Mathematical Modeling: Linear Functions

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

Big Idea

Linear functions review in the context of real life scenarios.

Warm Ups 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. 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?

Representations of a Linear Function

10 minutes

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. 

Identifying the Slope (Rate of Change) in a Linear Situation

10 minutes

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.

Modeling Linear Functions

17 minutes

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.

Exit Ticket

2 minutes

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.


1 minutes

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.