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 .)

In this activity, students work with data to determine if the situation represented by the data could be modeled by a linear function (MP4). First, students are given three different data points and use what they know about linear relationships to estimate when the candle will burn out. Next, students consider a situation that seems like it could be modeled with a linear function from the limited information given, but where a linear model fails with just a bit more data. While study of periodic functions is in high school, students' familiarity with the cyclical nature of the moon should be enough for them to make sense of the nonlinear nature of the situation.

The purpose of this discussion is for students to understand how and when a situation can be modeled by a linear equation and to understand that while we can create a linear function out of 2 data points, doing so does not always lead to accurate predictions. Consider asking the following questions:

• “How did you predict when the candle will burn out completely?” (Since it burns about 2 inches every hour, it will burn out about 6 hours after it was lit. After 3 hours the candle was already less than half of its original height, so it will take less than 6 hours.)
• “Was the candle data exactly linear? If not, what made you decide that you could treat it as such?” (I couldn’t draw a line exactly through all three points, so it isn’t linear, but my line is so close to going through all three that it seems like a decent estimate.)
• “The moon’s surface viewed from Earth is actually 100% illuminated on day 14. Does this agree with the prediction you made?” (No, I assumed a 4% increase each day which means the moon would be 100% illuminated on day 26.)
• “Is the percentage illumination of the moon’s surface a function of the day? If yes, is it a linear function?” (Yes, the amount of the moon's surface that is illuminated is a function of the day, Since it increases and decreases repeatedly over a year about every 28 days, it is not linear function .)

Tell students that if we only use two data points, it is always possible to model a situation with a linear function. For example, with the moon illumination, additional data points would have helped to determine whether a linear model was appropriate. Additionally, we have observed ourselves that the moon phases go in cycles that repeat over and over again, which isn't something a linear function is meant to model.

Then explain to students that when data is not precisely linear, such as with the candle, it can make sense to model the data with a linear function because the points resemble a line when graphed. We can then use different data points to help predict when the candle would burn out. Answers might vary slightly, but it results in a close approximation.

The purpose of this activity is for students to determine if a given set of data can be modeled by a linear function. Students first view a set of pictures and data for the length of a shadow at 0, 20, and 60 minutes. Then, they make a prediction about how long a shadow will be after 95 minutes. Students then compare their estimate with the actual length of the shadow and make conclusions about the model they used to make their estimate.

Monitor for students using different strategies to make their prediction. For example, students may use different pairs of points to make their prediction, or they may try to use all three. The discussion for this activity focuses on how if we are given two input-output pairs, we can always find a linear function with these inputs and outputs, but that doesn’t mean a linear function is actually appropriate for the situation. In this case, we can see when we get more data that a linear function is not appropriate.

In the digital version of the activity, students use an applet to estimate the length of the shadow. The applet allows students to plot points quickly and accurately. This activity works best when students have access to the applet because the points are close to linear but make a distinct downward curve.

Select groups that had different strategies for making their original prediction to share their reasoning about whether or not a linear model is a good fit for predicting the length of the shadow. In particular, make sure it is pointed out how much using the rate of change determined by the first two points overpredicts the length of the shadow after 95 minutes.

If not brought up by students, ask, “What will the length of the shadow be near sunset? After sunset?” (Near sunset the shadow will be very long, and after sunset there will not be a shadow.) Emphasize that over the short term, a linear model might be useful even for nonlinear data, but if the actual situation isn’t considered, like here where the sun will eventually set, conclusions will be wrong if we try to force a linear model.