This activity sets the groundwork for students to learn the techniques of graphing and the connections among graphs and other representations of functions.

Note: You are viewing an old version of this document. The latest version is available here.

This activity sets the groundwork for students to learn the techniques of graphing and the connections among graphs and other representations of functions.

Students begin to look at graphical representations of the relationship between two quantities, developing meaningful connections between a graph and the situation it represents. They interpret and create graph sketches for given situations. They recognize the distinction between discrete and continuous graphs and identify dependent and independent variables. This exploration sets the stage for the need to meaningfully scale graphs, followed by plotting points and graphing equations.

Students will be interpreting graphs in a qualitative way, focusing on rise and fall. They will also create sketches of graphs to reflect information describing various relationships. (The activity uses the term *graph sketches* to describe these more qualitative graphs.) In later activities, they will add scales, plot points, and draw graphs of equations on coordinate axes.

After the teacher introduces the activity, students work in groups, coming together as a class to share observations about the graphs. The teacher will use these observations to introduce such vocabulary as constant rate, linear, discrete, and continuous. The teacher also introduces independent and dependent variables and the graphing convention for placement of these variables on the *x-* and *y-*axes, respectively.

30 minutes

Individuals, followed by whole-class discussion

Transparencies of the graphs [Link to Wagon Train Sketches and Situations pdf, p. 1–2]

This activity assumes that students are generally familiar with this type of graph and the use of vertical and horizontal axes. You may want to mention that these graphs are quite different from the frequency bar graphs they worked with in *The Game of Pig*. During the activity, use the terms vertical and horizontal axes as students refer to the two perpendicular lines in their graph sketches.

To introduce the activity, either read through the brief introduction with students or do the example at the front of the room and introduce the task verbally.

You may want to acknowledge that the axes of the graph sketches do not have scales, but that students should still think of them as representing numeric information, with the numbers increasing as one moves up or to the right.

Ask students, What does the “shoes or boots versus people” graph tell you?

There are at least two observations they are likely to make.

- The more people there are, the more shoes or boots are needed.
- The relationship is linear. That is, the dots lie in a straight line.

For now, the term *linear* should be understood simply in its geometric sense: the graph follows a straight line.

Have students begin work in their groups on the activity*,* reminding them to keep a written record of their ideas. As you circulate, ask groups for explanations of the phenomena described in the graphs that students may have overlooked or what might be happening to cause a graph to “behave” in a certain way.

You need not discuss all of the questions, although it may be fruitful to have students give short presentations for some of them. Following are some ideas to look for in a discussion or as you circulate among groups.

*Question 1:* Students should be able to deduce from the linearity of this graph sketch that coffee consumption is the same each day. They may express this by saying that the graph “goes down” the same amount each day. Informally use the term *constant rate* in the context of this problem. The concept of rate is fundamental to the work of this unit, and the word will be used in many settings.

*Question 2:* Students may have various theories about what is happening in this graph sketch. One reasonable theory about the horizontal portions of the graph is that the wagon train group was traveling along a river and got water from the river rather than from the water barrels. So point *A* would represent arrival at the river, point *B* would represent people filling up their barrels, and point *C* would represent a time when they were traveling along another river.

*Questions 3:* Students should be able to articulate that the wagon train is moving fastest when the graph is “steepest.” You can ask what might cause changes in the wagon train’s speed (terrain, weather, and so forth). You can also ask what the graph sketch would look like if the wagon train had covered the same distance at a constant speed.

*Question 4:* This graph consists of individual points because the number of wagons must be a whole number. You might also point out that the points lie on a straight line and ask why that makes sense. Introduce the term discrete for a graph that consists of individual points.

You can also bring out the contrast between the graph in Question 4 and the graph in Question 3, which is unbroken because the time and distance concepts make sense for any nonwhole-number values. Introduce the term continuous for an unbroken graph. Reinforce the vocabulary by asking which of the other graphs in the activity are discrete and which are continuous.

Ask a couple of volunteers to present their graph sketches for each situation. Encourage students to make observations about the mathematics of their work.

*Questions 5: *Students might notice that this graph is discrete because the number of wagons is a whole number. The points on the graph should lie on a straight line because the wagons are “of a fixed size and type.”

*Question 6:* No specific information is given about the rate at which the buffalo population decreases as the number of settlers increases, so any decreasing graph is reasonable. Because the numbers are likely to be large, this graph might be treated as continuous, even though, technically, it would be discrete.

*Question 7:* The key idea here is that the graph should start positive, decrease to zero, and then increase. If the rider is assumed to be going at a constant speed, the graph should have the V shape of the absolute value function. As an extension, you might ask students to compare their graphs for this question with a graph of how far the rider has traveled as a function of time, and use that comparison to talk about the idea of absolute value.

One issue that may arise in the discussion of this activity is which axis to use for which variable—if not, raise it yourself. Explain that in many situations, one of the variables is *dependent* on the other. For example, in the graph introducing the activity, the number of pairs of shoes or boots needed *depends on* how many people there are in a family. (It may depend on other things as well, but that’s another issue.) Introduce the terms independent variable and dependent variable, and tell students that these concepts are essentially synonymous with the input and output of an In-Out table. Also mention that the convention in mathematics is to put the independent variable along the horizontal axis and the dependent variable along the vertical axis.

What does the “shoes or boots versus people” graph tell you?

*Spilling the Beans* (reinforcement or extension) is an activity with a western setting. Although the mathematics is more about reasoning—and specifically about proportional reasoning—the context makes this an appropriate place to use this activity.