The purpose of this Warm-up is for students to reason about two situations that can be represented with linear equations. Because 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.
Launch
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Activity
None
Student Lesson in Spanish
If you were babysitting, would you rather
Charge \$5 for the first hour and \$8 for each additional hour?
Or
Charge \$15 for the first hour and \$6 for each additional hour?
Explain your reasoning.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
Survey the class to determine 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.
If students do not use linear equations or graphs to choose a situation, and there is time, ask students for the equation and graph that could be used to model each situation.
The goal of this activity 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, , when the water heights are equal.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
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.
To introduce the context, tell students that many areas have water towers or tanks that are used to increase the water pressure of plumbing in the area. In some large cities, they can be seen on rooftops, and in other areas they stand alone, often with the town’s name written on the side.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe what is happening in each tank. Display words and phrases such as “rate of change,” “initial amount,” and “differences in rates.”
Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams. If students are unsure where to begin, suggest that they draw a diagram to help illustrate the information provided.
Activity
None
The amount of water in two tanks is recorded every 5 minutes in the table.
Describe what is happening in each tank. You can draw a picture, say it verbally, or write a few sentences.
time (minutes)
tank 1 (liters)
tank 2 (liters)
0
25
1000
5
175
900
10
325
800
15
475
700
20
625
600
25
775
500
30
925
400
35
1075
300
40
1225
200
45
1375
100
50
1525
0
Use the table to estimate when the tanks will have the same amount of water.
The amount of water (in liters) in Tank 1 after minutes is . The amount of water (in liters) in Tank 2 after minutes is . When is the amount of water in the 2 tanks equal?
Activity Synthesis
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.
Direct students’ attention to the reference created using Collect and Display. Ask students to explain how the expressions in the third question are related to what they noticed was happening. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond.
Consider asking:
“What does represent in the first expression? The second?” (In each expression, is the time in minutes since the tank’s water level started being recorded.)
“After we substitute a time in for 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.)
“How do you know from the expressions that there is only 1 value for that makes true?” (The coefficients of on each side are different.)
“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 increased, which means it must be Tank 1. The other is decreasing as increases, so it must be Tank 2.)
“How did you find the time at which the two water heights were equal using the expressions?” (Because 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 (MP2).
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Tell students to close their books or devices (or to keep them closed).
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the image, without revealing the questions.
For the first read, read the problem aloud, and then ask, “What is this situation about?” (elevators going up and down). Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured (the equations could be used to find the speed and initial heights).
After the third read, reveal the question: “When do the elevators pass one another?” and ask, “What are some ways that we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read (set the equations equal to one another to get the height when they pass, and then substitute that into the equation to get the time).
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 to 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 2 questions, 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, and follow that with a whole-class discussion.
Activity
None
A building has two elevators that both go above and below ground.
At a certain time of day, the travel time, in seconds, that it takes Elevator A to reach height in meters is given by the equation seconds.
The travel time for Elevator B is given by the equation .
Two elevators. One elevator is above a line labeled ground level with an arrow pointing down. Another elevator is below a line labeled ground level with an arrow pointing up.
What is the height of each elevator at this time?
How long does it take each elevator to reach ground level at this time?
If the two elevators travel toward one another, at what height do they pass each other? How long does it take?
Student Response
Loading...
Building on Student Thinking
Students may mix up height and time while working with these expressions. For example, they may think that at , 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. Ask them to explain in their own words what represents and then what represents, including using units, to help their understanding.
Activity Synthesis
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:
“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 and got 16, how would you help them correct their answer?” (I would remind them that is height and is the time, so when we start timing at 0 that means , not that .)
“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 know there would be only 1 time when the elevators are at the same height?” (After setting up the equation, the coefficients on each side of the equation are different.)
“Describe a situation in which the 2 elevators are never at the same height at the same time. What would their equations look like?” (If Elevator B is a little higher than Elevator A and they both move at the same rate. For example and .)
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code the different elevators to match the equations representing them. Supports accessibility for: Visual-Spatial Processing
Lesson Synthesis
Arrange students in groups of 2. Ask partners to think of other situations in which two quantities are changing and they want to know when the quantities are equal. Give groups time to discuss and write down a few sentences explaining their situation. Invite groups to share their situation with the class. (For example, in a race where participants walk at steady rates but the slower person has a head start, when will they meet?) Consider allowing groups to share their situation by drawing a picture, making a graph, explaining in words, or acting it out.
Then ask students if they could modify their situations so that there are either infinitely many or no solutions. (For example, in the same race, the person with the head start walks at the same pace as the person who didn’t get a head start.)
Student Lesson Summary
Imagine a full 1,500 liter water tank that springs a leak, losing 2 liters per minute. We could represent the number of liters left in the tank with the expression , where represents the number of minutes the tank has been leaking.
Now imagine at the same time, a second tank has 300 liters and is being filled at a rate of 6 liters per minute. We could represent the amount of water in liters in this second tank with the expression , where represents the number of minutes that have passed.
Since one tank is losing water and the other is gaining water, at some point they will have the same amount of water—but when? Asking when the two tanks have the same number of liters is the same as asking when (the number of liters in the first tank after minutes) is equal to (the number of liters in the second tank after minutes),
Solving for gives us minutes. So after 150 minutes, the number of liters of the first tank is equal to the number of liters of the second tank. We can check our answer and find the number of liters in each tank by substituting 150 for in the original expressions.
Using the expression for the first tank, we get which is equal to , or 1,200 liters.
If we use the expression for the second tank, we get , or just , which is also 1,200 liters. That means that after 150 minutes, each tank has 1,200 liters.
Standards Alignment
Building On
Addressing
8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
