Year 9: Generating polygons
Lesson plan 4a: Constructing regular polygons
60 minutes
A Year 9 lesson building on pupils’ previous experience of using LOGO, focusing on
writing procedures for constructing regular polygons.
15 minutes
Oral and mental starter
Objectives
•
•
Explain how to find, calculate and use:
– the sums of interior and exterior
angles of quadrilaterals, pentagons
and hexagons;
– the interior and exterior angles of
regular polygons.
Key vocabulary
Procedure
Equilateral
Regular polygon
Interior angle,
exterior angle
Resources:
Computer suite
with LOGO loaded
on all machines
Whole-class
display linked to
one computer
Find simple loci, both by reasoning and
by using ICT, to produce shapes and
paths, e.g. an equilateral triangle.
Starter activity
Ensure that pupils are grouped, if possible in pairs, at the computers, with blank LOGO
screen visible on each machine.
Share the objective for the starter, including the meaning of the word locus. Briefly
remind pupils of previous work using LOGO and ask them what shape would result from
this procedure:
REPEAT 4 [FORWARD 120 RT 90]
Establish that this is a square, ask pupils to explain why, and confirm by showing the
result.
Ask them how they would use LOGO to draw an equilateral triangle with the same side
length as the square.
Give pupils 3 minutes to agree and type in their procedures, and modify where
necessary. Any difficulties are likely to be encountered in determining the correct angle
of turn.
Establish by questioning that the turn is 120° to the right, asking one of the pupils to
demonstrate and explain, and that the correct procedure is
REPEAT 3 [FORWARD 120 RT 120]
Point to one of the internal angles of the equilateral triangle, and establish by
questioning that this angle is 60° and the reasons why this is the case. Ask pupils to
identify the link between the angle of turn at a vertex and the corresponding interior
angle of the equilateral triangle, and whether this relationship is also true for the square.
Ask whether it would be true for any regular polygon. Give pupils 1 minute to discuss
this in their pairs/small groups and ask individuals to share the reasoning behind their
responses with the rest of the class.
Finally, introduce the term exterior angle referring to the angle of turn, and establish the
result that the sum of the interior and exterior angles at a vertex of a polygon is 180°.
ICT in Mathematics
Year 9 Lesson 4: Generating polygons
© Crown copyright 2004
Key Stage 3 National Strategy
30 minutes
Main teaching
Objectives
•
•
Explain how to find, calculate and use:
– the sums of interior and exterior
angles of quadrilaterals, pentagons
and hexagons;
– the interior and exterior angles of
regular polygons.
Solve substantial problems by breaking
them down into simpler tasks, using a
range of efficient techniques, methods
and resources, including ICT; use trial
and improvement where a more efficient
method is not obvious.
Key vocabulary
Procedure
Similar
Lines of symmetry
Order of rotational
symmetry
Pentagon, hexagon,
heptagon, octagon,
nonagon, decagon,
unodecagon,
dodecagon
Resources
Computer suite
with LOGO
loaded on all
machines
Whole-class
display linked to
one computer
Teaching/learning activity
Tell pupils that, in the main part of the lesson, they are going to start by exploring how to
write LOGO procedures for any regular polygon, and relate this to the lesson objectives.
Emphasise to pupils that they are likely to find the result from the starter useful in
achieving the lesson objectives.
Ask them initially to work in their pairs/small groups to devise LOGO procedures for the
regular hexagon and regular octagon; then regular pentagon.
After about 20 minutes it may be useful to organise a mini-plenary to check on progress.
Pupils who have completed the task for the specified polygons could be asked to work
on generating a nonagon, decagon and dodecagon, and then the ‘missing’ shapes,
those with 7 sides (heptagon) and 11 sides (unodecagon?). At this stage, they should
be able to generalise the result for generating any regular polygon. Some pupils may be
able to generalise the size of an interior angle of a regular n-gon.
Plenary
15 minutes
Plenary activity
Gather the results generated by the class, asking for explanation
and justification of each response. Draw threads together, aiming
to use pupils’ responses wherever possible. Ask pupils to predict
procedures for generating 12-sided, 24-sided, 30 sided, 36-sided,
40-sided regular polygons.
Show the results of pupils’ conjectures to check for accuracy.
Ask pupils why you are choosing these numbers, and not others
like 23-sided or even 50-sided polygons.
If possible, draw out the response from the pupils that for an nsided regular polygon, the turn required (exterior angle) is
360°  n.
Resources
Whole-class
display from one
computer
ICT in Mathematics
Year 9 Lesson 4: Generating polygons
© Crown copyright 2004
Key Stage 3 National Strategy
Reinforce the connections between the number of sides of the
regular polygon and the sizes of its interior and exterior angles.
Pupils could use whiteboards, to respond to questions such as:
•
•
•
•
If the exterior angle of a regular polygon is 20°, what is its
interior angle? How many sides does it have? How do you
know?
A regular polygon has 36 sides. What are its interior and
exterior angles? Why?
A regular polygon has an interior angle of 156°. What is its
exterior angle and how many sides does it have?
A regular polygon has an interior angle of 162°. How many
sides does it have? Explain your reasoning.
ICT in Mathematics
Year 9 Lesson 4: Generating polygons
© Crown copyright 2004
Key Stage 3 National Strategy
Lesson plan 4b: Polygons and symmetry
60 minutes
A Year 9 lesson building on pupils’ experience of using LOGO to write procedures for
constructing regular polygons, focusing on amending those procedures to produce
polygons with the same symmetries and different symmetries from the regular polygons.
Oral and mental starter
Objectives
•
•
Explain how to find, calculate and use:
– the sums of interior and exterior
angles of quadrilaterals, pentagons
and hexagons;
– the interior and exterior angles of
regular polygons.
10 minutes
Key vocabulary
Equilateral
Regular polygon
Interior angle,
exterior angle
Resources
Computer suite
with LOGO loaded
on all machines.
Whole-class
display linked to
one computer.
Similar
Find simple loci, both by reasoning and
by using ICT, to produce shapes and
paths, e.g. an equilateral triangle.
Starter activity
As in the previous lesson, ensure that pupils are grouped, if possible in pairs, at the
computers, with blank LOGO screen visible on each machine.
Begin the lesson by making links to the previous lesson. Share the objectives, and
explain that the outcomes of this lesson will be to amend their LOGO procedures for
regular polygons to generate shapes with given numbers of lines of symmetry and orders
of rotational symmetry.
Ask pupils to supply LOGO procedures for generating specific regular polygons with side
120, e.g. with 6 sides, nine sides, 10 sides … . What would they look like if the side
length changed to 100, or to 80, or to 67? Where only the side length changes, reinforce
pupils’ use of the word similar to describe the shapes.
Type in pupils’ instructions for drawing the shapes, one at a time and, for each, ask the
individuals to tell the class the number of lines of symmetry and order of rotational
symmetry of each regular polygon. Ensure pupils recognise that a regular n-gon has n
lines of symmetry and order n rotational symmetry.
40 minutes
Main teaching
Objectives
•
•
Explain how to find, calculate and use:
– the sums of interior and exterior
angles of quadrilaterals, pentagons
and hexagons;
– the interior and exterior angles of
regular polygons.
Key vocabulary
Locus
Similar
Resources
Computer suite
with LOGO loaded
on all machines
Whole-class
display linked to
one computer
Solve substantial problems by breaking
them down into simpler tasks, using a
range of efficient techniques, methods and
resources, including ICT; use trial and
improvement where a more efficient
method is not obvious.
ICT in Mathematics
Year 9 Lesson 4: Generating polygons
© Crown copyright 2004
Key Stage 3 National Strategy
Teaching/learning activity
Tell pupils that, in the main part of the lesson, they are going to start by exploring what
happens to the symmetries of regular polygons when other regular polygons are added to
them.
On the whole class display, show this procedure
REPEAT 4 [FD 120 RT 90]
Check that pupils recognise that the shape produced will be a square before revealing it.
Now show this picture.
Explain that this is a square with a smaller square drawn on the centre of each side. Say
that the large square has side 120 and the small square has side 40.
Ask pupils they would describe how to walk round the outside of this shape, starting at
the bottom left-hand corner of the large square. Say that you want them to discuss this in
pairs for 5 minutes.
After 5 minutes, take responses, and note each step, using LOGO language and in
colour, on the diagram. Ask pupils how far they need to go before the can use REPEAT 4
in their explanations. Offer help in structuring responses where necessary.
A procedure like this is what you are aiming for.
REPEAT 4 [FD 40 LT 90 FD 40 RT 90 FD 40 RT 90 FD 40 LT 90 FD 40 RT 90]
or REPEAT 4 [FD 40 REPEAT 4 [FD 40 LT 90] FD 80 RT 90]
Ask pupils to agree in their pairs/groups about the symmetries of this shape. Take
responses and establish that they are the same as for the large square.
Now say that you want pupils, in their pairs, to write a procedure to draw an equilateral
triangle of side 120 with a regular polygon of side 40 drawn on the centre of each face,
like these.
ICT in Mathematics
Year 9 Lesson 4: Generating polygons
© Crown copyright 2004
Key Stage 3 National Strategy
After 5 minutes or so, collect responses again. This time the procedure for the first shape
may be
REPEAT 3 [FD 40 LT 60 FD 40 RT 120 FD 40 LT 60 FD 40 RT 120]
or REPEAT 3 [FD 40 REPEAT 3 [FD 40 LT 120] FD 80 RT 120]
Ask pupils about the symmetries of the shape. They should be the same as for the large
triangle. Ask pupils whether REPEAT 3 […] tells you anything about the symmetries.
Now ask them to start with a different regular polygon and construct triangles or squares
or … on each side, and to explore the symmetries of the shapes they have created.
Plenary
10 minutes
Plenary activity
Resources
Quickly review progress by asking pairs of pupils to model their
procedures to the class.
Whole-class
display from
one computer
Establish that the symmetries of the shape created are, in each case,
the same as those of the large regular polygon. Ask pupils whether the
symmetries would be the same if the polygon were not placed centrally
on each side.
Procedures for these are:
•
repeat 3 [fd 80 repeat 3 [fd 40 lt 120] fd 40 rt 120]
•
repeat 3 [fd 80 repeat 4 [fd 40 lt 90] fd 40 rt 120]
•
repeat 6 [fd 80 repeat 3 [fd 40 lt 120] fd 40 rt 60]
ICT in Mathematics
Year 9 Lesson 4: Generating polygons
© Crown copyright 2004
Key Stage 3 National Strategy
Through pupils’ responses, establish that, in these cases, only rotational
symmetry would be preserved.
Finally, show pupils a LOGO drawing of a regular hexagon. Say that, if
you constructed an equilateral triangle on each side and then continued
to build equilateral triangles on the sides of the new shape, then the
results could be illustrated by the following procedures.
to stel :side
fd :side lt 60 fd :side rt 120 fd :side lt 60 fd :side
end
to stellate :side
repeat 6 [stel :side lt 60 stel :side rt 120 stel :side lt 60 stel :side rt 60]
end
to pattern :side
repeat 6 [stellate :side rt 60]
end
Type:
cs stellate 12 Say that patterns like this are called snowflake patterns,
with the same symmetries as regular hexagons.
cs pattern 12 Rotating them provides other interesting patterns within
the shape.
ICT in Mathematics
Year 9 Lesson 4: Generating polygons
© Crown copyright 2004
Key Stage 3 National Strategy