Equable shapes
Purpose
Explore a rich activity and
 allow learners to make decisions
 involve learners in testing, proving, explaining, reflecting and interpreting
 promote discussion and communication
 encourage ‘what if’ and ‘what if not’ questions
 create connections between topics
Possible Mathematical goals
To enable learners to
 fix the difference between area and perimeter
 use trial and error to explore a problem
 use an analytical approach to solve a problem
 explore generalisation and proof
 practice solving simply equations
 practice solving quadratic and higher order equations
 practice solving trigonometrical equations
 practice various fraction operations
 plot coordinates to produce a graph
 interpret a graph as a solution to a complex problem
 interpret a graph in terms of real and unreal solutions
 find surface areas and volumes of 3D shapes

solve problems involving surface areas and volumes of 3D shapes
Materials needed
Pencil, ruler, calculator, graph paper
Suggested approach
Try this activity with a colleague and consider how you would use this with your
learners.
Introduction: An equable shape is defined as a shape whose area and
perimeter are numerically equal. Why do we have to say numerically?
This task can be tackled in a variety of ways. Here is one which has worked very
effectively many times:
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Consider equable regular shapes. How would you introduce the idea of regular
and irregular shapes?
It is fairly easy by trial and error to find the equable square 4x4, but in doing this
learners may draw a large number of squares and calculate area and perimeter
many times. Using an analytical approach leads to the equation:
a2 = 4a
and solving…
a2 - 4a =0
a(a-4) =0
a=0 or a=4
Why is it important to acknowledge the solution a=0?
This method may lead to a discuss about a formula for other regular shapes: an
equilateral triangle, pentagon, heptagon, circle etc. Why can we consider a circle
in this sequence of polygons?
How would you initiate this discussion with your learners?
Investigate alternative ways to generate equable equations for other regular
polygons. Can you generalize this for all regular polygons? What does the graph
look like?
How could you get your learners to do this?
How would you introduce this idea of an analytical approach?
Working in Groups
Use the guidelines for learners working in groups sheet in the post-16 module to
engage active learning in groups. Having discussed this whole approach to group
work chose one statement to focus on in each session until these become
second nature both for learners and teacher. Which statements in these
guidelines are most important for this activity?
Reviewing the Learning
Consider how this activity could fit into your curriculum and maps to your scheme
of work.
What are the advantages of starting with a rich task? What are the
disadvantages?
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Further Learning
This activity neatly links the basic geometrical topics of area and perimeter with
simple linear, quadratic and trigonometrical equations, fractions, graph plotting
and interpretation in the context of problem solving and mathematical modeling.
Extensions using a range of other 2D and also 3D shapes can lead to more
advanced challenges in all of these branches of mathematics up to A Level and
beyond.
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Taking Equable Shapes into Further Maths
Introduction: A equable 2D shape is defined as a shape whose area and perimeter are
numerically equal.
An equable 3D shape is defined as a shape whose surface area and volume are
numerically equal.
Taking these definitions of equable shapes explore:

Regular polygons. Can you generalise for all regular polygons? Can you
represent this graphically?

Different types of triangle. Can you generalise for all triangles? Can you
represent this graphically?

Different 3D shapes e.g. regular polyhedra, cones, pyramids and prisms
Consider how you might use this rich activity to link into the FM syllabus.
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