Products of Polynomial Functions

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Students will be able to multiply polynomial functions graphically and algebraically.

Big Idea

Students get the opportunity to explore and make conclusions about operations with functions using graphing technology.

Warm up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations.  The Video Narrative specifically explains this lesson’s Warm Up- Products of Polynomial Functions which asks students to find the area and perimeter of a rectangle with polynomial side lengths.

I also use this time to correct and record record any past Homework.  

Area Model Introduction

10 minutes

We begin this lesson with an area model involving a pool and concrete patio surround the pool (Math Practice 4).  This will help students gain a conceptual understanding before learning algebraic methods for solving polynomial inequalities.  Students are asked to draw a diagram and then write functions to model the length and width of the pool and patio.  We will then find the area by finding the product of the length and width functions using graphing technology and then algebra. 

Here are some notes on teaching this potion of the lesson using a TI-84: 

I remind them to check their MODE and FORMAT as well as clear out any functions from Y=. 

 f(x) should be typed into Y1 and g(x) into Y2.  Into Y3 I have them put (Y1) (Y2).  (To get to Y1 and Y2, press VARS, >(right arrow), and then ENTER.)  This does the addition for them with out them having to find the answer algebraically first.  Once they have graphed it and then found the answer algebraically, they can check their work by typing their polynomial into Y4.  

 This lesson is run as a guided investigation.  I have the students compete each task and then share with their partner.  We then have a class discussion.    This article, Rethinking Whole Class Discussion from Edutopia includes more information on holding effective class discussions.  

Exploring the Products of Functions

28 minutes

The remainder of the lesson will be spent investigating the properties of the product of functions of varying degree.  Students will be given the graph of two functions and asked to estimate the shape of their product, P(x) (Math Practice 7 and 2).  Once they have a graphical representation of  P(x), we will find it algebraically and check it using the calculator (Math Practice 5).  The goal here is that the students connect the graphical and algebraic versions of the product as well as deepening their understanding of the properties of the graphs and how they relate to each other.  Specific instructions for this portion can be found in the note section of the PowerPoint.

They will look at two linear functions, a square of a linear function, a linear function and a quadratic as well as two quadratics (if there is time).  

Exit Ticket

2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

Today's Exit Ticket asks students to find the product of two functions.


This Assignment reinforces the concepts explored in the lesson.  The students are asked to find the products of several pairs of functions.  They are also given the graph of a cubic function and a linear function and are asked to find the area graphically.  Finally, they are given an area scenario where they asked to find several functions related to the side lengths and area (Math Practice 4).  

note: I have students tape the half-sheet into their notebooks and work on separate paper. You can reformat the handout if you wanted to have students work directly on it.