symmetrical designs with letters and words

rotating capital letters through 90 degree turns

kite areas

what are the areas of these kites?

how much bigger is this kite?

which quadratic graph?

nonagon appreciation

a nonagon has several properties involving some nice simple angles

a ppt is here

show that a + b = c

ring alignments

if you have a ring of regular pentagons

it takes 10 to complete a full loop

why is that?

















if you have a ring of regular decagons

it also takes 10 to complete a loop

why is that?

















then one seems to fit snugly inside the other

how come they align so neatly?

hexagon fractions

what fraction of the
hexagon is
the orange shape?


















this work has been developed and discussed in detail by Derek Ball and Barbara Ball (MT March 2008)

the bottom right-hand picture is a useful starting place
folding bits inwards needs some additional justification - but is a helpful insight/start

the top left picture can also use this way of viewing (folding technique) to help establish the fraction

the other two pictures can be seen as repeated applications of this technique

pentagon angles

the three angles at each corner (vertex) seem to be equal
are they?

by symmetry the angles in  the yellow triangles must be the same

how can you show that the angle in the blue triangle (b) also = a?

write down an equation involving 'a' and 'b' for the yellow triangle

write down an equation involving 'a' and 'b' for this brownish triangle

use these two equations (solve them) to show that a = b

transforming expressions, snakes (and worms)

area and perimeter substitution

racetrack substitution game

substituting steps

relating charts to data sets

heights and age

compare the differences in median height for each age from 2 to 20

directed number arithmetic sped up

the rules for combining directed numbers with addition and subtraction are probably too burdensome for memory so some awareness needs to be cultivated, over time

some good graphics have been prepared by Shana McKay here
and dropbox versions here
many thanks to her

once some facility with negative number additions and subtractions has been attained, it is desirable to move towards automating (i.e. no need to think too much) these skills

this is an important step in the mathematics curriculum
many difficulties in algebra can sometimes seem more related to a lack of understanding of directed number techniques 

my view is that sophisticated directed number calculators recognise types of sum and can apply a technique for that general type

presenting students with a sheet of sums, they can work together on grouping the sums by type, explaining that the result (answer) and numbers involved are not important - it's the signs that need to be focused on (the 'form'):

it may be better that students choose how to group these sums themselves, with plenty of discussion,
but another sheet has them already grouped by types:

students can then be given or find some results
and then consider and create general rules for getting results for each type of sum, for themselves

e.g. for type (c) "you subtract the smaller from the larger and give it the sign of the larger (because this 'wins')"

e.g. for type (e) "two minuses don't make a plus; one story could be: if I take £7 from you and then take £12 from you, I've taken £19 from you altogether"

they can also appreciate, after a while, that types can be 'collapsed' to other types
e.g. (e) to (b) and (g) to (a)

and here is a test!

house to square

you could use pythagoras and the cosine rule to find out where P is located along the side of the square

but you don't need to


having decided where the point P goes
check that the angles and sides do actually create a square