a rule for (d)
Friday, 7 March 2014
the task: explore the relationships between the three variables
maybe an obvious 'family' to explore
see also 'octagon loops'
and 'hexagon loops'
Thursday, 6 March 2014
Tuesday, 4 March 2014
by Ivan Moscovich
the four shapes have to completely fill the space
some students might appreciate that each shape needs to have 2 squares (8 divided by 4) in the inner ring and 4 (16 divided by 4) in the outer ring - which makes this task a lot easier (thanks to Jenni Ingram, Oxford Uni. for this insight).
an interesting growing sequence, formed by overlapping two squares, based on one produced by Michael Fenton
different ways of viewing the patterns produces various generalisations
reasons for their equivalence (algebraically) can then be explored
in this case, the 'increases' for each stage can be identified
generalisations can involve quadratic expansions
these are two examples of families of a growing sequence of overlapping squares
what other families are possible?
what might the simplest case look like?
Monday, 3 March 2014
all five lines (of three squares) have the same total
can you find the one way to make lines total 13?
there is also only one way to make lines total 17
there are four ways (ignoring rotations etc.)
to make the lines sum to 12 and 18
totals of 14 and 16 can also be made in 4 different ways (each)
you can fairly easily prove that totals of 11 or lower and 19 or higher are impossible
15 has four solutions as well (I think...) although it appears that there should be more...
although a sytematic approach can be helpful, it is fairly easy to find solutions by playing around
the task provides practice in adding digits repeatedly
and results can be collected as a class
what is probably not obvious to many students is the complementartity: once you have one solution you have another solution by using individual complements to 10 (which is why the sets of results for a particular total are symmetrical)
4 key positions can determine the rest of the grid - it might be convenient to have these as the four corners
the digits in the top right and bottom left hand corner positions are key: they need three ways to combine with different pairs of digits to make the target total