www.teachernet.gov.uk/supportpack
Stage:
Subject
Secondary
mathematics
Year
Term
9
Exploring geometrical constructions
Module contents
Module focus
Curriculum focus
Using dynamic geometry software to explore perpendicular bisectors
and angle bisectors leading to the discovery of the circumcircle and the
in-circle.
Learning objectives
By the end of the lesson pupils will:

be able to find the perpendicular bisectors and angle bisectors of a
triangle;
 understand how identifying the bisectors of a triangle can lead to
finding the circumcircle and the in-circle.
Learning outcomes
Most pupils will:

find the perpendicular bisectors of each side of a triangle and
explain, using geometrical reasoning, why the point of intersection
of the perpendicular bisectors of AB and BC always lies on the
perpendicular bisector of side AC;
 identify that the distance between the point of intersection of the
sides of a triangle and the three points of a triangle are of equal
length and draw the circumcircle.
Pupils making slower progress will:

recognise that the point of intersection of the perpendicular
bisectors of AB and BC always lies on the perpendicular bisector of
side AC;
 draw the circumcircle of the triangle.
Pupils making faster progress will:

find the perpendicular bisectors of each side of a triangle and
explain, using geometrical reasoning, why the point of intersection
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.
www.teachernet.gov.uk/supportpack
of the perpendicular bisectors of AB and BC always lies on the
perpendicular bisector of side AC;
 identify that the segments from the point of intersection of the
perpendicular bisectors of the sides of a triangle to each of its three
vertices are of equal length and draw the circumcircle. Also identify
that the distance between the point of intersection of the angle
bisectors and each side of the triangle are of equal length and draw
the in-circle.
References
Strategy Framework references
Shape, space and measures
Geometrical reasoning: lines, angles and shapes

Solve problems using properties of angles, of parallel and
intersecting lines, and of triangles and other polygons,
justifying inferences and explaining reasoning with diagrams and
text; understand and apply Pythagoras' theorem.
Construction and loci

Find the locus of a point that moves according to a simple rule, both
by reasoning and by using ICT; extend to more complex rules
involving loci and simple constructions.
The Framework for teaching mathematics can be found at:
www.standards.dfes.gov.uk/keystage3/respub/mathsframework/forewo
rd.
National Curriculum references
Ma3 Shape, space and measures
Using and applying shape, space and measures
1 Pupils should be taught to:
Communication
e communicate mathematically, making use of geometrical diagrams
and related explanatory text;
Reasoning
j explore connections in geometry; pose conditional constraints of the
type 'If … then …'; and ask questions 'What if …?' or 'Why?'
Measures and construction
4 Pupils should be taught to:
Construction
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.
www.teachernet.gov.uk/supportpack
e use straight edge and compasses to do standard constructions,
including an equilateral triangle with a given side, the midpoint and
perpendicular bisector of a line segment, the perpendicular from a
point to a line, the perpendicular from a point on a line, and the
bisector of an angle;
Loci
j find loci, both by reasoning and by using ICT to produce shapes and
paths [for example, equilateral triangles].
The National Curriculum programme of study can be found at:
www.nc.uk.net/nc/contents/Ma-3-2-POS.html.
Use of ICT
Teacher use of ICT
This module will give you the opportunity to:

use dynamic geometry software to present and investigate the
properties of geometrical constructions with your pupils.
Requirements
Hardware


Computer
Data projector and large screen display or interactive whiteboard
(IWB)
 ICT suite
Software

Mathematics software, such as The Geometer's Sketchpad or Cabri
Geometry
Other

Plain paper, compasses and rulers
Trial or viewer versions of the software you need may be available
from the Software downloads page.
Lesson preparation
A key element of this module is a lesson for you to adopt and adapt to
meet the needs of your class. It is designed to help you evaluate the
impact of using ICT for learning and teaching. Display and discuss the
objectives and key vocabulary for this lesson with the pupils.
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.
www.teachernet.gov.uk/supportpack
Some examples of lesson resources have been created for you to use
and can be found in the download resources section. Review the
resources before the lesson and adapt if necessary. Ensure your
computer, large screen display or IWB are set up before the lesson
and that you have enough plain paper, compasses, pencils and rulers
for your pupils to use during the lesson. If you are working in an ICT
suite, ensure all pupils can access the resource Exploring geometric
constructions.
For ICT support, visit the ICT skills guidance tab.
Vocabulary
bisect, bisector, circumcentre, circumcircle, compasses,
congruent, construction lines, inscribed, perpendicular bisector,
straight edge
Health & Safety
All standard safety procedures with computers need to be in
place. Information can be found at http://schools.becta.org.uk.
ICT skills guidance
The guidance in this section supports the ICT skills described in the
Module contents.
Pupils' prior knowledge and skills
Pupils should already:

be able to use straight edge and compasses to construct:
the mid-point and perpendicular bisector of a line segment;
the bisector of an angle;
 know how to use ICT to explore these constructions;
 know how to solve geometrical problems using side and angle
properties of equilateral, isosceles and right-angled triangles and
special quadrilaterals, explaining reasoning with diagrams and text;
 classify quadrilaterals by their geometric properties.
Starter
Open the dynamic geometry software file Reviewing mathematical
constructions and display the page entitled Perpendicular bisector
construction. Tell pupils that this construction is based on the
properties of a rhombus.
Invite a pupil to drag the point B and ask the class to say what is
changing and what is staying the same. Demonstrate the effect of
dragging points A and B
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.
www.teachernet.gov.uk/supportpack
Ask pupils to draw a line segment on their paper and, by imagining that
this line represents a diagonal of a rhombus, use their compasses to
draw a rhombus around it. Use the dynamic geometry software file to
show the steps in the construction and drag the variables, such as the
lengths of the line segment AB and the circle radius, to show that the
construction holds true.
Display the page entitled Angle bisector construction (if you are using
Cabri Geometry software open the file Reviewing mathematical
constructions_2) Ask pupils to draw an (acute) angle on their paper
and, again, keeping the properties of the rhombus in mind, use their
compasses to bisect the angle.
Use the dynamic geometry software file to show the steps in the
construction and drag the variables, such as the size of the angle ABC
and the circle radius to show that the construction holds true.
Main
Ask pupils to open the dynamic geometry software file Exploring
geometric constructions and open the page Exploring perpendicular
constructions.
Give pupils a few minutes to explore the construction, dragging the
vertices of the triangle and observing what is changing and what is
staying the same. Establish that the point P represents the point of
intersection of the perpendicular bisectors of sides AB and BC. Ask
pupils to drag the point B and look very carefully at the locus of point P
as B moves. Encourage pupils to make a conjecture based on their
observation. The point of intersection, P can be outside of the triangle
ABC.
The next stage is to test this conjecture further by using the 'Trace'
facility on point P as point B moves. Refer to the relevant help sheet for
software specific instructions to do this.
Establish that the locus of P as B is moved is a straight line and
appears to be the perpendicular bisector of side AC. At this point ask
pupils to make a sketch of the construction on paper or print a copy to
annotate.
At this stage pupils will need to label the mid-points of each of the
sides of the triangle and identify congruent triangles within the
construction. For example, triangle BPC is an isosceles triangle. Their
task is to try to justify to a partner that the locus is the perpendicular
bisector of side AC.
Invite groups of pupils to present their reasoning to the class.
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.
www.teachernet.gov.uk/supportpack
The second page Exploring angle bisectors gives pupils the
opportunity to use a similar strategy focusing on angle.
Plenary
On the page Exploring perpendicular bisectors, construct the
perpendicular bisector of side AC and confirm to pupils that their
conjecture was correct.
Hide the perpendicular bisectors so that only the triangle and
circumcentre (point P) are visible.
Draw line segments from point P to the vertices A, B and C and ask
pupils to make a conjecture based on this, that is that the segments
PA, PB and PC are of equal length.
Ask pupils which geometric shape is defined as the locus of all points
equidistant from a fixed point. (A circle).
Use the circle tool to draw a circle, centre P and radius PA and drag
the features of the construction to give a visual proof.
Invite pupils to use the previously identified congruent triangles to
justify their observations. Alternatively, the page entitled Exploring
angle bisectors could be displayed adopting a similar approach leading
to the discovery of the in-circle.
Assessment
In assessing for learning you should consider the following points.
1. Ensure objectives are expressed in language that pupils
understand.
2. Give pupils clear success criteria related to these objectives.
3. Give pupils opportunities to discuss their successes and challenges
focusing on the objectives.
4. Provide oral and written feedback to pupils.
5. Encourage pupils to explain their thinking and reasoning in a secure
environment.
6. Provide time for pupils to reflect upon what they have learned and
understood and identify any difficulties.
You can find information on assessment for learning at:
www.standards.dfes.gov.uk/keystage3/respub/afl_ws.
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.
www.teachernet.gov.uk/supportpack
Adaptation
Pupils requiring additional support will benefit from using large paper
cut-outs of triangles, using folding techniques to generate the
construction lines and measuring lines and angles to make inferences.
Ask pupils the following questions.
What does it mean if you bisected a side of a triangle? …… and if
you then bisect an angle?
 How can you check by measuring?

Higher attaining pupils could be asked to formulate a written proof for
their verbal justification and to develop an argument as to why the
points P and T are both centres of (different) circles. To do this, pupils
will need to join the centre of each circle to the relevant points to
identify congruent isosceles triangles.
Evaluation
Lesson reflection
These prompts are designed to help you reflect on how the use of ICT
affected your teaching and pupils’ learning.
Prompts for reflection:
1. How did the use of ICT:
• help pupils to make better progress towards achieving the
learning objectives?
• affect the pace of learning?
• affect pupils’ motivation, interest and time spent on task?
• affect your ability to differentiate your teaching and personalise
pupils’ learning?
2. What knowledge or skills have you gained and extended in teaching
this lesson?
3. What adaptations would you make to the lesson and its resources
to suit the needs of your class?
You may wish to create a record of your evaluation and save it as
evidence of your professional development. If so, you can download a
template containing these prompts and spaces for your responses.
Materials evaluation
These prompts are designed to help you consider why, how and when
you would incorporate these lesson activities and resources into your
curriculum and teaching plans.
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.
www.teachernet.gov.uk/supportpack
Prompts for evaluation:
1. What are the benefits of using these teaching and learning
approaches and resources to achieve the subject objectives?
2. How do the suggested activities fit with your existing curriculum and
teaching plans?
3. What adaptations would be required to the activities or resources to
suit the needs of your class?
4. Are there any requirements for ICT equipment, other resources,
space, etc. that might limit how and where the lesson is taught?
You may wish to make a note of your thinking for your own records
and to share with your colleagues. If so, you can download a template
containing these prompts and spaces for your notes.
Download module
Download module
You can download a module pack containing the lesson plan,
resources, supplementary information and extension activities using
the link below. The pack is in a zipped file to minimise file size, but
unless you have a broadband connection, the download may be slow.
To extract the files within the module pack you will need either Winzip
or Microsoft Windows Extraction Wizard. This software is freely
available and can be downloaded from the Software downloads
page.
This document is part of a range of materials designed to help teachers teach using ICT.
For more information, log on to www.teachernet.gov.uk/supportpack
Please note - Permission to reproduce, distribute, adapt and use this document is subject to the terms and conditions
of the Practical Support Pack website, available at www.teachernet.gov.uk/supportpack/termsandconditions.aspx.