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## Quadratic Functions

###### Third Grade Lessons

**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.

###### Middle School Lessons

**Area of a Square as a Function** - Students will develop a quadratic equation to represent the area of a square.
**Biggest Output** - Students will decide on what linear and quadratic functions will result in the greatest output, starting from an algebraic expression, and using tables and graphs to help them make these decisions.
**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.
**Playground Construction** - Students will create a quadratic equation based on the context of building a playground referring to surface, fencing, and equipment needed, to create an equation of y = ax ^{2} + bx + c form.
**Relating Graphs and Equations - Linear and Quadratic Functions** - Students will generate graphs from given equations and equations from given graphs.
**Who Shares My Function? - Quadratics** - Students will make groups by finding other students who have the same quadratic or linear function in different representations.
**x2 and x** - Students will look at x squared and x as functions, and for which values of x one function value is greater than the other.
**Function Challenges - 20 Questions** - Students will compete in a game to generate equations for functions that meet certain criteria, as given by the instructor.
**Graphing Equations - Nonlinear Functions** - Students write equations for three graphs and examine their slopes by comparing and contrasting the graphs. Students also look at the same functions graphed on differently scaled coordinate planes.
**What Will Happen** - Students will work with equations of functions (both linear and non-linear) to find the y-intercept without graphing.

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