# Lesson 2

Understanding Points in Situations

These materials, when encountered before Algebra 1, Unit 4, Lesson 2 support success in that lesson.

## 2.1: A Day of Temperature (10 minutes)

### Warm-up

In this warm-up, students interpret points on a labeled graph. In the associated Algebra 1 lesson, students are introduced to function notation and understanding points in situations. This activity prepares students by thinking about the values in a situation given a visual graph.

### Student Facing

The temperature for a city is a function of time after midnight. The graph shows the values on a particular spring day.

- What does the point on the graph where \(x = 15\) mean?
- What is the temperature at 5 p.m.?
- What is the hottest it gets on this day?
- What is the coldest it gets on this day?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to talk about inputs and outputs of a function in context. Select students to share their responses. Ask students:

- "What is the temperature at 5:30 a.m.?" (It is not shown on the graph, so I do not know for sure. I would guess that it is around 12 degrees Celsius, though.)
- "On the news later, they say that the high temperature for the city on this day was 22 degrees Celsius. Is that reasonable? Explain your reasoning." (I believe it could happen. Sometime between 1 p.m. and 4 p.m. the temperature might have been up to 22 degrees Celsius, but not on the hour when the temperature was recorded in the graph.)
- "Would you believe the news if they say that the low temperature for the city on this day was 4 degrees? Explain your reasoning." (I do not believe it because the graph only shows temperatures going as low as 12 degrees Celsius, so it is unlikely to get that much colder and get back up to 12 degrees within the same hour.)

## 2.2: What Happens to -2? (15 minutes)

### Activity

In this activity, students practice solving an equation after substituting in a value. In the associated Algebra 1 lesson, students will do a similar task using function notation.

### Student Facing

For each of these equations, find the value of \(y\) when \(x = \text{-}2\).

- \(y = 3x - 4\)
- \(y = 10 - 2x\)
- \(y = \frac{3}{2}x + 5\)
- \(y = 2(x - 1) + 4\)
- \(y = \text{-}x + 19\)
- \(y = \frac{x - 3}{8}\)
- \(y = 0.3x + 5\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to understand how to compute the value of a variable after substituting in a value for another variable. Select students to share their solutions. Ask students:

- “What is the order in which you did the operations?” (Find the value inside the parentheses first, then multiply or divide, then add or subtract.)
- “The solution to the first equation can be said as, ‘When \(x\) is -2, \(y\) is -10.’ How does this look in a graph?” (A point at position \((\text{-}2, \text{-}10)\).)
- “How would your work change if you wanted to know the value of \(x\) if you knew \(y = \text{-}2\)?” (\(y\) would be replaced by -2 and operations would need to be done to both sides of the equation to solve for \(x\).)

## 2.3: It’s Heating Up! (20 minutes)

### Activity

In this activity, students interpret a function in terms of a situation and examine some values that do not make sense in the situation. In the associated Algebra 1 lesson, students are introduced to function notation and interpret points in situations. Students are supported by this activity providing them a chance to do all of the same work without the function notation so that they may focus on the new notation in the Algebra 1 lesson.

### Student Facing

The temperature, in degrees Fahrenheit, of a scientific sample being warmed steadily as a function of time in seconds after the sample is put in a machine can be represented by the equation \(y = 2.1x + 86\).

- What does it mean when \(x = 2\)?
- What is the temperature in that situation?
- What does it mean when \(y = 122\)?
- A graph of this equation goes through the point \((60,212)\). What does that mean?
- Give 2 values for \(x\) that do not make sense. Explain your reasoning.
- Give 2 values for \(y\) that do not make sense. Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to understand values in terms of the situation and recognize values that do and do not make sense for each variable. Select students to share their responses. For the values that do not make sense, record the responses for all to see. After responses are recorded, ask students if any of them might make sense in some situations.

Tell students, “Sometimes, values do not make sense because of the specific situation. For example, in this situation, the sample getting to 75 degrees does not make sense since it starts at 86 degrees and gets hotter. Other times, there are values that do not make sense at all. For example, -460 degrees Fahrenheit does not ever make sense since it is below absolute zero and, scientifically, matter cannot exist at that temperature. When there are realistic situations, it is usually important to think about whether solutions are realistic themselves.”