Modelling, Inquiry, Games & Datasets
'Teaching Mathematics via Video, Technology, Investigation, Modeling and Exploration '
There are lots and lots of great videos out on the net that make an engaging introduction to mathematical concepts and techniques but finding them can take time!
Session 1: Inspiration and inquiry - videos can bring the outside world into the classroom and provide the inspiration of where technology can take them. This is then backed up with a range of rich and engaging ready to go lesson activities based on an inquiry learning approach.
Session 2: Modelling using Tracker and graphing and spreadsheet packages such as Autograph, Geogebra, TiNspire etc.
Session 3: programming to understand mathematical concepts of variable and function in the context of simple game making.
A number of investigations and modelling opportunities result from the session above for which Autograph, Geogebra, Desmos, TiNspire [or other graphing/geometry software participants may have] really come into their own.
EXPLORATION ideas
Session 1: Inspiration and Inquiry
2. Sequences and 
Physical World Sequences
4. Wave Transformations 
How your ear hears (first 5m10 for IB trig classes, first 2m25 as an intro to Tri, or from 3m to 5m10 as a fractions starter!) and the amazing world of Waves, how they define physical realities. Here's another experiment you can do at home to produce waves (fizzics.org - watch from 2m48 to 3m15).
5. Audacity download. Dog Whistles and Varying Frequency and Wave Form website provide interesting sound files for students to analayse. Use Autograph to Model the wave. This use of audacity comes from Richard Wade - try this ready to go lesson resource with your classes: Trig Modelling - Sound
6. Twitter, Facebook, Edmodo, Geogebratube and Youtube channel recommendations below to get you started:
8. Normal Distribution and the 2008 Financial Crisis
9. Percentage Investments - Geogebra help videos below:
11. Olympic Records
11. UK house prices - Piecewise Functions Using Autograph
Lots of sources of exploration ideas (TED videos, podcasts, websites etc.) are listed on the TSM2014 Bringing the Outside World In page.
Optional Extensions
Create a magzinr page, create a classtools game and think of an activity idea to accompany it - maybe for one of your lessons next week - quick example here.
Create a Magzinr account
Magzinr is a book marking tool that allows you to create pages like Scoopit!, Diigo, Decliious etc. but with the added benefit of embedding your individual pages into a website e.g. 
Number Movies Library
Add magzinr to your tool bar now and create a page to embed.
Session 2: Modeling
Vernier Data Sensors
Data sensors offer a rich vein of hands-on-mathematics for the mathematics classroom (and experience of common practices in most scientific/mathematical careers.
Tracker
Great for modeling! (and hence for the International Baccalauréat Exploration at Higher and Standard level 16+ mathematics). The below video offers a quick overview to get you and your students started.
Download the software here: 
Tracker.
Volumes of Revolution - Autograph & Flipped Classroom
Ezvid download (FREE - scroll to bottom of the page for the download)
Snagit (£20 one off payment)
Modeling Real Data - Histograms and Normal Distribution
Students can analyse (and/or create) a set of normal data to represent this shoplifting scenario. This can be done in both Autograph and Geogebra, but it's a little quicker and easier for students to do using Autograph.
Desmos Functions Carnival
Session 3: Programming a Simple Game
Scratch
This activity from Richard Wade: Double Spinners makes a nice introduction to Scratch.
Online (no need to download) version of SCRATCH
Joining the scratch community online to see code, share and create!
Geogebra Games
Attaching an image, animating it and using Boolean operators are key building blocks to creating short "mathematical games" within which your students can explore a particular area(s) of mathematics: Inequality Pacman
Bootstrap / Racket
One application of programming in the mathematics classroom is as an engaging, and concrete, introduction to the concept of a function, defined by a set of inputs (domain) and a set of outputs (range). This is just a quick introduction to the US "Bootstrap" programme which is designed to introduce students to functions and coordinate geometry through the medium of the programming language Racket (a complete programming language).
Introduction to Functions and Wescheme (Racket)
Simple Boolean Operators - you will work from this file: CAGE
Introduction to "Piecewise Functions" in a programming environment - MONSTER game template.
Big Data sets and other statistical things . .
Websites such as Gapminder, Guardian Data provide huge data sets with which students can investigate the world they live in and developed a more representative view beyond their own personal experiences on a wide range of subjects. Google documents also provide huge opportunities for the development of data sharing projects, such as this one (founded by Jim Noble): International Student Survey which can be married with live skype presentations between participating schools. Once again, technology such as Autograph, Geogebra, TiNspire etc. provides the tools to analyse, investigate and explore the world we live in.
"Whoops" author John Lancaster provides some analogies (thanks to Primary colleague Simon Gregg for the book)What is the difference between a y on x and x on y regression?
The "Cross Validated" site for statisticians offers a good visual explanation and example, for why it is not logically consistent to use a y on x regression equation to predict x values for a given y. In the late 70s/early 80s a case was made in the US that women were being discriminated against in the workplace. The case used regression anlaysis to show that for men and women with the same qualifications (x = qualifications), women were paid less (y = salary). Those who opposed the discrimination argument countered that if this were true it should follow that for men and women with the same salary levels (y), women should, on average, have a higher qualification (y) - this was not found to be the case. However, an argument and its converse are not logically equivalent statements (IB maths studies opportunity!), "If you stay outside naked in a snowstorm, then you will catch a cold" and "If you catch a cold, you must have stood outside naked in a snowstorm".