• While graphing calculators or computer software are highly useful teaching tools, their use requires adherence to procedures and routines. Evaluate student acquisition of both procedures and concepts by assigning individual presentations to the class. In the presentations, students must demonstrate their knowledge of procedures and routines, and be able to explain concepts. Calculators or computer screens that can be projected for the whole class are particularly useful.
• Evaluate the Lesson 3 Exit Ticket problems 2 and 3 for the solutions that correspond to the correct ordered pairs for each equation. For the graphs, use a more global assessment: approximately correct slopes and x- and y-intercepts, and approximate point of intersection of the two graphs (29, 17).

Introduce systems of linear equations as a new topic. Ask, “What is a system of linear equations? A system of linear equations is a set of two or more linear equations that can be solved to find one particular value for a variable. Let’s look at some examples.” Write these on the board: [IS.4 - Struggling Learners]

The three systems shown can be solved for either the x- or y-variable through substitution, elimination, and graphing. The focus of this lesson is using the graphical approach to find the solution to systems of linear equations. “Now, let’s work on translating word problems into systems of linear equations and solving them.”

Problem 1

“We will start with a classic system of equations problem, Pigs and Chickens:”

Anna has a certain number of pigs and chickens on her farm. In counting the number of heads and feet, she counts 36 heads and 126 feet. How many pigs and chickens are on her farm?

“This problem translates into the following system of equations:

We will use a graphical approach to solve the system. What will the solution look like?” (The solution will be the point of intersection of the two graphs.)

“We will enter each equation into the graphing calculator, rewriting each equation in slope-intercept form. We will simply substitute x and y for c and p. We will first solve for one variable. Doing so gives:

Substituting y for p and x for c, we have:

We will enter each equation into our graphing calculator or a graphing calculator such as GeoGebra. The point of intersection is difficult to discern, so a check with the table is needed. We know the point of intersection occurs somewhere near an
x-value of 8 or 9. We find the y-value to be the same for each line at the point (9, 27). Thus, (9, 27) is our exact point of intersection, which is the solution.”

“What does this tell us? Since x represented c (number of chickens) and y represented p (number of pigs), we know Anna has 9 chickens and 27 pigs on her farm.”

Problem 2

“Another real-world problem is one involving a total number of people at an event and a fee per admission, where the number of people for each admission type must be determined:

The price for children under 12 to attend a movie is $5.25. The price for adults (or children over 12) to attend a movie is $7.50. On Friday night, the manager records an income of $1473.75, for a total of 210 ticket sales. How many children and how many adult tickets were purchased?”

“This problem translates into the following system of equations: [IS.5 - Struggling Learners]

a + c = 210

7.50a + 5.25c = 1473.75

where a represents number of adults and c represents number of children. Again, the solution will be the point of intersection for the two lines.”

“We will enter each equation into the graphing calculator, rewriting each equation in slope-intercept form. We will substitute y and x for a and c, respectively. We will first solve for one variable. We will solve for a. Doing so gives:

Substituting y for a, and x for c, we have: [IS.6 - Struggling Learners]

We will enter each equation into our graphing calculator or a graphing calculator such as GeoGebra. The point of intersection is difficult to discern, so a check with the table is needed. We know the point of intersection occurs somewhere near an
x-value of 43 to 46. Using the table, we find the y-value to be the same for each line at the point (45, 165). Thus, (45, 165) is our exact point of intersection, which is the solution.”

“What does this tell us? Since x represented c (number of children) and y represented a (number of adults), we know 45 children tickets and 165 adult tickets were sold on Friday night.”

Problem 3

“Sometimes, the graph gives us an approximation of the intersection point, and even the table, when set to the default setting, does not provide the exact answer we need. In this case, we simply change the table settings. Look at the following example:

On Monday, Elizabeth purchased a mixture of 5 apples and 4 bananas for $4.58. On Tuesday, she purchased 2 apples and 3 bananas for $2.49. What was the cost of one apple? Cost of one banana?”

“This problem translates into the following system of equations:

where a represents the cost of one apple and b represents the cost of one banana.”

“The solution will be the point of intersection for the two lines.”

“We will enter each equation into the graphing calculator, rewriting each equation in slope-intercept form. We will simply substitute y and x for a and b. Therefore, x will represent the cost of one apple and y will represent the cost of one banana. We will first solve for one variable. Doing so gives:

Substituting y for a and x for b, we have:

We will enter each equation into our graphing calculator or a graphing calculator such as GeoGebra. The graph of the two equations is shown below.”

“The point of intersection cannot be discerned from the graph alone, so a check with the table is needed. The default setting in the program for the increment by which x increases is usually equal to 1. However, for these two equations, the default setting of 1 for each subsequent x-value does not provide the level of exactness we need in order to see the point of intersection. We can modify the table setting by making the following adjustment to the program:”

1. Press 2^nd Window (TBLSET). [IS.7 - All Students]
2. Change the interval ΔTbl to another value. [IS.8 - All Students]

Explain, “‘Change ΔTbl’ simply means the change in x-values or the increment value for x. Since we are working with money amounts to the hundredths place, we will change our x-value increment to .01.”

“Using the table, we find the y-value to be the same for each line at the point (0.47, 0.54), where x represents the cost per banana and y represents the cost per apple. So each apple cost $0.54, and each banana cost $0.47.” Please show as many examples as needed.

Students should complete the Lesson 3 Exit Ticket as an activity (M-A1-3-3_Lesson 3 Exit Ticket and KEY.docx).

Review Activity

Design five linear-systems-of-equations word problems. Prepare a solution key, complete with a graph and any table processes used. Use PowerPoint to present the word problems. Use a separate “answer key and solution” PowerPoint to present the solutions. Each PowerPoint will be swapped with a classmate, using a random numbering process. Every student will solve five different word problems. Solutions will be compared to the author’s processes and solutions. Time for discussion and questions will be provided. [IS.9 - Struggling Learners]

To review the lesson, have a class discussion on any questions, concerns, or problems encountered during the lesson. In addition, ask students to create a list of at least ten “scenarios” where systems of linear equations are needed to solve a problem. For example, students might provide the example of needing to know how many red rose bushes and yellow rose bushes were purchased and the amount of money spent per rose bush type given a total number of rose bushes and money spent.

Extension:

• Ask students to make a list of consumer products they are familiar with, including an approximation of the cost of the product as they can best remember. Assign them to research the sales data for quarterly or yearly intervals and draw conclusions about relationships they are able to recognize between changes in the cost of each item in relation to its sales.

(Economics, as an academic discipline, teaches the law of supply and demand, which states that as the price of an item or a service increases, the demand for purchasing the item or service decreases, and correspondingly, as the price of an item or service decreases, the demand for purchasing the item or service increases. The law is also stated in reverse: as the demand for purchasing an item or service increases, the price of the item or service increases, and as the demand for purchasing the item decreases, the price for purchasing the item decreases. The dynamic system at work here is that consumers will buy more when the cost is low, buy less when the cost is high, and producers will sell more when the cost is low and sell less when the cost is high. The supply and demand model is crude, but it approximates the behavior of large numbers of individuals buying and selling large numbers of goods and services.)

• Assign students to make lists of other factors (variables) that affect the purchasing and selling of goods and services. A reasonable starting point can be asking whether or not buying certain categories of items is optional or mandatory. (Answers will vary, but sample answers include: needs such as food, clothing, housing, medical expenses; advertising; regulating laws; seasonal buying.)