Lesson 25

Students write expressions and equations representing total cost. The purpose of this activity is to support students in writing the equation for the “Ordering Fish” activity.

Arrange students in groups of 2. Display questions for all to see. Give 2 minutes quiet think time, followed by 2 minutes partner discussion then whole-class discussion.

Some students may not be sure how to approach writing the scenario as an equation. For these students, suggest that they make a table of possible prices based on the amount of fruit purchased.

The purpose of this discussion is to have students explain strategies for writing equations for real-world scenarios. Ask students to share what they discussed with their partners by asking:

• “What did each of the variables mean in the situations?” (Since we want a price based on the number of pounds of fruit, \(b\) and \(g\) represent the amount of bananas and grapes purchased. For the savings account, the \(x\) was the number of weeks, and the \(y\) was number of dollars.)
• “Was the slope for each of these equations positive or negative? Why does that make sense with the scenario?” (For the fruit, the slope was negative, which makes sense because if you buy more of one fruit, you have to buy less of the other. For the savings account, the slope is positive, which makes sense because the more weeks go by, the more money will be in the account.)

Given a graph with five lines representing changes in bank account balance over time, students write equations and interpret how points represent solutions. The activity also connects to and contextualizes students’ prior understanding of slope and intercepts, and lays the foundation for the coming unit on systems of equations by considering what points of intersection of lines and non-intersecting lines represent.

Display the image from the lesson for all to see and ask the students to consider line \(a\). Invite 2–3 students to describe in words what line \(a\) shows. If no students bring it up, tell students that they saw this line before in the warm-up, and they wrote an equation for it. Instruct students that for #1, they should not choose line \(a\).

Arrange students in groups of 3–4. Groups work for about 10 minutes, followed by a whole-class discussion.

Students should understand that points on a line show solutions to the equation of the line. Discuss with students:

• “What can we say about the points where two lines cross?” (The accounts had the same amount of money at the same time.)
• “How do the slopes of the lines help to tell the story from the graph?” (The slope tells us whether a person is spending or saving each week.)
• “What does your answer to question 3 tell us about their rates of saving?” (By knowing the value of the slope, we can compare who is spending or saving more quickly or more slowly.)

Students represent a scenario with an equation and use the equation to find solutions. They create a graph (either with a table of values or by using two intercepts), interpret points on the graph, and interpret points not on the graph (MP2).

Allow about 10 minutes quiet think time for questions 1 through 4, then have students work with a partner to discuss questions 4 and 5. Look for students who define the variables or label the axes differently. This can be an opportunity to discuss the importance of defining what quantities your variables represent and that different graphs can represent the same information.

If students let \(x\) be pounds of salmon, then the equation would be \(5x + 3y = 210\) and the coordinates would be reversed. The intercepts of the graph would be \((0,70)\) and \((42,0)\). This is a good place to mention the importance of defining what quantities your variables represent.

Ask students to share some strategies for graphing and features of their graphs. Invite 2–3 students to display their graphs. Consider asking:

• "Was the slope of the line positive or negative? Why does that makes sense in this situation?" (In this relationship, as one quantity increases the other must decrease in order to keep the sum \(3x + 5y\) constant.)
• "What was your strategy for graphing the relationship?" (Plotting points from question 1, using the table, figuring out the intercepts and connecting the line)
• "Why does it make sense for the graph to be only in quadrant I?" (You cannot purchase a negative amount of fish, so the \(x\) and \(y\) values cannot be negative.)
• "How is this situation different from the apples and oranges problem in a previous lesson?" (Buying \(\frac12\) pound of fish is reasonable while buying \(\frac12\) of an apple probably is not. )

Briefly reiterate key concepts: If a point is not on the graph of the equation then it is not a solution. The ordered pairs that are solutions to the equation all make the equation true and are all found on the line that is the graph of the equation.

A discussion could also include the detail that orders that are less than $210 can also be considered to work, because there is money left over. That gives an opportunity to discuss the shape of the graph of \(3x + 5y \le 210\).