# Lesson 9

When Are They the Same?

## 9.1: Which Would You Choose? (5 minutes)

### Warm-up

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.

### Launch

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

### Student Facing

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

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### Activity Synthesis

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.

## 9.2: Water Tanks (10 minutes)

### Activity

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.

### 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.

*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.

*Supports accessibility for: Conceptual processing; Visual-spatial processing*

*Representing, Conversing: MLR2 Collect and Display.*To begin the whole-class discussion, give students the opportunity to discuss their solutions to the first question in groups of 3–4. Circulate through the groups and record language students use to describe what is happening in each tank. Listen for language related to rate of change, differences between rates, initial amounts of water, etc. If groups are stuck, consider asking, “What happens to the amount of water in Tank 1 (or 2) as time goes on?” and “Can you draw a picture of Tank 1 (or 2) at 5 minutes and then another picture at 10 minutes? What do you notice?” Post the collected language in the front of the room so that students can refer to it throughout the rest of the activity and lesson. This will help students talk about the relationship between the two tanks prior to being asked to find the time when they are equal.

*Design Principle(s); Maximize meta-awareness*

### Student Facing

The amount of water in two tanks every 5 minutes is shown in the table.

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 |

- Describe what is happening in each tank. Either draw a picture, say it verbally, or write a few sentences.
- 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 \(t\) minutes is \(30t + 25\). The amount of water (in liters) in tank 2 after \(t\) minutes is \( \text-20t + 1000\). Find the time when the amount of water will be equal.

### Student Response

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### 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.

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.)

## 9.3: Elevators (15 minutes)

### Activity

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.

### Launch

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.

*Reading, Representing: MLR6 Three Reads.*Use this routine to support reading comprehension of this word problem, without solving it for students. Display only the diagram and the description of the situation, without revealing the questions. In the first read, students to read the problem with the goal of comprehending the situation (e.g., A building has two elevators that go above and below ground.). Use the second read to identify important quantities without yet focusing on specific values. Listen for, and amplify, the important quantities that vary in relation to each other in this situation: height of elevator A, in meters; height of elevator B, in meters; elapsed time, in seconds. For the third read, reveal the questions and ask students to brainstorm possible ways to use the given expressions to determine when the elevators will reach ground level at the same time. This will help students connect the language in the word problem and the reasoning needed to solve the problem.

*Design Principle(s): Support sense-making*

### Student Facing

A building has two elevators that both go above and below ground.

At a certain time of day, the travel time it takes elevator A to reach height \(h\) in meters is \(0.8 h + 16\) seconds.

The travel time it takes elevator B to reach height \(h\) in meters is \(\text-0.8 h + 12\) seconds.

- What is the height of each elevator at this time?
- How long would 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 would it take?
- If you are on an underground parking level 14 meters below ground, which elevator would reach you first?

### Student Response

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### Student Facing

#### Are you ready for more?

- In a two-digit number, the ones digit is twice the tens digit. If the digits are reversed, the new number is 36 more than the original number. Find the number.
- The sum of the digits of a two-digit number is 11. If the digits are reversed, the new number is 45 less than the original number. Find the number.
- The sum of the digits in a two-digit number is 8. The value of the number is 4 less than 5 times the ones digit. Find the number.

### Student Response

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### Anticipated Misconceptions

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.

### 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 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.)

*Representation: Internalize Comprehension.*Use color and annotations to illustrate student thinking. As students describe their strategies, use color and annotations to scribe their thinking on a display of the problem so that it is visible for all students.

*Supports accessibility for: Visual-spatial processing; Conceptual processing*

## Lesson Synthesis

### Lesson Synthesis

The work in this lesson is a prelude to a simple form of a system of equations, where each equation can be written in the form \(y=\) some expression (though students do not need to know the term "system of equations" at this point).

Arrange students in groups of 2. Ask partners to think of another situations where two quantities are changing and they want to know when the quantities are equal. Give groups time to 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 making a picture, a graph, in words, or by acting it out.

## 9.4: Cool-down - Printers and Ink (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

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 \(\text-2x+1,\!500\), where \(x\) 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 \(6x+300\), where \(x\) 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 \(\text-2x+1,\!500\) (the number of liters in the first tank after \(x\) minutes) is equal to \(6x+300\) (the number of liters in the second tank after \(x\) minutes),

\(\text-2x+1,\!500=6x+300.\)

Solving for \(x\) gives us \(x=150\) minutes. So after 150 minutes, the number of liters of the first tank is equal to the number of liters of the second tank. But how much water is actually in each tank at that time? Since both tanks have the same number of liters after 150 minutes, we could substitute \(x=150\) minutes into either expression.

Using the expression for the first tank, we get \(\text-2(150)+1,\!500\) which is equal to \(\text-300+1,\!500\), or 1,200 liters.

If we use the expression for the second tank, we get \(6(150)+300\), or just \(900+300\), which is also 1,200 liters. That means that after 150 minutes, each tank has 1,200 liters.