Once a set of learning objectives have been settled on for an activity, problem, or project, what should the problem’s context be? Since linear equations model situations where there is a constant rate of change, common contexts for linear equation projects often include the following:

- Steepness, height, angle

Examples: road grade, hillside, roof, skateboard park element, tide height over the two weeks before (or after) a full moon, sun angle at noon over a six month period - Estimating time to complete a task (setup plus completion)

Examples: mowing a lawn, painting a wall, writing a research paper - Purchase and delivery costs of bulk materials

Examples: mulch, gravel, lumber - Purchasing a service that charges by consumption

Examples: cell phone, electricity, water, movie rental, etc. - Total earnings over time from differing wage and bonus plan structures

Examples: hiring bonuses, longevity bonuses - Energy use over time

Examples: calories burned, electricity, heating oil, gasoline - Game points accumulated over time

Examples: by a professional athlete, a team, a video game player - Pollutant levels over time

Examples: atmospheric carbon dioxide levels, pH of rainwater - Distance or time remaining (or traveled) in a journey

Examples: by car, on foot, in a plane - Resale price of consumer goods over time

Examples: cars, video games, or appliances as they become dated and used - When will two separate objects meet or be in the same place

Examples: fast traveler overtaking a slower one, trains travelling in opposite directions meet

Non-linear situations can also often be modeled using linear equations for project purposes, either by choosing the interval to be modeled carefully so that it is nearly linear over that interval, or by using the modeling errors as a learning opportunity. Divergence between the linear model and the data provides good fodder for discussion, and can motivate both the discussion of modeling errors and confidence intervals. Divergences can also motivate the introduction of other types of mathematical relationships (inverse, exponential, logarithmic, quadratic, periodic, etc.).

While many linear equation tasks focus on student ability to follow a process (find the slope, write an equation, graph), I advocate that conceptual questions be used as the focus of the task:

- From the information given, and without completing any calculations, do you expect the graph of this situation to show a positive or negative slope, and why?
- What does the slope of the graph mean in the context of this problem?
- What is the “rate of change” in this situation? What does it represent or mean? Why is it positive or negative?
- Why does each number in the equation affect the result as it does? Explain the meaning of the slope and intercept values in the context of this situation.
- Predict one or more values. Explain why your prediction does or does not seem reasonable from an intuitive perspective.
- How would a change in the fixed cost, starting point, or initial height affect your prediction(s)?
- Pose several different kinds of questions that your model can answer, then explain how to use a graph, an equation, and a table of values to answer the questions.
- Compare two vendors or jobs: what circumstances will make each the preferred one?

Is one of your favorite activities, questions, or approaches missing from the above general lists? Please leave a comment, and I will gladly improve and expand the list!

Update 9/27/2012

A New York Times Education Blog post titled “N Ways to Apply Algebra With The New York Times” provides a list of interesting algebra applications that ask good open ended questions at the end of each, and base many of the activities on real-world data that students can research for themselves. I recommend reading it for examples of interesting data sets, student activities, and good questions that allow students to apply and extend their knowledge – providing good material for assessment.