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.
The prompt has intrigued class after class since it first appeared on the website. Students' responses typically start with, "It can't be true, can it?"
The inquiry that develops from the prompt involves the manipulation of surds, a search for other equations of the same type, an attempt to generalise from those examples, and, ultimately, the creation of an algebraic proof. It is suitable for 16- and 17-year old students with high prior attainment.
In the classroom, students are quickly hooked into the inquiry. They might assert the equation is true after using a calculator to check but require the teacher's support to reason step-by-step from the left-hand to the right-hand side. Sometimes, students use the regulatory cards to request assistance.
Depending on the prior knowledge, inquiry skills and disposition of the class, the teacher could provide more equations of the same type for verification or encourage students to search for them independently.
Genesis of the prompt
Andrew Blair describes his reaction on being shown the prompt:
"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."
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 the inquiry "opened many cans of worms."
The lesson finished with two lines of reasoning. The first 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."
In the second line of reasoning the students assume that which they have to prove - that is, the left-hand and right-hand sides are equal. Nevertheless, we know the two sides are equal so squaring both sides should maintain the equality.
The problem arises in squaring the right-hand side. As (2a)2 = 4a2, the right hand side gives 4 x 2/3 = 8/3 = 22/3. The mistake is to interpret the square as 4 + 2/3.
Lines of inquiry
1. Verify
The teacher might provide the next three cases after the initial case in the prompt and invite students to verify the equations are true. Alternatively, students might attempt to generate other cases themselves.
The inquiry then develops onto cube roots of the same type and beyond. Students aim to identify the mathematical structure of the equations.
2. Prove
Students in different classes have created two generalisations from the first set of equations (see the illustration).
With assistance from the teacher, the classes have gone on to show the generalisations are true with a formal algebraic proof. The mathematical notes include a proof of the first generalisation.
First publication
The prompt first appears in Tony Gardiner's Mathematical Puzzling (1987).
Gardiner says that "it just so happens" that the equation is true and invites readers to work out how many other cases 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.
