Normally, when I used to introduce locus my aim looked like this:
Aim: How do we sketch the locus of points a fixed distance from a point?

This year it looked like this:
Aim: How do cell phone towers determine the location of a customer?

Huge difference, and you can note the difference in the level of engagement the minute students read the aim walking into the room.

Before the break I found a really awesome lesson at Mathalicious.com. The lesson looked at locus, more specifically focused on the locus of points a fixed distance from a point. They really did a great job putting this lesson together. Below is a pic of a part of the lesson I did with my students.
Essentially the lesson first seeks to answer: How many cell phone towers does it take to determine a customer's location?
Students then go through a series of activities involving locus to determine this.

There are three main parts to the lesson but since I was using it to introduce locus, we just focused on that part of the lesson that looks at how many towers are needed to track a persons location. (Although we will probably later do the rest since the whole lesson is rich).
I tried to make the lesson more personal to my students by using a map of Brooklyn instead of the provided map of Albuquerque. This was easy, just copied and pasted a map of Brooklyn to sketchpad and had students download the pre-arranged file off my website. 

The lesson is a beauty, I really didn't have to change much. Overall, this really is a great lesson with very good questions. The lesson is also universally accessible, all of my students were engaged and I have the observation report to prove it!

With all the new changes now taking place (eg: Common Core) I believe the one that sticks out the most is: Learner-centered environment. It seems to me, that all of these changes seek to create a classroom environment where the student is doing math not watching the teacher do math. So what does a learner centered environment look like in a math class? I think that this is summarized in the 8 standards for mathematical practice:

1.       Make sense of problems and persevere in solving them

2.       Reason abstractly and quantitatively

3.       Construct viable arguments and critique the reasoning of others

4.       Model with mathematics

5.       Use appropriate tools strategically

6.       Attend to precision

7.       Look for and make use of structure

8.       Look for and express regularity in repeated reasoning

If you think about it, all of the above standards are learner-centered. They are all trying to get students to do something, not watch the teacher do something. Here is where the flipped classroom comes in.

The flipped classroom is a model of instruction in which the traditional lecture in class and homework at home model is flipped. Generally, in the flipped classroom students watch lectures at home while the class time is devoted to exercises, projects or discussions.

In a traditional 40 minute lecture setting classroom you have about 15 minutes, tops, of collaborative work amongst students, due to the fact that you’ll spend 15-20 minutes lecturing, 5-10 minutes on a Do now/warm up and 5-10 minutes going over hw. Is it possible to have any of the above standards taking place in 15 minutes? I suppose it is, but in the flip classroom, I believe the time for collaborative work is maximized.

The flip classroom eliminates the lecture time from the classroom by transferring it to the home thereby possibly doubling the time you devote to collaborative work in the class. In my own experience, I have seen the great benefit of devoting more of the chunk of time to collaborative work. It really makes the class a learner-centered environment and allows for things like constructing viable arguments and critiquing the reasoning of others to occur more naturally, rather than trying to speed up the process because you are pressed for time.

Much of the research surrounding the flipped classroom method is fairly new, but here are some statistics from a survey published in a study by Renner, Hamden and McKnight: 100% of educators using the flipped classroom agreed that after flipping their classrooms learning became more active and 80% of students agreed that the flipped classroom allows them to have more constant and positive interactions with their peers and teachers during class time. Also, 70% of the students surveyed agreed that because of the flipped classroom they are more likely to engage in critical thinking and problem solving.

One of the criticisms for the flipped classroom is that it eliminates the face to face time, or takes away the human element. However, flipping the classroom isn’t just about creating videos and having students watch them at home so that the teacher can sit back and watch students do homework in class. It’s about transferring the lecture to the home so that the classroom can be a community of collaborative learners, a place where students are at the very center of learning. It’s about maximizing the time in class where students are doing math not watching the teacher do math.