6.EE.9 8.F.1, 2, 3, and 4
When graphing linear functions, students frequently describe parallel lines being horizontal shifts of one another rather than vertical shifts. Although this conception is not incorrect, it is more useful to think of graphs in terms of vertical translations. A horizontal shift of a graph is a shift to the left or the right, whereas a vertical shift of a graph is a shift straight up or straight down. This entry will discuss how horizontal translations of linear equations can be represented mathematically and how to help students relate vertical shifts to algebraic equations.
|If you start with the equation y=x and then change it to the equation y=x+3, what would that do to the graph?*||Research indicates that many students respond that the graph shifts 3 to the left. Other students may say that the graph shifts three vertically, 3 to the right, or increase in steepness by three units.|
Graph the equation y=x+4. A student said that this line would go through the axis at (4,0) because in the equation you add 4 to x. Do you think that the student was right? Why or why not.
|S1:||The student said the equation would go through the axis at (4,0). Yes, I agree. Why? Because in the equation you add 4 to x... now graph it? … x plus 4... what? Oh, oh!|
|S2:||I don’t get this.... hmmm.|
|S1:||I forgot that it goes on this side. [points to the right]*
When students initially encounter graphical representations of parallel lines, they frequently describe the lines as horizontal shifts of one another. This is not incorrect thinking: because lines translate infinitely, they are both vertical and horizontal translations of one another. However, because the standard representation of lines is in the form y=mx+b, it is more helpful for students to see parallel lines as vertical translations of one another. In particular the “b” value represents exactly the amount that the line has been shifted vertically. In the form y=mx+b, a change in “b” also shifts the graph horizontally -b/m units, but the horizontal shift is impacted by both the “b” value and the slope of the line. See also The Missing x-intercept value in y=mx+b.
Graphing calculators can be an effective tool for allowing students to quickly graph multiple functions on the same set of axis. By varying the values of “m” and “b” students can explore how the coefficients in the algebraic equation affect the graphical representation of the function. Graphing calculators provide linked approaches (numeric, graphical, and algebraic) to the same problem and allow complex math to become more accessible to a greater number of students. So in using this app students may see parallel lines as vertical translations of each other first as they set their own submarine path and then as they hunt for the paths of their opponent.
Apple’s Grapher program (Included in the Utilities folder of the Apple OS) can also be an effective tool for this along with Web-based applications like Graph Sketcher by Shodor Interactivate (Use Apple Safari or Windows Internet Explorer for best results with this Java-based site). The app Submariner Algebra provides students with practice guessing lines in y=mx+b format as they hunt for a submarine, similar to the popular game battleship. In this app they will use a slider that shows what a change in slope does to a line as well as what a change in the y-intercept or b value will do to a line, which can help them visualize on a graph the impact of changes in these values.
Another instructional approach is to use tables as a transitional tool between graphs and algebraic representations. Given a table of data and a corresponding equation, have students explore how adding 5 to the equation affects the corresponding outputs, and link those outputs to the new points on the graph. For example, start with the equation y=2x+1. Have students make a table of y values for a set of x values. Next, consider the table of y-values for the equation y=2x+1 +3 and y=2x+1 +5. Have the students plot the outputs. Students can see point by point that adding 5 to an equation results in a vertical shift, because the y-value is increasing.
Noble, T., Nemirovsky, R., DiMattia, C., & Wright , T. (2002). On Learning to See: How Do Middle School Students Learn to Make Sense of Visual Representations in Mathematics?, TERC; Cambre
Hennessy, S., Fung, P., & Scanlon, E. (2001). The role of the graphic calculator in mediating graphing activity. International Journal of Mathematical Education in Science and Technology, 32(2), 267-290.
Penglase, Marina and Arnold, Stephen. (1996). The Graphics Calculator in Mathematics Education: A Critical Review of Recent Research. Mathematics Education Research Journal, 1996, Vol8, No.1, 58-90 Representation, Vision, and Visualization. Duval, Raymond. 1999. Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.
Moschkovich, J., Schoenfeld, A., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69-100). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers.
Moschkovich, J. (n.d.). "Students" use of the x-intercept as an instance of a transitional conception. Educational Studies in Mathematics, 37(2), 169*
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?