PATTERNS AND ALGEBRA: ZOOMING IN ON THE LEARNING PROGRESSIONS
In this section, we share a lesson study around patterns and algebra. The learning progressions that are addressed span across the fourth- through eighth-grade band where students generate numerical patterns using given rules, record them in a table, and graph the corresponding values of the patterns as ordered pairs on the coordinate plane. They extend the table by describing how the x-values and the y-values change simultaneously. This concept of covariation is used to describe the relationship between the two patterns. Describing a rule relating values in the first column to values in the corresponding second column is called a correspondence description of the relationship between the two patterns.
4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.
Across grades 4-8, a focus of learning progression is on reasoning about number or shape patterns, connecting a rule for a given pattern with its sequence of numbers or figures. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
- 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.
- 5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.
Developing algebraic reasoning in early elementary mathematics (NCTM, 2000) means that algebraic reasoning includes the following components: (1) understanding patterns, relations, and functions; (2) representing and analyzing mathematical situations and structures using algebraic symbols; and (3) analyzing change in various contexts (p. 37).
According to Carpenter, Franke, and Levi (2003), the artificial separation of arithmetic and algebra deprives students of powerful ways of thinking about mathematics, especially since the fundamental properties that children use in calculating are the basis for most of the symbolic manipulation in algebra. Greenes and Findell (1999) suggest that students develop algebraic reasoning when they are able to interpret algebraic problems by using pictorial, graphic, and verbal description, tables, and numeric representations.
As students evolve through their learning progression, it is important to help them start with an initial numerical understanding of iterative computing (e.g., “add 3”) within a single sequence of positive whole numbers. For example, having students extend the pattern
Once they are able to achieve this, the students must be exposed to spatial understanding that may involve representing, for example, the relative sizes of quantities (such as bars on a graph for a yearly census figures) and describe qualitative changes in the amount.
As the learning progression evolves from numerical to spatial then it is essential to elaborate on each of these understandings. An elaboration of the numerical understanding involves not only applying iteratively within a single sequence but also be able to related a sequence of numbers to generate a second sequence of numbers. For example,
The next step is often to help the students think about an algebraic expression (y = 3 x) or rule associated with this repeated operation. An elaboration of the special understanding involves using continuous quantities along the horizontal axis and recognizes emerging properties such as increasing or linear in the graph connecting different points.
Once students are able to achieve such elaborated understanding numerically and spatially, it is important to integrate them using correspondence relations between differences in the y-values in a table and the size of the step from one point to the next in a related graph. For example, for every 1 step increase in time, a constant 3 degree increase in the y-column of the table and the y-axis in the related graph generates a linear pattern (spatial) with a slope of 3 (numeric) with a corresponding linear relationship y = 3x.