Mathematical Modeling: Linear Functions Design Project

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Students will be able to create a graphical work of art using linear functions

Big Idea

Linear functions form the basis for the more complex functions covered in this course.

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 Linear Functions Day 2, which asks students to evaluate the work of a fictional student who is finding the slope between two points.

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

 At the beginning of this unit, my students switched partners.  For the first couple days in this unit, I provide my students with a get-to-know-you question to help them feel comfortable talking to each other.  Today's question is: What was your favorite thing to play with as a child? Why?

Finding a Linear Equation Review

10 minutes

The project at the end of this lesson relies on the fact that students can write the equations of lines confidently given two points.  This is a check point to make sure everyone is at that level.  I have included two examples.  The first is simpler while the second is not as pretty.   Please feel free to add to or take away from this section as needed by your students.  

Since writing equations of lines is not a goal in Common Core Algebra 2 and so much valuable information can be gleaned from slope-intercept form, I have chosen not to teach point-slope form.  This is an individual choice so please do what is best for your students.  

Vertical and Horizontal Lines

15 minutes

Horizontal Lines

We start the discussion of horizontal lines with this situation: Alvin’s pool stayed at a constant depth of 3 feet the month of August.  The goal is to model horizontal lines or situations where there is no rate of change.  I have the students sketch this as a graph in their notes and explain to their partner WHY the equation is f(x)=b.  I find it really helps when they realize that all of the ordered pairs have the same y value.  

Next, I ask them "What does a horizontal line look like if we were given two points?".  They find something like (3, 4) and (1, 4).  I pick one example and have them find the slope.  I ask them to discuss with their partner how they know that this has a zero slope.  We then discuss how the graph doesn't go up or down,  just right or left.  We can also write this equation like y = 0x + 4. 

Finally, they are asked to come up with a real life scenario to represent a horizontal line and we share as a class.

Vertical Lines

We are going to start by reminding the students of the formula for the equation of a vertical line, x=C (for example). I ask the students to identify the slope.  Vertical lines are different than the other lines we have studies.  Ask them, are these functions? It might be helpful to put up the class definition of a function and prompt students if they can think of a situation that might “work” (function) this way? No matter that the output the input is always the same?

Next I ask them "What does a vertical line look like if we are given two points?".  Students will come up with something like (3, 4) and (3, 6).  I pick an example and have the students find the slope algebraically, which results in a zero denominator.   Please watch my video on dividing by zero for information on how to explain this to students.  

Details and additional notes are in the second section of the lesson PowerPoint.

Linear Art Project

13 minutes

The goal of this activity is to give the students an opportunity to practice finding the equation of lines given two points.  I recommend sharing a sample.  Feel free to use the image of mine but it may help to have something that the students can examine.   I do ask that they use graph paper and a straight edge.  I will count this as a quiz grade since it is a small project. 

The instructions are located on the PowerPoint.  I have also included them in Activity - Design an Initial to give to the students.  

Exit Ticket

3 minutes

This Exit Ticket asks students to write the equation and draw a graph of a vertical and a horizontal line.  This was the new material for the day.  Although this is a simple skill, it is important for the art work project.