The prompt is suitable for all classes in secondary school, with the amount of a teacher's guidance and instruction contingent on the students' level of independent inquiry skills and mathematical knowledge. Younger students can substitute numbers into the formula and check whether it satisfies arrays of different sizes; older students can develop other formulae and extend the inquiry into series. The inquiry starts with questions and statements about the prompt:
- What do l, h and n stand for?
- The formula works because 62 + 32 = 45 and there are 45 lines.
- The length is twice the height.
- Does the formula hold true if the ratio of h:l is different?
- Is there a separate formula for square-shaped arrays?
- Does the formula work for other arrays?
- Are there separate formulas for the numbers of inside and outside lines?
- Could n stand for the number of gaps between the lines? (Can we count an 'outside' gap as the same as an 'inside' gap?)
- Does the formula have something to do with Pythagoras' Theorem where n stands for the square on the hypotenuse?
- We should draw another rectangle to check the formula works.
This prompt can intrigue and frustrate in equal measure. Students want the neat formula to 'work' in all cases, although they can realise very quickly that it does not. The formula in the prompt works for the array presented, but only because the length and height are consecutive triangular numbers. For other types of array, the formula does not give the total number of lines. However, this does not stop students from speculating about the conditions under which the formula works. Could it be when the length of the array is twice the height or both length and height are multiples of three? Only after a period of exploration might they acknowledge that the formula does not work far more often than it does, and start the search for an alternative.
At this point, students might suggest (or be guided towards) collecting their results in a table, from which they attempt to induce (or 'discover') a formula. Even when students are successful in this type of enterprise, however, very few can link the formula derived from a numerical pattern to the structure of the array. It is incumbent on the inquiry teacher to draw out an explanation or give an exposition (see box) on how to deduce the formula from the array in the prompt.
Ranelagh Maths Network's critical reflection on this prompt during an Inquiry Maths session in November 2013 was invaluable for the development of the inquiry. The group of maths teachers from Bracknell Forest, Reading and the Royal Borough of Windsor and Maidenhead (UK) meet for professional development activities over the school year in order to keep up to date with developments to the curriculum, assessment and pedagogy. Their organiser is Yvonne Scott (from Ranelagh School, Bracknell) who can be followed on twitter @DancingScotty.
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