INforM – Interactive Notebooks for MathematicsBook 9 – Transforming triangles

Transforming Triangles

The activities are related to the work in Ma3 Shape, space and measures:

Geometrical reasoning2 Pupils should be taught to:Properties of triangles and other rectilinear shapesd 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;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 coordinates3 Pupils should be taught to:Properties of transformationsb 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 SMART Notebook file `Transform triangles.notebook’ is designed to support lesson materials which are available for download from the Practical Support Pack area of the DfES’ TeacherNet website at:http://www.teachernet.gov.uk/supportpack/module.aspx?t=2&s=10&y=36&p=&m=18912 . The lesson plan for the activity is included in the attachments to the Notebook file as the MS Word document: `transforming_triangles.doc’.

INforM Page 1 of 6 August 2006

These materials include support for the use of dynamic geometry software (Cabri II Plus and the Geometers Sketchpad) – but since they were written, there is now a free web-based resource designed by the Mathematical Association and programmed in Flash by the Skoool team at Intel. This Mathematical Toolkit won the BETT 2006 award for Key Stage 3/4 mathematics. You and your pupils can download it from the Skoool area of the London Grid for Learning at: http://lgfl.skoool.co.uk/content.aspx?id=657 .

The Mathematical Toolkit is now included in the Mathematics resource area in the latest Gallery for SMART Notebook at: http://education.smarttech.com/ste/en-US/Ed+Resource/Software+Resources/Notebook+collections/Mathematics/General+resources/Mathematical+Toolkit+-+interactivity.htm

The attachments also include Cabri and Sketchpad files for you to use if you have the correct software already installed.

Organisation of the materialsThe Smart Notebook file is saved as `straight line graphs.notebook’.It consists of 10 pages of which the first is the title page, shown above.There are 8 pages to support the activity and its extension. Page 9 is a blank page.Page 10 contains teacher notes which are amplified here.

The first activityThe screen on page 2 shows a large blue capital letter “A”. The black vertical and horizontal line segments have been set up so that they can be dragged but not rotated. Invite pupils to come to the board and drag either line to divide the letter A symmetrically. They could also sketch a free-hand line with a pen or finger.

INforM Page 2 of 6 August 2006

Discuss what makes a line of symmetry. You can copy and paste the letter A to a different place, right click to bring up the drop-down menu and select Flip… Then you can try either Left/Right flip, or Up/Down flip.

Now you can ask pupils to suggest a letter (maybe one of their initials) and then to ask the class to suggest how many lines of symmetry it has. The next page is an aid to classifying all the letters of the alphabet.

The main task – classifying quadrilaterals by their symmetriesPage 4 shows an extract from a Mathematical Toolkit screen. It is from the 2D Shape creation tool, which has been used to define a blue triangle on a grid. Ask pupils to imagine what shapes will be created by reflecting the triangle in each of its sides. Ask volunteers to use the pens to show the possible reflections in different colours.

INforM Page 3 of 6 August 2006

You might like to use the Cabri or Sketchpad dynamic geometry files by clicking on them in the Attachments tab. We have chosen to illustrate the use of the Mathematical Toolkit for this activity, since pupils can download and use the software free at home. The next screen is there to give a feel of how this software can be used for work on geometric transformations. The triangle ABC has been defined by giving the coordinates of each of the three vertices. In order to reflect it in a line such as AC, say, you will need to enter the coordinates of any two points on the mirror line. In this case (0,0) and (0,4) might be one obvious choice. When you have carried out one reflection, to make a quadrilateral using the new triangle and its `ghost’, you can restore the original triangle by just reflecting it again in the same line.

The next page is there to help you make sure that you open the 2D Shape creation tool with the same grid as we’ve used in these notes.

INforM Page 4 of 6 August 2006

You now have a free tool which you can use to create any quadrilateral by clicking on Add Quad and entering coordinates for its four vertices. Since the software doesn’t use the mouse to create or drag points, but only to select shapes, once you have created and named your shape you can draw over it on the board with a pen – or use Notebook’s line tool. You may need to run over the names of some key quadrilaterals – square, rectangle, rhombus, parallelogram, trapezium, kite, arrow head – and get pupils to come to the board and sketch them, maybe on page 9 which is blank.

When the pupils have completed the activity you can use the table on page 8 to collect and share results. This also attached as an MS Word file `classifying_quadrilaterals.doc’.

INforM Page 5 of 6 August 2006

SMART specific issuesPage 10 (after the blank page 8) refers to the sources of useful supporting materials. The lesson plan from the TeacherNet site is attached to the Notebook as the file: `transforming_triangles.doc’.

INforM Page 6 of 6 August 2006