April 10, 2018, was “Equal Pay Day.” In the United States, on average, women earn less than men, which means that they must work more days to earn the same amount of pay. Equal Pay Day represents how many more days women must work to earn what men earned in the previous year.
Algebraic reasoning is often promoted through an analysis of and generalizations about patterns that appear in mathematics, in nature, or in everyday situations (Driscoll 1999; Kieran 2006; Lee 1996). In accordance with this tendency, the Common Core (CCSSI 2010) emphasizes finding patterns and expressing such regularity in repeated reasoning as an important mathematical practice. NCTM (2000) also recommends that students participate in patterning activities by asking them to describe numeric and geometric patterns; generalize patterns to predict what comes next while providing a rationale for their predictions; and represent patterns in multiple ways, including drawings, tables, symbols, and graphs.
The less you know about the patterns in Pascal's triangle, the more fun you will have discovering the triangle's many secrets. I am amazed at how few students and even teachers (especially at the middle school level) have ever explored Pascal's triangle. Although this famous triangle bears the name of Blaise Pascal (1623-1662), who saw many of the patterns when he was only thirteen years old, it had been around for centuries before he was born. See the ancient diagram in figure 1, which appeared at the front of a Chinese book in 1303 (Vakil 2008). Evidence suggests that the properties of the elements of Pascal's triangle were known before the common era. Students and teachers alike can enjoy exploring patterns through problem solving with Pascal's triangle.
Short, mathematical discussions can elicit students' reasoning and focus on foundational ideas.
This article explores three processes involved in attending to evidence of students' thinking, one of the Mathematics Teaching Practices in Principles to Actions: Ensuring Mathematical Success for All. These processes, explored during an activity on proportional relationships, are discussed in this article, another installment in the series.
