# Modeling Equations and Functions with Factoring

## Objective

Students will be able to model and solve equations with factoring.

#### Big Idea

Students will use multiple techniques to solve equations modeling area and the height of falling objects.

## 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- Modeling Equations and Functions with Factoring which asks students to find the two sides of a rectangle given the area.

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

## Area Model

13 minutes

This lesson begins with an Area Model (Math Practice 4).  The first thing the students need to do is draw a diagram and then write a function.  This is a good place to use the Note Card Activity as there is a good chance that there will be some errors and possibly some unusual ways to write these equations.

Once a function has been established, this will be graphed on the calculator and we’ll discuss the domain and range of our model.  The next problem simply asks them to evaluate the function at  A(10).

The next task is for the students to find the dimensions if the area is 680 inches2.  This problem is more involved.  They will be solving: 680 = (x+6)(2x+6) or 680 = 2x2 + 18x + 36.  I will give them some time to attempt this problem.  One method would be to solve this graphically using the calculator.  A student could use the sides of the equations as is or solve one side for zero and graph it that way.  I will give them an opportunity to solve it and then we will look at the methods that students used.  If no one remembered how to solve this algebraically, I will model this for them.

## Solving Equations Graphically and with Factoring

10 minutes

Now I will give my students several polynomial Equations that are factorable.  The first problem is 3r2 = 10r + 8.  The students will solve this problem in whatever method they choose and then I will ask them to find another way to solve it.  We will then pool answers, specifically looking at all the methods we can use to solve this graphically (Math Practice 7).  For example, we could graph 3r2 and 10r + 8 separately and find their intersection.  We could also graph 3r2 - 10r and 8 which would intersect at the same x values.

I want them to be able to solve these graphically, I also want them to be able to solve these algebraically as they may not always have a graphing calculator available.  There are two additional problems with similar steps.

## Height of a Falling Object Model

15 minutes

This lesson is the first of several times that will look at Falling Objects models.  I am limiting this one to factorable situations as this is what they can solve algebraically right now.  We will extend this later on.  I give them the function H(t) = -16t2 + 1600 and tell them that this models the height of an object at t seconds  that is dropped from a height of 1600 feet.   The first thing we do is look at the model and the students will talk in pairs about what each term of the model means (Math Practice 2).  Next, they evaluate the function at H(1.5) and then solve 0 = -16t2 + 1600. I then change the height to 144 feet, ask them to rewrite the function and answer several questions based on this new function.

Next, we discuss the limits of this function in modeling falling objects. The goal is that the students notice that the objects in the problems that we have done have no velocity of their own.  We figure out that this would add a simple rate x times.

I then give them several problems that include an upward velocity and ask the students to find the time when the object will hit the ground.   We do discuss the strengths and weaknesses of solving these graphically or algebraically (Math Practice 5) as one of my big goals for my students is that they can make informed decisions about which method to use rather than just depend on one for all problems that they solve.

## 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 solve an equations using factoring.

## Homework

This assignment begins with eight equations to be solved either graphically or with factoring.  The final two questions are modeling functions, one area and one height of a falling object.  Both of these can be solved by either factoring or graphing.

Data regarding the number of students who choose to graph these vs. solving them algebraically can help drive instruction and allow teachers to become more CCSS aware.