Connections to Teaching Mathematics
The data plan investigation could be used in middle and high school mathematics classrooms. As students can examine and use the patterns discovered from the recursive formulas to investigate and solve unit rate problems as identified in the following student expectation from the Alabama Course of Study (ACCRS) (Alabama Course of Study, 2016):
- 6-RP3b – Solve unit rate problems including those involving unit pricing and constant speed.
Using formulas within spreadsheets also build students’ understanding of variables in expressions. It may be easier for students to describe the pattern and relationship between the values in the spreadsheet, but it is also important to ensure students understand how the cell notation is represented by a variable. This aligns with the following student expectation from the ACCRS:
- 6-EE6 – Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or, depending on the purpose at hand, any number in a specified set.
While I defined the patterns recursively, students could also represent the context of the situation using closed formulas. For example, the Pay-Go plan represents a direct variation. We could use the spreadsheet to show how the cost varies directly with the number of gigabytes purchased. Students can use the expression 30x to represent the cost and then prove why the equation y=30x also represents the same situation. The closed formula for the Select plan is an example of an expression which could be represented as 15x + 100 or 100 + 15x. This discovery and discourse allow for the following student expectation from the ACCRS to be addressed:
- 7-EE2 – Understand that rewriting an expression in different forms in a problem context can shed light on the problem, and how the quantities in it are related.
Another extension for grade 8 and Algebra 1 would be to ask students to use the data within the spreadsheet to create a graph comparing the data plans to understand the connection between proportional relationships, lines and linear equations, also leading into discussions about functional relationships. From this vantagepoint, the following student expectations from the ACCRS are addressed:
- 8-EE5 – Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
- 8-F2 – Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
- 8-F4 – Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of linear function in terms of the situation it models and in terms of its graph or a table of values.
- F-IF2 – Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F-IF3 – Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
While using technology to determine solutions to this situation, helping students to be precise about their mathematical knowledge is critical. In Algebra 1, as students begin to discover the domain of linear functions are all real numbers, understanding the restrictions for real world situations will be important. The following student expectation from the ACCRS could be addressed:
- F-IF5 – Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes
Citations
Alabama State Department of Education (2016). Revised Alabama Course of Study: Mathematics. Montgomery, AL
Beigie, D. (2017). Solving optimization problems with spreadsheets. Mathematics Teacher, 111(1), 26-33.
Dick, T.P. & Hollebrands, K.F. (2011). Focus in high school mathematics: Technology to support reasoning and sense making (pp. xi-xvii). Reston, VA: National Council of Teachers of Mathematics
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reson,VA
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