Surds inquiry

The prompt

Mathematical inquiry processes: Verify; reason; extend to other cases and generalise; prove. Conceptual field of inquiry: Surds; mixed numbers and improper fractions.

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 the mathematical notes for an algebraic approach to the general case and for an extension to cube roots.)

Andrew Blair

Alternative prompts

Richard Mills and Tung Tran, secondary school teachers in north London, devised the prompt during a school training day when they were collaboratively planning an inquiry lesson. Once the class verifies the truth of the prompt, the inquiry could develop into finding similar statements. For example, √(27) is a third of √(243), √(8) is a quarter of √(128) and √(216) is two-thirds of √(486).

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.)

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."

True or false?

Stuart Price, a teacher of mathematics at Hurtwood House (Surrey, UK), 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."

Resources

First publication

The first publication of the prompt we have found is in Gardiner's Mathematical Puzzling (1987). He says that "it just so happens" that the prompt is true and invites readers to work out how many other equations there are like the one in the prompt. In his 'commentary', Gardiner asserts there are lots and reminds us that √(22/3) means 'root two and two thirds', which is √(2 + 2/3), whereas 2√(2/3) means 2 x √(2/3). Later in the book Gardiner suggests four ways to tackle the problem (see Extract pp. 140-141).

Acknowledgement

Jason King contacted Inquiry Maths with the reference.