Linear functions are defined as relationships between two variables that have a constant rate of change. Linear functions are an extension of arithmetic sequences where successive terms are generated by adding a constant number (rate of change) to the previous term.
Activity
Begin by working through the Beach Problems. Download the document and either mark it up on your screen with your finger or a touch-pen or print it and use a pencil. Please pay particular attention to the number patterns in the table.
1.2_-Algebraic-Reasoning-SWTDownload Beach Problems.
Reflect
On your own paper or digital document (or in your professional learning log!), answer these two questions.
- What do you notice about the differences in values for successive table rows for both the independent and dependent variable?
- What relationship exists between the ratio of the differences in the dependent variable to the differences in the independent variable and the equations that you have written?
Why does that work?
Next, watch the video to see how finite differences relate to slope and linear functions.
Connections and Teaching Hints
Here’s a video showing connections to Grade 8 and Algebra 1 as well as other hints for teaching this topic with your kids from the author’s perspective. In this video, you will learn more about:
- vertical connections from middle school, Algebra 1, and Algebraic Reasoning
- connections between arithmetic sequences and linear functions
- connections between the constant first difference and slope
- calculator hint for using lists to determine first differences
