median
don steward
mathematics teaching 10 ~ 16

Friday, 10 March 2017

in the limit

puntmat posted an interesting idea (7th August 2014) involving various functions and exploring their limit as n gets very large indeed (tends to infinity)

these resources involve two of their functions plus another one

what fractions are shaded with a colour (out of the whole rectangle) at each stage?
what happens to this fraction as the shape (n) grows bigger and bigger?

all give rise to a common set (or subset) of fractions
with the same limit...
why?



















the puntmat post helpfully included some animated gifs for two of these sequences:




cuboidal parts

this problem varies one posed by Suman Saraf (found here) involving areas

this task considers cuboid volumes (and factors)
a cuboid is cut into 8 cuboids (with 3 cuts, parallel to the original faces)
their volumes (apart from one) are as shown
dimensions are integers
choose a solution that looks about right...


fractions of rectangles

this is practice in areas of shapes
with the question reversed - given the area, what could the shape look like?

students will hopefully seek suitable base and height dimensions
(possibly forgetting that in a triangle the area involves halving)

some questions have two solutions
this can create opportunities to discuss why e.g. triangles with the same base and between two parallels have the same area


it's been suggested (thanks Tom) that you need blank grids on the back of each sheet




these could involve finding areas by dissecting a rectangle

the bottom left two can be justified visually

or you could involve surds

Sunday, 5 March 2017

getting the same number (ii)

extending 'getting a positive whole number' (here) to two expressions

could be a start to solving simultaneous equations





Thursday, 2 March 2017

mean of a frequency distribution, with algebra

my apologies, a previous version of this task did not work...


in questions (13) and (14) the sum of the frequencies is also given (n = 20)

Friday, 24 February 2017

same volume and surface area, different cuboids

it's likely to be a bit of a surprise that you can have two different cuboids (rectangular prisms) with the same volume and surface area

but these are quite rare creatures
[thanks to www.mathspad.co.uk for the two that both have integer dimensions]




Wednesday, 22 February 2017

volume doubling

this is a problem posed by James Tanton
to find solutions, students can use trial and improvement with maybe some thinking about factors







area doubling

students use trial and improvement to try to find values
should all be integer L,
exploring the function




new £1 coin

a new £1 coin is due to be introduced on 28th March 2017
it's a dodecagon
the powerpoint is here
thanks for the insights of Birmingham University ITT students (21st Feb 2017)






thanks to David Wells, 'curious and interesting geometry'

also thanks to David Wells




see chapter 22 of Pierre van Hiele's 'structure and insight'

Saturday, 18 February 2017

45 degree angles

a companion to isometric angles

a restricted set of angles form the interior angles of polygons

to practice multiples of 45 degrees angles:

tap on the end points
students call out the angles
(chanting in unison...)

decide what each interior angle is, then sum them