Step 1 : Identify lines of symmetry in simple shapes and recognise shapes with no lines of symmetry. Shade in two more squares to make this design symmetrical about the mirror line. You may use a mirror or tracing paper. What do you look for when trying to decide whether a shape has at least one line of symmetry? Talk me through what you notice in these shapes. How do you go about finding lines of symmetry in a shape? Take digital pictures of various everyday objects, identifying symmetries.
Step 2 : Classify polygons, using criteria such as number of right angles, whether or not they are regular, and symmetry properties. Show me a polygon that is regular and has at least one right angle. Are there any others? Show me a regular polygon. How do you know it is regular? What do you look for? A regularpolygon's sides are all of the same length and its angles are the same size. Is this polygon regular? Why not? SQUARE A polygon with four equal length sides, four right angles, and parallel opposite sides. If a polygon is not a regular polygon, then it is said to be an irregularpolygon.
Step 3i : Recognise perpendicular and parallel lines, and properties of rectangles. How would you check whether two lines are parallel, or perpendicular? Tell me some facts about rectangles. All squares are rectangles.Some rectangles are not squares. Give me some instructions to help me to draw a rectangle. Is it possible for a quadrilateral to have only three right angles? Why? RECTANGLES: The rectangle is one of the most commonly known quadrilaterals. A Rectangle is a parallelogram : opposite sides are equal, opposite angles are equal Each angle is 90 degrees. Diagonals of a rectangle are equal. Diagonals bisect each other What is the same about a square and a rectangle? What might be different? Parallel lines are the same distance apart. They are straight and never meet Perpendicular lines are at right angles to each other parallel perpendicular intersecting horizontal vertical quadrilateral bisect
Step 3ii : Recognise perpendicular and parallel lines, and properties of rectangles. Look at the 2D shapes. Which ones have parallel sides? Pick out the rectangle. Write down three things you know about rectangles. • How many sides does a rectangle have? • How many pairs of equal sides does a rectangle have? • How many right angles? • Any parallel sides? • How many lines of symmetry does a rectangle have? • How do you work out the area of a rectangle? • Can a triangle have sides that are a pair of perpendicular lines? Explain. Which two lines are parallel? Which line is perpendicularto both the red and blue lines? Draw a line which isparallel to the orange line. • Can parallel lines be curved? • Give me some examples of shapes that have pairs of parallel lines.
Step 4a : Recognise and visualise the transformation and symmetry of 2-D shapes, including reflection in given mirror lines and line symmetry. Give me instructions to reflect this shape into this mirror line. How do you decide where to position each point in the image? • A reflection is made with the use of a mirror line. • Each corner of the shape is reflected to the opposite side of the mirror line. • The reflected image of each corner is perpendicular (at right-angles) to the mirror line and is the same distance from the mirror line as the original corners. Make up a reflection that is easy to do. Make up a reflection that is hard to do. What makes it hard? Line of symmetry reflective symmetry mirror line transformation
Step 4bi : Construct 3-D models by linking given faces or edges. Construct a 3-D shape with given properties, e.g. at least two sets of parallel faces and at least two triangular faces. Given the shape on the cross-section (e.g. an L-shaped hexagon), how many faces would the corresponding prism have? What shape would the faces be? Show any net of a cube or a cuboid. Where would you put the tabs to glue the net together? Cross-section Cuboids Net prism
Step 4bii : Construct 3-D models by linking given faces or edges. • Imagine a triangular prism. How many faces does it have? Are any of the faces parallel to each other? Look at these diagrams. Which of them are nets of a square-based pyramid? Explain how you know. • How many pairs of parallel edges has a square-based pyramid? How many perpendicular edges? Is this a net for an open cube? Explain why not • Describe the properties of 3-D shapes, such as parallel or perpendicular, faces or edges • e.g. Look at this cube. • How many edges are parallel to this one? • How many edges are perpendicular to this one? Here are 4 nets, which ones will make a net of a cube? Add a square to complete the net to make a closed cube
Step 5a : Identify parallel and perpendicular lines; know the sum of angles at a point, on a straight line and in a triangle and recognise vertically opposite angles. What do you understand by perpendicular lines? Can a triangle have sides that are a pair of perpendicular lines? Why? How do you go about identifying parallel lines? Parallel lines are always equidistant Perpendicular lines intersect at right angles Give me some examples of shapes that have pairs of parallel lines. Is it possible to draw a triangle with: one acute angle? two acute angles? one obtuse angle? two obtuse angles? Why? Vertically opposite angles Give an example of each triangle, suggesting the sizes of the three angles, if it is possible. If it is impossible, explain why. Angles on a straight line Remember an obtuse angle is more than 90 but less than 180. Parallel Intersect Perpendicular Obtuse Right angle Acute angle Isosceles Equilateral
Step 5b : Use a ruler and protractor to measure and draw lines to the nearest millimetre and angles, including reflex angles, to the nearest degree. Why is it important to estimate the size of an angle before measuring it? What important tips would you give to someone about using a protractor? Always guess the angle first. Is it acute or obtuse? Line up the protractor so the 'cross hair' is exactly on the angle. Line up one of the lines with the 0 line on the protractor. See which numbers the angle comes between. If it is between 30 and 40, the angle must be thirty something degrees. Count the small degrees up from 30. How would you draw a reflex angle, using a 180° protractor? • Protractors usually have two sets of numbers going in opposite directions. • Be careful which one you use! • When in doubt think "should this angle be bigger or smaller than 90° ?" Draw angles of 36o, 162o and 245o Decide whether these angles are acute, obtuse or reflex, then measure them each to the nearest degree. Construct Draw Sketch Measure Perpendicular Distance Rules Set Square Degree Acute angle Obtuse Angle Reflex angle
Step 5c : • Recognise and visualise the transformation and symmetry of a 2-D shape: • reflection in given mirror lines and line symmetry; • rotation about a given point and rotational symmetry What clues do you look for when deciding whether a shape has been formed by reflection or rotation? Sketch me a quadrilateral that has one line of symmetry; or two lines, three lines, no lines, etc. Can you give me any others? How many lines of symmetry does this shape have? What is the order of rotational symmetry of each of the quadrilaterals you sketched? This shape has a rotational symmetry of ____ because it maps onto itself in ____ different positions under rotations of ____ degrees. A rotation is specified by a centre of rotation and an (anticlockwise) angle of rotation The centre of rotation can be inside or outside the shape Transformation Image Object Reflection Rotation Symmetry Congruent Mirror line Translation Centre of rotation Order of rotation
Step 6 : Transform 2-D shapes by simple combinations of rotations, reflections and translations, on paper and using ICT; identify all the symmetries of 2-D shapes. What information do you need to do a reflection? A rotation? A translation? What stays the same and what is different when you reflect a shape? When you rotate it? When you translate it? What is the order of rotational symmetry for each of these quadrilaterals? If I had a shape and rotated it and then reflected it would that be the same as reflecting it and then rotating it? Which of the following are translations of A? Transformation Image Object Reflection Rotation Symmetry Congruent Mirror line Translation Centre of rotation Order of rotation Line of symmetry
Step 7 : Use a straight edge and compasses to construct: the midpoint and perpendicular bisector of a line segment; the bisector of an angle; the perpendicular from a point to a line; the perpendicular from a point on a line. Construct a triangle, given three sides (SSS); use ICT to explore these constructions. Why are compasses important when doing constructions? For which constructions is it important to keep the same compass arc (distance between the pencil and the point of your compasses)? Construct the mid-point and perpendicular bisector of a line segment AB. Construct the bisector of an angle. Construct the perpendicular from a point P to a line segment AB. What do you know about a rhombus? How can this be used to help you construct the Rhombus? Construct the perpendicular from a point Q on a line segment CD. • Construct a ABCwith∡36°, ∡B=58° and AB=7cm • Construct a rhombus, given the length of a side and one of the angles. • Use ruler and protractor to construct triangles: • given two sides and the included angle (SAS) • given two angles and the included side (ASA) Compass Bisector Construction Rhombus Mid-point Perpendicular Segment Equidistant Protractor
Step 7b : Identify alternate and corresponding angles; understand a proof that the sum of the angles of a triangle is 180° and of a quadrilateral is 360°. How could you convince me that the sum of the angles of a triangle is 180°?? Why are parallel lines important when proving the sum of the angles of a triangle? How does knowing the sum of the angles of a triangle help you to find the sum of angles of a quadrilateral? Will this work for all quadrilaterals? Why? Intersect Parallel Corresponding Alternate Quadrilateral Exterior angle Complementary Equidistant Prove Proof
Step 7c : Classify quadrilaterals by their geometric properties How could you convince me that a rhombus is a parallelogram but a parallelogram is not necessarily a rhombus? What properties do you need to know about a quadrilateral to be sure it is a kite; a parallelogram; a rhombus; an isosceles trapezium? Why can't a trapezium have three acute angles? Which quadrilateral with one line of symmetry has three acute angles? Isosceles Trapezium Parallelogram Rhombus Kite Delta Quadrilateral Symmetry Angles Acute Obtuse
Step 7d : Enlarge 2-D shapes, given a centre of enlargement and a positive whole-number scale factor. What changes when you enlarge a shape? What stays the same? If someone has completed an enlargement how would you find the centre and the scale factor? What is the scale factor of enlargement in this diagram? When drawing an enlargement, what strategies do you use to make sure your enlarged shape will fit on the paper? What information do you need to complete a given enlargement? What is the scale factor of enlargement in this triangle? Enlarge Enlargement Scale factor Scale Map Plan Drawing Ratio
Step 8a : Know that translations, rotations and reflections preserve length and angle and map objects on to congruent images. If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent. What changes, and what stays the same, when you: translate; rotate; reflect; enlarge a shape? After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths. When is the image congruent? How do you know? X-Axis If the mirror line is the x-axis, just change each (x,y) into (x,-y) Rotation Translation Reflect Resize Congruent Similar Axis Mirror line Co-ordinates Angle Y-Axis If the mirror line is the y-axis, just change each (x,y) into (-x,y)
Step 8bi : Visualise and use 2-D representations of 3-D objects; analyse 3-D shapes through 2-D projections, including plans and elevations The following are shadows of solids. Describe the possible solids for each shadow (there may be several solutions) Starting from a 2-D net of a 3-D shape, how many faces will the 3-D shape have? How do you know? What will be opposite this face in the 3-D shape? How do you know? Which side will this side join to make an edge? How do you know? How would you go about drawing the plan and elevation for the 3-D shape you could make from this net? For each shape, identify the solid shape. Draw the net of the solid. Given this plan and elevation, what can you know for sure about the 3-D object they represent? What can you not be sure about? Here are three views of the same cube. Which letters are opposite each other? Plan View Isometric Elevation Cross-section Plane Projection Net Model
Step 8bii : Visualise and use 2-D representations of 3-D objects; analyse 3-D shapes through 2-D projections, including plans and elevations The following diagrams are of solids when observed directly from above. Describe what the solids could be and why? This diagram represents a plan of a solid made from cubes, the number in each square indicating how many cubes are on that base. Make an isometric drawing of the solid from the chosen viewpoint. Is it possible to slide a cube so that the cross section is: A triangle A rectangle A pentagon A hexagon If so describe how it can be done. Write the names of the polyhedra that could have isosceles or equilateral triangle as a front elevation. Construct a solid, based on the views Sit back to back with a partner. Look at the picture of the model. Don’t show it to your partner. Tell you partner how to build the model.
Step 7di : Explain how to find, calculate and use the interior and exterior angles of regular polygons The formula for calculating the sum of the interior angles of a regular polygon is: (n - 2) × 180° where n is the number of sides of the polygon. How can you use the angle sum of a triangle to calculate the sum of the interior angles of any polygon? If the polygon is regular, what else can you calculate? This formula comes from dividing the polygon up into triangles using full diagonals. We already know that the interior angles of a triangle add up to 180°. For any polygon, count up how many triangles it can be split into. Then multiply the number of triangles by 180. This quadrilateral has been divided into two triangles, so the interior angles add up to 2 × 180 = 360°. What is the sum of the interior angles for each shape? This pentagon has been divided into three triangles, so the interior angles add up to 3 × 180 = 540°. Formula Exterior Interior Angles Polygons Quadrilateral
Step 7dii : Explain how to find, calculate and use the interior and exterior angles of regular polygons We know that the exterior angles of a regular polygon always add up to 360°, so the exterior angle of a regular hexagon is Remember : The interior angle and its corresponding exterior angle always add up to 180°. (For a hexagon, 120° + 60° = 180°.) The interior angles of a regular polygon are each 150°. Calculate the number of sides. The interior angles of a regular polygon are each 120°. Calculate the number of sides.
Step 9b : Enlarge 2-D shapes by a positive whole-number or fractional scale factor. Enlarge triangle ABC with a scale factor 1/2, centred about the origin. If someone has completed an enlargement how would you find the centre and the scale factor? What changes when you enlarge a shape? What stays the same? Enlarge the rectangle WXYZ using a scale factor of - 2, centred about the origin. When drawing an enlargement, what strategies do you use to make sure your enlarged shape will fit on the paper? What information do you need to complete a given enlargement? If you enlarge a shape by scale factor 3, what scale factor will take your image back to the size of your original? What can you say about the centre of the enlargement if the final image is in exactly the same position as the original object? What are the coordinates of B after an enlargement, scale factor 1/3, centre (0, 3)? Enlarge Enlargement Scale factor Scale Map Plan Drawing Ratio Fractional
Step 9c : Find the locus of a point that moves according to a given rule, both by reasoning and by using ICT. A locus is a path. The path is formed by a point which moves according to some rule. Give me an example that you find more difficult to visualise. What makes it harder? How does this work relate to your earlier work on construction using compasses? P and Q are two points marked on the grid. Give me an example of a given rule that you find easy to visualise. How do you go about finding a locus? Two points X and Y are 10cm apart. Two adjacent sides of a square pass through points X and Y. What is the locus of vertex A of the square? Construct accurately the locus of all points that are equidistant from P and Q. Locus Loci Compasses Visualise Equidistant Vertex
Step 7c : 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 What clues do you look for when finding a missing angle for a geometrical diagram? What's the minimum information you would need in order to be able to find all the angles in this diagram? ABCDE is a regular pentagon. A regular pentagram has been formed, and the intersections marked as P, Q, R, S, T. • How many pairs of parallel lines can you find? • How many pairs of perpendicular lines? • Can you find all the angles in the diagram? • How many different types of triangle can you find? • How many different types of quadrilateral? Be prepared to explain your reasoning. Repeat for regular hexagrams and octograms. Can you explain why the exterior angle of a triangle is equal to the sum of the two interior opposite angles? Talk me through the information that has been given to you in this diagram. How do you decide where to start in order to find the missing angle(s) or to solve the geometrical problem? How would you convince somebody that the exterior angles of a polygon add up to 360°?