What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
A 'doodle' is a closed intersecting curve drawn without taking
pencil from paper. Only two lines cross at each intersection or
vertex (never 3), that is the vertex points must be 'double points'
not 'triple points'. Number the vertex points in any order.
Starting at any point on the doodle, trace it until you get back to
where you started. Write down the numbers of the vertices as you
pass through them. So you have a [not necessarily unique] list of
numbers for each doodle. Prove that 1)each vertex number in a list
occurs twice. [easy!] 2)between each pair of vertex numbers in a
list there are an even number of other numbers [hard!]
How many different cubes can be painted with three blue faces and
three red faces? A boy (using blue) and a girl (using red) paint
the faces of a cube in turn so that the six faces are painted in
order 'blue then red then blue then red then blue then red'. Having
finished one cube, they begin to paint the next one. Prove that the
girl can choose the faces she paints so as to make the second cube
the same as the first.
There is an important configuration of four congruent
right-angled triangles which make a square in two ways. Either the
sides of the square are formed from each triangle's hypotenuse, or
from putting together the two lengths adjacent to the right angle
in the triangle. This problem uses two of the four matching points
from each configuration. The other two points are not shown and so
exploring this problem will draw students into discovering this
arrangement and also provides experience in how a situation may
sometimes present as only part of a more meaningful or useful
Ask students to guess whether the four points indicated do or do
not lie on a straight line, especially once the triangles are free
to move as a general right-angled triangle.
There is scope for an activity to draw accurately the
configuration shown in the problem, but with right-angled triangles
which students form for themselves.
Alternatively use a Dynamic Geometry package, but in either case
move the group into a discussion about whether these four points
form one straight line or not.
The aim, as the exploration continues, is to cause some
high-energy discussion shot-through with geometric reasoning, where
students challenge each other to say why they are sure that this is
one line. And as they do that the configuration illustrated above
is discovered along the way.
What does this diagram show ?
Are the four points aligned or just close to it, and would that
answer change if the arrangement used right-angled triangles of
another proportion ?
How can you be sure of that ?
What perspective would you try to share with others to help them
see and enjoy the relationships within this