8.1: The Secret Club (10 minutes)
Display a table where 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. Example:
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.
In 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?
The outcome of this discussion should be: 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 that 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.
For any first name, is there only one possible secret name? Yes! Each person was only born in one month. The completed table might look something like:
But let’s say you were trying to figure out which first name went with which month:
We don’t know whether “January” refers to Mai or Diego. Therefore, first name is not a function of secret name.
Ask students to summarize, “What does it mean to say that a relationship is a function?” It’s that for any input, there is only one possible output (each person was only born in one month, so birth month is a function of person). If you have a relationship where an input has more than one possible output, the relationship is not a function (for any possible birth month, there are many possible people born in that month. So person is not a function of birth month).
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,” or “Secret name is a function of name.”
8.2: Examples of Functions (15 minutes)
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.
Allow quiet think time to consider the situations in the first question and answer yes or no. Ensure students understand which ones are functions, and which are not. Then, give students time to complete the table.
- For each question, answer yes or no.
- It is 50 miles to Tucson. Can we figure out how many kilometers it is to Tucson?
- It is 200 kilometers to Saskatoon. Can we figure out how many miles it is to Saskatoon?
- A number is -3. Can we figure out its absolute value?
- The absolute value of a number is 8. Can we figure out the number?
- A circle has a diameter of 8 cm. Can we figure out its circumference?
- A circle has a circumference of \(10\pi\) cm. Can we figure out its diameter?
- A square has a side length of 6 units. Can we figure out its perimeter?
- A rectangle has a perimeter of 30 meters. Can we figure out its width?
- Which of the relationships are functions?
- For each function definition in the table, match it with the situation, write a statement explaining which variable depends on which, and write an example using function notation. An example is done for you.
function definition situation statement example \(m(x) = 0.62x \) You know kilometers and want to find miles. Distance in miles depends on distance in kilometers, or, distance in miles is a function of distance in kilometers. \(m(100)=62\)
\(f(x) = x \boldcdot \pi\)
\(g(x) = 1.6x\)
\(h(x) = 4x\)
\(k(x) = |x|\)
Invite students to share their responses in the table, and how they know. Emphasize that there are many correct examples one could choose—the thing to understand is how function notation communicates an input and output to the function (or a value of the independent variable and corresponding value of the dependent variable.)
8.3: Matching Representations (15 minutes)
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).
Arrange students in groups of 2. Partners take turns identifying representations that match, and explaining to their partner how they know they match. If necessary, demonstrate the protocol before students start working, including productive ways to agree or disagree, for example, by explaining your mathematical thinking or asking clarifying questions.
Your teacher will give you a set of representations. Sort them so that in each group there is a table, graph, equation, and example that all represent the same function.
Much of the discussion will happen within groups. Ensure that students found correct matches, and invite selected students to explain how they know they are matches. Here are some possible discussion questions:
- “Which matches were tricky? Explain why.”
- “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”