Intent

This activity will help students draw connections between what they know about reading information from graphs and what they know about In-Out tables and rules. In addition, several standard terms and conventions are introduced. The activity prepares students to reflect on how to move from rules to graphs, the focus of the next activity.

Mathematics

This activity will further the connections students are building between situations, graphs, tables, and rules. Students read data from scaled graphs of linear situations and find rules for the data, pausing to consider that the straightness of a graph corresponds to a constant rate of change. They also review the four-quadrant coordinate system, using the terms rectangular (or Cartesian) coordinate system and quadrant.

Progression

Students work on the activities in groups, sharing ideas about how to respond to the questions. Sometime after groups have completed Question 2, the teacher brings the class together to discuss observations, highlighting the mathematical connections students observe between representations and introducing new vocabulary.

Approximate Time

40 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Transparencies of the graphs [link to Out Numbered pdf, p. 6–7]

Doing the Activity

This activity requires little introduction. Emphasize that the focus of this activity, rather than “finding the answers,” is to engage in discussion of the problems posed. Through such interaction, students will begin to create meaning for the relationships among graphs, tables, rules, and situations and, in part f of each question, how the concept of rate is connected to all the representations.

For example, if a group notices that a graph “goes up the same amount for each wagon,” ask questions that might help the students notice how this amount appears, or should appear, in the table or rule. How does the fact that 8 more people can be carried by each additional wagon affect your table? Can this number tell you how many people could be carried by 100 wagons? What would you do to calculate the number of people for w wagons?

Discussing and Debriefing the Activity

When groups are ready, ask them to prepare transparencies to lead the discussion for parts d through f of each question. You may want to omit Question 1 to save time, as Questions 2 and 3 are more likely to need discussion. Also, you can begin the discussion before all groups have finished Question 3.

Use the presentations as an opportunity to connect the rules to the situations, which is the main idea in part f. Students may have noted that the numbers they found in answering Questions 1f and 2f showed up in the rules, and the discussion can emphasize the reason behind this. For example, in Question 1, students may see that the rule is something like Out = 8 · In and that there are 8 people per wagon. Ask why the number 8 appears both in the rule and as the number of people per wagon.

The geometric linearity of the graphs reflects the fact that each situation involves a constant rate. If you haven’t yet used the phrase constant rate, now is a good time to work this language into the conversation. Encourage students to use the terms as well.

Question 2 involves a nonzero “starting value.” The discussion of Question 2f should address this initial amount, which represents pounds of coffee on Day 0, as well as the rate.

Focus on how the rules offer a way to describe the situations, rather than on any formal procedure for getting such rules from In-Out tables. Although students may be able to find a rule by examining the table itself, help them to see the value of connecting the rule back to the situation. Some students may actually develop their rules directly by thinking through what’s happening in the situation.

Introduce the synonymous terms rectangular coordinate system and Cartesian coordinate system for the standard graph setup with vertical and horizontal axes and equal-interval scales.

Remind the class that although their examples so far have involved only positive coordinates, axes are generally viewed as complete number lines, including negative as well as positive values. If you think it is needed, you might give students some coordinate pairs, including negative values, to plot.

Also be sure students know that unless otherwise indicated, x is used for the independent variable, which is represented on the horizontal axis, and y for the dependent variable, which is represented on the vertical axis.

Review the term quadrant and the standard numbering system as shown below. You might also remind students that the point where the axes meet, with coordinates (0, 0), is called the origin. Students will graph functions throughout their mathematics work, and you need not worry about whether they memorize the quadrant numbering system now. Casual references in context—for example, Which quadrants does this graph use?—will gradually familiarize them with the system.

Figure 1

Use an equation from this activity to give meaning to the common phrase “graph of an equation.” For example, the graph in Question 2 is the “graph of the equation y=25−12xy=25−12x size 12{y="25"- { {1} over {2} } x} {}.”

Also mention that the set of all points that fit a rule is called the “graph of the equation” (or rule) and that the process of putting these points together to form an overall picture is called “graphing the equation.”

Key Questions

How does the fact that 8 more people can be carried by each additional wagon affect your table?

Can this number tell you how many people could be carried by 100 wagons?

What would you do to calculate the number of people for w wagons?

Why does the rule Out = 8 · In make sense in Question 1?

How much coffee is consumed each day in Question 2?