Senior Cycle - Geometry & Trigonometry (Strand 2)

Particularly at Foundation level and Ordinary level, geometrical results should first be encountered by students through investigation and discovery. At all levels the use of a dynamic geometry software package is expected. Students are asked to accept these results as true for the purpose of applying them to various contextualised and abstract problems. They should come to appreciate that certain features of shapes or diagrams appear to be independent of the particular examples chosen. These apparently constant features or results can be established in a formal manner through logical proof. Even at the investigative stage, ideas involved in mathematical proof can be developed. Students should become familiar with the formal proofs of the specified theorems (some of which are examinable at Higher level). Students will be assessed by means of problems that can be solved using the theory.

Click here to view Senior Cycle syllabus, English | Gaeilge

Resources for strand 2.
A number of resources have been developed by the Project Maths Development Team to complement Strand 2 for Senior Cycle. These are listed below, click on the relevant resource to access it.
Teaching & Learning Geometry: The Van Hiele Model	T&L Plan: Vectors
Possible sequence to teaching Geometry	T&L Plan: The triangle law for Vector addition
T&L Plan on trignometric functions	T&L Plan: Coordinate geometry of circle
T&L Plan on Radians	The Geometry Appendix
T&L Plan on the unit circle	Synthetic Geometry : Guide to Axioms, Theorems and Constructions for all Levels.

Teaching & Learning Geometry: The Van Hiele Model

The work of two Dutch educators, Pierre van Hiele and Dina van Hiele-Geldof, have provided a vision around which to design geometry curriculum. Through their research they have identified five levels of understanding spatial concepts through which students move sequentially on their way to geometric thinking. There are four characteristics of these levels of thought:

The Van Hiele levels of geometric reasoning are sequential. Students must pass through all prior levels to arrive at any specific level.
These levels are not age-dependent in the way Piaget described development.
Geometric experiences have the greatest influence on advancement through the levels.
Instruction and language at a level higher than the level of the student may inhibit learning

To view a presentation exploring the Van Hiele model, click here

Possible sequence for teaching Geometry

A comprehensive document which offers up a suggested approach to the teaching and learning of geometry to second level students. The document contains several links to online student activities as well as paper based activities which can be used in the classroom with students.

To view document, click here

T&L Plan: The unit circle.

Through investigation students will become familiar with the unit circle, and the use of the unit circle to evaluate the trigonometric functions sin, cos and tan for all angles. Students will then be able to use their knowledge of the unit circle to find all the solutions of equations.

To download this T&L plan, click here

T&L Plan: Trignometric functions.

Starting with examples from real life of the use of trigonometric functions, students will then use the unit circle to investigate the graph of the function y = sin x by projecting the y coordinate onto an x-y plane. They will proceed to use tables of values to plot the graphs of sin x, cosx and tanx. Students then proceed to investigate and draw functions of the type "a sin bx", "a cos bx" where a, b are natural numbers and "tan x" leading to an understanding of the effect of changing the variable a and b. These investigations will also be carried out using dynamic geometry software.

To download this T&L plan, click here | To view interactive file on Sine & Cosine graph, click here

T&L Plan: Radians

For Leaving Certificate Higher Level students. Its purpose is to introduce radian measure from a “Teaching for Understanding” perspective. The plan takes an explorative approach to investigate the properties of radian measure. Students will be provided with a range of activities and teacher support to generate a deep understanding of radian measure so that students will be able to apply its use in problems.

To download T&L plan click here | To download student activity 1, click here

To download student activity 2, click here | To view interactive file on Radians, click here

T&L Plan: Vectors

Leaving Certificate Higher level students are introduced to the topic of vectors. Through a discovery process students find out that vectors have both magnitude and direction. Then the concept of vector notation, the null vector, equal vectors and negative vectors are introduced. Everyday applications of vectors are used where possible. Accompanying this plan is an interactive file to demonstrate the concepts of a vector alternatively for those with access to the software package Autograph the appendix of the plan contains instructions on how to use this package to demonstrate the concepts of a vector.

To view T&L Plan, click here | To view student activity click here

To view interactive on vectors file, click here

T&L Plan: The triangle law for Vector addition.

This is a continuation of the Vectors teaching and learning plan, it introduces the concept of the addition of two vectors by the Triangle Law. Examples of vectors from everyday situations are used where possible to enable students to gain both an understanding and appreciation of the topic, these include travelling to different locations, winds and currents. Accompanying this plan is an interactive file to demonstrate the concept and the appendix of the plan contains instructions on how to use the software package Autograph to do the same.

To view T&L Plan, click here | To view student activites, click here

To view interactive file for Triangle Law of addition for vectors, click here

To view interactive file for parallelogram addition of vectors, click here

T&L Plan: Coordinate geometry of circle.

This Teaching and Learning plan is for both Leaving Certificate Ordinary level and Higher Level students. It takes the students through a series of activities which help them to construct the General equation of a circle with centre (h,k). The activities help students to visualise the loci of points on various curves and hence determine the locus of a point on a circle. For Higher level students the Teaching and Learning plan contains an activity by which they can arrive at the General equation of a circle of centre (h,k) by an alternative
method. The Higher level students, using an Activity Sheet, will also derive the general equation of a circle of centre (-g,-f).

To download T&L plan, click here

To download;

Student activity 1, click here | Student activity 2, click here

Student activity 3, click here | Student activity 4, click here

The Geometry Appendix.

The Geometry section in both syllabuses refers to Appendix 1, which lists all of the relevant information needed to complete this topic. The Geometry Appendix can be downloaded by clicking the link below.

Appendix 1: Geometry course for post-primary school mathematics.

Synthetic Geometry : Guide to Axioms, Theorems and Constructions for all Levels.

This document is intended as a quick guide to the various axioms, theorems and constructions as set out in the Appendix to Strand 2 -Geometry and Trigonometry. It is not intended as a replacement for that appendix, merely as an aid to reading at a glance which material is required to be studied at various levels.
As stated in the heading, these theorems and constructions are underpinned by 46 definitions and 20 propositions which are all set out in the appendix, along with many undefined terms and definable terms used without explicit definition.

To view guide, click here

Files:

(01 Jan 1970, 0B)