Here are images from a handout that serves as an introduction to piecewise functions. The focus is to develop conceptual understanding of piecewise before attempting to graph independently. The work is divided into chunks to reduce the level of task demand at a given time in the process.
The first section has an application and introduces the idea of pieces addressed together. The graph is discrete which allows the students to see the points in lieu of looking at lines.
Pages 2 and 3 provide scaffolding for graphing. Page 2 presents the sections separately. Page 3 pulls them together, but with the intervals physically separated into columns over the graph.
Finally, the students are introduced to the function notation, with additional scaffolding. They are also asked to identify y-values for given x-values in function notation to help connect x-values with different pieces.
Making math meaningful and maybe interesting is a challenge. The photo above refers to a real life application for triangles and trigonometry (see photo below) that is found in a news story about Russian jets and a US destroyer. The jet was flying at an altitude of 100 yards and within 200 yards of the destroyer. Topics that could be addressed:
Altitude (and perpendicular)
Trigonometry: e.g. find angle of elevation or depression
Vectors (include velocities)
A relevant, real life application is a method to make information meaningful. When talking about the altitude of a triangle (the up and down part shown in the photo below) the vocabulary term of altitude becomes more meaningful both in terms of context and with the visual below.
Here is the agenda I would follow to use this application as an activity.
I would show the video (show on the webpage linked at bottom of handout) and explain what a destroyer and the jets are.
Discuss the situation with Russia (age appropriate discussion)
Show the picture and ask the students to draw a sketch.
Review the sketch and refer to the parts of the triangle in real life terms, e.g. altitude.
Task the students with a problem related to this problem – create your own, e.g. find the angle of elevation or use Pythagorean Theorem to find length of missing side.