median
don steward
mathematics teaching 10 ~ 16

Sunday, 24 April 2016

combined enlargements

what happens (in general) if you do one enlargement scale factor 2 from one centre and then enlarge the resultant shape from another centre, also scale factor 2?

the ppt is here
it needs to be downloaded for the animations to work

enlarge the brown triangle, scale factor 2, to obtain the blue triangle


then enlarge the blue triangle from a different centre, to obtain the red triangle

the two enlargements, one after the other

students could either do their own or use these resources


















the third triangle can also be obtained from the first via an enlargement

what will the scale factor be and where will the centre of enlargement be?

it seems as if the three centres have some relationship

check this with the ones drawn above
















what happens with different scale factors?






















the proof of the relationship can be done using vectors

this was done by James Pearce, of Mathspad fame

many thanks for the very neat vector proof, which he did generalise



Thursday, 21 April 2016

presentation

this is a copy of my LIME presentation after school today in Oldham (21/04/2016)
please download it to get the animations

many thanks to Lindsey Bennett and the Radclyffe School for organising this
it's great that such clustering events happen

Wednesday, 20 April 2016

barn roof angles

involving angles in polygons
creating a generalisation




Tuesday, 19 April 2016

randomly generated numbers

numbers generated using Excel 2007 (=randbetween(1,6))
concerns have been expressed about this random number generator passing standard tests for randomness but it is deemed to be good enough for illustrative purposes
and is so much quieter than dice rolling





Sunday, 17 April 2016

scatter graphs with algebra

using the fact that the line of best fit passes through
the mean of the x and the y numbers
find an added point from the old and new line of best fit equations 

involving straight line graph rules as a possible extension to work on scattergraphs




averaging expressions

extending finding the mean of numbers to this average of expressions
simplification and substitution, involving subtracting negatives

students could be asked to create their own sets of expressions

eating olives

thanks to professor smudge (@ProfSmudge) for featuring and providing this context for a % question by John Mason (RME 17.2) where he used the third question on the classroom resource as an example of mathematical unexpectedness

they can be solved in a variety of ways

 a mixture of olives
 a simpler view






Wednesday, 13 April 2016

pyramid pile of oranges

Roelof Louw's conceptual art involves viewers in eating the fruit

as Gavin Wraith points out (in a letter to the i paper, 13/04/2016) it would have been (numerically) more interesting if the square based pyramid had been 24 tiers high

at 25 pence per orange, how much would a pyramid this size cost?
if each orange weighs, on average, 80 grams how much would a pyramid this size weigh?




Sunday, 10 April 2016

generalising GCSE questions (16)

a past problem from Eduqas asks students to find a set of five numbers fitting given rules about the averages and range

it is interesting that there are two such solutions

















what happens for other values for the averages and range?

 prove an impossibility
explore (by substitution) some general rules that work for a limited set of values

set up inequalities for 'k' and 'n' for which these general rules work

generalising GCSE questions (15)

an AQA question about finding nth term values for numbers that are in two (linear) sequences

a more general question might be: what numbers are in the (infinite) sequences
2n + 1 and
3n - 1?

other 'overlapping' sequences can be explored

some don't have an overlap - when?




some examples




a rule for finding the coefficient of 'n' for the 'overlap' sequence (12 in the above examples) is reasonably easy to discern

a rule for the constant term in the expression is slightly more difficult to sort out

it needs to be appreciated that an infinite linear sequence (indicated by ...) has an infinity of general (nth terms)
e.g. the nth term rules: 3n - 1 and 3n + 2 and 3n + 5 and 3n - 4 etc. all give the same (infinite) sets of numbers
they just start in different places

for the given nth term rules, create other nth term rules that give the same sequence of numbers









any of the common constant terms of the two expressions can form the constant term of the 'overlap' nth term rule

e.g. the nth term for the overlap of nth term sequences 3n - 2 and 4n + 1 is        12n + 13 or 12n - 11 or...