Simplifying surds inquiry

The prompt

Mathematical inquiry processes: Verify; reason; extend to other cases and generalise; prove. Conceptual field of inquiry: Simplifying surds.

Richard Mills and Tung Tran, secondary school teachers at Haverstock School (London, UK), devised the prompt during a school training day. They set themselves the aim of collaboratively planning an inquiry lesson on simplifying surds. 

Finding an intriguing prompt was the starting point. The statement they settled on has hooked classes into inquiry ever since. Surely, students argue, √8 cannot equal half of √32 because 8 is a quarter of 32. Others reason that √8 is just below three and √32 is between five and six so the prompt could be true. 

Yet others might turn to a calculator and report that √8 = 2.828427125 and √32 = 5.656854249. So √8 is just a tiny amount over half of √32. 

To reach a conclusion in the discussion, the teacher should intervene to demonstrate how students can simplify a surd.

Students do not need to know how to simplify surds before embarking on the inquiry. Indeed, it is better they do not know because then the new knowledge comes at a meaningful and necessary time to resolve the intrigue.

Lines of inquiry

Once the class verifies the truth of the prompt, the inquiry could develop into verifying and creating similar statements. 

In another line of inquiry students explain why 'families' of statements cannot exist. In the set below, for example, √8 does not equal a third of64. That is because the prime factorisation of 64 is made up of an even number of twos.

Alternative prompts

Kelly Anne Garner devised this prompt for her middle school students at the Frankfurt International School. She hoped it would lead students to wonder about negative integers, cube roots, exponents within the roots and the roots of non-square numbers. Kelly posted the picture of the students' questions and observations on twitter. She reported that during the inquiry there was "lots of discussion, learning and sharing."  

Aine Carroll‏ posted this picture on twitter. She used the prompt with her year 9 class, reporting that the inquiry involved "lots of amazing conversation and discovery of simplifying surds." Aine described the lesson as an "excellent first inquiry" with the students. (You can read the record of a discussion about the prompt that occurred in a year 10 class here.)