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# Mathematical Modeling: Linear Functions

Lesson 3 of 24

## Objective: Students will be able to write the equation of a line given information (two points) for a linear relationship.

*50 minutes*

#### Warm Ups and Homework Review

*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. This lesson’s Warm Up- Modeling Linear Functions asks students to come up with three real-life examples of slope, as a rate of change. My goal is to get students to recall what they know about slope as well as prepare them for the activity later on in the lesson.

I also use this time to correct and record the previous day's Homework.

NOTE: At the beginning of this unit, my students switched partners, so for the first couple lessons in the unit I provide my students with get-to-know-you questions to help them feel comfortable talking to each other. Today's question is: *What was the weirdest food you've ever eaten?*

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Is It Linear?

I begin this lesson by having the students do a think-pair-share about the question "How can I tell if a function is linear?". It is important that they know when a function is linear graphically, algebraically and even in terms of a real-life scenario. Algebraically, it is important that they can differentiate between a linear and a non-linear function (we will be practicing this later in the lesson). If any of these representations aren’t brought up, I pose them as questions ("How can I tell from the equation?").

Slope-Intercept Form

Once the students have a decent concept of what makes something linear, we look at a real-life situation. "Alvin has a pool that is filled to a depth of 80 cm of water. He is emptying it at a rate of 12 cm per minute." This was a problem that we looked at in the previous lesson, and we use it to discuss two forms of a linear equation, slope-intercept form and standard form. We also discuss what the y-intercept and the slope mean to a real life situation.

This key here is that the students link the concepts of slope and y-intercept to the situation. This will make a huge difference as we work through problems later in this lesson. I have them verbalize the pattern. For example, “There is 80 cm of water and each minute there is 12 cm less.” I have found that students really struggle with the idea of slope as a repeated addition or subtraction. Associating slope to the real-life situations and focusing on the idea of repeated addition/subtraction will help **(Math Practice 8).**

Next, they write down a description of two forms of a line, slope-intercept and standard. I have chosen not to include point-slope form as slope-intercept works just as well when using points to find an equation.

#### Resources

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In the previous lesson, each pair of students wrote some two variable situations. I pick out a variety of these problems, both linear and non-linear, and read them to the class. Using a thumbs up/thumbs down approach, they identify whether each situation is linear, and then if it is linear, identify the slope. The needs of the class dictate the number of examples that I use. This activity can also be done using personal white boards.

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#### Modeling Linear Functions

*17 min*

In this section, students review writing the equations of linear functions using two ordered pairs from a real-life scenario. The first scenario says "Ann is saving money for a senior trip. In 2 months, she has $150 and in five months she has $315".

They start by finding the slope. I chose to embed review of finding slope into this section on writing equations rather than have a re-teach lesson. I give them a chance to find the slope with no introduction and then judge how much review (if any) is needed. As appropriate add additional practice of finding slope here. ** **

Next, they find the y-intercept. Again, I allow the students to find the y-intercept with no initial help. I chose not to include point-slope form but if someone brings it up, I encourage them to share it. If no one shares it, show them how to put the slope and one point into slope-intercept form to find the y-intercept.

Finally, they use the slope and y-intercept to write the equation of the line.

There are two additional scenarios where the students write the linear equation in slope-intercept form.

For all three problems, the students identify what the slope and the y-intercept mean in the context of the problem scenario.

Please see Section 2 for the PowerPoint.

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#### Exit Ticket

*2 min*

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

Today's Exit Ticket, located in the PowerPoint, asks the students to find the mistake in a slope problem. While not the center of the lesson, it is an important skill and could be covered again in the next lesson if a lot of the students are struggling.

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#### Homework

*1 min*

The assignment consists of four linear scenarios. They are given two pairs of data for each problem and must write an equation in slope-intercept form. The goal of this assignment is to give the students a chance to practice finding equations of lines on their own. This is a direct application of **Math Practice 4 **and for many students will also allow them to work on **Math Practice 1**.

#### Resources

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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