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    • Leaving Certificate ›› LC Geometry & Trigonometry
    The synthetic geometry covered at Leaving Certificate is a continuation of that studied at junior cycle. It is based on the Geometry for Post-primary School Mathematics, including terms, definitions, axioms, propositions, theorems, converses and corollaries. 

    It is also envisaged that, at all levels, learners will engage with a dynamic geometry software package.

    In particular, at Foundation level and Ordinary level learners should first encounter the geometrical results below through investigation and discovery.

    Even at the investigative stage, ideas involved in mathematical proof can be developed. Learners should become familiar with the formal proofs of the specified theorems (some of which are examinable at Higher level). Learners will be assessed by means of problems that can be solved using the theory.

    -Leaving Certificate Syllabus page 23
    General Resources for Geometry and Trigonometry
    The following teaching and learning plans are available:

    The Unit Circle Trigonometric Functions
    The Unit Circle (PPT) Trigonometric Functions (PPT)
    Exploring Radian Measure (PDF)  
    Resources from the Junior Cycle page may be useful for teaching the Foundation Level Syllabus.
    Visit our dedicated Geometry page with resources outlining the key Van Hiele Theory of Geometric Understanding along with examples that you can use in the classroom.
    Some useful resources for teaching Geometry and Trigonometry can be found on the Workshop and Event Resources Page, in particular, Teaching Geometry for Understanding (2019 – 2020), NQT Workshop 3 (2018 – 2019), Project Maths Workshop 2 , Modular Course 2 (2008 – 2015) and Seminar (2014 – 2015).
    GeoGebra tutorials on using the Geometry App and may useful for the teaching of Geometry and Trigonometry.
    An instructional video explaining how the use of Digital Geoboards  may be useful for the teaching of Geometry and Trigonometry.
    This series of posters highlighting Geometry in the world around us. Posters can also be downloaded as individual files. This poster demonstrates the use of circle theorems, to judge a freehand circle-drawing competition.
    There are a range of lesson plans developed by teachers available in our Lesson Study Library.
    There are a range of resources created by teachers available on Scoilnet to support teaching and learning of Geometry and Trigonometry.
    Resources by Syllabus Outcome

    Outcome 2.1

    Synthetic Geometry

    Proof
    A Glance Card list of axioms and theorems for all levels.
    Student activity on describing theorems in their own words (Leaving Certificate Ordinary Level)
    Student activity on describing theorems in their own words (Leaving Certificate Higher Level)
    Student activity on Theorem 7 – The angle opposite the greater of the two sides in a triangle is greater than the angle opposite the lesser.  
    Student activity on Theorem 8- The angle opposite the greater of the two sides in a triangle is greater than the angle opposite the lesser. Accompanying Excel File and GeoGebra file for investigating Theorem 8.
    This Presentation investigates the proof of Theorem 12.
    Student activity on Theorem 13 – If two triangles are similar then their sides are proportional in order.
    This Presentation investigates proof by contradiction and the converse of a statement.
    This Presentation highlights geometry in the world around us and the notion of proof in geometry and number.

    Constructions
    Glance Card list of constructions for all levels.
    This teacher activity guide investigates a suggested approach to introducing the circumcenter.
    Instructional videos on locating the circumcentre and the incentre of a circle.
    Instructional video on construction a 60 degree angle.
    Student activity using paper folding for constructing the circumcentre of different types of triangles.
    Teacher file and student activity demonstrating how to use problem-solving as a strategy for teaching circle constructions.
    Teaching and learning plan developed by teachers on using GeoGebra to teach constructions for understanding.

     The circle
    A student activity showing that what seems like a reasonable generalisation of a pattern for small values of n may not hold for all values of n. In this case the pattern of the powers of 2 breaks down at n=6
    Instructional Video showing how to construct the centre of a circle, given an arc of the circle, using paper folding. 
    Student activity on Theorem 21 – The perpendicular bisector of a chord passes through the centre of its circle.

    Area 
    Teaching and learning plan, developed by teachers to investigate the properties of geometric shapes.

    Outcome 2.2

    Co-ordinate Geometry

    Area of a Triangle
    This teacher activity guide outlines a suggested approach to exploring suitable strategies for finding the area of a triangle.

    The Circle
    Teaching and learning plan, developed by teachers on the equation of a circle.
    Instructions on how to use GeoGebra to investigate changes to the equation of a circle with radius.
    Sequence of lesson ideas for teaching the circle.
     

    Outcome 2.3

    Trigonometry

    Trigonometric Ratios
    Presentation and student activity on trigonometric ratios.
    Tutorial on how to use GeoGebra to explore trigonometric ratios.

    Trigonometric Graphs
    Teaching and learning activities on exploring trigonometric graphs in context.
    Instructional video showing students investigating trigonometric graphs.

    3d Trigonometry
    Instructional video showing students using trigonometry to solve 3D problems.
     

    Outcome 2.4

    Transformation geometry, enlargements

    Enlargements
    Presentation and Student activity on making connections in geometry based on triangles and enlargements.
    Teaching and learning plan, developed by teachers, on enlargements.
     

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