This unit bundles student expectations that address the constant rate of change, the constant of proportionality, and various representations of linear relationships. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.”
Prior to this Unit
In Unit 02, students modeled and solved one-variable, two-step equations and inequalities with concrete and pictorial models and algebraic representations. In Unit 03, students examined proportional reasoning with ratios and rates through the lens of constant rates of change. Students represented constant rates of change given pictorial, tabular, verbal, numeric, graphical, and algebraic representations. In Grade 6, students identified independent and dependent quantities from tables and graphs, wrote an equation that represents the relationship between independent and dependent quantities from a table, and represented a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. Students also compared two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.
During this Unit
Students use bivariate data, data with two variables, to reexamine constant rates of change given pictorial, tabular, verbal, numeric, graphical, and algebraic representations and extend their understandings of the constant of proportionality. Students are formally introduced to the slope intercept form of equations, y = mx + b, to represent linear relationships. Although students are not formally introduced to slope or y-intercept in linear proportional and non-proportional relationships until Grade 8, students are expected to relate the constant rate of change to m, and the y-coordinate, when the x-coordinate is zero, to b in equations that simplify to the form y = mx + b. This relationship is examined through the ratio of rise to run and the change in the y-values to the change in the x-values. Students represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
After this Unit
In Grade 8, students will solve problems involving direct variation, distinguish between and represent proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0. Students will formally develop the concepts and applications of slope and y-intercept in proportional and non-proportional functions. Additionally, students will study systems of linear equations as they identify and verify values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.
In Grade 7, representing constant rates of change is identified as STAAR Readiness Standard 7.4A, determining the constant of proportionality (k = y/x) is identified as STAAR Supporting Standard 7.4C. These standards are a foundational block of the Grade 7 Texas Response to Curriculum Focal Points (TxRCFP): Representing and applying proportional relationships. Representing linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b is identified as STAAR Readiness Standard 7.7A and is a part of the Grade 7 Focal Point: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems (TxRCFP). All of these standards are subsumed within the Grade 7 STAAR Reporting Category 2: Computations and Algebraic Relationships. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, II. Algebraic Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, and X. Connections.
According to research, “Students in middle grades should develop an understanding of the multiple methods of expressing real-world functional relationships (words, graphs, equations, and tables). Working with these different representations of functions will allow students to develop a fuller understanding of functions.” (Van de Walle, Lovin, 2006, p. 284). As students begin to relate constant rates of change within multiple algebraic representations as a prerequisite for future coursework with slope and y-intercept, it should be noted that “Children need to learn that, in mathematics as in most subject areas, they should not do something a certain way because someone tells them to; rather they need to understand why doing it that way makes sense (or doesn’t make sense)” (Reyes, Lindquist, Lambdin & Smith, 2012, p. 318). Students need to develop the ability to move among algebraic representations flexibly, “Students are often not made aware of the power of mathematics. The realization of how the transfer of an equation to graphic form can reveal a whole set of possible solutions may be an eye-opening motivating factor for learning mathematics” (Solomon, 2007, p. 200).
Reyes, R. E., Lindquist, M., Lambdin, D. V., & Smith, N. L. (2012). Helping children learn mathematics. (10th ed.). Hoboken, NJ: Wiley.
Solomon, P. (2006). The math we need to know and do in grades 6 – 9. Thousand Oaks, CA: Corwin Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 5 – 8. Boston, MA: Pearson Education, Inc.