Squares and products inquiry
Mathematical inquiry processes: Identify patterns; conjecture, generalise and prove; extend to other cases. Conceptual field of inquiry: Squares, products, algebraic notation, terms and expressions.
Claire Lee, a teacher at Ecolint in Geneva (Switzerland), designed the prompt for her year 6 class. It encourages students to identify, describe and generalise a pattern: The square of a number equals the product of one more and one less than the number add one.
The inquiry involves extending the pattern. Either students can change the left-hand side of the equation and square different numbers (for example. 112 = 12 x 10 + 1); or they can change the right-hand side and explore the product of terms that are more than one away from 10 (for example, 102 = 12 x 8 + 4).
The inquiry is ideal for making the transition from numerical to algebraic reasoning by using symbols to express the generalisation and, ultimately, by developing a proof. (Other inquiries that facilitate this transition are the number line inquiry and steps inquiry.)
In her original inquiry, Claire wanted to make the prompt more intriguing for her class. She chose the more challenging equation 402 = 41 x 39 + 1.
Once the students had verified the truth of the equation, they completed the sentence "I would like to ..." on post-it notes (see the picture below). Their responses included:
Do 40 x 40 = (38 x 42) +? with other numbers;
Work on square numbers;
Use a square root because I don't know a lot about square roots; and
Do the same with 20 x 20.
Lines of inquiry
1. Extend patterns
Students extend the prompt in two ways:
Case 1: They follow the pattern by increasing (or decreasing) the square on the left-hand side, while following the rule for the right-hand side - that is, find the product of the numbers one greater and one less than the number squared, then add one.
Case 2: They find alternative expressions on the right-hand side that also equal 102. This can be achieved by working out the sum and difference of 10 and another number (n); finding the product of the sum and difference; and, then, adding the square of n to the product.
2. Visual representation
A visual representation helps students to understand the structure of the equation in the prompt.
In the first diagram an 11 x 9 array (yellow) is laid over the 10 x 10 array (blue). The overlap (green) does not cover nine yellow and 10 blue squares. In the second diagram, the nine yellow squares have been cut off and positioned over the blue squares. One square remains (red in the diagram).
To promote structural reasoning, the teacher might direct students to create a representation of, for example, 52 = 6 x 4 + 1.
Proofs of the two cases above and the general case, which subsumes the first two, are shown here.
Students will need to know the role of a variable and how to expand two binomials to follow each proof. They could try to construct a proof of n2 = (n + 2)(n - 2) + 1 as a separate case.
The teacher might initially define n and k as any positive whole number to reduce the cognitive demand on students' thinking.
However, once the class is willing to accept that definition, the teacher could orchestrate a discussion about broadening the definition to include negative and rational numbers.
Ann Macdonald, a teacher of mathematics at Longhill High School (Brighton, UK), devised the prompt for her year 10 class. She aimed to develop the students' ability to generalise from a pattern, express the generalisation algebraically, and, ultimately prove it is always true.
In Ann's version of the prompt (see the picture below), the students are given two lines to identify the pattern more quickly. Students have done just that in their initial responses to the prompt. One pair has given an expression for 552, while another uses variables in an attempt to generalise the relationship between the left-hand and right-hand sides of the equation.
The prompt with only one case (the square of 45) is ambiguous. Students soon realise that the type of equation does not work with all two-digit numbers and restrict their search for another example to squares of numbers ending in five.
However, they often go on to generalise wrongly. By noticing that the digits 4 and 5 appear again on the right-hand side of the equation, they conjecture that other examples will follow the same rule. Thus, the square of 35 becomes 3 x 5 x 100 + 25. Further exploration is required to develop the general pattern and its algebraic expression: (10n +5)2 = n(n + 1) x 100 + 25. (See the PowerPoint in Resources for a proof.)