hollow rectangles

an exploration of surrounding paths
the powerpoint is here

this work could follow on from hollow square

for hollow squares see also the approach nrich 11257

some problems involving paths around lawns
the powerpoint for these is here

equable orthogonal octagons

an equable shape has the same value for the area as for the perimeter

set up an equation
from A = P
and solve it

maybe consider different ways to chop up the octagons to ascertain their areas

equable orthogonal hexagons

an 'equable' shape has the same value for the perimeter as the value for the area
this work can involve setting up and solving linear equations and forming linear relationships
and generalisations (that can be proved)

an orthogonal polygon just has right angles and 270 degree angles (that I like to call 'left' angles) inside it

[sometimes these are called 'rectilinear polygons'
but 'rectilinear' idetifies shapes that are 'bounded by straight lines' rather than those involving multiples of 90 angles]

equable parallelograms

 value of perimeter = value of area

equable shapes

'equable shapes' can be an interesting study

these resources practice finding areas and perimeters of common and compound shapes, leading possibly to solving equations

the powerpoint is here

for rectangles, it can be beneficial to appreciate the function/relationship by looking at a full range of values (not just integer values, as is usually a restriction for equable shapes)

the values for 'a' are selected so that the corresponding values for 'b' can be calculated, maybe without a calculator
and a graph of (a, b) values could be plotted - or at least considered

square is best

this work involves the area and perimeter of rectangles
(i) fixing the perimeter to be 16cm
(ii) fixing the area to be 36 carpet tiles
in this way two important functions (quadratic and reciprocal) can be considered
and number work practiced along the way...

the geogebra applet by Anthony Or is helpful, with a perimeter of 20 rather than 16

area and perimeter of rectangles

this task could be initiated by asking students to draw several rectangles on squared (or dotty) paper
they find the perimeter ('P' number) and area number ('A' number) (using the size of square provided as a unit)

then invite one of them out to let the class know the P and A numbers for one of their rectangles
the class try to work out the dimensions
etc.

discussions might focus on the two sizes of rectangles (including squares) for which P = A
and then for which A = 2P

finding examples that fit various relationships

linking P with A for various widths

continuing a pattern

providing reasons why rules work from a diagram

reasons can become difficult...

maybe focus attention on:
the four corners
the (non-corner) left and right edges
the lengths

plotting the rules for particular fixed widths

what are the intersection points?

forming and solving simultaneous equations in P and A

various hard questions....