Linear Equations
These are the lessons I use in my classroom to teach linearitypretty much the 1st semester of Algebra 1 and a couple weeks of Algebra 2. As soon as I can, I push my students towards experiments and messy data because I like the conversations surrounding the value judgments we have to make. Lessons with blog posts associated with them have hyperlinked names.
Ferrari Ride
using D=RT 
Cute 3 Act lesson by Mr. Stadel. My Algebra 1 kids did great at it. It's a good intro to asking mathematical questions of concepts we want to explore together as a class.
<http://vimeo.com/album/2019063> 
Pixel Pattern

Yet another 3 Act lesson, this time by Dan Meyer. I needed to give a little more support on this one. Up to this point, we've not done a lot with creating equations from patterns. If I had to do it over, I'd start the chapter with a review pattern recognition and how to create equations from tables. <http://threeacts.mrmeyer.com/pixelpattern/>

InNOut Burger

I suggest really big group whiteboards for this lesson

Equations of Attack

From NCTM Illuminations. Great activity for helping students build intuition for what the slope and yintercept do in an equation.

Tiling a Patio

One of the tasks from Five Practices for Orchestrating Productive Mathematics Discussions: tiling a garden. It's a task that boils down to a visual pattern.

Tiling a cafeteria 3 ACT task from Dan Meyer with a bit of remodeling. Again, another visual pattern with some support structure built in for the 1st couple weeks of Algebra 1.

I liked some of my book's projects but I present them in slightly different format. Students get to choose from one of the following 3 projects.

Styrofoam Cup Stacking
(y=mx+b) 
If you have the time, order a bunch of 8oz styrofoam cups off Amazon and do this with actual manipulatives. Take a moment and read Dan Meyer's post. It's a great activity for talking about rate of change and yintercept. If you don't have the time to get the cups, I suggest using Andrew Stadel's 3 Act version of this lesson.

This is one of those problems that I can't wait to let kids loose on. I'll push solutions through multiple methods on this one, though I do need to make sure we talk about graphing to cover my objectives for the lesson.

Talk is Cheap activity

All in all the lesson does a really good job helping students see solutions to systems using multiple representations. It does need some updating based on current cell phone rates.

Cup Stacking
(ax+b = cx+d) 
Here's another cup stacking problem that would be best done with actual cups. Get two different sized cups and start stacking them in front of the studentssay 8oz Styrofoam cups and some red Solo cups. At the beginning, the red Solo cup will be much taller. How many cups do you need to stack before they are the same height? Systems of equations? Check.
If you can't get your hands on cups for your students, or if you just don't feel like storing a bunch of junk in your classroom, here's a 3 Act version complements of Andrew Stadel

My search for a new Washer and Dryer turned into a math lesson. Systems of equations.

Music industry revenue

Comparing linear expressions and inequalities using graphs, tables, and equations.
https://clopendebate.wordpress.com/2013/10/09/mathdebatingoverviewmtbosmission1/

Regular or Hybrid?

Another brilliant lesson by Robert Kaplinsky. I've found the amount of detail Robert goes into in his lesson is too much for my Algebra 1 students. I go to the Toyota website and cue up the Camry. We pick a model (usually the SE) and then start the process of comparing the regular Camry to the Hybrid. We use GasBuddy.com to talk about how we plan to estimate gas prices.

Systems of equations. This is the showmeyouunderstandwhatwe'vejustspentthelasttwoweeksdoingandcanapplyittoasituation task.

Inequalities Card Sort

A review for solving inequalities in a card sort format.

Design Inequalities

Students create a design out of inequalities. You can choose to increase the level of the task by including domain restrictions.

This is much like the Barbie Bungee Jump except instead of varying the length of the bungee cord (adding more rubber bands) we change the weight of the jumper (adding washers to a large butterfly paper clamp).

This was a lot of fun but I would only use glass containers next time. The reflections off the metal surface made it too hard for students to get consistent measurements.

Exploring the relationship between the number of spaghetti pieces and the weight the bridge can hold. I've used this for over a decade and the kids still love it.

Lines of Bestfit, Equations from points (PointSlope possibly).
Thanks to Fawn Nguyen: http://fawnnguyen.com/2013/10/10/vroomvroom.aspx

This is probably my favorite lesson to teach. It's such a rich context and you can usually get some good student conversations going.
Equations from points, Lines of best fit, Linear Regression, Correlation, Rational Functions
This last year, I wrote a program for the TI84 which automatically loads the data into student calculators. Just put this program on the calculators and, viola, hundreds of data points!

At this point in the year I expect students to gather data on their own without my help or suggestions. Students were given: 7 rubber bands (including the one tied around Barbie's feet), a Barbie or other dolllike figure, a meter stick, and about 30 minutes of class time. The next day groups asked me for the number of additional rubber bands they wanted and then we threw Barbie of the cliff. It's worth taking a look at the blog post on this one.

Racing toy cars with a CBR and Fines for Speeding.

Linearity and Piecewise linear.
We started the class with a demonstration of how police radar guns work using a CBR and one of my children's toddler toy cars. We talked about the graph, slope, etc. Then I broke out another car and we used the CBR on it. Then we took bets as to which one would win in a race... and then we raced them. This led right into the following task/situation for speeding tickets and linear piecewise functions. Great discussions.

Lego furniture
 linear programming 
Thanks to:
Bob Lochel http://mathcoachblog.wordpress.com/2013/04/29/linearprogrammingwithfriends/ and Fawn Nguyen http://fawnnguyen.com/2012/12/31/legopiecesandfeasibleregion.aspx

Linearity and Inverse Variation

I did a project with my Algebra 2 students early on in the year looking at variation functions (direct and inverse). The flashlight portion of this activity would work quite well for studying linearity.
The next time I try it, I think I'll attempt a 3 Act format.
