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Surds inquiry

     
I could not believe this prompt was true when I first saw it. The equation comes from Rachael Read, a teacher I worked with in 2006-07 on a national programme run by the Leading Edge schools (UK). One of her year 10 students had stumbled across it. We can only wonder at the levels of enthusiasm and excitement that must have been generated in a lesson that gives a student the freedom to 'stumble' upon something like this. The prompt that I have subsequently developed has caused many audiences (students and adults alike) similar doubts to mine. Typical responses start with, "It can't be true ... can it?"
   
This prompt is suitable for students with high prior attainment in years 10 and 11. In the classroom, students are quickly hooked in to the prompt, particularly when one of them claims it 'works' after 
checking on a calculator. My classes have usually selected the regulatory card "Ask the teacher to explain", thereby inviting the teacher to instruct them in how to manipulate surds. The inquiry might then follow one of two pathways. Either students opt to practice the manipulations in order to become fluent or, if confident, they will explore the peculiarities of the prompt in order to produce more examples of the same type (see mathematical notes below).
Andrew Blair
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Alternative prompts
This prompt has led to a short inquiry by students with high prior attainment in years 10 and 11. The record of a discussion that occurred in a year 10 class can be read here.

James Pearce (a secondary school teacher of mathematics in London, UK) posted this picture on social media, suggesting that the triangles share a "wonderful property". In the discussion that followed the posting, a teacher shared a proof from her year 11 class using Pythagoras' Theorem. If the hypotenuse is 2x√(x), then the length of the two short sides is x√(2x).
True or false?
Stuart Price used the surds prompt for his first lesson with a new year 12 A-level class. He reports that it "opened many cans of worms." The lesson finished with two lines of reasoning (below). One line shows the prompt to be true; the other shows it to be false. Stuart added that "the best part was ending the lesson with the students still looking for the mistake."

Stuart is a teacher of mathematics at Hurtwood House (Surrey, UK). You can follow him on twitter @sxpmaths.
  
It can’t be true ... can it?
The prompt received widespread coverage on social media in April 2016. Richard Green (a mathematician) posted it for his 115 000 followers on Google+ under the title 'It can’t be true ... can it?' More than 150 comments were posted with some arguing about a perceived ambiguity in the presentation of the prompt. Others debated its educational value. You can read the original post and comments here.