The purpose of this activity is for students to recognize proportional relationships as linear functions. They also practice writing equations and using the language of functions to describe two proportional situations . A key part of this activity is students making connections between the situation and the possible equations and graphs that can be created to represent the situation (MP2).

Monitor for students who assign the independent and dependent variables in opposite ways. For example:

• For the first problem, some may have written , and some may have written .
• For the second problem, some may have written , and some may have written .

The goal of this discussion is for students to understand that proportional relationships are functions and to connect the parts of functions to what they know about proportional relationships.

Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the graphs have in common? How are they different?” (Each graph uses the same variables, but the axes are switched.)

• “What do the equations have in common? Which is more useful?” (Both equations for Jada describe the same thing: time spent mowing lawns and money earned. The equation is more helpful if Jada knows how much money she wants to earn and is trying to figure out how much time it will take while,  is more helpful if Jada knows how long she spends mowing laws and is trying to calculate what she is owed.

Conclude the discussion by asking, “How do we know that each of these situations are represented by functions?” (For each valid input, there is only one output. For example, no matter which equation I use for the relationship between feet and yards, a specific number of feet will always equal the same number of yards.)

The purpose of this activity is to connect features of an equation representing a function to what that means in a context (MP2).

Students start with two functions about water tanks represented in different ways. They identify key facts about the situations, write an equation for one of the tanks, and practice interpreting information about contexts from equations and graphs. This thinking gives students the opportunity to connect initial value and slope, which they learned about in a previous unit, to the general form of the linear equation and to the fact that linear relationships are functions.

The goal of this discussion is for students to make connections between and linear functions.

Consider asking some of the following questions to begin the discussion:

• “When you were writing an equation for Tank B, how did you decide if the value for in was positive or negative?” (Since the water in Tank B is decreasing, it has a negative rate of change.)
• “What is similar and what is different between the equation for Tank B and the equation for Tank C? How would their graphs be similar and different?” (Both equations have negative rates of change and positive vertical intercepts. The initial value and rate of change for Tank C are greater than those for Tank B. Graphed, the vertical intercept of Tank C would be higher than Tank B, and the slope for Tank C would decrease steeper than the slope of Tank B.)
• “For the last question, what in the graph tells you that Tank D is draining out? What would a graph that has a tank filling up look like? What would be different?” (I know it is draining out because the graph is going down from left to right. If the tank were filling up, the graph would be going up from left to right.)

Tell students that a linear function can always be represented with an equation of the form . The slope of the line, , is the rate of the change of the function, and the initial value of the function, , is the vertical intercept.

If time allows, give students the following scenario to come up with a possible equation for Tank D:

“Tank D started out with more water than Tank B but less water than Tank C. The water is draining from Tank D faster than from Tank B but slower than Tank C. What is a possible equation for the graph of the function for the amount of water in Tank D over time ?” (Students should choose an initial value between 400 and 800 and a constant rate of change between -7 and -5. One possible such equation might be .)

The purpose of this activity is for students to connect their work with linear equations to functions. The two linear functions in this activity are represented differently, and students are asked to compare various features of each representation.

Identify students who use different methods to answer the questions. For example, students may write an equation to represent Noah’s account, and others may make a table to show the value in each account at different numbers of weeks by reasoning about the rate of change and the amount in each account when they were opened.

Display the graph of Noah’s savings over time and the equation for the amount of money in Elena’s account for all to see. Select students previously identified to share their responses.

Consider asking the following questions to help student make connections between the different representations:

• “How did you determine the amount Noah saved in a year?” (I used the graph to figure out that Noah saves \$5 each week and multiplied that by 52 weeks.)
• “What equations could you use to solve the last question?” (I could use the equation for the amount of money in Noah’s account after weeks. When , Noah has \$320. If I solve the equation for , I would know how many weeks it would take Elena to have \$320 in her account.)“How could you solve the last question without using an equation?” (I could extend the graph out to 52 weeks and plot the value of each account over the year.)