®
Recommended prior knowledge
All of Core and particularly Core 5. 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. There is a link to Core 7 (7.2) and Pythagoras Core 8 (8.1) so these need to have been covered particularly.
Context
There are five Core geometry units and this is the second of five Extended Geometry units. Once Core 5 and the other prior experience for Core 5, Core 7 and Core
8 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 5.
Outline
The unit extends the knowledge of Core 5 so be aware that examination questions that relate to aspects of Core 5 may have a greater degree of challenge as they combine with other areas of mathematics. This unit covers finding a vector, the effects of adding and subtracting vectors, finding the magnitude of a vector, multiplying a vector by a constant, stretches, and inverse and combined transformations.
Syllabus ref Learning objectives Suggested teaching activities
5.1
CCSS:
N-VM1
N-VM2
5.2
CCSS:
N-VM2
Vector Notation: use appropriate symbols for vectors and their magnitudes
Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point
Notes and exemplars e.g., v , | v |
General guidance
This needs practicing throughout the unit rather than being treated as a separate component. However it is necessary to be rigorous with student use of symbols for vectors and to understand the different forms used in text and handwritten mathematics.
Notes and exemplars
See also section 5.6, translations using column vectors.
General guidance
Students will have already met the idea of finding a vector for transformations in Core 5 (5.6) but make need reminding it applies to points to make the connection here.
Learning resources www.bbc.co.uk/schools/gcsebitesize/math s/shapes/vectorshirev1.shtml
http://nrich.maths.org/2390 http://nrich.maths.org/7453 http://nrich.maths.org/6632 http://nrich.maths.org/4890 v1 2Y01 Cambridge IGCSE Mathematics (US) 0444 1

Syllabus ref
5.3
CCSS:
N-VM5
5.4
CCSS:
N-VM4
5.5
CCSS:
N-VM5
Learning objectives Suggested teaching activities
Use position vectors Use of position vectors needs practice particularly when connected with geometric reasoning problems.
Learning resources
Past Paper 22 June 2011 Q16
(syllabus 0580)
Past Paper 43 June 2011 Q10
(syllabus 0580)
Past Paper 42 June 2011 Q8
(syllabus 0580)
Past Paper 21 June 2011 Q18
(syllabus 0580) www.bbc.co.uk/schools/gcsebitesize/math s/shapes/vectorshirev1.shtml
Calculate the magnitude of a vector x
  as x + y 2
Add and subtract vectors
Multiply a vector by a scalar
General guidance
This can be linked to Core 7 (7.2) and to Core 8 (8.1) and needs developing as a rule and then practicing.
Notes and exemplars
Both algebraic (component) and geometric (parallelogram rule) addition/subtraction
Teaching activities
The discussion for the game in Core 5 (5.1) can be extended to the addition of the vector cards of the two players for consecutive moves and finding the vector from start to each end of a move by either player.
Give a grid with the points of a polygon and ask students to find the vectors for moving from any point to the next until they return to the start point.
Different students can start at different points. They can then add the total set of vectors and explain the result.
Notes and exemplars e.g., 3
 
 
= 3 5 = 15 c x
 
= cx
 
If c| v | ≠ 0, the direction of c v is either along v (for c > 0) or against v (for c <
0).
General guidance
Link this to the stepping pattern in Core 7 (7.5) and gradient. www.bbc.co.uk/schools/gcsebitesize/math s/shapes/vectorshirev1.shtml
www.youtube.com/watch?v=2dHk_yJ9ntQ http://nrich.maths.org/1317
See questions for 5.2 above. v1 2Y01 Cambridge IGCSE Mathematics (US) 0444 2

Syllabus ref Learning objectives Suggested teaching activities
5.6
CCSS:
G-CO2
G-CO3
G-CO4
G-CO5
G-SRT1
G-SRT2
5.7
CCSS:
G-CO5
Transformations on the cartesian plane: stretch
Inverse of a transformation
Teaching activities
Once the skill has been practised relate to scaling problems and splitting lines in ratios on a coordinate grid.
Notes and exemplars
Representing and describing transformations.
General guidance
Students need to understand the difference between an enlargement and a stretch. They need to understand that enlargement is the special case where the horizontal and vertical scale factors are the same.
Link to the magnitude of vectors for the effects on the horizontal and vertical change.
Teaching activities
Ask students to complete a number of stretches recording start and finish coordinates and to explain the general pattern on the coordinates.
Look at the stretches that take squares to rectangles, rectangles to parallelograms, kites to rhombi etc. and the effects on lengths of diagonals and sides.
General guidance
Students need to understand:
1. the meaning of an inverse operation as one that takes you back to where you started
2. that reflection is self inverse
3. that translations require a negative of the vector
4. that enlargements require 1 over the original scale factor and the centre doesn’t change and link to the inverses of stretches.
5. Rotations have take the angle back the other way so clockwise to anticlockwise or vice versa. Discuss the difference between this and continuing on 360 ° - the original angle of rotation. i.e. it takes you back to the original position but doesn’t reverse the movement.
Teaching activities
Students should find the rules above for themselves by drawing the transformation and describing the transformation from image to object. This could be used as revision for describing transformations completely.
Learning resources www.haeseandharris.com.au/samples/igc se_20.pdf
(section 5.4) http://nrich.maths.org/4958
Past Paper 41 June 2011 Q5a and b
(syllabus 0580)
Past Paper 42 June 2011 Q8
(syllabus 0580)
Past Paper 43 June 2011 Q8 ignoring matrix parts
(syllabus 0580) v1 2Y01 Cambridge IGCSE Mathematics (US) 0444 3

Syllabus ref Learning objectives Suggested teaching activities
5.8
CCSS:
G-CO5
Combined transformations
Notes and exemplars e.g. find the single transformation that can replace a rotation of 180° around the origin followed by a translation by vector


4
− 2


General guidance
Students need to understand that order matters and to complete several examples first one way around and then the other to see this in action.
Teaching activities
In Core 5 this activity was used - On page 205 of the framework document there is a grid of L shapes and an activity that can be used for the transformations that produce congruent outcomes.
It or something similar can be adapted here to create a competition. Split the class into groups. Each student in the group describes transformations between any two of the shapes by a combination of two transformations. 2 points for each one correct as judged by the rest of the group, bonus 1 point for any that do not include translation as one of the moves. Minus 3 points for any incorrect.
Discussion with whole class at the end. A translation and vertical / horizontal reflection will generally be described by using the axes but any line parallel to the axes will work with different translations. Finding several of these could be another challenge and will revise naming vertical and horizontal lines.
Learning resources www.counton.org/resources/ks3framewor k/pdfs/transformations.pdf page 205 v1 2Y01 Cambridge IGCSE Mathematics (US) 0444 4