• Observe students during the three activities, monitoring how much each student is contributing to the group. Ask individual students to describe their own contributions and also how each contribution advanced the group’s understanding.
• Observe students’ performance on detailing the steps necessary to solve problems of this type. By carefully examining student responses during the self-evaluation and expression of their understanding of the lesson, educators can derive a solid grasp of where gaps exist in a student’s (and class’s) understanding.

Pose the following question to students: “Suppose that you have $20 to spend and you can purchase chocolate candy for $5 a pound or fruit candy for $1 a pound. What do you purchase and why?” Allow students to offer possibilities for what they can purchase and their reasons (for example, 4 pounds of chocolate candy because I don’t like fruit candy.) After allowing students a few minutes to explore the problem, point out that there are numerous combinations of chocolate and fruit candy that they can purchase. Also, remind students that they can purchase candy in a half-pound, quarter-pound, or any other fraction of a pound they like.

Ask students, “Is there a particular number of different combinations of chocolate and fruit candy you can buy, or are the possibilities endless?” After discussion that leads to the realization that there are an infinite number of possible combinations (with a “perfect” scale capable of measuring miniscule fractions of a pound), explain: “We will try to figure out if there’s a pattern to the solutions to these kinds of problems. This type of problem, in which people have various choices about how to spend their money, time, or energy, comes up frequently, and it’s important to be able to examine possible choices in order to make the best choice possible. As you came up with the different possibilities for how many pounds of each kind of candy to purchase, you also noticed that your personal preferences came into play. It’s important to consider these when examining your options in situations like these.”

Note: The activities in this lesson begin with a large amount of guidance and simply explore the various solutions to a problem, seeing what the solutions look like when graphed. The activities then move into a more abstract and slightly more self-guided realm, having students work with the realization that the solutions, since there are an infinite number of them, are best represented as points on a line (or line segment).

Activity 1 (Small groups)

This activity continues with other candy combinations: “Suppose that you have $20 to spend and you can purchase chocolate candy for $5 a pound or fruit candy for $1 a pound. What do you purchase and why?”

On the board or on an overhead, draw the first quadrant of a Cartesian plane and label the axes by ones. [IS.4 - Struggling Learners]

Divide students into groups of three or four each, and ask them, “How many pounds can you get if you decide to just buy chocolate candy?” (4 pounds)

Draw a point on the Cartesian plane at (4, 0) and explain to students that this point represents 4 pounds of chocolate candy and 0 pounds of fruit candy.

• “How many pounds of fruit candy can you get if you decide to buy 1.5 pounds of chocolate candy?” (12.5 pounds)

Guide the groups through the solution, showing them how to determine the answer, as needed. Then ask,

• “How much for one pound of chocolate candy?” ($5)
• “How do you find out how much for 1.5 pounds of candy?” (multiply 1.5 by $5 = $7.50)

If students show difficulty working with this concept, ask them how much for 2 or 3 pounds of candy and how they got their answer. The realization that total cost is number of units (lbs) times unit cost ($/lb) is a key to setting up the problem; [IS.5 - Struggling Learners] all students need to grasp this concept clearly to understand the rest of the lesson. Ask further,

• “After buying 1.5 pounds of candy, how much money do you have left from your original $20?” ($12.50)
• “How many pounds of fruit candy can you buy with $12.50?” (12.5)

Again, make sure that students understand the process to arrive at this answer: Divide the money you have by the unit cost.

Once all students understand how to arrive at various solutions for this situation, have each group record four different solutions to the problem, clearly noting how many pounds of each kind of candy the group would purchase and also representing their solutions as ordered pairs of the form (x, y) where x represents the pounds of chocolate candy and y represents the pounds of fruit candy.

Instruct each student group to make sure students have at least one solution where they buy just whole numbers of pounds, and at least one solution where they buy fractional amounts of each candy. Allow students sufficient time to work on the multiple solutions; while they are working, check with each group to answer any questions students have and spot-check that they are doing it correctly.

A Sample Table for Activity 1

Once all groups finish, ask each group to select a spokesperson. Each group should make sure its spokesperson has all the ordered pairs that the group decided upon. Have each spokesperson come to the board or overhead projector and choose one of his/her group’s solutions to plot on the Cartesian plane. Rotate through all the spokespersons, with each one plotting a point to represent one of the solutions his/her group came up with.

Once the spokesperson’s solutions are represented (either because s/he graphed it or another group did), the spokesperson can sit down. Once all the group solutions have been plotted, ask the class, “Do the points we have graphed represent all the solutions?” (no) Then ask, “What do you notice about all the solutions we graphed?”

Students should note that if you connect them all, you get a line segment. Ask students, “Do you get a line or a line segment?” Make sure students realize that the solutions do not go on forever because you only have $20. Connect the points with a line segment that goes from (4, 0) to (0, 20). Ask students,

• “Does this line segment represent all the solutions to the problem?” (yes)
• “How many solutions are there?” (an infinite number)

Guide the class in a discussion of whether, in reality, there are an infinite number of solutions or if there are real-world constraints that limit the number of actual solutions in real life. Possible constraints include the accuracy level of the scale and the fact that the smallest unit of currency we have is the penny, both of which make certain calculations impossible (or requiring rounding).

Graph of Money Spent on Chocolate and Fruit Candy, per pound

Activity 2 (same groups as Activity 1)

Begin this activity by posing a new problem to the class, providing each group with graph paper. Ask the class,

“Suppose that you have to read a total of 50 pages this week for your English class, but you get a choice between two different books. Book A is a little easier, and you can read 2 pages per minute. Book B is a little more difficult, and you can only read one-half page per minute. What are all the possible combinations of the books we can read?”

Begin by asking students to think about Book A:

• “How many pages in Book A can you read in 1 minute?” (2 pages)
• “How about in 6 minutes?” (12 pages)
• “How about in 20 minutes?” (40 pages)
• “How about in x minutes?” (2x pages)

If students show difficulty in understanding the leap to the concept of abstract x here, ask them how they got their previous answers, i.e., “How did you arrive at 40 pages for 20 minutes?” Students should recognize that they took the number of minutes and multiplied it by 2. [IS.6 - Struggling Learners] Use this realization to guide them toward the unit “2x pages” for x minutes. Write 2x on the board, and ask,

• “Now, let’s say you can read only  page in Book B in 1 minute. How many pages can you read in 6 minutes?” (3 pages)
• “How about in 20 minutes?” (10 pages)
• “How about in y minutes?” ( * y)

Activity 2, Book A and Book B Comparison

Book A		Book B
Minutes	Pages	Minutes	Pages
1	2	6	3
6	12	20	10
20	40
x	2x	y

Ask students, “How many total pages can we read if we read Book A for x minutes and Book B for y minutes?” Remind students that the word total implies addition, and we already have expressions for the pages read in each book.

Once students have derived , write it on the board, and then write an equal sign after it. If necessary, remind students of the original problem, and then ask them what goes on the other side of the equal sign. (50)

Number of Minutes to Read 50 Pages

Next, have the groups work on graphing  on their graph paper. Ask them to think of as many different methods to graph the line as possible. Remind them to be as accurate as possible. While groups are working, walk around and answer questions, providing hints to students; the two methods they should be using are making an xy-chart and putting the equation in slope-intercept form. Ask,

• “What does the graph of  look like?” (a line)

• “At what point does it start? At what point does it end?” [(0, 100) and (25, 0)]
• “What does the point (0, 100) represent in terms of our original problem?” (reading Book A for 0 minutes; reading Book B for 100 minutes)

“The next version of this graph shows only the interval between (0, 100) and (25, 0).”

If students have difficulty plotting the numbers on a graph, remind them to think about what the variables stood for (the number of minutes reading Book A and the number of minutes reading Book B, respectively). [IS.7 - Struggling Learners]

Note: Realistically, we are only interested in the segment from (0, 100) to (25, 0) because we aren’t looking at negative minutes spent reading either book. For example, if we read Book A for 26 minutes, we would need to read Book B for −4 minutes, according to our linear equation. This doesn’t make sense in the context of the problem. We are not constraining the minutes, but negative minutes do not make sense here. [IS.8 - Struggling Learners]

Ask students which point on the graph represents the solution they might choose—remind them that they do not have to choose one of the endpoints. Have a discussion about factors that might affect which solution is the best one for particular students—factors could be how interesting the book is, what the book is about, and so on.

Activity 3 (pairs)

Tell students that in this activity, they’re going to come up with their own problems, but before they do that, the class is going to look back at similar situations, and see what kinds of variables go into scenarios that may be familiar to them. If they are not familiar, it gives them an opportunity for a greater perspective.

Ask students to think about the two problems in Activity 1 and Activity 2 (the one about fruit/chocolate candy and the one about reading Book A and Book B). “What do the two previous problems have in common?” Guide students toward the recognition that in each problem, there are two variables (number of pounds of each candy and number of pages of each book read). Also, the variables should be something that can be broken down into fractional parts. Point out that letting the variables represent, for example, a number of people, changes the solution because we can’t divide people up into fractional parts. Ideal variables represent things like time and money. Also note the differences between the nature of the two variables in the previous example. The number of pages is a discrete quantity (see the definitions of discrete and continuous data in the glossary). The number of minutes is also a discrete number, though time can be thought of and considered as continuous. [IS.9 - Struggling Learners]

Also, students should recognize that each problem has a constraint—something that limits how much of each of the two options you can choose. Have students identify the constraints in each problem (the amount of money in the first problem, and amount of required pages in the second problem).

Have students work in pairs to develop their own problems. As students are working, walk around the classroom and ask students what their two variables are and what their constraints are. Encourage students to think about situations in their lives―they can think about sports, or different jobs they have, different things they can be, different ways they can spend time, etc. Students in each group should write the group’s word problem neatly on a sheet of paper.

Once each group has written a word problem, students should pass the word problem to a nearby group so each group has a new word problem.

“Before starting to set up the problem and graph the solution, your group should identify the three important parts of the problem: the two variable quantities and the constraint.” [IS.10 - Struggling Learners] Have each group clearly identify each of the three important parts of the problem and then write an equation that represents all the possible solutions to the problem.

Each group should then graph the solution for the problem the students were given and list the endpoints.

Students need to be reminded of where they were at the beginning of the lesson and what they learned through the lesson. By quickly reviewing the scope and sequence of the activities, students can reflect upon the progress they made and new skills they learned.

Have students think back to the problem about chocolate and fruit candy. “When we started with that problem, how many different possibilities did you think there were for choosing the candy?” Ask students how many possibilities there actually are (if we had a scale of infinite accuracy and could pay fractions of a penny): “What are the important parts of these kinds of problems?” (two variables and a constraint)

Then ask, “Why do the solutions end up being lines or line segments when we graph them?” Help students make a connection between the fact that there are two variables in the problem and a line can be represented by an equation with two variables. Ask students if any variables were squared or cubed, or if they were just multiplied by “regular” numbers. Remind students of the definition of linear functions that they have encountered earlier, and how we can identify linear functions (two variables, each to the first power, with special exceptions for lines like x = a and y = b).

Pose the following two problems to the class: “Suppose that we consider our original problem, for which chocolate candy is $5 per pound, fruit candy is $1 per pound, and we have $20 to spend. Also, consider our problem about reading books, but let’s change it a little. Suppose that we can read 5 pages per minute in Book A, 1 page per minute in Book B, and we only have 20 minutes to spend reading. In the first problem, we want to find the possible combinations of candy we can buy; in the second problem, we want to find the possible combinations of books we can read. How are these two problems similar and different?”

Students should note the obviously different situations, but realize that the problems are really exactly the same. Have students write what each variable represents for each problem, as well as the constraint for each problem. Then, have students write the equations to represent each situation (the first one was done earlier in the class). Ask, “What do you notice about the equations (and hence the solutions) for the two problems?”

Explain to students that this ability to generalize different problems and reduce them to their most basic elements is one of the most important ideas in mathematics and is one of the things that makes mathematics such a powerful tool—the ability to examine what appear to be very different situations using precisely the same tools.

When presented with new problem types, students need to not only be able to solve the problems, but have a clear plan of how to approach problems of the new type. In this section of the lesson, students are encouraged to look back and evaluate themselves, asking themselves how they solved each problem. Not only should they examine their own steps, but they should think about how they could improve their solution method and make sure that they included all the necessary information and steps.

Have students work in pairs and tell each group, “Describe the steps you should go through when presented with a problem that you think might have a solution that is represented by a line.” Make sure that student groups use the same scales for both x and y axes.

When each group has a clear, ordered list of steps, ask the class what the first step should be; work with the class to find a first step that everyone can agree upon. Proceed through the rest of the steps required to solve the problem until the class has a step-by-step guide to solving problems whose solutions are linear equations. A possible guide is:

1. Read the problem.
2. Identify the two variables and decide what letter will represent each quantity.
3. Identify the constraint.
4. Write an equation that relates the two variables and the constraint.
5. Graph the equation by putting it into slope-intercept form or by making an xy-chart.

Once students are done with this part of the lesson, hand out the Lesson 1 Exit Ticket (M-A1-4-1_Lesson 1 Exit Ticket.docx and M-A1-4-3_Lesson 3 Exit Ticket KEY.docx) for students to work on after class. Have students share and discuss answers or review outcomes before leaving the class.

Extension:

• Students can begin exploring concepts like slope and the y-intercept, as well as using algebra to convert their standard-form equation to slope-intercept form (or the other way around: students can write the slope-intercept form equation based on their graph, and then convert it to standard form).
• Students can work on a method to convert a problem situation directly into an equation in slope-intercept form to facilitate graphing.