Relaxing at Christmas, it must be time for a puzzle! To play Diffy, draw a square and label each of the corners with some whole number. At the midpoint of each square write the (positive) difference between the numbers at the corresponding vertices. Then draw a new square through the midpoints and repeat the process. Note we ended up with four zeros.

Change the starting numbers, do we always end up with four zeros, and how many steps does this take?

Diffy Excel

Or perhaps it is rather easier to set this out as follows and use a spreadsheet to explore many possibilities.
I was introduced to Diffy at an excellent lecture by Rob Eastaway and have used this successfully since with students from Year 7 (UK age 11-12) to my Year 13 Further Mathematicians!

Having done a little further research I found an excellent post by Don Steward on Diffy where he has numerous excellent questions for students to explore.

Diffy - Don Steward
Diffy – Don Steward

Note The PowerPoint included in Don Steward’s post. We can practise our Algebra too!

Do we need to start with integers?

Diffy - Irrational
For further reading and extension, a very thorough analysis, try Diffy Boxes (iterations of the Ducci four number game) by Peter Trapa, September 27, 2006 and Joshua Zucker on Circle of Differences in Numberplay from the New York Times.

Correlation & Regression

Studying Statistics with my Further Mathematicians I thought I would put some resources together. The use of Technology can really help with understanding here.

This GeoGebra applet allows students to move points and watch the effect on the line of best.
GeoGebra Correlation

This can be used in class by asking students to plot the points, draw their lines of best fit and then comparing with the computer. This worked really well on my phone, I simply sent myself an email with the link and was able to move points easily. This could also be used with younger classes when talking about lines of best fit.

We can also demonstrate correlation coefficients and lines of best fit with this PhET simulation on Least Squares Regression.

Correlation & Line of best fit

PhET – Least Squares Regression

Choose from a range of examples or choose Custom to add your own points and guess then check the correlation coefficient. You can also draw your own line of best fit and compare it to the theoretical line of best fit. Note the option to include residuals for both your own attempt and the line of best fit.

We can check Regression Calculations using this Linear Regression calculator from Social Science Statistics.

Linear Regression

Social Science Statistics – Linear Regression

On the subject of correlation coefficients, we can play a game to see how well we can guess the correlation coefficient! Guess the Correlation Coefficient.


Guess the Correlation – Omar Wagih

From Cambridge PhD student, Omar Wagih ‘Guess the Correlation‘, a rather addictive game with a purpose – Omar Wagih is collecting the data on the guesses collected and using it to analyse how we perceive correlations in scatter plots. Select About to read the rules and further details.

We also need to look at ranked data and students must be able to calculate Spearman’s rank correlation coefficient from raw data or summary statistics. Again, Social Science Statistics, offers us a calculator which will be useful for checking work.

Spearman's RanK Calculator

Social Science Statistics

Calculation details provide a useful check on work.

Note Social Science Statistics also has a calculator for calculating the Pearson correlation coefficient.