## Non-Linear Functions

###### Third Grade Lessons

**Comparing Different Functions**- The students will discuss, represent, and solve a verbal problem involving the choice between two functions.

###### Fourth Grade Lessons

**Comparing Functions**- This lesson is split into two days. In the first class, the students will analyze eight basic graph shapes and will represent and solve a verbal problem involving the choice between two functions. In the second one they will be asked to choose, among the eight basic graph shapes, the ones that matches specific situations.**Fourth Grade Assessment IV**- This is a written assessment where children will compare two students. One of the students' speed can be represented linearly while the other's speed is represented by a non-linear graph.**Intervals**- Students reason about graphs showing growth over time. They compare heights of children and heights of two animals at different time intervals.**Linear vs Quadratic Functions**- The students will use two functions (a linear and a quadratic) that are represented as a sequence of patterns and create a sequence of hops on the number line and an algebraic expression to express the functions.**Running Race I**- Compare a race between two students: one who runs at a constant pace, the other who tires out as the race proceeds.**Running Race II**- Compare a race between two students: one who runs at a constant pace and one who changes pace as the race proceeds.**Swimming Pools I**- Compare how two swimming pools fill up with water over several hours.**Varying Speed**- Children are asked to tell a story about a trip depicted through a graph that has varying slopes/speeds.**Varying Velocity**- Children are asked to tell a story about a trip depicted through a graph that has varying slopes/velocities.

###### Fifth Grade Lessons

**Basic Function Shapes**- In this lesson, the students will (a) discuss, represent, and solve a verbal problem involving the choice between two functions; (b) choose, among 8 basic graphs (7 distinct shapes), the one that matches specific situations; and (c) write stories to match a specific graph shape.**Review on Graphs and Equations**- In this lesson, the students will solve individually or in small groups the set of problems. For each problem, the teacher will lead a discussion based on the students' work (the teacher should identify strong and weak points in the students' work). The class is organized around four main problems. Within each problem students will answer different questions.**Varying Rates of Change**- Students will compare three functions, two of which are nonlinear, that tell the story of three cousins who all save $1,000 in one year. One saves a lot the first day and less and less each day as time goes on; one saves very little the first day and more and more each day throughout the year; the last cousin saves the same amount each day. Students will be asked to predict the shape of the graph for each function and, later, to look at and describe graphs of all three cousins' savings.

###### Middle School Lessons

**Area of a Square as a Function**- Students will develop a quadratic equation to represent the area of a square.**Box Extremum**- Students will start by finding average rates of change for a non-linear function over increments of the independent variable. The size of the increments will decrease to introduce the idea of using tangent lines to find instantaneous rates of change of linear and non-linear functions. Students will see what a tangent looks like at the extrema of a graph. Students will then create a box that maximizes the volume and see how determining the extrema of a graph can help to find the maximum volume.**Can We Predict Differences?**- Students will predict, produce, and compare linear and non-linear function graphs used to represent the number of punches on a balloon.**Function Challenges - 20 Questions**- Students will compete in a game to generate equations for functions that meet certain criteria, as given by the instructor.**Guess My Rule - Non-Linear**- Students will produce algebraic expressions starting from non-linear graphs produced by other students in the class.**Lotto Winnings**- Students will generate a graph for a nonlinear function, point by point, in order to realize that there are different types of functions that they might not know about yet.**Sound Loudness**- Students will examine a non-linear function depicted in a graph and generate the corresponding function table and equation.