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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 4: Geometry (Core)
Recommended prior knowledge
Students should have a working knowledge of the vocabulary of 2D shape, simple solids (cube, cuboid, prism, cylinder, pyramid, and sphere) angles, and an
intuitive understanding of parallel and perpendicular lines and be able to identify them in their classroom and surroundings and symmetries.
Context
This is the first of five geometry units. This unit must be taught before unit 5 and 7. If split into blocks some could be taught after unit 1. It could be split into smaller
blocks and taught between other units.
• Block 1 - 4.1, 4.2 and 4.3
• Block 2 - 4.1, 4,2 and 4.4
• Block 3 - 4.5 which could be could be split so that a few constructions are taught in one or two lessons blocks to break up other areas of mathematics.
• Block 4 - 4.6
• Block 5 - 4.7 which could be taught with proportionality in unit, or at the beginning of unit 7
Students who are following the extended syllabus will move through this faster but need to have all these skills in place before working on the extended units, or
applying them in other areas of mathematics.
Outline
Within the suggested teaching activities ideas are listed to identify and remediate misconceptions and to pull learning through to the required standard. The learning
resources give both teaching ideas, summaries of the skills and investigative problems to develop the problem solving skills and a depth of understanding of the
mathematics, through exploration and discussion. This unit encourages the correct use of terms to describe shapes, their properties and their symmetries. It looks at
the methods for finding missing angles, and to construct accurate diagrams with compass and straight edge, rulers and angle measures, circle properties and angle
in a semi circle. It also looks at similarity.
Syllabus ref
v1 2Y01
Learning objectives
Suggested teaching activities
Learning resources
General guidance
Documents 1 and 2 give a good overview of the coverage of this unit.
Document three is designed to deliver geometric reasoning. It aims to develop
logical reasoning, deeper understanding and as a stepping stone towards
www.counton.org/resources/ks3framewor
k/pdfs/geometrical.pdf
Cambridge IGCSE Mathematics (US) 0444
www.counton.org/resources/ks3framewor
1
Syllabus ref
4.1
Learning objectives
Vocabulary:
acute, obtuse, right
angle, reflex,
equilateral, isosceles,
congruent, similar,
regular, pentagon,
hexagon, octagon,
rectangle, square,
kite, rhombus,
parallelogram,
trapezoid,
and simple solid
figures
Suggested teaching activities
Learning resources
formal proof through the use of student explanations. It builds geometry
through a series of overlays. This can be effective, but the whole document
needs to read and understood as a whole. It cannot be cherry picked.
k/pdfs/transformations.pdf
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-year-9-geometrical-reasoningmini-pack
General guidance
Ensure students use the vocabulary correctly throughout the unit.
www.excellencegateway.org.uk/pdf/Tand
LMathematicsHT281107.pdf page 51 and
52
Teaching activities:
Card sorts with diagrams of angles or polygons or triangles (with all angles
marked), or quadrilaterals (with angles marked) and relevant vocabulary
cards, can be an effective way of checking understanding.
Include additional that is sorted into a pile of unneeded cards.
Past Paper 11 June 2011 Q4
(syllabus 0580)
Past Paper 13 June 2011 Q17
(syllabus 0580)
Odd one out activity – sets of three, angles/ polygons/ triangles/
quadrilaterals. Either with an obvious odd one out or with no obvious odd one
out but students can note it one is odd one out because it holds a property
others don’t. See page 51 and 52 of the excellence gateway document.
Use a 4 x 3 pin board and rubber bands or square spotty paper and mark of
blocks of 12 spots (4 x 3). Try to identify as many triangles as possible using
the spots as vertices (should be 20) and identify them by type.
Use a 3 x 3 pin board and rubber bands or square spotty paper and mark of
blocks of 9 spots (3 x 3). Try to identify as many quadrilaterals as possible
using the spots as vertices (should be 16) and identify them by type.
Use a 4 x 4 pin board and rubber bands or square spotty paper and mark of
blocks of 16 spots (4 x 4). Try to identify at least one 3 sided, 4 sided, 5 sided
up to 15 sided polygon using the spots as vertices – it is possible and label
those that have known names.
4.2
CCSS:
G-CO1
v1 2Y01
Definitions:
Know precise
definitions of angle,
circle, perpendicular
line, parallel line, and
line segment, based
General guidance
Give the definitions and ask pupils to use the definitions when explaining their
reasoning.
www.mathleague.com/help/geometry/geo
metry.htm
Teaching activities
Matching/sort of definitions and vocabulary with some missing words in the
Cambridge IGCSE Mathematics (US) 0444
2
Syllabus ref
4.3
CCSS:
G-CO3
Learning objectives
Suggested teaching activities
on the undefined
notions of point, line,
distance along a line,
and distance around
a circular arc
Line and rotational
symmetry in 2D
definitions - also supplied on the cards.
Notes and exemplars
e.g., know properties of triangles, quadrilaterals, and circles directly related to
their symmetries.
General guidance
Ensure students can both recognise lines of symmetry and rotational
symmetry and its order and can transform a shape by reflection of rotation.
Learning resources
http://nrich.maths.org/5369
http://nrich.maths.org/6987
http://nrich.maths.org/6742
Past Paper 11 June 2011 Q3
(syllabus 0580)
Ensure students know that colour is also preserved as part of symmetry.
Teaching activities
Create a worksheet with a triangular flag on a pole that is rotated about the
base of the pole 30°, repeat 12 times. e.g.
Ask students to colour it in so that, it has rotational
symmetry order 1, 2, 3, 4, 6, and 12 and ask them to explain why these are
the only possibilities.
Have a set of cut out quadrilaterals available and ask students to fold them,
turn them and identify the rotational and reflective symmetry.
Ask students to draw polygons in the cells of a two way grid that has no lines
of symmetry, 1 line of symmetry, 2 lines of symmetry, 4 lines of symmetry as
the headers in one direction and order of rotational symmetry, 1, 2, 4 in the
other and then to explain why some cells cannot be filled.
Get pictures of different car alloy wheel designs and identify which have
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
3
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
rotational and which have reflective symmetry.
Put students in pairs and 2 pairs work as competing teams. They have a set
of square tiles (you specify the number of tiles they can have). The first pair
arranges the tiles in a pattern which has a line of symmetry or rotational
symmetry (tiles must meet on whole edges). The second pair moves one tile
so that the arrangement has no line of symmetry and the moved tile still
touches at least one other tile on a full side. The first pair tries to restore
symmetry moving the first tile or any other tile (just one) and restores the
symmetry, but not back to its original place. The game continues until one pair
cannot continue without repeating a previous arrangement. Rules can be
changed so that more than one tile can be moved. You may find each pair of
teams needs a fifth student to act as an adjudicator.
4.4
Angles around a point
CCSS:
G-CO9
G-CO10
Angles on a straight
line and intersecting
straight lines
Vertically opposite
angles
Alternate and
corresponding angles
on parallel lines
Angle properties of
triangles,
quadrilaterals, and
polygons.
Interior and exterior
angles of a polygon
Link to Unit 5 (5.6)
Notes and exemplars
Proof of properties will not be tested, but candidates should be able to use
these properties to find unknown angles.
General guidance
Ensure students have the facts to learn, and regularly test their knowledge of
them. You might want to model how one fact is deduced from others after
getting students to explore the idea first. The geometric reasoning pack has
some guidance on how to develop this type of reasoning leading to proof.
Past Paper 32 June 2011 Q5
(syllabus 0580)
Past Paper 33 June 2011 Q6a,b,c
(syllabus 0580)
www.teachfind.com/nationalstrategies/interacting-mathematics-keystage-3-year-9-geometrical-reasoningmini-pack
Show how to solve sets of problems for each fact separately but also mix up
the facts so that students have to choose the appropriate fact to solve a
problem.
Students also have difficulty identifying angles at a point when the lines are at
a vertex of one of more meeting polygons. So practice at seeing where
various facts can be applied is required. Distinguish between regular and
irregular polygons.
Teaching activities
Have a proof cut up as separate line statements and a diagram. Ask students
to reconstruct the proof in a logical order.
Draw seven intersecting lines on a page (no more than four intersecting at
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
4
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
one point). Have at least one pair of lines parallel and one line perpendicular
to another. Give one angle and ask students to find all the other angles on the
sheet but to identify their route around the diagram and to give reasons to
justify the answers. They will need the ends of the line segments and
intersecting points lettered so that they can refer to different parts of the
diagram, by two letter line segment names and three letter angle names.
Draw polygons and using the method of joining 1 vertex to all the others, to
create (n-2) triangles where n is the number of sides of the polygon, create a
table for the angle sum of the polygons, check answers using the exterior
angle of a regular polygon to find the internal regular angle and hence the
total for the interior angle for the regular polygon.Create tables and ask
students to generalise.
4.5
Construction.
CCSS:
G-CO12
G-CO13
G-C3
G-C4
Make formal
geometric
constructions with
compass and straight
edge only.
Copy a segment;
copy an angle; bisect
a segment; bisect an
angle; construct
perpendicular lines,
including the
perpendicular bisector
of a line segment
Construct an
equilateral triangle, a
square, and a regular
hexagon inscribed in
a circle
There is a bank of problems to solve at the end of the geometric reasoning
unit.
General guidance
All of these need to be practised – ensure students have reliable compasses
and sharp pencils to avoid frustration. Many are shown in the first two
resources given for this unit.
Relate the bisector of an angle, construction of perpendicular lines, etc to the
properties of a rhombus.
http://nrich.maths.org/5357
Past Paper 33 June 2011 Q2
(syllabus 0580)
Past Paper 32 June 2011 Q8
(syllabus 0580)
Use 360° angle measures to link to definition of angle and to make
construction and measurement of reflex angles easier.
After a few have been given to pupils, ask them how they would complete
others so that they have a means of remembering how some are done by
building from others.
Link skills to the construction of bearings diagrams and practice some of
these as accurate scale diagrams. (link to proportionality/ratio models in Unit
1).
Construct the
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
5
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
General guidance
Ensure students can correctly use the names, radius, diameter,
circumference, centre, arc, sector, chord and segment.
http://nrich.maths.org/6007
inscribed and
circumscribed circles
of a triangle.
Construct a tangent
line from a point
outside a given circle
to the circle
4.6
Angle measurement
in degrees.
Read and make scale
drawings
Vocabulary of circles
CCSS:
G-C1
Properties of circles:
• tangent
perpendicular to
radius at the point of
contact
• angle in a semicircle
4.7
Similarity
CCSS:
G-SRT2
G-SRT3
Calculation of lengths
of similar figures
Notes and exemplars
Use scale factors and/or angles to check for similarity.
Past Paper 11 June 2011 Q18
(syllabus 0580)
Past Paper 33 June 2011 Q6e
(syllabus 0580)
General guidance
Use the proportionality model in unit 1 to find missing lengths.
Test for similarity by checking corresponding pairs of lengths have the same
multiplier to get from one to the other. Students sometimes have difficulty
identifying corresponding pairs and keeping all the ratios the same way
around. This needs practicing as a separate skill first.
Link to Enlargement (dilation) Note that angles are preserved under
enlargement. Link to Unit 5 (5.6).
Set up diagrams and draw a fan of lines from the centre of enlargement to the
object and extend to find the vertices of the image. Using
scaling/understanding of symmetry to find these lengths. Then prove the
object and the image are similar.
v1 2Y01
Cambridge IGCSE Mathematics (US) 0444
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