9.1: Notice and Wonder: What Do You See? (5 minutes)
The purpose of this warm-up is to elicit the idea that graphs can be discrete or continuous, which will be useful when students interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson. While students may notice and wonder many things about these graphs, the fact that one is a line and one is a discrete set of points are the important discussion points.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Display the table and the 2 graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Here is a table of values of data that was collected.
Here are two graphs of the data. What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the graphs. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the situation represented does not come up during the conversation, ask students to discuss this idea. What are some situations that might be appropriate to represent with just points, versus a connected line? (Students will see many examples in this lesson, so it’s not necessary to dwell on this question for long.)
9.2: Connect . . . or Not (20 minutes)
The purpose of this activity is to elicit the idea that graphs can be discrete or continuous based on the context, which will be useful when students interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson. When students pay close attention to the appropriate domain and range, both restricted by the context, they are engaging in work that is important in modeling with mathematics (MP4).
Arrange the students in groups of 4. Each member of the group should choose one of the questions and work individually. After quiet work time, ask students to share their responses within their group and decide if they are correct. Follow with a whole-class discussion.
In order to find the table of values, students may choose to use graphing technology strategically (MP5), if available.
Here are descriptions of relationships between quantities.
- Make a table of at least 5 pairs of values that represent the relationship.
- Plot the points. Label the axes of the graph.
Should the points be connected? Are there any input or output values that don’t make sense? Explain.
A cab charges \$1.50 per mile plus \$3.50 for entering the cab. The cost of the ride is a function of the miles, \(m\), ridden and is defined by \(c(m)=1.50m+3.50\).
The admission to the state park is \$5.00 per vehicle plus \$1.50 per passenger. The total admission for one vehicle is a function of the number of passengers, \(p\), defined by the equation \(a(p) = 5 + 1.50p\).
A new species of mice is introduced to an island, and the number of mice is a function of the time in months, \(t\), since they were introduced. The number of mice is represented by the model \(b(t)=16 \boldcdot (1.5)^t\).
When you fold a piece of paper in half, the visible area of the paper gets halved. The area is a function of number of folds, \(n\), and is defined by \(A(n)=93.5\left(\frac12\right)^n\).
The goal of this activity is for students to interpret functions presented in a context and represent the function in a table and with a graph. The focus is on making sense of whether the graph should be discrete or continuous. If it would be helpful to have words that refer to the ideas, introduce the terms continuous and discrete. (Students won’t be assessed on knowing these terms.)
9.3: Thinking Like a Modeler (15 minutes)
The purpose of this activity is to show how a context can impose restrictions on the domain and range of a function. Given a description of a context, students determine the restrictions on the domain and range imposed by the context. Considering how a context restricts the domain or range of a function modeling it is an important part of mathematical modeling (MP4). This prepares students to interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson.
Remind students that domain refers to possible values of the independent variable, and range refers to possible values of the dependent variable. Use a few examples from the previous activity to illustrate. For example, for the cab ride, the domain is all numbers that are greater than 0, and the range is all numbers that are greater than 3.5. This presumes that a ride must have some distance (it can’t be 0 miles long), and can be any number of miles long. A modeler might decide not to consider any cab rides that are longer than, say, 100 miles, in which case the domain would be all numbers between 0 and 100. Sometimes restrictions on the domain are a decision made by the modeler. In contrast, for the function modeling the admission price to the park based on number of people, the domain only contains whole, positive numbers, since the number of people must be a whole, positive number.
Arrange students in groups of 2. After a few minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they look different. Follow with a whole-class discussion.
To make sense in a given context, many functions need restrictions on the domain and range. For each description of a function
- describe the domain and range
- describe what its graph would look like (separate dots, or connected?)
- weight of a puppy as a function of time
- number of winter coats sold in a store as a function of temperature outside
- number of books in a library as a function of number of people who live in the community the library serves
- height of water in a tank as a function of volume of water in the tank
- amount of oxygen in the atmosphere as a function of elevation above or below sea level
- thickness of a folded piece of paper as a function of number of folds
Much discussion takes place between partners. Invite students to share how they determined the domain and range of the function representing the context. Ask if other groups have different responses. Choose as many different groups as time allows. Attend to the language that students use to describe their pairing, giving them opportunities to describe their relationships more precisely.