Graphs+of+non-linear+function

Unit: Graphs of nonlinear functions Unit Goals: Student apply their experience with linear and exponential functions to build a basic understanding of graphing and interpreting non-linear functions
 * 1) Given a written real world scenario, represent it with a graph.
 * 2) Given data, represent it with a graph
 * 3) Given a graph, represent it with a real world scenario.
 * 4) Given a graph, classify as linear, exponential, quadratic, square root, cube root, absolute value or piecewise.
 * 5) Given linear, exponential, quadratic, square root, cube root, absolute value or piecewise graphs, relate to a real world scenario.
 * 6) Given a graph, identify intercepts and their meaning in a real world scenario.
 * 7) Given a graph, identify maxima/minima and their meaning in a real world scenario.
 * 8) Given a graph, identify domain/range and their meaning in a real world scenario.
 * 9) Graph linear, quadratic, square root, cube root, absolute value or piecewise functions by hand or with technology.
 * 10) Compare and contrast linear, quadratic, square root, cube root, absolute value or piecewise graphs.
 * 11) Identify linear, quadratic, square root, cube root, absolute value or piecewise graphs from an equation.

First learning cycle goal: Identify graphical representations of real world situations
 * 1) Given a written real world scenario, represent it with a graph.
 * 2) Given data, represent it with a graph
 * 3) Given a graph, represent it with a real world scenario.
 * 4) Given linear, exponential, quadratic, square root, cube root, absolute value or piecewise graphs, relate to a real world scenario.

Develop Understanding Task
 * 1) Review of previous material: functions relate to two variables, distinguishing situations that can be modeled with linear functions and exponential functions.
 * 2) Questions to bring up: Can all situations be modeled with a line.
 * 3) Modeling real world situations with functions that are not lines.

Task #1 Supplies: handout entitled Graphic stories

Launch: Have students read the directions. Check for any questions before beginning task.

Explore: If there are any students who are faster than others, have them do the extension and create their own stories for one or both of the remaining graphs.
 * 1) Students perform the task.
 * 2) Ensure that students are labeling axes correctly. Use teacher guided questions to prod students towards conventional axes norms.
 * 3) Encourage student to produce justifications that are complete thoughts.

Discuss: Ideas and connections to be discussed
 * Not all situations are linear
 * Why labeling is important (axes )
 * What’s more important – matching the graph or the justification?
 * Sharing of stories from the extension section on the handout
 * What other types of situations could the given graphs on the handout represent?

Task #2 Supplies: handout entitled Stories to Graphs Launch: Hand out Stories to Graphs and allow students five minutes to work independently. Allow students to partner up after five minutes.

Explore: If students are having difficulty with one problem, encourage them to move on to another problem. Monitor the student groups to make sure they are on task. This is a good time to prod students to consider positive and negative numbers on an x/y axis.

Discuss: Ideas and connections to be discussed
 * Labeling axes (units are critical)
 * Axes measurements – appropriate scale factors
 * Why might student’s graphs look similar but not the same?
 * After reading each story, what were the visual clues and terms that lead you to determine the shape of your graph?
 * Specifically question 3 – the idea of graphing below the x axis

Solidify understanding

Task #3 Supplies: handout entitled graphic stories 2

Launch: Display one of the problems from the Graphic Stories 2 on the board. As a class, have students brainstorm ideas for scenarios that could model the graph that is being displayed

Explore: Students will partner up to complete the rest of the task. Using creative writing practices and their knowledge from the previous two tasks, instruct students to create a story that will model each of the graphs. Remind students to consider labels, axes, position, etc. Extension: If students finish early, have them classify each graph with their appropriate name (parabola, cubic, absolute value, etc.)

Discuss: Ideas and connections to be discussed
 * Using units in their stories – Are they always needed?
 * Labeling axes
 * Does the picture match the story? If you used units does your graph accurately display that information?
 * Do you think there are more real world scenarios that can be modeled with linear or with non-linear functions? Why or why not?
 * What was hard about the activity? Easy?
 * Any other student observations that are brought up during the activity that need to be discussed.

Practice understanding

Homework option: Each student will create their own graphic model and exchange with another student. Students will create a story to match the graph. Next class period share stories and graphs. How did the student’s answer relate to what the graph drawer had in mind?