GCSE OCR GCSE (91) Maths
Section 09  Congruence and Similarity
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Introduction
Overview
Delivery guides are designed to represent a body of knowledge about teaching a particular topic and contain:
 Content: a clear outline of the content covered by the delivery guide;
 Thinking Conceptually: expert guidance on the key concepts involved, common difficulties learners may have, approaches to teaching that can help learners understand these concepts and how this topic links conceptually to other areas of the subject;
 Thinking Contextually: a range of suggested teaching activities using a variety of themes so that different activities can be selected that best suit particular classes, learning styles or teaching approaches.
Curriculum Content
Overview
GCSE content Ref.  Subject content  Initial learning for this qualification will enable learners to…  Foundation tier learners should also be able to…  Higher tier learners should additionally be able to…  DfE Ref. 

OCR 9
 Congruence and Similarity
 
9.01  Plain isometric transformations  
9.01a  Reflection  Reflect a simple shape in a given mirror line, and identify the mirror line from a shape and its image.  Identify a mirror line x = a, y = b or y = ± x from a simple shape and its image under reflection.  G7
 
9.01b  Rotation  Rotate a simple shape clockwise or anticlockwise through a multiple of 90° about a given centre of rotation.  Identify the centre, angle and sense of a rotation from a simple shape and its image under rotation.  G7
 
9.01c  Translation  Use a column vector to describe a translation of a simple shape, and perform a specified translation.  G7, G24
 
9.01d  Combinations of transformations
 Perform a sequence of isometric transformations (reflections, rotations or translations), on a simple shape. Describe the resulting transformation and the changes and invariance achieved.  G8
 
9.02  Congruence
 
9.02a  Congruent triangles  Identify congruent triangles.  Prove that two triangles are congruent using the cases:
3 sides (SSS) 2 angles, 1 side (ASA) 2 sides, included angle (SAS) Right angle, hypotenuse, side (RHS).  G5, G7  
9.02b  Applying congruent triangles  Apply congruent triangles in calculations and simple proofs.
e.g.The base angles of an isosceles triangle are equal.  G6, G19  
9.03  Plain vector geometry  
9.03a  Vector arithmetic  Understand addition, subtraction and scalar multiplication of vectors.  Use vectors in geometric arguments and proofs.  G25  
9.03b  Column vectors  Represent a 2dimensional vector as a column vector, and draw column vectors on a square or coordinate grid.  G25  
9.04  Similiarity
 
9.04a  Similar triangles  Identify similar triangles.  Prove that two triangles are similar.  G6, G7  
9.04b  Enlargement
 Enlarge a simple shape from a given centre using a whole number scale factor, and identify the scale factor of an enlargement.
 Identify the centre and scale factor (including fractional scale factors) of an enlargement of a simple shape, and perform such an enlargement on a simple shape.  Perform and recognise enlargements with negative scale factors.  R2, R12, G7

9.04c  Similar shapes
 Compare lengths, areas and volumes using ratio notation and scale factors.
 Apply similarity to calculate unknown lengths in similar figures.
[see also Direct proportion, 5.02a]  Understand the relationship between lengths, areas and volumes of similar shapes.
[see also Direct proportion, 5.02a]  R12, G19 
 reflection, rotation, translation and their combinations
 congruent triangles
 similarity and enlargement
 vector arithmetic and the use of column vectors.
The following activities are designed to give learners practice in each of the basic skills.
[9.01a] Reflection Questions Worksheet KS3/GCSE (TES)
[9.01b] Rotations
[9.01b] Rotation Practice and Revision (TES)
[9.01c] Vectors
[9.02a] Conditions for Congruence (TES)
[9.03a] Translation Vectors
[9.04a] [9.04c] Ratio of Length, Area and Volume of Similar Shapes (TES)
[9.04a] [9.04c] Similarity (TES)
[9.01a] [9.01b] [9.01c] [9.04b] Transformations of Shapes (STEM Learning)
Thinking Conceptually
Overview
Approaches to teaching the content
Learners will need to be confident with the Cartesian coordinates (x, y) before starting this topic and know that the coordinates refer to the line intersections and not the squares. Translations are usually the best starting point for this topic, as this allows a recap of coordinates, and writing vectors in column form is a natural extension of this. Make sure that learners recognise the difference between an absolute position and the vector displacement.
Reflections and rotations are trickier and whilst some learners can ‘see’ the transformations easily, others will need to rely on tracing paper and these resources should be prepared beforehand. Emphasise that for rotations, the angle, direction and centre should be specified. For reflections the line of reflection needs to be specified. Learners often confuse vertical and horizontal mirror lines; some initial time spent completing results tables and plotting mirror lines can be helpful in the long term. Another common mistake is to reflect into all three of the other quadrants rather than carrying out just the one axis reflection requested.
There are four types of classification for congruency in triangles. Two triangles are congruent if: all their side are the same (SSS); 2 angles and the included side are the same (ASA); 2 sides and the included angle are the same (SAS) or they are rightangled and the hypotenuse and another side are the same (RHS). Whilst the SSS and RHS classes are relatively straightforward the other two classes are trickier. The included side has to be between the two angles given for ASA whilst the included angle has to be between the two included sides for SAS.
Before attempting similar shape calculations, learners need to understand proportional reasoning as this is the basic tool required for these types of calculations. Encourage learners to work systematically, working from one shape to the other, getting all of the information written down in an equation and then solving the equation for the unknown side length.
Common misconceptions or difficulties
When carrying out transformations, learners often:
 think vectors represent a translation from the origin (as in (x, y) coordinates) rather than from ANY point
 have difficulty finding the centre of rotation – encourage the use of tracing paper to help with this
 reflect a shape by just translating it ‘across’ the mirror line – emphasise the ‘mirror’ and get learners to imagine what the shape would look like as a reflection in a mirror (a good word to use is that the shape gets ‘flipped’).
When thinking about the congruency of triangles, learners often:
 get confused with the relative position of angles and sides – emphasise that included angles and sides are ‘between’ the given sides or angles in ASA and SAS classifications
 think that a hypotenuse exists for any triangle and not just rightangled triangles – emphasise that the hypotenuse is the side opposite the right angle to help learners avoid this.
When carrying out similarity/enlargement calculations, learners often:
 forget that for negative scale factor enlargements, the shape gets ‘flipped across’ the centre of enlargement – encourage learners to imagine that if a shape is enlarged by a scale factor of \(\displaystyle \frac{1}{2}\)then the new coordinates are \(\displaystyle \frac{1}{2}\) multiplied by the translation from the centre to the original shape
 use the wrong sides in similarity calculations – emphasise that they must use corresponding sides in the shapes to form the ratio; they cannot choose sides randomly.
Conceptual links to other areas of the specification – useful ways to approach this topic to set learners up for topics later in the course
Symmetry of shapes is a useful precursor for the work on transformations and congruency.
Proportional reasoning is essential for the similarity calculations and should be mastered in the contexts of distance conversions, currency conversions and ratio problems before attempting this topic.
Mensuration for area and volume is a useful topic to teach concurrently or before the work on similar shapes as the formulae can be revised in this topic.
[9.01d] Mirror Mirror (NRICH)
[9.01d] On The Wall (NRICH)
[9.04b] Who Is The Fairest Of Them All? (NRICH)
[9.01d] Transformation Game (NRICH)
[9.01d] Transformation Golf (STEM Learning)
[9.02a] Congruent Shapes (BBC Bitesize)
[9.04c] Similar Shapes Worksheet (TES)
[9.04c] Similar Shapes Worksheet 2 (TES)
[8.03b] [9.02b] Angle Proofs (NRICH)
[8.04a] [9.02a] Congruence Proofs (Maths Warehouse)
Thinking Contextually
Overview
Tessellations and fractal patterns link geometry and transformations. Escher tessellation patterns also have links to congruency.
Reflections on coordinate grids link equations of lines and reflections.
Similarity in shapes brings together proportional relationships and enlargement scale factors. Scaled diagrams use similarity as do model making and packaging problems. There is a direct link to multiples of the values of Pythagorean triples, and the trigonometry ratios can be introduced through the use of enlargement.
[9.01d] Surprising Transformations (NRICH)
[9.01d] Combination of Transformations
[9.02a] Categorising and Using Venn Diagrams with Congruent Triangles (Regents Exam Prep Centre)
[9.04a] Measuring Height in Durham Cathedral (STEM)
[9.04c] Star Wars Day (Transum)
Check In Tests
Overview
Check In test 9.01
Check In test 9.02
Check In test 9.04
Acknowledgements
Overview
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