www.inquirymaths.com © 2007 Andrew Blair

PROMPT

24 x 21 = 42 x 12

Solving

Exploration, Planning

Orientation

These notes are not meant to suggest a prescribed path through the inquiry. Instead they offer an ‘exploration zone’ of possible constituent parts. Each class will negotiate its own path, emphasising different parts, by-passing others and developing new ideas. The teacher asks for a question or comment about the prompt (written on the board) from each pair or group. This leads to, for example:  Is the sum correct?  How do you multiply two 2-digit numbers?  One number on the right is half one on the left and the other is double.  The digits on the right have been reversed to make the numbers on the right.  Do these rules always work?  Is it something to do with the digits doubling? (1, 2, and 4) The teacher asks: ‘What shall we do next?’ Students offer suggestions as to how the class can proceed. The teacher might guide the class towards ordering the questions and comments from the orientation phase on the basis of difficulty. Obviously, the teacher should ensure all students can verify the accuracy of the statement by knowing a method for multiplying two 2-digit numbers. Students often want to go straight into cognitive processing (by, for example, ‘finding more sums like this one’), rather than discussing how to regulate the lesson. During a period of exploration, students begin to realise that the ‘doubling-halving rule’ is always correct (and can even be extended infinitely) and suspect that the ‘reverse rule’ is rarely so. They might also begin to realise that if the class does not plan to share results and approaches, then individuals could become involved in lengthy and fruitless exploration. The difficulty students often have in setting collaborative goals and in directing their own learning (in negotiation with the teacher) in the planning phase reflects the ‘strangeness’ of being asked to think metacognitively. It is particularly important in this inquiry that students monitor and regulate their progress if they are not to get bogged down in computations and lose sight of the ‘big picture’.These metacognitive processes must be made explicit by the teacher if students are to learn how to direct their own learning. In the solving phase, the teacher might propose a systematic method (perhaps using a spreadsheet) for finding further cases. One method is to start with a 2-digit number (let’s say, 23) and halve or double it, giving 23 x __ = 46 x __. Now reverse the digits: 23 x 64 = 46 x 32. This phase has highlighted misconceptions, such as students’ attempts to find further cases by adding the same amount to each number (e.g. adding two to get 26 x 23  44 x 14), thus confusing a multiplicative relation for an additive one.

Reflecting, Evaluating

Students might use algebra to develop the inquiry and find a complete set of results, keeping the ‘reverse rule’ and extending the ‘doubling-halving rule’ (1:2, 2:1) to incorporate other ratios. A general equation might be: (10a + b)(10c + d) = (10b + a)(10d + c), simplifying to ac = bd. The inquiry could be extended further to examine the relationships between a, b, c and d for pairs of 2-digit numbers with different ratios.

The class reflects on and evaluates the inquiry process at a cognitive and metacognitive level. In reviewing the ‘pages’ of interactive boards, classes have discussed the (in)efficiency and (in)effectiveness of their methods, and proposed improvements for future inquiries. Notably, in one case, students appointed peers to alert the class if frustration levels rose during future inquiries (after it had proved difficult to find further cases that followed both ‘rules’). This inquiry has spawned supplementary questions:  If you add instead of multiplying, the digits in the answer are reversed (24 + 21 = 45, 42 + 12 = 54). Why does this work? Is it always the case?  Do the two ‘rules’ work with 3-digit numbers? And 4-digit numbers?