This page is dedicated to the conjecture that modelling work doesn’t need to take place in a context. What’s important is the direction of the work: that it asks students to build a model to specifications before turning around and employing the model to solve for unknowns. The activity of designing custom models to fit specific constraints is what I term “Upstream Thinking,” and this design process fits quite nicely in our algebraic-minded courses if we use task structures that encourage students to make a high number of design decisions and analyze the fallout from said decisions. Here, you will find lists of constraints that pertain to specific types of functions you and your class may be studying. They are the building blocks for three types of contextless modelling tasks that I have used with my students: boxes tasks, wanted tasks, and menu tasks. Each type of task is described below. The hope is you use the lists of constraints to design tasks of these types to encourage students to build custom models to specifications–to think upstream. (These lists are a starting point; please get creative and feel free to pass along any inventions!)
A boxes task takes the parameters of the equation of a function and replaces them with empty boxes. The students are then asked to fill in the boxes with values to satisfy a set of constraints (Typically 3-4 constraints are given to the learners all at once). The numbers available for entry into boxes can be restricted to sponsor further thinking. (This type of problem has become popularized by the work of openmiddle.com).
A wanted task (name inspired by Chris Hunter) asks students to build and shift the design of a model as a list of constraints is released one at a time. After each constraint, students need to verify that their current model satisfies the new constraint, or alter their function to meet this new requirement. The model is appropriate if it satisfies every constraint. Teachers can pre-sequence a list of constraints or choose them randomly–but this makes it more likely that a model satisfying each constraint is impossible to build. This is an important conversation, but impossibility is not the ideal result for every wanted task. Examples of wanted tasks asking for quadratic models can be found in this post.
A menu task gives students a longer list of constraints (around 8-10) and asks them to build as many functions as necessary so that each constraint is satisfied with at least one of their custom functions. There is no restrictions for the values of the parameters, and all the menu’s constraints are given at once. Rich opportunity arises as students attempt to resolve the menu with as few functions as possible. They begin to analyze which constraints pair well together and which cannot pair together. A more detailed rationale can be read in this post, and a complete set of the online community’s submission is crated on this site under #MenuMath tab.