Before students can sensibly analyze data in two variables they must have a strong foundation in algebra, particularly linear functions. Thus in this unit we return to Algebraic Thinking to address sense-making in the context of equations in one variable, and subsequently functions in two variables. Multiple representations are essential to build strong conceptual understanding. Equations in one variable can be represented in verbal description, in diagrams, and as algebraic statements. Functions can be represented as graphs in the x-y plane, as well as in all of these other ways. When linear functions are well-understood, particularly the role of slope in the equation, students are ready to consider scatterplots and best-fit lines in a meaningful way.
In addition to the resources listed below, many excellent activities and readings are included in the book Fostering Algebraic Thinking, by Mark Driscoll.
Algebraic Thinking
AT3.a. Understand solving an equation as a process of answering a question: which values from a specified set, if any, make the equation true?
AT3.b. Solve linear equations by successively transforming the given equation into simpler forms by applying the same operation to each part of the equation; use physical or pictorial models where appropriate.
AT3.c. Solve systems of linear equations graphically (finding point of intersection) and algebraically (substitution, elimination).
Data
D3.e. Explore patterns of association by using values of one variable to predict values of another variable.
AT3.a.b
Having students build equations to represent situations encourages them to make robust connections between the verbal description of relationships and the mathematical equations of same.
Using concrete representations of equations such as a pan balance or strip diagrams can reinforce the focus on relationships vs procedures. It can also support the reasoning that allows us to simplify equations in order to find the solution(s).
AT3.c
Within the transition to systems of linear equations is the introduction of a linear function. Typically each line represents a collection of points that fit a specified relationship, e.g. the cost of a cell phone plan depending on the # of GB of data used. This is a shift in the use of symbols, as well: the variables are now quantities in relationship, rather than a variable representing a fixed unknown. This shift should be made explicit, with a discussion about the varying ways that symbols are used in algebra. One reference that can be used as a starting point for this discussion is Kieran, Carolyn, "Helping to Make the Transition to Algebra," Arithmetic Teacher, March 1991, pp 49 - 51. Students with more extensive math backgrounds might also benefit from reading "The Many Uses of Algebraic Variables," by R. Philipp, in Mathematics Teacher, Oct. 1992, pp 557 - 561.
D3.e. With a robust understanding of linear functions, students are ready to make sense of best-fit lines in the context of scatterplots. In particular students should make sense of the slope (and, when appropriate, the y-intercept) of the best-fit line in the context of the situation represented in the scatterplot, while simultaneously noting the variability visible in the plot.
Algebraic Thinking
Equations
AT3 Activity "Building and Solving Linear Equations"
AT3.a. Activity "What Are the Solutions of These Equations?" 9J in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed.
AT3.a.b. Activity "Solving Word Problems with Strip Diagrams and with Equations," Class Activity 9K in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed
AT3.a.b. Activity "Solving Linear Equations"
AT3.b Activity "Solving Equations Algebraically and with a Pan Balance," Class Activity 9I in Beckmann, Mathematics for Elementary Teachers with Activities, 5th Ed
Functions and Linear Functions
AT3.c Activities from Beckmann, Mathematics for Elementary Teachers, 5th Ed
9R What Does the Shape of a Graph Tell Us About a Function?
9S Graphs and Stories
9V Modeling Linear Relationships with Variables and Equations
9W Interpreting Equations for Linear Relationships
AT3.c NCTM Online Activities: Exploring Linear Data (several activities)
AT3.c Book with activities: Navigating Through Algebra in Grades 6 - 8
Systems of Linear Equations
AT3.c. Activity "Acrobats, Grandmas, and Ivan"
AT3.c. Activity "Kimi and Jordan"
AT3.c. Activity "Summer Swimming"
AT3.c. Activity "Cell Phone Plans"
AT3.c. Activity "Fixing the Furnace"
AT3.c. Activity "How Many Solutions?"
AT3.c. Activity. "Estimating a Solution via Graphs"
AT3.c Activity "Splat" problem with missing coefficients.
Data
Scatterplots and Best-Fit Lines
D3.e Book with Activities Navigating Through Data Analysis in Grades 6 - 8, pp 67-80 and associated activities
D3.e Article with Activity "The Median Median Line," Mathematics Teacher, Nov 2010
D3.e Online Applet / Investigation for Least Squares Line:
D3.e NCTM Online Activities: Exploring Linear Data (several activities)
D3.e Online Applet with Examples of Scatterplots and Least Squares Line
D3.e Activity: Non-linear Correlation (Open Middle)
D3.e Activity: Line of Best Fit (Open Middle)
D3.e Activity: Barbie Bungee