Functions with Multiple Parts
These materials, when encountered before Algebra 1, Unit 4, Lesson 12 support success in that lesson.
12.1: Notice and Wonder: Ticket Price (10 minutes)
The purpose of this warm-up is to elicit the idea that boundary conditions on intervals are important, which will be useful when students write domain intervals for piecewise functions in a later activity. While students may notice and wonder many things about this table, the price of a ticket for people on the boundary between 2 intervals are the important discussion points.
Display the table 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.
What do you notice? What do you wonder?
|55 and up||$5.00|
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 table. 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 cost of a ticket for people whose ages are on the boundary of the age ranges does not come up during the conversation, ask students to discuss this idea.
12.2: Group Ticket Cost (15 minutes)
In this activity, students use a step function to determine the price of tickets for groups composed of people in different age groups. In the associated Algebra 1 lesson, students examine piecewise functions and their graphs. Although this activity does not mention piecewise functions, understanding how to compute the ticket price for the group will help students think about this kind of function.
A community orchestra charges different amounts for tickets to shows based on the age of the person attending. A sign in front of the box office where tickets are sold shows the prices.
|55 and up||$6.00|
- How much does each group need to pay for their tickets?
- 2 adults aged 40 and 36, and 2 kids aged 4 and 1
- 3 adults aged 74, 37, and 36
- 5 adults in their 30s and 25 students aged 15 and 16
- 1 adult aged 25 and 4 kids aged 1, 9, 13, and 16
- A mother arrives and tells the box office clerk that her child is 35 months old. How much should the clerk charge for the child?
- If there is a rule that uses the age of a person attending the orchestra concert as the input and outputs the ticket price for that person, is that rule a function? Explain your reasoning.
- What is the domain for the rule?
- What is the range for the rule?
The purpose of the discussion is to get students thinking about functions for which there are different rules for different domains. Select students to share their solutions. Ask students,
- “Is there an equation for the function that connects the input and output?” (There is not a single, nice equation that works for all the ages.)
- “Are there people for whom the price of a ticket would be confusing or unknown?” (No. The convention for ages is that if someone is turning 14 tomorrow, they will still be charged the $4 price since they are still currently 13. So, everyone should fit into one of the categories listed.)
- “If we graphed the information in the table, can you visualize what that graph would look like?” (There would be horizontal lines for each range of ages to look a little like steps.)
12.3: A Light Trip (15 minutes)
In this activity students analyze graphs representing situations to select the intervals within the domain in which certain events are happening. In the associated Algebra 1 lesson students examine piecewise functions and use the function notation that includes domain restrictions. Students are supported by examining scenarios and noting what is happening on different parts of the domain.
Noah leaves his home, sometimes running, sometimes walking, sometimes stopping until he remembers that he doesn’t have his wallet, then he goes back home. A graph representing his journey is shown in the graph.
- Describe what is happening on the domain \(210 < x < 400\).
- What are the domain intervals that represent the times when Noah was running?
- What are the domain intervals that represent the times when Noah was stopped?
- What are the domain intervals that represent the times when Noah was walking away from home?
- The amount of light in a room is shown as a function of the number of hours after midnight. Describe what might be happening in the room. Be sure to use intervals within the domain in your description.
The purpose of the discussion is to recognize that using the domain is important when describing graphs of situations where different things may be happening. Select students to share their solutions. If students describe the domain for the light levels in words (like “between 9am and 4pm”), ask students how they might write that as an inequality.