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# Mathematical Modeling: Properties of Functions Day 1 of 2

Lesson 1 of 24

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

*50 minutes*

#### Warm Up

*10 min*

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?*"

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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.

#### Resources

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#### Exit Ticket

*3 min*

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

This Exit Ticket asks students to describe the difference between the independent and the dependent variables of a function.

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Mathematical Modeling: Properties of Functions Day 1 of 2
- LESSON 2: Mathematical Modeling: Properties of Functions Day 2 of 2
- LESSON 3: Mathematical Modeling: Linear Functions
- LESSON 4: Mathematical Modeling: Linear Functions Design Project
- LESSON 5: Finding Equations of Parallel and Perpendicular Lines
- LESSON 6: Inverse Functions
- LESSON 7: System of Equations Day 1 of 2
- LESSON 8: Systems of Equations Day 2 of 2
- LESSON 9: Modeling Systems of Equations
- LESSON 10: Function Review
- LESSON 11: Function Mid Test
- LESSON 12: The Tortoise and The Hare Project Day 1
- LESSON 13: The Tortoise and The Hare Project Day 2
- LESSON 14: The Tortoise and The Hare Project Day 3
- LESSON 15: The Tortoise and the Hare Project Day 4
- LESSON 16: The Tortoise and the Hare Piece-wise Functions
- LESSON 17: Step Functions
- LESSON 18: Absolute Value Equations
- LESSON 19: Absolute Value Inequalities
- LESSON 20: Absolute Value Functions: Transformations Day 1
- LESSON 21: Absolute Value Functions: Transformations Day 2
- LESSON 22: Functions Test Review Day 1
- LESSON 23: Functions Test Review Day 2
- LESSON 24: Functions Test