Mathematical Modeling: Properties of Functions Day 1 of 2

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Objective

Students will be able to model functions and identify their important features.

Big Idea

Student created situations populate this modeling functions lesson plan.

Lesson Author

Amelia Jamison

Caldwell, ID

Grade Level

Eleventh grade

Subjects

Math

x-intercept

Algebra

Standards

HSA-CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HSA-CED.A.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

HSF-IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

HSF-IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

HSF-IF.C.7a

Graph linear and quadratic functions and show intercepts, maxima, and minima.

MP2

Reason abstractly and quantitatively.

MP3

Construct viable arguments and critique the reasoning of others.

MP7

Look for and make use of structure.

Warm Up

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. As seen in the video narrative, this lesson’s Warm Up- Modeling Functions asks students to model a linear situation which is the introductory example in today's lesson.

Since this is the first day of a new unit, my students are placed with new partners. Since they will be having a lot of conversations with this person, I provide them with a conversation starter at the beginning of the first several lessons in this unit. Today's starter is: "What was your favorite toy growing up?"

Introduction to Modeling and Parts of a Function

37 minutes

The warm up is the introduction to today's lesson. I begin by talking about the expression from the warm up: Alvin has a pool that has 80 cm of water. He is emptying it at a rate of 12 cm per minute. This is a similar problem to the ones done in the first unit. I explain that we can rewrite this as a two variable situation: y = 80 – 12x where x represents minutes and y represents the depth of the pool.

I then introduce function notation as a different method for writing the same scenario and relate that f(x) is another method of writing y.

Now we review evaluating functions with function notation. I show them how to find f(1). Then the students try the second one, f(3), and the third one, f(10). I ask them to relate each answer back to the situation as this keeps their mind on the scenario which will deepen their understanding of how function notation works (Math Practice 7).

The students then make a table of the first five minutes of this situation and graph the ordered pairs. I make sure to have them label the intercepts with both the variable and the unit. Either a student that I choose or myself graph it on the white board to help with next portion of the activity.

Once they have a graph, we talk about some of the important terms associated to the graph of this function including x- intercept, y-intercept, domain, range, and independent/dependent variables. We define each of them and relate them back to both the graph and the situation (Math Practice 2).

I present them with a formal definition for a function and then we look at a non-linear model. Write an equation that relates the side of a square to its area. The students write an equation, make a table and graph it, and then identify the x-intercept, y-intercept, domain, range, and independent/dependent variables of both the graph and the situation. This helps them solidify some of these introductory concepts discussed in the first example.