Some Starters using Dynamic Geometry
1. Properties of Reflections & Lines of Symmetry
Y7 Teaching Objectives
· Understand and use the language and notation associated with reflections
· Recognise and visualise the transformation of a 2-D shape
- Reflection in given mirror line and line symmetry
Associated vocabulary:
Line of symmetry, mirror line, object, image, perpendicular
Used to support pupils understanding of properties of reflections (e.g. constructed line between object and image point is perpendicular to and equidistant from mirror line) Helps to address the common misconception which arises when mirror lines are limited to “vertical” and “horizontal” lines
Start with a simple shape such as a triangle and a skew mirror line projected on to the IWB or ordinary WB:
Computer image can be generated and compared. Pupils can be invited to suggest how we might adjust our original triangle or mirror line to ensure the computer image matches the drawn (blue)image. Line segments can be drawn between object and images (A to A’) etc. and questions asked about the relation between these and the mirror line. A, B and/or C can be moved to illustrate the invariability of this.
Additional Actvities/extensions:
Objects and mirror lines could be drawn on a square dotty grid with/without coordinates and with/without equations of mirror lines/lines of symmetry. Invite pupils to create a dynamic Rangoli pattern (see GSP example for simple Rangoli pattern, “Foursym” created on square dotted grid)
Aditional Objectives:
· Use conventions and notation for 2-D coordinates in all four quadrants, find coordinates of points determined by geometric information
2. Lines of Symmetry-Properties of 2-D Shapes
Teaching Objectives
Y7
· Use correctly the vocabulary, notation and labelling conventions for lines angles and shapes
· Identify parallel and perpendicular lines
Y8
· Explaining reasoning with diagrams and text, classify (triangles and) quadrilaterals by their geometric properties
· Know that if two 2-D shapes are congruent corresponding sides and angles are equal.
Associated vocabulary:
Line of symmetry, mirror line, object, image, perpendicular, parallel, quadrilateral, arrowhead, kite, parallelogram, intersect, bisect, mid point, equidistant, diagonal, equal sides (angles), congruent, adjacent, triangle, equilateral, isosceles, vertex, vertices
The blue line is a mirror line. Imagine moving any of the points D, C or E. Describe the shapes you can/cannot make. Sketch them in your book/on your white board. Give reasons for your answers.
Invite pupil(s) to drag point D to create the shapes that they have drawn. If pupils have drawn their shapes on acetate these could be projected onto the screen and superimposed on a GSP diagram which could be manipulated to fit/not fit their drawing. Line segments and angles can be measured to check/validate properties of the shape. Pupils can be questioned/prompted with, for example “Why can a parallelogram or rectangle not be made and a rhombus and square can”.Link to diagonals of these shapes and their lines of symmetry (Note pupils often harbour the misconception that the diagonal of a rectangle is a line of symmetry). Link this with possible classification of quadrilaterals through properties of their diagonals.
3. Rotations, Lines of Symmetry- Properties of 2-D Shapes
Y7 Teaching Objectives
· Understand and use the language and notation associated with rotations and reflections
· Identify parallel and perpendicular lines
· Recognise and visualise the transformation of a 2-D shape
- Reflection in given mirror line and line symmetry
- Rotation about a given point and rotation symmetry
Associated vocabulary:
Angle, degrees, centre of rotation, line of symmetry, mirror line, object, image, parallel perpendicular, equal sides angles, line segment, vertex, square
Challenge pupils to create a square using transformations of a line segment. This line segment could be defined as one side of our square or could be left ambiguous i.e it could also be one of the diagonals of the square.
Display the vocabulary of transformations on the board or on large cards or alternatively give them a copy of the words on a sheet of A4..Encourage pupils to use mathematical terminology and precise, unambiguous instructions, for example ask pupils what they would need to define a rotation (viz: centre of rotation, angle and direction). Pupils might do this initially in groups/pairs and write out their instructions on paper. Then challenge different groups to recreate their square using DGS such as GSP (whole class activity). The DGS requires precise non-ambiguous “mathematical” instructions and offers useful non judgemental feedback to the pupils.
Compare and contrast the different methods i.e some might do it by rotation only, others by a combination of rotation and reflection. Tease out the how the method used to create their square is linked to the property of the square eg equal sides, adjacent sides at right angles or perpendicular, opposite sides are parallel. (You might relate this to the fact that in one of the methods the square is obtained by rotating through say +90 then –90 about different points and that two 90 rotations about different points produces a non-coincident parallel line segment). Point out the invariant qualities of their constructions i.e each shape remains a square no matter how we drag different segments or points.
Additional Activities/extensions
Challenge the pupils to generate a tile design (by rotations, reflections and/or translations based on a single tile. DGS such as GSP offers pupils the opportunity for pupils to explore dynamic alterations to their designs (see enclosed GSP file “Floors”).
e.g
Etc.
Which simple single “dynamic” tile design offers a good range of overall designs
Other activities such as creating a Kaleidoscope can be used to develop pupils understanding of rotational symmetry.
4. Properties of 2-D Shapes, Constructing a Rectangle
Y7 Teaching Objectives
· Understand and use the language and notation associated with rotations and reflections
· Identify parallel and perpendicular lines
· Explore constructions using ICT
· Recognise and visualise the transformation of a 2-D shape
- Reflection in given mirror line and line symmetry
Y8
· Explaining reasoning with diagrams, classify quadrilaterals (rectangles and squares) by their geometric properties
· Know that if two 2-D shapes are congruent corresponding sides and angles are equal.
Similar to starter 3 but this time ask pupils to create a rectangle from a line segment by parallel and perpendicular constructions. Again emphasis is on precise unambiguous instruction (and some knowledge of how the software works), eg creating a perpendicular line to the line segment requires a given point.
A rectangle can be created in a variety of ways. Note how we can use the interior polygon facility to illustrate symmetry properties of a rectangle and countering the misconception of a line of symmetry along a diagonal. Dragging points, line segments illustrates the invariant nature of their construction. Note how we can create a square by dragging an end point of our original line segment. i.e a square is a special kind of rectangle and in this case the square has a line of symmetry on the diagonal. The diagrams below illustrate one of the ways that pupils might construct a rectangle and develop these points.