Lesson 9

The purpose of this warm-up is for students to reason about two situations that can be represented with linear equations. Since the number of babysitting hours determines which situation would be most profitable, there is no one correct answer to the question. Students are asked to explain their reasoning.

Give students 2 minutes of quiet work time followed by a whole-class discussion. 

Poll the class on which situation they would choose. Invite students from each side to explain their reasoning. Record and display these ideas for all to see. If no one reasoned about babysitting for less than 5 hours and therefore chose the second option, mention this idea to students.

Students may not use linear equations or graphs to decide which situation they would choose. If there is time, ask students for the equation and graph we could use to model each scenario.

The goal of this activity, and the two that follow, is for students to solve an equation in a real-world context while previewing some future work solving systems of equations. Here, students first make sense of the situation using a table of values describing the water heights of two tanks and then use the table to estimate when the water heights are equal. A key point in this activity is the next step: taking two expressions representing the water heights in two different tanks for a given time and recognizing that the equation created by setting the two expressions equal to one another has a solution that is the value for time, \(t\), when the water heights are equal.

Give students 2–3 minutes to read the context and answer the first problem. Select students to share their answer with the class, choosing students with different representations of the situation if possible. Give 3–4 minutes for the remaining problems followed by a whole-class discussion.

The purpose of this discussion is to elicit student thinking about why setting the two expressions in the task statement equal to one another is both possible and a way to solve the final problem.

Consider asking the following questions:

• “What does \(t\) represent in the first expression? The second?” (In each expression, \(t\) is the time in minutes since the tank’s water level started being recorded.)
• “After we substitute a time in for \(t\) and simplify one of the expressions to be a single number, what does that number represent? What units does it have?” (The number represents the amount of liters in the water tank.)
• “How accurate was your estimate about the water heights using the table?” (My estimate was within a few minutes of the actual answer.)
• “If you didn’t know which expression in the last problem belonged to which tank, how could you figure it out?” (One of the expressions is increasing as \(t\) increased, which means it must be Tank 1. The other is decreasing as \(t\) increases, so it must be Tank 2.)
• “How did you find the time the two water heights were equal using the expressions?” (Since each expression gives the height for a specific time, \(t\), and we want to know when the heights are equal, I set the two expressions equal to each other and then solved for the \(t\)-value that made the new equation true.)

In this activity, students work with two expressions that represent the travel time of an elevator to a specific height. As with the previous activity, the goal is for students to work within a real-world context to understand taking two separate expressions and setting them equal to one another as a way to determine more information about the context.

Give students 1 minute to read the context and the problems. You may wish to share with the class that programming elevators in buildings to best meet the demands of the people in the building can be a complicated task depending on the number of floors in a building, the number of people, and the number of elevators. For example, many large buildings in cities have elevators programmed to stay near the ground floor in the morning when employees are arriving and then stay on higher floors in the afternoon when employees leave work.

Arrange students in groups of 2. Give 2–3 minutes of quiet work time for the first two question and then ask students to pause and discuss their solutions with their partner. Give 3–4 minutes for partners to work on the remaining questions followed by a whole-class discussion.

Students may mix up height and time while working with these expressions. For example, they may think that at \(h = 0\), the height of the elevators is 16 meters and 12 meters, respectively, instead of the correct interpretation that the elevators reach a height of 0 meters at 16 seconds and 12 seconds, respectively.

This discussion should focus on the act of setting the two expressions equal and what that means in the context of the situation.

Consider asking the following questions:

• “If someone thought that the height of Elevator A before we started timing was 16 meters because they substituted 0 for the variable of the expression \(0.8h+16\) and got 16, how would you help them correct their answer?” (I would remind them that \(h\) is height and \(0.8h+16\) is the time, so when we start timing at 0 that means \(0.8h+16=0\), not that \(h=0\).)
• “Which of the elevators in the image is A and which is B? How do you know?” (A is the elevator on the right since at time 0 the height is negative, while B is the elevator on the left since at time 0 the height is positive.)
• “How did you find the height when the travel times are equal?” (Since each expression gives the time to travel to a height \(h\), I solved the equation \(0.8h+16=\text- 0.8h+12\), which gives the value of \(h\) when the two expressions are equal.)