 n 
Showing that  r    r 
r 1
 r 1 
n
2
3
n
1. Print off the two pages below and stick them on to separate sheets of
sturdy card. The card used on a typical pad of A4 paper works well.
n
rr  r
2
r 1
3
r 1
2. Using the perimeter outlines, cut out these two pieces of card. For each
shape, cut along the diagonal and then take the piece above the diagonal
and make the two vertical cuts down to diagonal so that you are left with
four shapes from each piece: the big piece and three trapezia.
Place paper-side down on the table.
4. Stick the two large pieces of card back-to-back and leave to set.
5. Insert the three pins into the corresponding point in the large shape
between the two pieces of card. If this is done correctly it should be
possible to rotate the trapezia so that the top shape (on the right) is
transformed into the bottom shape.
 n 
r
 r 1 
2
2
 4 
This demonstrates  r    r  . This could be done numerically –
r 1
 r 1 
what do you need to show to turn this demonstration into a general proof?
4
3
© MEI 2007
1+2+3+4
4×42
3×32
2×22
1×12
3. Place one pin as shown on one trapezium so that it perpendicularly bisects
the diagonal edge. Using wood glue or other strong glue, stick together the
two trapezia of the same size so that the pin is between the card and the
paper is outermost. Repeat for the other two pairs of trapezia.
© MEI 2007
© MEI 2007