Thought for the day: Sometime, a Creative Math Prompt can come from an image straight out of a textbook. The difference comes in how you use the image!
Concepts: linear inequalities, linear functions, linear programming
Examples of noticing and wondering
I notice a two-dimensional region bounded by line segments in quadrant I.
I notice a function of two variables that appears to be defined on the region (its domain).
I notice that constant values of the functions occur on line segments within the region.
I notice that the outputs of the function appear to increase predictably as you move upward and to the right.
I notice that it looks as if the greatest value of the function will occur at the vertex (6, 10) of the region.
I wonder what the greatest value of the function is within the region.
I wonder if the greatest value of a function defined over a two-dimensional polygonal region will always occur at a vertex.
I wonder what happens if the region is not polygonal.
I wonder how I could redraw the region so that the maximum value would occur at a different vertex.
I wonder if I could think of a real-world situation to match the function on its given domain.
Creating something new
Students can create a problem or story to fit the graph (a very challenging task!). They could also search for ways to modify the objective function, C(x), and/or the region over which it is defined in order to accomplish predetermined goals such as making the maximum or minimum of C(x) occur at a particular vertex. (Alternatively, by trying to make it occur at a non-vertex point, they could deepen their understanding about why it must occur at a vertex!) They may even explore domains that are not bounded by line segments in order to see if it is possible to draw conclusions.
The graph represents a linear programming situation—a linear objective function defined on a domain in the coordinate plane bounded by line segments. The blue segments belong to lines of constant values of C. It appears that each time you shift a segment three units to the right (and consequently 2 units up) the value of C increases by 6. (If students are skeptical that this pattern will continue, they may figure out why this happens by looking at the formula for C.)
The image is designed to help students understand (at least intuitively) why maximum or minimum values of C for bounded regions occur at vertices.
This type of image may already appear in some algebra textbooks. The idea of including it as a Creative Math Prompt is to suggest shifting the focus of instruction from explaining (on the part of the teacher) to thinking (on the part of the student). By beginning with "noticing and wondering, " you are helping students to build their understanding from a starting point based on their own ideas.