The goal of this opening activity is to activate, through a familiar context, what students know about functions from middle school.

Students first encounter a relationship in which two quantities—the number of bagels bought and price—do not form a function. They see that for some numbers of bagels bought, there are multiple possible prices. The relationship between the number of bagels bought and the best price, however, do form a function, because there is only one possible best price for each number of bagels.

Students contrast the two relationships by reasoning about possible prices, completing a table of values, and observing the graphs of the relationships.

Consider displaying a table to summarize the different possibilities for calculating the price of 13 bagels (or the prices for 6 or more bagels) and a table showing the best price for each number of bagels bought.

Ask students,

• “‘Number of bagels’ and ’price’ do not form a function, but ’number of bagels’ and ’best price’ do form a function. Why is this? What do you recall about functions?”
• “If we graph the relationship between ’number of bagels’ and ’price,’ what do you think the graph would look like?”
• “If we graph the relationship between ’number of bagels’ and ’best price,’ what would the graph look like?”

After students make some predictions, display the two graphs for all to see. In the first graph, solid blue dots represent the shopkeeper’s price, open green circles represent Jada’s price, red squares represent Priya’s price, and yellow triangles represent Han’s price. In the second graph, each X represents the best price for each number of bagels.

Because there are multiple possible prices when the number of bagels is 6 or greater, price is not a function of the number of bagels bought.​​​​​​

Because there is only one best price for a particular number of bagels, best price is a function of the number of bagels bought.

Emphasize that a function assigns one output to each input. Clarify that the word “function” in mathematics has a very specific meaning that does not necessarily agree with how “function” is used in everyday life (for instance, as in the sentence “The function of a bridge is to connect two sides of a river”).

This activity further refreshes students’ understanding of functions through contextual examples. Here students are prompted to reason graphically about the relationship between the two quantities in the situation—time in seconds and the distance of a dog from a post. Students also recall the meaning of independent and dependent variables.

Students are given descriptions of the dog’s movement while it was attached to a post and asked to sketch corresponding graphs. Along the way, students interpret each point on the graph to mean a particular point in time and a particular distance from the post.

Students later reason that the relationship between time and distance is a function because the dog can be in only one location at any given time. For instance, the dog could not be both 5 feet and 1.7 feet away from the post at the same exact time.

As students work, monitor for students that create a graph with a vertical value greater than 5 or a graph that does not go through the two given points. If many graphs show these features, discuss with the class why the former is not possible and why the latter does not match the given descriptions.

This activity gives students a chance to use mathematical language to describe relationships that are functions and to practice sketching a graph of a function given a description. The context is the same as in the previous activity, but the quantities are different.

Deciding on which variable is independent and which is dependent, as well as sketching a graph of the relationship, engages students in aspects of modeling (MP4). It requires students to make sense of quantities, consider how they are related, and think about what values are reasonable. Using the language of “function” to articulate the relationship between variables is an opportunity to attend to precision (MP6).

When sketching a graph for the function that defines the number of barks, students are likely to create either a discrete graph or a continuous graph. (Because the total number of barks cannot be fractional, creating a step graph would most accurately represent it as a function of time, but students are not expected to do so at this point.) Identify students who sketch each kind of graph. If time permits, select them to share their thinking during the whole-class discussion.