Unit 6, Lesson 8
Representing Functions
Extra Support Materials for Algebra 1
8.1
Warm-up
Standards Alignment
Building On
Addressing
8.F.A.1
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.
Building Toward
HSF-IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function gives the number of person-hours it takes to assemble engines in a factory, then the positive integers would be an appropriate domain for the function.
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students analyze a situation to determine if it or its inverse is a function. The class works together to complete a table before analyzing the situation. As students make sense of the situation and apply their knowledge of functions, they make sense of problems and persevere in solving them (MP1).
Launch
Display a table in which one row contains some first names of students in your class. (If possible, select a few students you know were born in the same month.) There should be no ambiguity in the name row, so if more than one student shares the same first name, also write their last initial, or make the names distinct in another way. For example:
| name | Mai | Jada | Tyler | Diego |
|---|---|---|---|---|
| secret name |
Complete the table with some “secret names” (the month each student was born). Ensure that there is at least one duplicate month among the secret names. Then, invite students to read and complete the rest of the task.
Activity
NoneIn a secret club, everyone is known by the month they were born. So, Diego would be called “January,” and Tyler would be called “August.”
- What would be the name of some people in your class in the secret club?
- Why might club meetings get kind of confusing?
- Can you think of a better system for assigning club members new names?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to determine that the problem with deciding which “January” you are referring to is that the rule is not a function—the same input (month) can match to different outputs (people). Hopefully for the last question, rules were suggested where everyone gets a unique secret name. That way, a person is a function of a secret name.
Continue to display the table from the Launch.
| name | Mai | Jada | Tyler | Diego |
|---|---|---|---|---|
| secret name |
For any first name, is there only one possible secret name? Yes! Each person was born in only one month. The completed table might look something like:
| name | Mai | Jada | Tyler | Diego |
|---|---|---|---|---|
| secret name | January | May | August | January |
But let’s say you were trying to figure out which first name went with which month:
| name | ||||
|---|---|---|---|---|
| secret name | January | January | May | August |
We don’t know whether “January” refers to Mai or Diego. Therefore, a person’s first name is not a function of their secret name.
Here are some questions for discussion:
- “What does it mean for something to be a function?” (Each input results in only one output.)
- “Why is birth month a function of a person?” (Each person was born in only one month.)
- “Why is a person not a function of their birth month?” (Many people are born in the same month, so the input of a month (like January) can have multiple outputs (like Mai and Diego).)
Remind students that the input is sometimes called the “independent variable” and the output the “dependent variable.” We could say things like “A person’s secret name depends on their name” and “Secret name is a function of name.”
8.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
Students continue their review of functions in this activity. First, students examine several examples of functions and non-functions from the perspective of “Can we figure out . . . ?” Then, they decide which of the functions can be represented by a given expression. As students refine their language to decide what can be figured out they attend to precision (MP6). They reason abstractly and quantitatively (MP2) when using context to write equations.
Launch
Allow quiet think time to consider the situations in the first question and to answer yes or no. Ensure students understand which ones are functions, and which are not. Then, give students time to complete the table.
Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language
Supports accessibility for: Language
Activity
None8.3
Activity
Materials
To Copy (from Blackline Masters)
Matching Representations Cards
Activity Narrative
In this partner activity, students take turns finding a set of representations that represent the same function. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Launch
Tell students that the cards contain either a graph, table, equation, or example and that they will take turns matching the cards. Explain how to set up and do the activity. If time allows, demonstrate the steps with a student as a partner. Consider demonstrating productive ways to agree or disagree, for example, by explaining mathematical thinking or asking clarifying questions.
Arrange students in groups of 2. Give each group a set of 16 slips cut from the blackline master.
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed , so I matched .” Encourage students to challenge each other when they disagree.
Advances: Speaking, Listening
Advances: Speaking, Listening