Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Notes for Session 7

In this session, we'll explore nonlinear functions and situations in which these functions arise. People tend to think of functions mostly in terms of linear functions, but exponential, quadratic, and other nonlinear functions are also common in the world, and they're also important to understand.

 Part A: Exploring Exponential Functions Part B: Exponential Growth Part C: Figurate Numbers Part D: Quadratic Functions

Another important idea in this section is that families of functions can be described by certain characteristics. For example, exponential functions are characterized by a constant ratio between successive outputs. Emphasizing these common characteristics helps us to think about functions as objects of study in and of themselves, and not just as rules that transform inputs into outputs.

Materials Needed: Computers with spreadsheet program, graph paper, toothpicks.

Groups: If it's easy to move back and forth to computers, you may want to use the spreadsheet program throughout the session. If not, you may use it just for Part A, and work with a calculator and graph paper after that time.

Review
Groups: Discuss any questions that came up on the homework. Share solutions to the mobile problems. Talk about the thinking used in solving the problems. Did you think "algebraically?" Did you use any symbols to solve the problems?

Move into today's session by reviewing what you know or remember about linear functions. Some key points are:

 • There is a constant difference between successive outputs of a linear function • Linear equations look like y = ax + b, where a and b are numbers and x is the input • The graphs of linear functions are lines • The slope of a linear function measures how much the output changes for each change of 1 in the input; in other words, it measures rise over run • Slope is constant everywhere on a line • There is only one linear equation that fits any two given points • Rate problems are related to linear functions • Direct variation is a kind of linear function that takes the form y = ax

Groups: You may want to record your comments on an overhead, flip chart, or blackboard.

Linear functions are important in mathematics, but there are a host of other mathematical functions used to model both abstract and real-world phenomena. This session will introduce just a few of those functions.

Groups: Brainstorm any functions that are not linear. You can leave the list on a board or paper posted in the room and add to it throughout the session.

 Session 7: Index | Notes | Solutions | Video