Squaring numbers inquiry
Mathematical inquiry processes: Verify; extend the pattern; generalise and prove. Conceptual field of inquiry: Square numbers; algebraic notation; rearrangement of expressions.
Claire Lee devised this inquiry for her year 6 class. Using the prompt 40 x 40 = 39 x 41 + 1, she reports that the inquiry involved high levels of motivation, rich activities and the introduction of many new topics:
I had seen someone's work about asking students to start with 2 x 2 then look at 1 x 3 and it built from there. One of the things I loved was the way students started to change the prompt. They predicted that moving two numbers, for example 40 x 40 = 38 x 42, would mean you had to add 2 but then quickly picked up that there would be a square number pattern. We got a lot of new material out of it: square roots, exponents, some simple multiplication of two digit numbers and place value. I am really enjoying using the idea of a prompt to start the lessons.
Arithmetic to algebra
The prompt can develop into proving that the difference between n2 and (n - 1)(n + 1) is always one. It is possible to bridge between number and algebra through the same approach to multiplication. The grid method for finding the product of two double-digit numbers can be used to find the product of two algebraic expressions. The method has the advantage of showing the structure of the calculation.
It can also be shown that the difference between n2 and (n - 2)(n + 2) is 4 and, in general, the difference between n2 and (n - k)(n + k) is k2.
The prompt designed by Ann Macdonald (a maths teacher in Brighton, UK) is made up of two equations. The picture shows the initial questions and observations from year 7 students. They have tried to identify, extend and generalise a pattern and have wondered about other cases (such as, negative numbers and cubes). When the square is in the form (10n + 5)2 students can verify the equation is true for different types of values of n. From the general form of the equation
(10n + 5)2 = n x (n + 1) x 100 + 25
they can show that both sides equal 100n2 + 100n + 25.
A more open prompt uses only the second equation:
452 = 4 x 5 x 100 + 25
This might lead to a longer period of exploration and different conjectures. One conjecture, which turns out to be false, is that the form of the equation works for all double-digit numbers (for example, 242 = 2 x 4 x 100 + 16 with the addend being the square of the '4').
In his book Getting the Buggers to Add Up, Mike Ollerton discusses the prompt in the following form:
(115)2 = 11 x 12
He explains that the '25' in the square brackets emerges from 52 and this forms the tens and units digits when combined with the product of 11 and 12. Thus the answer becomes 13 225. In fact, placing the '25' at the end is equivalent to multiplying the product of 11 and 12 by 100 in the original prompt. Mike explored the prompt with a fellow participant at a conference session. They asked why the prompt worked and whether it would work in a similar way when squaring numbers that end in six, seven or any other digit.