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Scheme of work – Cambridge IGCSE® Mathematics (US) 0444
Unit 4: Geometry (Extended)
Recommended prior knowledge
All of Core and particularly Core 4. Only those parts of the learning objectives or notes and exemplars not included in the Core units are itemised, so this document
should be read alongside the Core document.
Context
There are five Core geometry units and this is the first of five Extended geometry units. Once Core 4 unit and the other prior experience for Core 4 is completed,this
unit can be slotted in at any point. It is probably best taught as a whole but used to revise some of the Core 4 unit.
Outline
The unit extends the knowledge of Core 4 so be aware that examination questions that relate to aspects of Core 4 may have a greater degree of challenge as they
combine with other areas of mathematics. This unit covers understanding the definitions of vocabulary, symmetry in 3D, the additional circle theorem properties,
similarity as it affects area and volume, and congruence.
Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
4.1
Vocabulary:
Know precise
definitions of acute,
obtuse, right angle,
reflex, equilateral,
isosceles, congruent,
similar, regular,
pentagon, hexagon,
octagon, rectangle,
square, kite,
rhombus,
parallelogram,
trapezoid, and simple
solid figures
Line and rotational
symmetry in 3D
General guidance
The difference between this and the core unit is the ‘Know precise definitions’.
www.mmlsoft.com/index.php?option=com
_content&task=view&id=9&Itemid=10
Ensure students have the definitions and check throughout the unit that they
use vocabulary correctly.
www.mmlsoft.com/index.php?option=com
_content&task=view&id=11&Itemid=12
4.3
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Teaching activities
Use the ‘Tarsia’ software found at ‘mmlsoft’ to create a domino set of
definitions and vocabulary and ask students to complete it periodically as a
lesson starter.
Notes and exemplars
Recognize symmetry properties of the prism and the pyramid.
Cambridge IGCSE Mathematics (US) 0444
www.youtube.com/watch?v=gBg4lJ19Gg&feature=related
1
Syllabus ref
Learning objectives
CCSS:
G-GCO3
Suggested teaching activities
Learning resources
General guidance
Some students find it very hard to visualize the 3D symmetries. The use of
models that can be split or rotated on an axis are vital for them to see what is
happening.
www.youtube.com/watch?v=cEXx_8FWC
sE&feature=related
Teaching activities
View these two videos and discuss the implications of moving from symmetry
in 2D to 3D (points to lines, lines to planes).
4.6
CCSS:
G-C1
G-C2
4.7
CCSS:
G-SRT2
G-SRT3
G-SRT5
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Vocabulary of circles
Properties of circles:
• tangents from a
point
• angles at the centre
and at the
circumference on the
same arc
• cyclic quadrilateral
Use the following
symmetry properties
of a circle:
• equal chords are
equidistant from the
centre
• the perpendicular
bisector of a chord
passes through
the centre
• tangents from an
external point are
equal in length
Similarity
Area and volume
scale factors
General guidance
Showing students the proofs of the circle properties/theorems will add depth
to their understanding. However the main requirement is to solve problems
that relate to those properties. It is experience and practice that is required,
both to select the required facts and to sequence a justification for an answer.
http://nrich.maths.org/6007
http://nrich.maths.org/6624
http://nrich.maths.org/2845
http://nrich.maths.org/833
http://nrich.maths.org/599
Teaching activities
Students can be given jumbled up lines to a solution and asked to order them.
Or in some cases if a series of angles are required students can be
challenged to provide more than one route to the complete set and asked to
decide which is neatest.
www.timdevereux.co.uk/maths/geompage
s/index.html
The ‘teachnet’ resources provide some sample files that will promote
discussion but do not give the theorems themselves in an obvious form.
Notes and exemplars
Use of the relationships between areas of similar figures and Extended to
volumes and surface areas of similar solids.
Past Paper 43 June 2011 Q4
(syllabus 0580)
Past Paper 22 June 2011 Q13
(syllabus 0580)
Past Paper 21 June 2011 Q17
(syllabus 0580)
Past Paper 23 June 2011 Q20
(syllabus 0580)
www.cimt.plymouth.ac.uk/projects/mepres
/book8/bk8i19/bk8_19i3.htm
http://nrich.maths.org/6923
General guidance
The most difficult aspect for students to grasp is to decide whether the
Cambridge IGCSE Mathematics (US) 0444
www.maths4scotland.co.uk/GHS%20Exa
2
Syllabus ref
4.8
CCSS:
G-GCO6
G-GCO7
G-GCO8
G-SRT5
Learning objectives
Congruence
Use the definition of
congruence to show
that two triangles are
congruent if, and only
if, corresponding pairs
of sides and
corresponding pairs
of angles are
congruent
Suggested teaching activities
Learning resources
particular case is about an area or a volume as the problems can be about 3D
objects but the scaling to do with 2D, either because one of the dimensions is
fixed or because it is the surface of the object that is the crux of the problem,
not its volume.
Teaching activities
Draw a large triangle. Find the midpoints of two sides (vertex to vertex and
pinch the midpoint) and fold the triangle along this line connecting the
midpoints. The vertex should touch the opposite side and model nicely that
the area of the smaller triangle fits into the larger four times. Ask students to
split the sides into thirds along two sides and ask them to fold the top triangle
over and construct other lines to show the equivalent numbers of triangles.
Ask the general case ‘Can they make diagrams that show this effect for other
polygons?’.
m%20Revision/GHS%20Credit/Similar%2
0shapes%20%20area%20&%20volume.swf
Take students through cases of simple objects like cubes of length 3cm, etc.
to prove the squaring and cubing of lengths principal.
Notes and exemplars
Justify why two triangles are congruent with geometric reasons and reference
to ASA, SAS, SSS, or RHS. Justify why two triangles are congruent with
geometric reasons and reference to ASA, SAS, SSS, or RHS.
General guidance
Students need to prove the equal facts and identify them in the two triangles
using geometric reasoning and then to show that the facts fit one of the four
criteria.
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/congruencysimilarityrev4.shtml
www.mathwarehouse.com/classroom/wor
ksheets/congruent_triangles/Triangle_pro
of_ASA-SAS.pdf
www.bbc.co.uk/schools/gcsebitesize/math
s/shapes/congruencysimilarityrev3.shtml
Teaching activities:
Ask students working in groups to construct a variety of triangles with the
following criteria. Some will leave them with choices, or prove impossible. Do
not tell them they are going to check if they are identical until the end.
1. Sides of 4cm, 5cm, 7cm constructed with any one of those as the
base.
2. A base of 5cm with a line at 75° at one end and a line of 6cm at the
other end.
3. A base of 6cm with a side drawn at 75° to this that is 5cm long.
4. A base of 8cm, another line of 7cm and the angle opposite to the
base of 55°.
5. Draw a parallelogram with sides 5cm and 7cm and cut in half to form
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Cambridge IGCSE Mathematics (US) 0444
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Syllabus ref
Learning objectives
Suggested teaching activities
Learning resources
two triangles.
6. Draw a right angle and sides forming the right angle of 5cm and 9cm.
7. Draw a base of 6cm, a right angle at one end and the hypotenuse at
the other end of 9cm.
8. A triangle with angles 40°, 65°, 75°, base of any length.
Finally students cut out the triangles and decide when they are identical (even
if flipped over) and when they are not, and if any are similar. Are there any
other combinations they could invent?
Discuss results and the difficulty of constructing some with or without extra
decisions.
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