) coordinate system of geometry, the model of a cube uses the length, width, and height as physical spaces. The x-axis points to the observer, the y-axis is horizontal, and the z-axis is vertical.