pentagons and angles

a ppt is here

wombats and diprotodons

full skeletons of a giant marsupial, the diprotodon have been discovered in Australia
they were calculated to be around 3.2 m long
they are related to the wombat

in 2012, a significant group of about 40 diprotodon remains was found in South-West Queensland

a male wombat grows to around 1 metre long and weighs 28kg





using a length ratio of 1 : 3.8 estimate the weight of the diprotodon


(1 tonne = 1000kg)

which fit better?

which fit around a point better (i.e. with a smaller gap)?

kites and polygon angles

four regular octagons
surround four kites
in a design that forms the basis for a tessellation

what are the angles in the kites?

what are the angles in the kites?

a square
with a regular octagon inside it

what are the angles in the kites?

sampling circles

this task was introduced to me by Mary Rouncefield, Chester University and my understanding is that she and Peter Holmes devised it.

following a discussion about why people sample for statistical work (more economical)
provide students with a copy of this sheet and a ruler

their task is to estimate the mean diameter of all the circles on the sheet by picking a smallish sample (e.g. five circles)

these circles should all have integer diameters, the smallest being 1cm
(it may need to be scaled)

ask students to pick 5 circles at random, measure their diameters and then work out the mean of their 5 circle diameters (so some work on mentally dividing by 5 could take place)

collecting individual's mean results can show some interesting variation

the moral, to be explored in the course of the lesson, and not introduced just yet, is that humans are not very good at picking at random

the next stage is to introduce what the actual mean is, (100 divided by 50 = 2cm)

they can then think about why their estimates based on their samples are too large and consider a better way to select 5 circles at random

this might be, for example, numbering them all and using a random number generator (on a calculator) or in Excel

allowing repeats is important because the sampling procedure would be biased if extreme samples could not occur (the same circle chosen five times)

they can check that a more mechanical method for choosing a sample gives a more accurate estimate of the actual mean and also explore the variation

students should also consider why closing your eyes and picking circles with a pen usually leads to a biased sample (maybe by trying this out)

to emphasise the point that you need a gadget to produce a a set of random numbers, like the lottery, ask students to write down ten digits at random then compare their sets with those generated by e.g. using Excel's random number generator [=randbetween(1,9) dragged down]

divisibility rules

isometric angles

when tackling angle work there seems to be merit in restricting the angles to those found on an isometric grid

with an isometric grid the basic shape of an equilateral triangle can be used to determine other angles

see potentially earlier work on 7 pins

later work can then focus on deciding upon and establishing (i.e. justifying) what various angles are - from a restricted set, to work towards establishing the interior angle sum for various polygons

the 'other' angle to make 360 degrees might also be of interest
'explementary' seems to be a (rarely used) term for this

a multiple number

what is the smallest number such that when you

• subtract 4 from it you get a multiple of 4, 
• when you subtract 5 you get a multiple of 5
• when you subtract 6 you get a multiple of 6?

baby socks

baby socks, looking like shoes, come in three colours
there are lots of these in a drawer, all jumbled up
and it's dark, very dark

(i) how many socks do I need to pick out to be sure of having 2 of one kind?

how many to be sure of having (ii) 3 of one kind? (iii) 4 of one kind? (iv) n of one kind?

car lock

a five digit key pad is set to a number such that the number with a 1 after it is 3 times larger as it is with a 1 before it

what is the code number?

[Moscow puzzles 253]

decreasingly speedy

two part journeys (legs)
average speed over both legs

add one square

the extra square must join along the whole of one edge

not just a corner

three, two, one

2 ways with one dot
3 ways with two dots
4 ways with three dots
then the 'opposites'

on average...

star polygons

the 'youtube' clip on doodling with stars makes an excellent introduction to this topic

the geogebra applet is helpful for checking results

there are several varieties of star polygon on Commons.Wikimedia if larger versions are needed

several other applets are helpful, including this one
or this one

if you mark off 'n' points equally around a circle and then join a point to the one 2 along you get the (n , 2) family of star polygons

and joining 3 points along (as in the diagram) gives the (n, 3) star polygons

students who have studied exterior and interior angles of regular polygons can be asked to find the angles at the 'pointy' bits of the stars

some might then work towards and possibly derive a generalisation (using the rule for the external angle of any regular polygon) for each pointy angle of a regular (n , 2) star polygon is :

then, if we still used LOGO in classrooms, students could be asked to draw the nested star polygons, using sub-routines:

for the (n , 3) family of star polygons:

the task is a little more complex, probably involving angles in kites (or symmetrical arrowheads)

the generalisation for the pointy angles of this family (n , 3) can involve the exterior and interior angle of a regular polygon and is closely related to and can involve the (n , 2) rule:

what happens for other star polygons?

the (n , 4) family:

as might be anticipated, has a generalisation

successive generalisations can be developed either from an appreciation that (n , p) stars have (n , p - 1) stars within them:       

the (12 , 5) star has a (12 , 4) star within it

and the (12 , 4) star has a (12 , 3) star within it etc....












or from using one of the circle theorems

or by considering the numbers of whole turns when you go around a star polygon:

area of any triangle

lengthy expressions

mini times tables

number fitting

vegetable relationships