Make shapes using all 7 pieces and find their perimeters. What

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Tangrams: making shapes
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Tangrams: making shapes
Use all 7 shapes each time; they can be rotated
and flipped over if needed.
Individual pieces must not overlap each other.
Try to make the following:
• A rectangle
• A large isosceles triangle
• A parallelogram
• An isosceles trapezium
• An irregular pentagon …with a line of symmetry
• An irregular hexagon …with 2 lines of symmetry
Tangrams: Square
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Tangrams: Square
Use all 7 shapes each time; they can be rotated
and flipped over if needed.
Individual pieces must not overlap each other.
How many different ways can the large square be
made?
Tangrams: area
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Find the area of each of the individual shapes.
Tangrams: fractions
Look at the tangram below.
What fraction of the original square is each of the
individual shapes?
Tangrams: perimeter
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Find the perimeter of each of the individual
shapes.
Tangrams: perimeter 2
Make a tangram by drawing an 8 x 8 square and
cutting out the 7 shapes as shown below.
Tangrams: perimeter 2
Use all 7 shapes each time; they can be rotated
and flipped over if needed.
Individual pieces must not overlap each other.
Make several different shapes and find their
perimeters.
What is the largest perimeter you can make?
What is the smallest perimeter you can make?
Tessellations
A tessellation is a regular pattern made of tiles
placed so that it can continue in all directions.
Every bit of the pattern should be repeated.
Anyone looking at the pattern should be able to
see exactly how it will continue.
Some tessellations are created using just one
shape, others use two or more shapes
Which of the patterns on the following slides are
tessellations?
Quadrilateral Tessellations
Which quadrilaterals tessellate?
Try using one of the following as a tile:
• Square
• Rectangle
• Parallelogram
• Rhombus
• Trapezium
• Kite
• ‘Irregular’ quadrilateral
Regular Tessellations
A regular tessellation is one in which only one
regular polygon is used. Tiles must all be the
same size and have to be placed edge to edge.
How many regular tessellations are there?
Prove that there can be no others.
Semi-regular tessellations
A semi-regular tessellation is
one in which more than one
regular shape is used.
How many different ones
can you find?
Given that the largest
number of sides of a polygon
in a semi-regular tessellation
is 12, can you prove that no
others are possible?
Teacher notes
The idea of this month’s classroom resource is to take a simple starting
point and use it in several ways to address different aspects of the
curriculum.
It is not expected that this is a sequence of lessons as different
activities will be most suitable for different groups of pupils.
Additionally, many of these activities would make for good sources of
display work, just in case anyone’s thinking about school ‘open days’…
Tangrams: overview
This ancient puzzle can be used in a variety of ways, traditionally:
• To recreate given shapes
• To create new shapes
However, additional activities and questions are:
• Area: Given that the side length of the original tangram is 8 units,
find the area of each of the smaller shapes making up the tangram
• How many different ways can all 7 shapes be used to make the
square. Rotations and reflections not permitted.
• What fraction of the original shape are each of the 7 pieces?
• If the original tangram has side length 8 units, what are the side
lengths (and/or perimeters) of each of the 7 pieces?
• Make shapes using all 7 pieces and find their perimeters. What is
the smallest perimeter that can be made using all 7 shapes? What’s
the maximum perimeter that can be made using all 7 shapes
Tangrams
This ancient puzzle can be used in a variety of ways, traditionally:
• To recreate given shapes
• To create new shapes
The usual rule is that pieces cannot overlap, sometimes an additional
rule is given that all adjacent shapes must meet at an edge/ partial
edge.
This activity is accessible for almost all pupils
Some answers, although there are probably several possibilities for
each:
Tangrams
Area: Given that the side length of the original tangram is 8 units, find the
area of each of the smaller shapes making up the tangram.
Suitable for most KS3 pupils, can be
solved by:
• Counting squares
• Reasoning - using the smaller
square and deducing that each of
the smaller triangles is half the area
and then using combinations of
these to physically recreate the
larger shapes
• Finding what fraction of the whole
each piece represents
• Using formulae
Tangrams
How many different ways can all 7 shapes be used to make the
square.
Suitable for most KS3 pupils
Just one arrangement, but there are several rotations and reflections
that can be created.
Ask pupils to make the square in as many different ways as they can
and use their responses to initiate discussions about ‘same and
different’ and transformations as an introductory activity for reflections
and rotations.
Tangrams
What fraction of the original shape are each of the 7 pieces?
Tangrams: perimeters
If the original tangram has side length 8 units, what are the side lengths
(and/or perimeters) of each of the 7 pieces?
•
•
•
•
•
•
•
A
B
C
D
E
F
G
Tangrams: perimeters 2
Make shapes using all 7 pieces and find their perimeters. What is
the smallest perimeter that can be made using all 7 shapes?
What’s the maximum perimeter that can be made using all 7
shapes?
Mathematics required: surds
Encourage pupils to leave answers in surd form for addition purposes,
although they may have to convert to decimals to compare some.
Pupils can experiment with shapes of their own or be given the shapes
on slides 12 & 13 to begin with. They might also be given a limitation of
only using certain shapes i.e triangles and quadrilaterals.
Tangrams: perimeters 2
Long thin shapes will have a larger perimeter.
A square might be expected to have the minimum perimeter, but
shapes such as the hexagon shown actually have smaller ones.
This could lead to a discussion about the relationship between area
and perimeter. For a fixed area, the closer a shape is to being circular,
the smaller its perimeter will be.
Tangrams: perimeters 2
A selection of answers are shown
Tessellations: overview
Tessellations make for engaging activities, which are accessible to
most pupils.
In Key Stage 3 the activities might begin with creating tessellations
However, they also provide opportunities for utilising dynamic geometry
software and also for reasoning and proving.
Activities:
Is it or isn’t it a tessellation
Quadrilateral tessellations
Regular tessellations
Semi-regular tessellations
Tessellations: is it or isn’t it?
•
•
•
•
•
1 – yes
2 – yes
3 – no: not a regular pattern
4 – yes
5 – no: this is a pattern, but it’s not repeated
Tessellations: Which quadrilaterals tessellate?
All quadrilaterals tessellate
This can be explored by pupils cutting out a template and drawing
round it to create a tessellation.
Another way to demonstrate this is to use Dynamic Geometry Software,
using the Geogebra file ‘Quadrilateral Tessellation’. (free Geogebra
software required). The quadrilateral in the top left hand corner of the
page is the driver. Move the vertices of this shape and all others will
change with it, maintaining a tessellation. Hence it can be
demonstrated that all quadrilaterals tessellate.
Tessellations: Regular tessellations
Moving into reasonably simple proof, it can be shown that there are only 3
possible regular tessellations. It would be helpful for pupils to be given time to
think about how they could prove this and perhaps have a class discussion
rather than telling them how to prove it.
Tiles are fitted edge to edge and hence meet at points.
Since the angle sum must be 360°, the interior angle of the regular polygon
must be a factor of 360.
There are (at least) two ways to approach this.
• Find all the factors of 360 and work out which ones are interior angles of
regular polygons
or
• For a tessellation there must be 3 shapes meeting at a point - 2 wouldn’t be
a point. Therefore the largest angle it could be would be 120° (hexagon)
and the smallest regular polygon is a triangle 60° . This means that only 4
shapes need to be checked to determine if their interior angles are factors
of 360°
Tessellations: Semi-regular tessellations
There are 8 semi-regular tessellations, although there is a 9th if a mirror image
is permitted.
Proof by exhaustion can be used to prove that there are no others, but the
entire proof would be daunting. Providing the information that the largest
number of sides for any regular polygon in a semi-regular tessellation is 12
makes the problem more accessible.
Finding a logical and systematic way to identify combinations of interior angles
of regular polygons which have a sum of 360° allows another proof by
exhaustion.

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