facets

a set of growing shapes made up with squares and equilateral triangles

[a dodecagon after step 2]

with number patterns that aren't too easy to discern
but are regular

what will the next two shapes look like, for n = 5 and for n = 6?

for n = 5,

• why will there be 26 extra triangles?
• why will there be 14 extra squares?

a block (side) of n squares goes to 2n + 1 triangles in the next ring

m triangles goes to? squares in the next ring?

it might also be helpful to focus on the shapes that surround an 'interior' point

a ppt is here

number braids

an article by Richard Bennett appeared in the June 1976 edition (075) of Mathematics Teaching, published by the ATM

he called them 'number plaits' rather than braids

the 'triangle totals' must be in smallest to largest order going from left to right

a variety of 'triangle totals' are (usually) possible

why is there no variety of 'triangle totals' for the 1 to 6 case?














and of course you could carry on...

6 totals occur due to the braids...

square symmetries (i)

these resources link with the fine task 'attractive tablecloths' developed by Charlie Gilderdale for Nrich

initially you could ask students to place 8 crosses on a 4 by 4 grid so that there is a symmetrical pattern for the overall square




















students could be asked to produce e.g. three patterns using 8 crosses so that the whole square has just rotational symmetry, order 2 etc.

can they produce a pattern with just two lines of symmetry (i.e. not with rotational symmetry)?

can they produce a pattern just with rotational symmetry, order 4 (i.e. not with any lines of symmetry)?



describe the symmetries of these patterns for the whole 4 by 4 square

describe the symmetries of these patterns for the whole 5 by 5 square













the NRICH tasks develop this work to consider the maximum numbers of colours that can be used for patterns with various symmetries and for squares of different sizes - a clever link to well known sequences

NRICH provide a set of interactivities for 5 by 5 grids, that can be made whole screen

but... these use flash and may well be blocked in most (all) browsers
[flash player is due to be discontinued in 2020]

one line of symmetry
just rotational symmetry, order 4
two lines of symmetry, rotational symmetry, order 2
two diagonal lines and rotational symmetry, order 2
four lines of symmetry, rotational symmetry, order 4

even sized grids

stack

assuming things continue in the same pattern,
how many cubes are there altogether?

image from Moebius Noodles

Damien Hirst art

Damien Hirst's spot paintings are well known

how would you easily count the dots in these artworks?

circle cutting problem

there's maybe some danger that standards/classics might get lost...

dividing a circle into regions with straight line arcs
with an increasing number of nodes on the circumference
[sometimes known as Moser's circle problem]

successively add points on the circumference of a circle
see how many lines (the maximum number) can be drawn joining these points

and count how many regions the interior of the circle is divided into

try to avoid losing a small region
by not having three lines coincide in the middle

1 dot, 2 dots, 3 dots ...
1 region ,  2 regions  , 4 regions , ...
a clear pattern
does it continue?

wikipedia

















not the same problem
cutting a pancake/pizza into the most number of pieces with straight cuts:

fibonacci patterns

trapped semicircle

another problem, slightly extended, from aplusclick

a semicircle is inscribed in a right angled triangle with sides
( a , b,  c )










what is the radius of this semicircle if ( a , b , c ) =

1)   ( 6 ,  8 , 10 )
2)   ( 15 , 36 , 39 )
3)   ( 28 , 96 , 100 )

what patterns are there in these triples and the resulting radius?
what would the next few be?

Eudoxus' ladder

Eudoxus of Cnidus (now the tip of SW Turkey) was a renowned mathematician (amongst other things) who lived around 400 BC

one of the methods he (reportedly) developed or adopted when studying irrationals was to use a number series (called his 'ladder') in order to approximate to the square root of 2:

how is the 'ladder' formed?
how can it be used to approximate to the square root of 2?









'Reaching the Core of AS Mathematics', available from the ATM, interestingly links this blended recursion 'ladder' to the expansion of:

work out and simplify this expression for n = 2, 3, 4 etc

what has it got to do with Edoxus' ladder and why?

in the limit,

how does this provide an approximation to the square root of 2?

a sequence

a number sequence starts
60 , 120 , 210 , 336 , ....

there is a definite pattern/rule to this sequence

what is it?