NOTE: This is taken from the Teacher's Edition of Variables and Patterns. ©2002

Variables and Patterns mathematical and problem solving goals:

Variables and Patterns was created to help students:

The overall goal of the Connected Mathematics curriculum is to help students develop sound thematical habits. Through their work in this and other algebra units, students learn important questions to ask themselves about any situation that can be represented and modeled mathematically, such as: How can mathematics be used to show how quantities change over time? What does it mean when we see regular and predictable changes in a table of data or a graph? How can we use these predictable changes to find out about other possible data? Where in the world around us can we find these patterns? Why do some straight-line graphs rise as x increases, while others fall? When can an equation describe the information in a table? In a graph? When is a graphing calculator helpful in analyzing data? Other than a graph, what information can be found with a graphing calculator?

The Mathematics in Variables and Patterns

The relationship between two variables—in particular, the way in which one variable changes in relation to another—is an important idea in mathematics. it is central to understanding func- tions and concepts in calculus. This unit develops methods for representing these relationships and patterns of change. Verbal descriptions, tables, and graphs are the central representations in this unit. Toward the end of the unit, written and symbolic rules are introduced. This unit pro- vides the basis from which to study other algebra units, such as Say It with Symbols, in which the focus is on symbolic reasoning, or Moving Straight Ahead, in which the focus is on linear relationships—that is, as one variable changes, the other variable changes by a constant amount.

Each representation has its advantages and disadvantages in promoting understanding of rela- tions hips and patterns of change.

Verbal Descriptions

Verbal descriptions of a relationship are useful because they are descriptions in students’ every- day language. This helps the students form mental pictures of the situations and the relation- ships among the variables. The disadvantages of verbal descriptions are that they are sometimes ambiguous, making it difficult to get a quick overview of the situation and the relationships among variables in the situation.

Tables

Tables are easy to read. From a table, it is easy to see how a unit change in one variable affects the change in the other variable. Students can recognize whether the change is additive, multi- plicative, or unpredictable. Once students recognize the pattern of change, they can apply it to the variables to get the next entry. For example, consider the following two tables.

In Table 1, as the variable x changes by one unit, the variable y changes by 3 units. The table could be continued by adding 1 to the previous entry in the x column and 3 to the previous entry in the y column. If x is 3, then y is 9 + 3, or 12. The table can be generated backwards by reversing the pattern of change. If x is —2, then y is 0 minus 3, or -3. The particular change pattern in Table 1 is indicative of all linear relations. It is an additive pattern because the rate of change between the two variables is always constant.

The change pattern in Table 2 is characteristic of an exponential relation. It is a multiplicative pattern because the variable y is doubling or is increasing by a factor of 2 as the variable x increases by one unit.

In some tables, the patterns of change are not regular. For example, Table 3, which occurs in the second investigation, (does not show a pattern of change that is regular; that is, there is no way to predict the change from one point to the next.

Graphs

Graphs are another way to view the relationship and the patterns of change between the variables, such as in Tables 1 and 2. Graphs 1 and 2 can be thought of as pictures of the relationships shown in those tables.

The linearity or constant rate of change is represented by a straight-line graph, while the exponential relation or multiplicative rate of change is characterized by a curved line. These relationships can be represented symbolically as y = 3x + 3 (Table 1 and Graph 1) and y = 2x (Table 2 and Graph 2). Both of these patterns are explored in future units. Linear relationships are the focus of Moving Straight Ahead, and exponential growth is studied in Growing, Growing, Growing.

This unit provides a firm foundation to continue to study important patterns of change. Only simple linear expressions are explored in this unit. For example, the distance, d, a cyclist can cover depends on time, t, and the rate, r, at which the cyclist peddles. If a cyclist rides at 10 miles per hour, then d = 10t. This is a linear relation—its graph is a straight line.

Connecting Points

Many situations are discrete relationships, such as the number of sweatshirts sold and the revenue. If the shirts sell for $5.50, then the revenue, r, for selling n shirts is r = 5.50n. In this situation, it does not make sense to connect the points. Points (1, 5.50) and (2, 11) are on the graph; however, if these two points are connected it would imply that 1½ or part of a shirt could be sold. Other situations, such as the distance/time/rate relation, are not discrete; they are continuous. For example, if a bicyclist peddles at a rate of 10 miles per hour, then distance, d, after t hours is d = 10t. In the graph of d = 10t, it is reasonable to connect the points (1, 10) and (4, 40) since one can travel 1½ hours and go a distance of 15 miles; it makes sense because time is a continuous quantity

In a distance/time/rate situation, students are asked to decide whether the points can he connected and if so, how they can be connected. For example, the points representing the time and distance of a cyclist can be connected in many ways. A straight line connecting the starting and ending points for a day’s ride implies that the cyclist traveled at a constant rate. The four graphs below show other ways these two points may be connected.

Selecting a Scale

Another aspect of graphing is that of scale. This is closely connected to the range of values for each variable. To represent a relation graphically, students must have a good feel for the range of values. Students must select an appropriate scale so that the relevant pieces of the graph can be displayed. The effects of the scale can often lead to distortion as shown in the graphs below For example, suppose students select a scale of 1 to 10 for both axes when they graph the equation d = 10t, as in Graph A. Only the information about the first hour would be shown on this graph. This may not be enough information fOr students to understand the relationship. The scale in Graph B may lead students to believe that the distance covered in three hours is minimal. Graph C more nearly mirrors the situation.

While the relationship between the variables is the most important idea in this unit, it is the representation of these relationships that is the dominant theme. It is important for students to move freely among the various representations. it may not be obvious initially to students how the entries in a table relate to points on a graph or to solutions of a symbolic statement; and conversely, how solutions to an equation or a graph relate to the other representations; however, these connections are explored in depth in this unit.

By the end of the unit, students should feel very comfortable with tables and graphs and with some simple symbolic rules. Students should also have an appreciation of the advantages and disadvantages of each representation. The value of a symbolic rule is that it is brief and represents a complete picture of the pattern, while tables and graphs can represent only parts of the relationships.