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Secondary Maths ITE

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Stage         Secondary                                     Year             7
Subject       mathematics                                   Term


Transforming triangles
Module contents

Module focus
      Curriculum focus

          Using dynamic geometry to explore angles, side and symmetry in
          triangles and quadrilaterals made from triangles.


          Learning objectives

          By the end of the lesson pupils will:

              understand where a shape will be after a reflection;
              begin to understand the geometrical properties of quadrilaterals.



Learning outcomes

          Most pupils will:

              draw a mirror line along one side of an equilateral, isosceles and scalene
               triangle and accurately reflect them to create a quadrilateral;
              identify the angle and symmetrical properties of the quadrilaterals they
               have constructed.

          Pupils making slower progress will:

              draw a mirror line along a suggested side on an equilateral triangle and
               accurately reflect the triangle to create a quadrilateral;
              identify the line of symmetry and pairs of equal angles in the quadrilateral
               they have constructed.

          Pupils making faster progress will:

              create a triangle and investigate a number of ways to reflect that triangle
               to form a range of quadrilaterals;
              identify the angle and symmetrical properties of the quadrilaterals they
               have created and suggest quadrilaterals which do not have a diagonal
               line of symmetry.




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References

          Strategy Framework references

          Shape, space and measures

          Geometrical reasoning: lines, angles and shapes

              Begin to identify and use angle, side and symmetry properties of triangles
               and quadrilaterals; solve geometrical problems involving these properties,
               using step-by-step deduction and explaining reasoning with diagrams and
               text.

          Transformations

              Understand and use the language and notation associated with
               reflections, translations and rotations.
              Recognise and visualise the transformation and symmetry of a 2D shape:
               - reflection in given mirror lines, and line symmetry;
               - rotation about a given point, and rotation symmetry;
               - translation;
               explore these transformations and symmetries using ICT.



          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

          Geometrical reasoning

          Pupils should be taught to:

          Properties of triangles and other rectilinear shapes

          d use angle properties of equilateral, isosceles and right-angled
          triangles; understand congruence, recognising when two triangles are
          congruent; explain why the

          angle sum of any quadrilateral is 360 degrees




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          f recall the essential properties of special types of quadrilateral,
          including square, rectangle, parallelogram, trapezium and rhombus;
          classify quadrilaterals by their

          geometric properties

          Transformations and coordinates
          Pupils should be taught to:
          Properties of transformations
          b recognise and visualise rotations, reflections and translations,
          including reflection symmetry of 2-D and 3-D shapes, and rotation
          symmetry of 2-D shapes;
          transform 2-D shapes by translation, rotation and reflection,
          recognising that these transformations preserve length and angle, so
          that any figure is congruent to its
          image under any of these transformations

          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 select and reflect objects and drag the
               resulting images;
              move between pages within dynamic geometry software files;
              save files to and import files from the shared area of the computer
               network.

          Pupil use of ICT

          It will support your pupils to:

              explore reflection using dynamic geometry software.



Requirements

          Hardware

              A computer

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              Data projector and large screen display or interactive whiteboard (IWB)
              An ICT suite

          Software

              Dynamic geometry software, such as The Geometer's Sketchpad and
               Cabri Geometry Software

          Other

              Sheets of paper, rulers, compass and pencils

          You can download the viewers needed for these files on 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.

          Make sure this equipment is set up prior to the lesson, taking into
          account strong light sources and potential obstructions. Dynamic
          geometry software is used in this lesson and you will need to install the
          resource files on to the school network for the pupils to access them
          during the main activity. (This lesson is best carried out in an ICT
          suite).

          Some resources have been pre-prepared for this lesson and you will
          find these in the Download resources section. Familiarise yourself
          with the resources before the lesson. The resources Transforming
          triangles and Find the mirror line have been provided in two different
          formats, a Geometer's Sketchpad version and a Cabri Geometry
          version.

          Before the lesson begins ensure the key vocabulary is displayed where
          pupils can easily refer to it.

          Vocabulary

          congruent line of symmetry, equilateral, images, isosceles, mirror line,
          object, perpendicular bisector, reflect, reflection, scalene triangles,
          symmetrical, transformation, vertex, vertices

          Health & Safety

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          All standard safety procedures with computers need to be in place.
          Information can be found at www.ictadvice.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 identify and name an equilateral and isosceles triangles;
              recognise parallel and perpendicular lines, and properties of rectangles.

          ICT skills

          Pupils should already:

              know how to open up a file from a shared location on the network.



Starter

          Display resource Find the mirror line and give pupils some time to
          look carefully at the object and image. Invite a pupil to suggest where
          the mirror line would be.

          Join some corresponding points on the object and image making sure
          that pupils understand that these lines will always be parallel. Inform
          pupils that the line of symmetry must pass through the midpoint of this
          set of lines and that it is called the 'perpendicular bisector'.

          As dynamic geometry software is being used, any of these shapes can
          be dragged to convince pupils that these geometrical features are
          always true.



Main

          Using a data projector and interactive whiteboard, open the resource
          Transforming triangles. Alternatively, ask pupils to open this file from


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          a shared location on the school network. Firstly look at the page
          Equilateral triangle.

          Invite a pupil to drag a vertex of the equilateral triangle and explain to
          pupils what is meant by equilateral. (All three angles and all three sides
          are equal and therefore the triangle has three lines of symmetry.)

          Model how to reflect an object, so that pupils are able to define the line
          of symmetry as one of the sides of a triangle and reflect the remaining
          sides and vertex to form a quadrilateral (i.e. any shape with four edges.
          Quad - is the Latin name for four). Demonstrate to the class how the
          resulting shape can be dragged and ask them to describe the features
          of the resulting quadrilateral. Inform pupils that the two triangles are
          congruent shapes and that the corresponding angles and sides are
          equal.

          Give pupils the opportunity to do this for themselves and conclude that
          there is only one possible quadrilateral, a rhombus that can be
          constructed from this starting point. ( This rhombus is unique and has
          opposite interior angles of 60º and 120º.)

          Then, keeping with the resource Transforming triangles ask the
          pupils to click the page titled Isosceles triangle. Give pupils the
          opportunity to drag the vertices and deduce for themselves that the
          triangle is, by construction, isosceles. Ask the pupils to explore
          reflecting it onto a chosen side to produce a quadrilateral and explore
          the features of the resulting shape. You may want pupils to sketch
          shapes and record observations or conclusions on paper as they are
          working. The pupils can repeat this reflection using a different side.
          (Only four named quadrilaterals are possible: square, kite, arrowhead
          and a rhombus.)

          Many pupils share a misconception that irregular quadrilaterals are
          those without a specific name. Establish that pupils understand that
          there is only one regular quadrilateral. (Square)

          Finally, ask pupils to click on the page titled Scalene triangles and
          continue their investigation. No new quadrilaterals can be constructed,
          but each of those previously discovered can be 'rediscovered' as the
          scalene triangle appears to be equilateral or isosceles. Show pupils
          how to use the text tool in the right hand toolbar, to add names,
          descriptions and any observations to pages.

          Discuss the outcomes of the investigation with the class and invite
          pupils to show and discuss their findings.




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Plenary

          Display the resource table Classifying quadrilaterals and give pupils
          a few minutes to discuss, in pairs, into which cells in the table they
          would place the quadrilaterals that they have produced.

          Remind pupils that each quadrilateral was constructed from two
          congruent triangles. Ask pupils to discuss which quadrilaterals it was
          possible to construct using this approach and why.

          Similarly, ask pupils which quadrilaterals were impossible to construct
          and why. (For example, parallelograms and non-square rectangles as
          the diagonals are not also lines of symmetry.)

          In both cases encourage pupils to use correct mathematical names
          and terms in their descriptions.



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.



Adaptation

          Pupils making faster progress could begin with a new dynamic
          geometry software file and be asked to construct equilateral and
          isosceles triangles using their knowledge of their respective properties,
          consequently exploring the construction features of the software.

          Ask questions such as:



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              Can you find more than one strategy to construct an equilateral triangle?
               What about an isosceles triangle?
              Can you use the same approach to draw an equilateral triangle using a
               pencil, compasses and straight edge?

          Pupils making slower progress may benefit from spending more time
          exploring the nature of the shapes by dragging the vertices and getting
          a feel for which side lengths and angles can vary and which are
          constant. Further support could be provided by having keywords on
          cards available and encouraging pupils to use them in their discussion
          and their contributions to the plenary session.

          Ask questions such as:

              When a shape is reflected, what can you say about the appearance of the
               new shape?

          In the plenary, may benefit from being given a version of the resource
          Classifying quadrilaterals which contains some completed cells.



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:
          o      help pupils to make better progress towards achieving the learning
             objectives?
          o      affect the pace of learning?
          o      affect pupils’ motivation, interest and time spent on task?
          o      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.




This document is part of a range of materials designed to help teachers teach using ICT.
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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.

          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
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