Education Leeds 20090212 VJC Identify lines of symmetry in
simple shapes and recognise shapes with no lines of symmetry. Step
1 : Talk me through what you notice in these shapes. Take digital
pictures of various everyday objects, identifying symmetries. 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? How do you go about finding lines of symmetry in
a shape?
Slide 2
Education Leeds 20090212 VJC Classify polygons, using criteria
such as number of right angles, whether or not they are regular,
and symmetry properties. Step 2 : Is this polygon regular? Why not?
Show me a regular polygon. How do you know it is regular? What do
you look for? Show me a polygon that is regular and has at least
one right angle. Are there any others? A regular polygon's sides
are all of the same length and its angles are the same size. If a
polygon is not a regular polygon, then it is said to be an
irregular polygon. SQUARE A polygon with four equal length sides,
four right angles, and parallel opposite sides.
Slide 3
Education Leeds 20090212 VJC Recognise perpendicular and
parallel lines, and properties of rectangles. Step 3i : Give me
some instructions to help me to draw a rectangle. Tell me some
facts about rectangles. How would you check whether two lines are
parallel, or perpendicular? What is the same about a square and a
rectangle? What might be different? Is it possible for a
quadrilateral to have only three right angles? Why? parallel
perpendicular intersecting horizontal vertical quadrilateral bisect
Parallel lines are the same distance apart. They are straight and
never meet 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 Perpendicular lines are at right angles to each other All
squares are rectangles. Some rectangles are not squares.
Slide 4
Education Leeds 20090212 VJC 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? Which
two lines are parallel? Which line is perpendicular to both the red
and blue lines? Draw a line which is parallel to the orange line.
Give me some examples of shapes that have pairs of parallel lines.
Can parallel lines be curved? Can a triangle have sides that are a
pair of perpendicular lines? Explain. Recognise perpendicular and
parallel lines, and properties of rectangles. Step 3ii :
Slide 5
Education Leeds 20090212 VJC Recognise and visualise the
transformation and symmetry of 2-D shapes, including reflection in
given mirror lines and line symmetry. Step 4a : Make up a
reflection that is easy to do. How do you decide where to position
each point in the image? Give me instructions to reflect this shape
into this mirror line. Make up a reflection that is hard to do.
What makes it hard? Line of symmetry reflective symmetry mirror
line transformation 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.
Slide 6
Education Leeds 20090212 VJC Construct 3-D models by linking
given faces or edges. Step 4bi : Cross- section Cuboids Net prism
Construct a 3-D shape with given properties, e.g. at least two sets
of parallel faces and at least two triangular faces. Show any net
of a cube or a cuboid. Where would you put the tabs to glue the net
together? 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?
Slide 7
Education Leeds 20090212 VJC Look at these diagrams. Which of
them are nets of a square-based pyramid? Explain how you know. 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. Imagine a triangular prism. How
many faces does it have? Are any of the faces parallel to each
other? How many pairs of parallel edges has a square-based pyramid?
How many perpendicular edges? 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 Construct 3-D models by linking given
faces or edges. Step 4bii :
Slide 8
Education Leeds 20090212 VJC 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. Step 5a : Parallel Intersect Perpendicular Obtuse Right
angle Acute angle Isosceles Equilateral How do you go about
identifying parallel lines? Give me some examples of shapes that
have pairs of parallel lines. What do you understand by
perpendicular lines? Can a triangle have sides that are a pair of
perpendicular lines? Why? Is it possible to draw a triangle with:
one acute angle? two acute angles? one obtuse angle? two obtuse
angles? Why? Give an example of each triangle, suggesting the sizes
of the three angles, if it is possible. If it is impossible,
explain why. Parallel lines are always equidistant Vertically
opposite angles Angles on a straight line Perpendicular lines
intersect at right angles Remember an obtuse angle is more than 90
but less than 180.
Slide 9
Education Leeds 20090212 VJC Use a ruler and protractor to
measure and draw lines to the nearest millimetre and angles,
including reflex angles, to the nearest degree. Step 5b : Construct
Draw Sketch Measure Perpendicular Distance Rules Set Square Degree
Acute angle Obtuse Angle Reflex angle What important tips would you
give to someone about using a protractor? How would you draw a
reflex angle, using a 180 protractor? Why is it important to
estimate the size of an angle before measuring it? Decide whether
these angles are acute, obtuse or reflex, then measure them each to
the nearest degree. Draw angles of 36 o, 162 o and 245 o
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 ?" 1.Always guess
the angle first. Is it acute or obtuse? 2.Line up the protractor so
the 'cross hair' is exactly on the angle. 3.Line up one of the
lines with the 0 line on the protractor. 4.See which numbers the
angle comes between. If it is between 30 and 40, the angle must be
thirty something degrees. 5.Count the small degrees up from
30.
Slide 10
Education Leeds 20090212 VJC 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 Step 5c : Transformation Image Object
Reflection Rotation Symmetry Congruent Mirror line Translation
Centre of rotation Order of rotation What clues do you look for
when deciding whether a shape has been formed by reflection or
rotation? What is the order of rotational symmetry of each of the
quadrilaterals you sketched? Sketch me a quadrilateral that has one
line of symmetry; or two lines, three lines, no lines, etc. Can you
give me any others? 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 This shape has a rotational symmetry
of ____ because it maps onto itself in ____ different positions
under rotations of ____ degrees. How many lines of symmetry does
this shape have?
Slide 11
Education Leeds 20090212 VJC 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. Step 6 :
Transformation Image Object Reflection Rotation Symmetry Congruent
Mirror line Translation Centre of rotation Order of rotation Line
of symmetry 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?
What information do you need to do a reflection? A rotation? A
translation? 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?
Slide 12
Education Leeds 20090212 VJC 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. Step 7 : Compass Bisector Construction Rhombus
Mid-point Perpendicular Segment Equidistant Protractor What do you
know about a rhombus? How can this be used to help you construct
the Rhombus? For which constructions is it important to keep the
same compass arc (distance between the pencil and the point of your
compasses)? Why are compasses important when doing constructions?
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. Construct the
perpendicular from a point Q on a line segment CD. Use ruler and
protractor to construct triangles: given two sides and the included
angle (SAS) given two angles and the included side (ASA) Construct
a ABC with 36, B=58 and AB=7cm Construct a rhombus, given the
length of a side and one of the angles.
Slide 13
Education Leeds 20090212 VJC 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. Step 7b :
Intersect Parallel Corresponding Alternate Quadrilateral Exterior
angle Complementary Equidistant Prove Proof Why are parallel lines
important when proving the sum of the angles of a triangle? How
could you convince me that the sum of the angles of a triangle is
180?? 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?
Slide 14
Education Leeds 20090212 VJC Classify quadrilaterals by their
geometric properties Step 7c : Isosceles Trapezium Parallelogram
Rhombus Kite Delta Quadrilateral Symmetry Angles Acute Obtuse How
could you convince me that a rhombus is a parallelogram but a
parallelogram is not necessarily a rhombus? Why can't a trapezium
have three acute angles? What properties do you need to know about
a quadrilateral to be sure it is a kite; a parallelogram; a
rhombus; an isosceles trapezium? Which quadrilateral with one line
of symmetry has three acute angles?
Slide 15
Education Leeds 20090212 VJC Enlarge 2-D shapes, given a centre
of enlargement and a positive whole- number scale factor. Step 7d :
Enlarge Enlargement Scale factor Scale Map Plan Drawing Ratio If
someone has completed an enlargement how would you find the centre
and the scale factor? What information do you need to complete a
given enlargement? What changes when you enlarge a shape? What
stays the same? When drawing an enlargement, what strategies do you
use to make sure your enlarged shape will fit on the paper? What is
the scale factor of enlargement in this diagram? What is the scale
factor of enlargement in this triangle?
Slide 16
Education Leeds 20090212 VJC Know that translations, rotations
and reflections preserve length and angle and map objects on to
congruent images. Step 8a : Rotation Translation Reflect Resize
Congruent Similar Axis Mirror line Co-ordinates Angle When is the
image congruent? How do you know? What changes, and what stays the
same, when you: translate; rotate; reflect; enlarge a shape? X-Axis
If the mirror line is the x-axis, just change each (x,y) into
(x,-y) Y-Axis If the mirror line is the y-axis, just change each
(x,y) into (-x,y) After any of those transformations (turn, flip or
slide), the shape still has the same size, area, angles and line
lengths. If one shape can become another using Turns, Flips and/or
Slides, then the two shapes are called Congruent. Congruent If
you...Then the shapes are...... only Rotate, Reflect and/or
Translate Congruent... need to ResizeSimilar
Slide 17
Education Leeds 20090212 VJC Visualise and use 2-D
representations of 3-D objects; analyse 3-D shapes through 2-D
projections, including plans and elevations Step 8bi : Plan View
Isometric Elevation Cross-section Plane Projection Net Model 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? Starting
from a 2-D net of a 3-D shape, how many faces will the 3-D shape
have? 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? 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? The following are shadows of solids. Describe the possible
solids for each shadow (there may be several solutions) For each
shape, identify the solid shape. Draw the net of the solid.
Slide 18
Education Leeds 20090212 VJC Visualise and use 2-D
representations of 3-D objects; analyse 3-D shapes through 2-D
projections, including plans and elevations Step 8bii : The
following diagrams are of solids when observed directly from above.
Describe what the solids could be and why? Write the names of the
polyhedra that could have isosceles or equilateral triangle as a
front elevation. 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.A triangle b.A rectangle c.A pentagon d.A hexagon
e.If so describe how it can be done. Construct a solid, based on
the views Sit back to back with a partner. Look at the picture of
the model. Dont show it to your partner. Tell you partner how to
build the model.
Slide 19
Education Leeds 20090212 VJC Explain how to find, calculate and
use the interior and exterior angles of regular polygons Step 7di :
Formula Exterior Interior Angles Polygons Quadrilateral If the
polygon is regular, what else can you calculate? How can you use
the angle sum of a triangle to calculate the sum of the interior
angles of any polygon? 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. 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. This pentagon has been divided into three triangles, so
the interior angles add up to 3 180 = 540. What is the sum of the
interior angles for each shape?
Slide 20
Education Leeds 20090212 VJC 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
120. Calculate the number of sides. The interior angles of a
regular polygon are each 150. Calculate the number of sides.
Explain how to find, calculate and use the interior and exterior
angles of regular polygons Step 7dii :
Slide 21
Education Leeds 20090212 VJC Enlarge 2-D shapes by a positive
whole-number or fractional scale factor. Step 9b : Enlarge
Enlargement Scale factor Scale Map Plan Drawing Ratio Fractional If
someone has completed an enlargement how would you find the centre
and the scale factor? What information do you need to complete a
given enlargement? What changes when you enlarge a shape? What
stays the same? When drawing an enlargement, what strategies do you
use to make sure your enlarged shape will fit on the paper? Enlarge
triangle ABC with a scale factor 1 / 2, centred about the origin.
Enlarge the rectangle WXYZ using a scale factor of - 2, centred
about the origin. What are the coordinates of B after an
enlargement, scale factor 1 / 3, centre (0, 3)? 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?
Slide 22
Education Leeds 20090212 VJC Find the locus of a point that
moves according to a given rule, both by reasoning and by using
ICT. Step 9c : Locus Loci Compasses Visualise Equidistant Vertex
Give me an example that you find more difficult to visualise. What
makes it harder? How do you go about finding a locus? How does this
work relate to your earlier work on construction using compasses?
Give me an example of a given rule that you find easy to visualise.
A locus is a path. The path is formed by a point which moves
according to some rule. P and Q are two points marked on the grid.
Construct accurately the locus of all points that are equidistant
from P and Q. 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?
Slide 23
Education Leeds 20090212 VJC 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 Step 7c : How would you convince somebody that
the exterior angles of a polygon add up to 360? What's the minimum
information you would need in order to be able to find all the
angles in this diagram? 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? What clues do you look for when finding a missing angle
for a geometrical diagram? Can you explain why the exterior angle
of a triangle is equal to the sum of the two interior opposite
angles? 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.
