Linear equations and functions is one of the largest Algebra 2 units.  It is also one of the easiest to teach, because students are familiar with the concepts from Algebra 1.  The goal of the unit is to refresh and reinforce students' understanding of these concepts and to take their knowledge to the next level.  There are three big ideas to shape the unit:

1. How we represent functions
2. Graphing linear equations and inequalities in two variables
3. Writing linear equations and inequalities in two variables (including modeling).

The full unit takes students from an inquiry review of functions through absolute value functions.  I include:

What are functions?
Domain and Range
Discrete and continuous functions
Slope (or rate of change)
Graphing linear functions including parallel and perpendicular lines
Equations of a line
Variations
Linear inequalities
Absolute value functions

My favorite of the above concepts is probably variations, because they are great for modeling i.e. applying.  Variations are funny in that we introduce these ideas of direct, inverse and joint variation, but still they really are just linear equations.  It's funny how often students don't see these readily as just linear equations.  We start with the basics:  graph y=3x.  This is a direct variation.  y varies directly by x. As x changes y changes by a factor of 3 (in this case).    Or sometimes we don't know the amount by which y varies but we know that y varies directly to x and goes through a point such as (2,-4).  In this case, we just substitute and solve the linear equation for a (y=ax).  -4=a(2) so a =-2.  The variation equation is simply y=-2x.  Inverse and joint variations are examined similarly.

Once we get through the basics, it is on to modeling.  An easy way to apply direct variations is with a job scenario:

You (or insert on of your student's names) get a job as a master storyteller (yes, we like to be silly.  This page has a good list of wacky job titles if you need some inspiration).  For each 5 minute story you tell, you receive \$12.05.  Write an equation for the amount you will be paid for telling x stories.

This one is pretty easy.  If students are still struggling with what makes variations distinct from other linear equations, I follow the above with the following scenario:

You (or insert on of your student's names) get a job as a master storyteller.  You receive \$20 when you arrive plus \$12.05 for each 5 minute story.  Write an equation for the amount you will be paid for telling x stories.

The second problem has the equation y=12.05x + \$20 so now y does not directly vary with x.  This tends to be the "aha moment" for students who had not yet seen the difference between variations and other linear equations yet.

To wrap up variations, I have students complete a short project.  The parameters: present six related real-life problems on variations (2 direct, 2 joint, and 2 inverse).  The problems need to be expressed in words, graphically, and in equation form.  An answer key is included.  Beyond this I just ask them to be creative.  I have had picture books featuring the Three Little Pigs, cubes with music related problems on each face, and Prezi presentations on animal population changes.  Students love the open-endedness of this project.  I love that by thinking about, writing, and solving their own problems, students have a solid grasp on variations and how to apply them.

My second favorite is probably our cost of college project that we do for linear equations (read about it here).  I usually save this one until Unit 3: Systems of equations, because there is a systems component to the project.  Other times we start with the linear equation section and then revisit when we get to systems of linear equations.  Either way, this is one of the most impactful project for students not only to apply math concepts but also to start thinking about and learning about how college costs vary and how depending on your situation a state school might be more expensive than a private school.

This is one of several posts on teaching Algebra 2.  Topics in the series include:
Algebra 2 Day 2
Linear Equations and Functions
Systems of Equations
Logarithms and Exponents
Rational Functions
Probability
Permutations and Combinations

Math Mondays is a bi-weekly blog post (2nd and 4th Monday of each month) sharing tips, ideas, resources, and products for teaching math.  If you have questions or think there is something I should include, you can leave me a message in the comments section below or at the store in the question and answer section.