Teacher Handbook LC OL

Teacher Handbook LC OL
Senior Cycle
TEACHER HANDBOOK Senior Cycle Ordinary Level th
th
5 & 6 year Based on the 2015 syllabus SeniorCycleOrdinaryLevel
Contents
Page
Introduction
4
Section 1
Strand 3
Number systems
10
Section 2
Strands 3, 4, 5
Patterns, Functions, Algebra
14
Section 3
Strand 2
Co-ordinate Geometry (Line)
21
Section 4
Strand 2
Synthetic Geometry 1
26
Section 5
Strand 1
Probability and Statistics 1
36
Section 6
Strand 3
Length, area and volume
40
Section 7
Strand 5
Functions & Calculus
41
Section 8
Strand 1
Probability & Statistics 2
44
Section 9
Strand 2
Synthetic Geometry 2
48
Section 10
Strand 2
Co-ordinate Geometry (Circle)
52
Section 11
Strand 2
Trigonometry
53
Section 12
Strand 3
Complex numbers
54
Appendix A
Geometry: Thinking at Different Levels: The Van Hiele Theory
Appendix B
Guide to Theorems, Axioms and Constructions at all Levels
Appendix C
Investigations of quadrilaterals and triangles
Appendix D
How to use CensusAtSchool
The strand structure of the syllabus should not be taken to imply that topics are to be studied
in isolation. Where appropriate, connections should be made within and across the strands
and with other areas of learning. (NCCA JC syllabus page 10 and LC syllabus page 8)
2 SeniorCycleOrdinaryLevel
Resources which will allow teachers plan lessons, easily access specific learning outcomes in
the syllabus and relevant support material such as “Teaching & Learning Plans” and
suggested activities to support learning and teaching are available on the Project Maths
website www.projectmaths.ie
3 SeniorCycleOrdinaryLevel
Introduction
Student Learning
While this is a handbook for teachers, it must be emphasised that student learning and the process of
mathematical thinking and building understanding are the main focus of this document.
Information and Communications Technologies are used whenever and wherever appropriate to help
to support student learning. It is also envisaged that, at all levels, learners will engage with a dynamic
geometry software package.
Students with mild general learning disabilities
Teachers are reminded that the NCCA Guidelines on mathematics for students with mild general learning
disabilities can be accessed at
http://www.ncca.ie/uploadedfiles/PP_Maths.pdf
This document includes
 Approaches and Methodologies (from Page 4)
 Exemplars (from page 20).
Note: Synthesis and problem solving listed below must be incorporated into all of the Strands.
The list of skills below is taken from Strand 1of the syllabus but, an identical list is given at the beginning of
each Strand in the syllabus.
4 SeniorCycleOrdinaryLevel
Useful websites
http://www.projectmaths.ie/
http://ncca.ie/en/Curriculum_and_Assessment/PostPrimary_Education/Project_Maths/
http://www.examinations.ie/
Literacy and Numeracy Strategy
The National Strategy to Improve Literacy and Numeracy among Children and Young People 20112020
Numeracy encompasses the ability to use mathematical understanding and skills to solve problems and
meet the demands of day-to-day living in complex social settings. To have this ability, a young person needs
to be able to think and communicate quantitatively, to make sense of data, to have a spatial awareness, to
understand patterns and sequences, and to recognise situations where mathematical reasoning can be applied
to solve problems.
Literacy includes the capacity to read, understand and critically appreciate various forms of communication
including spoken language, printed text, broadcast media, and digital media.
Colour coding used in the suggested sequence below:
Strand 1
Strand 2
Strand 3
Strand 4
Strand 5
Statistics and
Probability
Geometry and
Trigonometry
Number
Algebra
Functions
5 SeniorCycleOrdinaryLevel
Suggested sequence of topics
Section
number
Section 1
Strand
(Syllabus
section)
3.1
3.2
Section 2
Corresponding
Lesson Number
Title of lesson
Idea
Page
number
LCOL.1
Number systems
10
LCOL.2
Indices
10
3.2
LCOL.3
3.1
LCOL.4
3.3
LCOL.5
3.3
LCOL.6
3.3
LCOL.7
3.1
LCOL.8
3.1
LCOL.9
3.1
LCOL.10
5.1
LCOL.11
5.1
LCOL.12
4.1
LCOL.13
4.1
LCOL.14
4.1
LCOL.15
4.1
LCOL.16
4.2
LCOL.17
4.3
LCOL.18
Adding, subtracting,
multiplying and
dividing real
numbers
Factors, multiples
and primes
Arithmetic:
Percentages,
estimation, % error
Financial Maths
Ratio and proportion
in various contexts
Scientific notation
Relations approach
to algebra
Arithmetic
sequences and sum
to n terms of an
arithmetic series
12
Graphing functions
15
Composition of
functions
Revision of JC
algebra
Factorising in
algebra
Rearranging
formulae
Adding algebraic
fractions
Solving linear and
quadratic equations
Inequalities
6 11
11
12
13
13
14
15
16
16
17
18
18
18
19
SeniorCycleOrdinaryLevel
Section
number
Strand
(Syllabus
section)
Corresponding
Lesson Number
LCOL.19
Section 3
Section 4
Title of lesson
Idea
Relations without
formulae (JC 4.5)
Co-ordinate
geometry:
JC Revision Coordinating the
plane, distance and
midpoint formulae
Slope, parallel and
perpendicular lines
21
20
2.2
LCOL.20
2.2
LCOL.21
2.2
LCOL.22
Area of a triangle
22
2.2
LCOL.23
Equation of a line
23
2.2
LCOL.24
2.1
LCOL.25
2.1
LCOL.26
2.1
LCOL.27
2.1
LCOL.28
2.1
LCOL.29
2.1
LCOL.30
2.4
LCOL.31
2.1
LCOL.32
Intersection of two
lines
Synthetic geometry:
plane and pointsrevision of
preliminary concepts
Revision - Angles,
Axiom 3,Theorem 1,
Constructions 8 & 9
Revision:
Constructions 5, 6,
10,11,12 &
Theorem2
Revision: Theorems
3, 4,5,& 6
Theorems 7 & 8
Revision:
Constructions 1,2, &
4
Revision: JC
Transformation
geometry
Revision of JC
synthetic geometry:
Quadrilaterals &
parallelograms and
Theorems 9 & 10,
Construction 20
7 Page
number
21
23
26
26
27
28
28
29
29
30
SeniorCycleOrdinaryLevel
Section
number
Section 5
Strand
(Syllabus
section)
Corresponding
Lesson Number
Title of lesson
Idea
Page
number
2.1
LCOL.33
Revision of JC
synthetic geometry:
More on
quadrilaterals
32
2.1
LCOL.34
Theorem 11
33
Theorem 12 and
revision of Theorem
13
Revision:
Constructions
13,14,& 15 and
Theorems 14 & 15:
Pythagoras’
Theorem & converse
of same;
Proposition 9
Theorems 16, 17 &
18
2.1
LCOL.35
2.1
LCOL.36
2.1
LCOL.37
2.4
LCOL.38
Enlargements
35
1.1
LCOL.39
Fundamental
principle of counting
36
1.2&1.3
LCOL.40
Concepts of
Probability
36
1.2&1.3
LCOL.41
Rules of probability
369
1.2&1.3
LCOL.42
1.4 &1.5
LCOL.43
1.6 & 1.7
LCOL.44
The purpose of
statistics and the data
handling cycle
Data handling cycle
and sampling
Analysis of and
drawing inferences
from data
33
34
34
37
38
40
Proposed beginning of 6th year programme
Section 6
3.4
LCOL.45
Length, nets, area
and volume
40
3.4
LCOL.46
Trapezoidal rule
41
8 SeniorCycleOrdinaryLevel
Section
number
Strand
(Syllabus
section)
Corresponding
Lesson Number
Section 7
JC 4.5
LCOL.47
LCOL.48
5.2
LCOL.49
LCOL.50
Section 8
1.3
LCOL.51
1.2
LCOL.52
LCOL.53
1.6
LCOL.54
LCOL.55
Section 9
Title of lesson
Idea
Relations without
formula
Revision of
functions
Differential Calculus
Revision of counting
and probability
concepts from 5th
year
Bernoulli trials
Random variables
and expected value
Revision of 5th year
statistics
Bivariate data,
scatter plots and
correlation
Revision of 5th year
geometry
Corollaries 3 & 4
Theorem 20,
Corollary 6 &
Construction 19
41
42
44
44
45
45
45
46
48
2.1
LCOL.56
2.1
LCOL.57
2.1
LCOL.58
Construction 18
49
2.1
LCOL.59
Theorem 21
49
2.1
LCOL.60
Construction 21
50
2.1
LCOL.61
Section10
2.2
LCOL.62
Section11
2.3
LCOL.63
Trigonometry
Section12
3.1
LCOL.64
Complex numbers
Constructions 16 &
17
Coordinate geometry
of the circle
9 Page
number
48
48
50
52
53
SeniorCycleOrdinaryLevel
Lesson Ideas
Section 1: Number
Lesson Idea LCOL.1
Title
Number systems
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

The operations of addition, multiplication, subtraction and division (emphasising the order of
operations, including brackets) in the following domains:
N, Z, Q and representing these numbers on a number line

Decimals as special equivalent fractions and strengthening the connection between these numbers
and fractions and place value understanding
Fractions, decimals (that have a finite or a repeating decimal representation) and percentages as
different representations of rational numbers
Rounding of decimals
Terminating and non-terminating decimals
Irrational numbers R \ Q
Number system R , appreciating that R \ Q and representing real numbers on a number line




Lesson Idea LCOL.2
Title
Rules for indices
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

The rules for indices (where a, b  R, p, q  Q; a p , a q  Q; a, b  0 )
a p a q  a pq
ap
 a p  q
q
a
a0  1
a 
p q
 a pq
10 SeniorCycleOrdinaryLevel
a p 
1
ap
 ab 
 a pb p
p
p
ap
a

 
bp
b
Lesson Idea LCOL.3 (Possibly for 6th year depending on the class)
Title
Adding, subtracting, multiplying and dividing real numbers
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:



Irrational numbers, including surds
How to apply the following rules:
1
q
q
p
q
q
a  a,
a  ap =

q  Z, q  0, a  0
 
q
a
p
p, q  Z, q  0, a  0
Adding, subtracting, multiplying and dividing surds
Lesson Idea LCOL.4
Title
Factors, multiples and primes
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:



Factors, multiples and prime numbers in N
Expressing numbers in terms of their prime factors
Highest Common Factor and Lowest Common Multiple
11 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.5
Title
Percentages, estimating using real world contexts, % error
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:





Make estimates of measures in the physical world around them
How to make and justify estimates and approximations of calculations
How to check a result by considering whether it is of the right order of magnitude and by working
the problem backwards; round off a result
How to calculate percentage error and tolerance
How to calculate accumulated error ( due to addition or subtraction only)
Lesson Idea LCOL.6
Title
Financial Maths
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

How to solve problems involving
o Cost price, selling price, loss, discount
o Mark up (profit as a % of cost price)
o Margin (profit as a % of selling price)
o Income tax and net pay including other deductions
o Compound interest investigated using multi-representations i.e. table, graph and formula (link
to exponential functions and geometric sequences)
o Compound interest rate terminology such as AER, EAR and CAR
o Depreciation (reducing balance method) investigated using multi-representations i.e. table,
graph and formula (link to exponential functions and geometric sequences)
o Costing: materials, labour and wastage
o Currency transactions
12 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.7
Title
Ratio and proportion
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:


Ratios as comparing two quantities of the same kind by division (no units)
Contexts involving ratio and proportion
o Metric system; change of units; everyday imperial units
(conversion factors provided for imperial units)
o Rates as the comparison of two quantities by division but with different units
o Average rates of change with respect to time and units
o Diagrams drawn to scale
Lesson Idea LCOL.8
Title
Scientific notation involving real world contexts
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:





How to express non -zero positive rational numbers in the form a 10n , where n  Z and
1  a  10 involving real world contexts
How to enter very large numbers on the calculator
How to enter very small numbers on the calculator
How to perform arithmetic operations on numbers in scientific notation
How to solve problems involving numbers in scientific notation
13 SeniorCycleOrdinaryLevel
Section 2: Patterns, Functions and Algebra
Lesson Idea LCOL.9
Title
Relations approach to algebra- revision and extension of Junior Cycle material
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:



That processes can generate sequences of numbers or objects
How to investigate and discover patterns among these sequences
How to use patterns to continue the sequence

How to develop generalising strategies and ideas, present and interpret solutions, in the following:
o The use of tables, diagrams, graphs and formulae as tools for representing and analysing
linear patterns and relations

Discuss rate of change and the y - intercept. Consider how these relate to the context
from which the relationship is derived and identify how they can appear in a table, in
a graph and in a formula

Decide if two linear relations have a common value (decide if two lines intersect and
where the intersection occurs).

Recognise that the distinguishing feature of a linear relationship is a constant rate of
change

Recognise discrete linear relationships as arithmetic sequences
o The use of tables, diagrams, graphs and formulae as tools for representing and analysing
quadratic patterns and relations

Recognise that a distinguishing feature of quadratic relations is that the rate of change
of the rate of change is constant
o The concept of a function as a relationship between a set of inputs and a set of outputs where
each input is related uniquely to only one output
o The use of tables, diagrams, graphs and formulae as tools for representing and analysing
exponential patterns and relations

Recognise that a distinguishing feature of exponential relations is a constant ratio
between successive outputs

Recognise discrete exponential relationships as geometric sequences
14 SeniorCycleOrdinaryLevel
o Recognise whether a sequence is arithmetic, geometric or neither
Lesson Idea LCOL.10
Title
Arithmetic sequences and series
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:



The link between linear relations and the formula for the general term (Tn) of an arithmetic sequence
How to find the sum (Sn) of n terms of an arithmetic series
How to apply the formula for the nth term of an arithmetic sequence and the formula for the sum to n
terms of an arithmetic series to different contexts.
Lesson Idea LCOL.11
Title
Functions -interpreting and representing linear, quadratic and exponential functions in graphical form
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:


That a function assigns a unique output to a given input
Domain, co -domain and range of a function

The use of function notation f ( x)  ,

How to graph functions of the form:
o ax  b where a, b  Q, x  R
f : x , and y 
o
ax 2  bx  c, where a, b, c  Z, x  R
o
ax3  bx 2  cx  d , where a, b, c, d  Z, x  R
o
ab x where a  N, b, x  R

How to interpret equations of the form f ( x)  g ( x) as a comparison of the above functions

How to use graphical methods to find approximate solutions to
15 SeniorCycleOrdinaryLevel
o
f ( x)  0
o
f ( x)  k
o
f ( x)  g ( x)
where f ( x ) and g ( x) are of the above form, or where
graphs of f ( x) and g ( x) are provided

How to find local maximum and minimum values of quadratic functions from a graph

The relationship between the graphs of f  x   x 2 , g  x   ax 2 ,
h  x   x 2  c, k  x   ax 2  c, and l ( x)  ( x  c)2
Lesson Idea LCOL.12
Title
Composition of functions
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:


How to form composite functions (including notation used)
Composite functions in context
Lesson Idea LCOL.13
Title
Revision of basic JC algebra - evaluating and expanding expressions
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

The role of the variable (as unknown, representing quantities that vary and in expressing generality)

Indices in algebra (exponents  N )

Terms, coefficients and expressions

How to generate algebraic expressions from simple contexts
16 SeniorCycleOrdinaryLevel

Evaluate expressions given the value of the variables

Expand and regroup expressions

How to add and subtract expressions of the form:
 ax  by  c   ...   dx  ey  f 
 ax
2
 bx  c   ...   dx 2  ex  f  where a, b, c, d , e, f  Z
 How to use the associative and distribution properties to simplify expressions of the form:
o
o

a  bx  cy  d   ...  e  fx  gy  h 
where a, b, c, d , e, f , g , h  Z
 x  y  w  z 
How carry out operations of the form :
ax 2  bx  c  dx  e, where a, b, c, d , e  Z
Lesson Idea LCOL.14
Title
Factorising in Algebra - revision and extension of JC material
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:


Factors and multiples
Factorising expressions of the form:
ax, axy where a  Z
abxy  ay where a, b  Z
ax 2  bx where a, b  Z
sx  ty  tx  sy where s, t , x, y are variable

Factorise expressions of order 2
17 SeniorCycleOrdinaryLevel
Idea LCOL.15
Title
Rearrangement of formulae
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

How to rearrange formulae using familiar context
Lesson Idea LCOL.16
Title
Adding algebraic fractions - revision and extension of JC material
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Operations of the form:
ax  b dx  e

where a, b, c, d , e, f  Z
c
f
a
q

where a, b, c, p, q, r  Z
bx  c px  r
Lesson Idea LCOL.17
Title
Solving linear and quadratic equations
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

How to select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions
to equations of the form:
18 SeniorCycleOrdinaryLevel
o
f  x  g  x
with f  x   ax  b, g  x   cx  d and where a, b, c, d  Q
f  x   g  x  with
a
p

bx  c qx  r
e
g  x 
f
where a, b, c, e, f , p, q, r  Z
f  x 
o
o

f ( x )  k with f  x   ax 2  bx  c (and not necessarily factorisable),
a, b, c  Q and interpret the result
o How to solve simultaneous linear equations with two unknowns and interpret the results
o How to solve simultaneous equations where one is linear and the other is of order two with
two unknowns (restricted to the case where either the coefficient of or the coefficient of
is  1 in the linear equation) and interpret the results.
How to form quadratic equations given whole number roots
Lesson Idea LCOL.18
Title
Inequalities
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:
How to select and use suitable strategies (graphic, numeric, algebraic, mental) for finding solutions to
inequalities of the form:

g  x  k, g  x  k,
g  x   k , g  x   k where g  x   ax  b, a, b, k Q

Graph solution sets on the number line for linear inequalities in one variable
19 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.19
Title
Relations without formulae (See 4.5 JCOL)
Resources
Online resources on the Project Maths website
Dynamic software package
Motion sensor
Content
These lessons will involve the students in investigating and understanding:




Graphs of motion
Quantitative graphs and drawing conclusions from them
The connections between the shape of a graph and the story of a phenomenon
Quantity and change of quantity on a graph
20 SeniorCycleOrdinaryLevel
Section 3: Co-ordinate Geometry of the line
Lesson Idea LCOL.20
Title
Review of Junior Cycle ordinary-level coordinate geometry - coordinating the plane, distance and midpoint
formula
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:



The coordination of the plane
The distance formula
The midpoint formula
Lesson Idea LCOL.21
Title
Review of slope including application to parallel and perpendicular lines
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:
Rise
 The idea of slope as
Run
 The slope formula
 The meaning of positive, negative, zero and undefined slope.
 Use of slopes to investigate if two lines are parallel
 The use of slopes to investigate if two lines are perpendicular or not

That 3 points on the coordinate plane are collinear if and only if the slope between any two of them
is the same
21 SeniorCycleOrdinaryLevel
Lesson Idea LCOL. 22
Title
Area of a triangle
Resources
Online resources on the Project Maths website
Dynamic software package
Content
This lesson will involve the students in investigating and understanding:
 How to calculate the area of a triangle using coordinates
 The connection between this formula ,the geometric approach to the area of a triangle and the
formula used in trigonometry for finding the area of a triangle
Lesson Idea LCOL.23
Title
Equation of Line
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:
 The equation of a line in the forms :
y  y1  m  x  x1 
y  mx  c






ax  by  c  0
(The significance of the variables m and c)
Whether or not a point is on a line
Where a line intersects the axes and why these points might be of interest to someone trying to
interpret or plot a graph
The interpretation of the intercepts in context
How to find the slope of a line given its equation
How to solve problems involving slopes of lines
The link between coordinate geometry of the line and linear functions
22 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.24
Title
Intersection of two lines
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:
 A graphical approach to the intersection of two lines
 An algebraic approach to the intersection of two lines, using simultaneous equations
 The comparison and verification of both of these methods above
23 SeniorCycleOrdinaryLevel
Section 4: Synthetic Geometry 1
The following is a suggested sequence for teaching the Leaving Certificate Course. In teaching these
lessons, teachers and students can draw from the Teaching and Learning Plans and student activities on the
website at www.projectmaths.ie
As outlined at the workshops, the use of learning materials such as “geostrips”, “anglegs”, geo-boards etc.
can make the learning so much more enjoyable for students of all perceived abilities.
While proofs are not the issue as regards informal introduction, it is important that students are kept
aware that the theorems build logically.
The lesson divisions which follow are for guidance only. The initial lesson ideas give the students a chance
to revisit the material they met in the Junior Cycle. This can be done at a pace that is appropriate to the
student’s needs. It is recommended that new activities and challenges be introduced during this revision so
that students do not see it as too much repetition and that they can see new ways of investigating familiar
situations.
Note on experimentation and experimental results:
With experimentation, involving measurement, the results are only approximations and won’t agree exactly.
It is important for students to report faithfully what they find e.g. for a triangle they could find the sum of
the angles to be 1790 or 181° etc. The conclusion is that the angles appear to add up to 1800 .This is a
plausible working assumption. There is a distinction between what you can discover and what you can
prove.
See below: Section 8.2 (From Discovery to Proof) of Geometry for Post-primary School Mathematics”
24 SeniorCycleOrdinaryLevel
25 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.25
Title
Revision of preliminary concepts - Plane and points
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:
 Plane, points, lines, rays, line segments, collinear points, length of a line segment
 What is meant by the terms “axiom” and “implies”
 Axiom 1: [Two points axiom]: There is exactly one line through any two given points.
 Axiom 2: [Ruler Axiom]: The properties of the distance between points
 Angles as a rotation, angles in different orientations

How to estimate angles in degrees, naming angles

Terms: Perpendicular, parallel, vertical, horizontal
Lesson Idea LCOL.26
Title
Revision - Angles, Axiom 3, Theorem 1, Constructions 8 & 9
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Measurement of angles using a protractor
(Possible misconceptions: Students thinking that size of an angle varies with arm or arc-length;
failure to recognise equal angles in different orientations
Common error: Reading from the incorrect scale on the protractor)

Axiom 3 (The protractor axiom)

That a straight angle measures 180⁰

Supplementary angles

Vertically opposite angles

What is meant by the term “theorem”

Theorem 1: Vertically opposite angles are equal in measure.
26 SeniorCycleOrdinaryLevel

What is meant by “proof”

The use of the compass

Construction 8:
Line segment of a given length on a given ray

Construction 9:
Angle of a given number of degrees with a given ray as one arm
Lesson Idea LCOL.27
Title
Revision of JC synthetic geometry
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Construction 5: Line parallel to a given line, through a given point;

Axiom 5: Given any line l and a point P, there is exactly one line through P parallel to l.

Construction 6: Division of a line segment into 2 or 3 equal segments without measuring it

Triangles and congruent triangles
o
Triangles: scalene, isosceles, equilateral, right-angled
o
Construction 10:
Triangle given SSS – Congruent triangles (Axiom 4)
o
Construction 11:
Triangle given SAS - Congruent triangles (Axiom 4)
o
Construction 12:
Triangle given ASA - Congruent triangles (Axiom 4)
o
By construction, show that AAA and ASS are not sufficient conditions for congruence.
o
What is meant by the term “converse”
o
Theorem 2: (i) In an isosceles triangle the angles opposite the equal sides are equal.
(ii) Conversely, if two angles are equal, then the triangle is isosceles
Suggested class activities
Students might engage in activities in relation to scalene, equilateral, isosceles, right-angled and obtuseangled triangles
o
o
o
o
o
Describe each in words
Draw three examples of each in different orientations
Determine the sum of the angles in each
Determine which triangles, if any, contain two or more equal angles
Establish if they can say for certainty the size of the angles in each
27 SeniorCycleOrdinaryLevel
o Establish if any of the triangles can belong to more than one category
Lesson Idea LCOL.28
Title
Revision of JC synthetic geometry
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Alternate angles by examples and measuring

Theorem 3:
(i) If a transversal makes equal alternate angles on two lines, then the lines are
parallel.
(ii) Conversely, if two lines are parallel, then any transversal will make equal alternate
angles with them.

Theorem 4:

Corresponding angles explained by examples and measuring

Theorem 5:
The angles in any triangle add to 180⁰.
Two lines are parallel if and only if for any transversal, corresponding angles are
equal.

Theorem 6:
Each exterior angle of a triangle is equal to the sum of the interior opposite angles.
Lesson Idea LCOL.29
Title
Theorem 7 and Theorem 8
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Theorem 7: In a triangle, the angle opposite the greater of two sides is greater than the angle opposite
the lesser. Conversely, the side opposite the greater of two angles is greater than the side opposite
the lesser angle.
28 SeniorCycleOrdinaryLevel

Theorem 8:
Two sides of a triangle are together greater than the third.
Lesson Idea LCOL.30
Title
Revision - Constructions 1, 2, & 4
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Construction 1: Bisector of a given angle, using only compass and straight angle

Construction 2:
Perpendicular bisector of a segment using only compass and straightedge

Construction 4:
Line perpendicular to a given line l, passing through a given point on l
Lesson Idea LCOL.31
Title
Revision of translations, central symmetries, axial symmetries and rotations
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:







Translations
Axial symmetry
Central symmetry
Rotations
Suggested class activities
Students might engage in the following investigations:
Does a translation preserve length?
Does a translation preserve angle size?
Does a translation map a line onto a parallel line?
Does a translation map a triangle onto a congruent triangle?
Does an axial symmetry preserve length?
Does an axial symmetry preserve angle size?
Does an axial symmetry maps a line onto a parallel line?
Does an axial symmetry map a triangle onto a congruent triangle?
How many axes of symmetry does an isosceles triangle have?
How many axes of symmetry does an equilateral triangle have?
29 SeniorCycleOrdinaryLevel

How many axes of symmetry does a circle have?
(Draw examples of the above.)
Does a central symmetry preserve length?
Does a central symmetry preserve angle size?
Does a central symmetry map a line onto a parallel line?
Does a central symmetry map a triangle onto a congruent triangle?
Does an isosceles triangle have a centre of symmetry?
Does an equilateral triangle have a centre of symmetry?
Which types of triangle have a centre of symmetry?
Does a circle have a centre of symmetry?
Note: quadrilaterals are investigated in the lessons following.
Lesson Idea LCOL.32
Title
Revision of quadrilaterals, parallelograms, Theorems 9 & 10, Construction 20
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Theorem 9: In a parallelogram, opposite sides are equal, and opposite angles are equal
Conversely, (1) if the opposite angles of a convex quadrilateral are equal, then it is a parallelogram;
(2) if the opposite sides of a convex quadrilateral are equal, then it is a parallelogram.
Remark 1: Sometimes it happens that the converse of a true statement is false. For example, it is true
that if a quadrilateral is a rhombus, then its diagonals are perpendicular. But it is not true that a
quadrilateral whose diagonals are perpendicular is always a rhombus.
Remark 2: The converse of Corollary 1 is false: it may happen that a diagonal divides a convex
quadrilateral into two congruent triangles, even though the quadrilateral is not a parallelogram.
 Theorem 10: The diagonals of a parallelogram bisect one another.

Conversely, if the diagonals of a quadrilateral bisect one another, then the quadrilateral is a
parallelogram.
Construction 20: Parallelogram, given the length of the sides and the measure of the angles

The properties of different quadrilaterals
Suggested class activities
Students might engage in the following activities which lead to an informal proof of theorem 9:
Draw a parallelogram ABCD which is not a rectangle or a rhombus
Draw in one diagonal BD
30 SeniorCycleOrdinaryLevel
Mark in all the alternate angles – they should find 2 pairs
Establish that triangles ABD and BCD are congruent and explain their reasoning
Establish what this means about the opposite sides of parallelogram ABCD
Make a deduction about the opposite angles of parallelogram ABCD
The students can determine:
If the diagonal bisects the angles at the vertex
The sum of the four angles of parallelogram ABCD
The result if two adjacent angles of the parallelogram are added together
Students might engage in the following activities which lead to an informal proof of
theorem 10 (In all instances they should be encouraged to explain their reasoning):
Draw a parallelogram ABCD which is not a rectangle or a rhombus
Draw in the two diagonals AC and BD intersecting at E
Determine if the two diagonals equal in length. (Measure)
Mark in all the equal sides and angles in the triangles AED and BEC
Explain why triangles ADE and BEC are congruent (Give a reason.)
Possible further investigations:
The students can determine:
If the triangles AEB and DEC are congruent
If the diagonals perpendicular
If the parallelogram contains 4 two pairs of congruent triangles
If the diagonals bisect the vertex angles of the parallelogram
The number of axes of symmetry the parallelogram has
If the parallelogram has a centre of symmetry and its location if it does exist
Students might engage in the following activities about a square, rhombus, parallelogram and
rectangle: (In all instances they should be encouraged to explain their reasoning.)
Describe each of them in words.
Draw three examples of each in different orientations.
Determine which sides are equal in length.
Determine the sum of the angles in each case.
Determine which angles are equal.
Determine the sum of two adjacent angles in each case.
Establish whether or not a diagonal bisects the angles it passes through.
Establish whether or not the diagonals are perpendicular.
Determine whether or not a diagonal divides it into two congruent triangles.
Calculate the length of a diagonal given the length of its sides, where possible.
Establish whether or not the two diagonals are equal in length.
Determine whether or not the diagonals divide the different shapes into 4 congruent triangles.
Establish if the diagonals bisect each other.
31 SeniorCycleOrdinaryLevel
The students should determine the number of axes of symmetry each of the shapes has and which ones have
a centre of symmetry.
An interesting option would be to conduct the activities above on a KITE.
Lesson Idea LCOL.33
Title
Revision: More Quadrilaterals – Investigating a Square
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:


The properties of a square
The properties of a rectangle(which is not a square)
Suggested class activities
Students might engage in the following activities relating to a square: (In all instances they should be
encouraged to explain their reasoning.)
Draw a square ABCD.
Draw in the two diagonals AC and BD intersecting at E.
Determine whether or not the two diagonals are equal in length.
Mark in all the equal sides and angles in the triangles AED and BEC.
Establish that triangles ADE and BEC are congruent.
Determine if the triangles AEB and DEC are congruent.
Determine if there are two pairs of congruent triangles in the square.
Show that the diagonals perpendicular. Give a reason.
Establish whether or not the diagonals bisect the vertex angles of the square.
Find how many axes of symmetry the square has.
Determine whether or not the square has a centre of symmetry and if it does, what is its location.
Students might engage in the following activities about a rectangle:
(In all instances they should be encouraged to explain their reasoning.)
Draw a rectangle ABCD which is not a square
Draw in the two diagonals AC and BD intersecting at E and establish if the two diagonals are equal in
length
Mark in all the equal sides and angles in the triangles AED and BEC.
Establish that triangles ADE and BEC are congruent.
Determine whether or not the triangles AEB and DEC are congruent.
Determine whether or not there are two pairs of congruent triangles in the rectangle.
32 SeniorCycleOrdinaryLevel
Show that the diagonals are perpendicular.
Determine whether or not the diagonals bisect the vertex angles of the rectangle.
Find how many axes of symmetry the rectangle has.
Determine whether or not the rectangle has a centre of symmetry and if it does, find its location.
Possible extra activity:
Repeat these activities for the rhombus ABCD
Lesson Idea LCOL.34
Title
Theorem 11
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Theorem 11: If three parallel lines cut off equal segments on some transversal line, then they will
cut off equal segments on any other transversal.
Lesson Idea LCOL.35
Title
Theorem 12 and revision of Theorem 13
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Theorem 12: Let ∆ABC be a triangle. If a line is parallel to BC and cuts [AB] in the ratio
: , then it also cuts [AC] in the same ratio.
Converse to Theorem 12: Let ∆ABC be a triangle. If a line cuts the sides AB and AC in the same
ratio, then it is parallel to BC.

The meaning of similar triangles and the difference between similar and congruent triangles.

Theorem 13: If two triangles ∆ABC and ∆ ′ ′ ′’ are similar, then their sides are proportional, in
order:
| |
| |
| |
|
| |
′| | ′ |
33 SeniorCycleOrdinaryLevel
Converse: If
|
|
|
|
|
|
|
|
|
|
|
|
,then the two triangles ∆ABC and ∆ ′ ′ ′’ are similar .
.
Lesson Idea LCOL.36
Title
Revision of right-angled triangles and Pythagoras’ Theorem
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:


Construction 13:
Construction 14:
cases)
Right-angled triangle, given length of hypotenuse and one other side.
Right-angled triangle, given one side and one of the acute angles (several

Construction 15:
Rectangle, given side lengths

Theorem 14:
Theorem of Pythagoras

Theorem 15:
[Converse to Pythagoras] If the square of one side of a triangle is the sum of
the squares of the other two, then the angle opposite the first side is a right angle.

Proposition 9:
If two right-angled triangles each have hypotenuse and one other side equal
in length respectively, then they are congruent. (RHS)
Lesson Idea LCOL.37
Title
Introduction to area
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Theorem 16: For a triangle, base times height does not depend on the choice of base.

Definition 38: The area of a triangle is half the base multiplied by the height.

Theorem 17: A diagonal of a parallelogram bisects the area.

Theorem 18: The area of a parallelogram is the base by the height.
34 SeniorCycleOrdinaryLevel
Suggested class activities
Students might engage in the following activities:
In the case of each of these types of triangles: equilateral, isosceles, right-angled and obtuse-angled:
draw three diagrams for each type of triangle showing each side as a base and the corresponding
perpendicular height.
Students investigate the validity of the following statement and its converse: “Congruent triangles have
equal areas”.
Lesson Idea LCOL.38
Title
Enlargements
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

Enlargements, paying attention to
o Centre of enlargement
o Scale factor k, 0  k  1, k  1 ,k  Q
o How to draw an enlargement of a figure given a scale factor when the centre of enlargement
is outside the figure to be enlarged
o How to draw an enlargement of a figure given a scale factor when the centre of enlargement
is inside the figure to be enlarged
o How to draw an enlargement of a figure given a scale factor when the centre of enlargement
is a vertex of the figure to be enlarged or is a point on the figure
o How to find the scale factor

That when a figure is enlarged by a scale factor k , the area of the image figure is increased by a
factor k 2

How to solve problems involving enlargements
35 SeniorCycleOrdinaryLevel
Section 5: Probability and Statistics 1
Lesson Idea LCOL.39
Title
Fundamental Principle of Counting and Permutations
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:



The Fundamental Principle of Counting
How to count the arrangements of n distinct objects (n!)
How to count the number of ways of arranging r objects from n distinct objects
Lesson Idea LCOL.40
Title
Concepts of probability
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

JC learning outcomes for probability
Lesson Idea LCOL.41
Title
Rules of probability
Resources
Online resources on the Project Maths website
Dynamic software package
36 SeniorCycleOrdinaryLevel
Content
These lessons will involve the students in investigating and understanding:


How to use set theory to discuss experiments, outcomes, samples spaces
The basic rules of probability (AND/OR), mutually exclusive events, through the use of Venn
diagrams
The use of the formulae:
1. Addition Rule (for mutually exclusive events only): P ( A  B )  P ( A)  P ( B )
2. Addition Rule: P ( A  B )  P ( A)  P ( B )  P ( A  B )
3. Multiplication Rule( for independent events): P ( A  B )  P ( A)  P ( B )

Use of tree diagrams
Lesson Idea LCOL.42
Title
The purpose of Statistics and the Data Handling Cycle
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

The purpose and uses of statistics and possible misconceptions and misuses of Statistics

How to design a plan and collect data on the basis of the above knowledge

The data handling cycle (Pose a question, collect data, analyse data, interpret the result and refine the
original question if necessary)

The Census at School (CAS) questionnaire as a means of collecting data

Questionnaire designs

Populations and samples

That sampling variability influences the use of sample information to make statements about the
population
 The importance of representativeness so as to avoid biased samples
 Sample selection (Simple Random Sample)
 The extent to which conclusions can be generalised
 Primary sources of data ( observational (including sample surveys) and experimental studies) and
secondary sources of data

The different ways of collecting data
37 SeniorCycleOrdinaryLevel

How to summarise data in a diagrammatic form
The students will also engage in analysing spreadsheets of data for example the spreadsheet of class
data returned from the Census at School questionnaire to include:
o Recognising different types of data – category (nominal /ordinal), numerical (discrete/
continuous)
o Recognising univariate/bivariate data
o Discussing possible questions which might be answered with the data
Lesson Idea LCOL.43
Title
Analysing data graphically and numerically, interpreting and drawing inferences from data
Resources
Online resources on the Project Maths website
Dynamic software package
Content
These lessons will involve the students in investigating and understanding:

The concept of the distribution of data and frequency distribution tables

The selection and use of appropriate graphical methods to describe the sample(univariate data only)
taking account of data type: bar charts, pie charts, line plots, histograms(equal class intervals), stem
and leaf plot (including back to back)

The distribution of numerical data in terms of shape (concepts of symmetry, clustering, gaps,
skewness)

The selection and use of appropriate numerical methods to describe the sample:

o The distribution of data in terms of centre (mean, median and mode and the
advantages and disadvantages of each)
o The relative positions of mean and median in symmetric and skewed data
o The distribution of numerical data in terms of spread (range, inter-quartile range)
o The concept of inter-quartile range as a measure of spread around the median
o The distribution of data in terms of spread (standard deviation)
o The concept of standard deviation as a measure of spread around the mean
o The use of a calculator to calculate standard deviation
How to analyse plots of data to explain differences in measures of centre and spread

How to interpret a histogram in terms of distribution of data and make decisions based on the
empirical rule (based on a normal distribution)
38 SeniorCycleOrdinaryLevel

The effect on the mean and of adding or subtracting a constant to each of the data points and of
multiplying or dividing the data points by a constant

Outliers and their effect on measures of centre and spread

The effect on the mean of adding or subtracting a constant to each of the data points and of
multiplying or dividing the data points by a constant
Proposed end of fifth year
39 SeniorCycleOrdinaryLevel
Section 6: Length, Area and Volume
Lesson Idea LCOL.44
Title
Length of the perimeter and area of plane figures, nets of solids, surface area and volume
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:

How to solve problems involving the length of the perimeter and the area of plane figures: disc,
triangle, rectangle, square, parallelogram, trapezium, sectors of discs and figures made from
combinations of these

The nets of prisms, cylinders and cones

How to solve problems involving the surface area and volume of the following sold figures:
rectangular block, cylinder, right cone, triangular based prism (right angle, isosceles and equilateral),
sphere, hemisphere and solids made from combinations of these
Lesson Idea LCOL.45
Title
Trapezoidal rule
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:


How to use the trapezoidal rule to approximate area
How to calculate percentage error involved in using trapezoidal rule in e.g. the circle
40 SeniorCycleOrdinaryLevel
Section 7: Functions and differential calculus
Lesson Idea LCOL.46
Title
Relations without formulae
Resources
Online resources on the Project Maths website
Dynamic software package
Motion sensor
Workshop 4 booklet (graph matching)
Content
These lessons will involve the students in investigating and understanding:

Graphs of motion
 Quantitative graphs and drawing conclusions from them
 The connections between the shape of a graph and the story of a phenomenon
 Quantity and change of quantity on a graph
Lesson Idea LCOL.47
Title
Revision of function concepts
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:

Revision of function concepts
41 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.48
Title
Calculus
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:
Rate of change, average rate of change, instantaneous rate of change, the derivative
This will include:

Calculus as the study of mathematically defined change How to use graphs and real life examples to
analyse rates of change for:
o Linear functions f  x   k where k is a constant
o Linear functions in general - links should be established to the slope of a line from
coordinate geometry
o Functions where the rate of change varies









Instantaneous rate of change as opposed to average rate of change say over the course of a journey
The equality of the instantaneous and average rates of change for linear functions
How to find the rate of change in situations where it is not constant and the need to define it at every
point
o The idea of average rate of change between two points on f ( x)= x2 and its calculation as the
slope of the secant connecting the two endpoints of the interval under consideration
o That the instantaneous rate of change is not the same as the average rate of change between
two points for f ( x)= x2
o That the average rate of change approaches the instantaneous rate as the interval under
consideration approaches zero (the concept of a limit)
o That the instantaneous rate of change is the slope of the tangent at the point
The meaning of the first derivative as the instantaneous rate of change of one quantity relative to
dy
another and the use and meaning of the terms “differentiation” and notation such as
and f '  x 
dx
How to find the first derivatives of linear functions using the equation y  mx  c and observing that
the slope the first derivative
Use the observation of patterns of change in the slopes of quadratic and cubic functions at points on
their graphs to come up with a rule for differentiating these functions.
How to find the first derivative of linear, quadratic and cubic functions by rule
How to apply differentiation to real life examples of rates of change
What it means when a function is increasing/decreasing/constant in terms of the slope, rate of
change, derivative
42 SeniorCycleOrdinaryLevel

How to apply an understanding of the change in
dy
from positive to zero to negative around a local
dx
maximum in order to identify a local maximum
How to apply an understanding of the change in

minimum in order to identify a local minimum
The meaning of the second derivative as the rate of change of a rate of change at any instant






Real life examples of the rate of change of a rate of change, for example acceleration as a rate of
change of velocity
How to match a function with its first and second derivatives
How to find second derivatives of linear, quadratic and cubic functions by rule
The application of the second derivative to identify local maxima and local minima
The second derivative and its connection with “concave up” and “concave down” sections of curves
How to apply differentiation to curve sketching
43 dy
from negative to zero to positive around a local
dx

SeniorCycleOrdinaryLevel
Section 8: Probability and Statistics 2
Lesson Idea LCOL.49
Title
Revision of counting and probability concepts from fifth year
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:






The Fundamental Principle of Counting
How to count the arrangements of n distinct objects (n!)
How to count the arrangements of n distinct object taking r at a time
Concepts of probability
Rules of probability
Use of tree diagrams, set theory
Lesson Idea LCOL.50
Title
Bernoulli Trials
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:



Find the probability that two independent events both occur
Bernoulli trials
(A Bernoulli trial is an experiment whose outcome is random and can be either of two possibilities:
“success” or “failure”. How to solve problems involving up to 3 Bernoulli trials
How to calculate the probability that the 1st success occurs on the nth Bernoulli trial where n is
specified
44 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.51 (option to switch lessons 51 and 50 as perhaps
random variables should be introduced before Bernoulli trials. See HL
lessons 48 and 49.)
Title
Random variables and expected value
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:




Random variables, discrete and continuous, which lead to discrete and continuous probability
distributions
Expected value
of probability distributions
The calculation of expected value and the fact that this does not need to be one of the outcomes
The role of expected value in decision making and the issue of fair games
Lesson Idea LCOL.52
Title
Revision of statistics concepts from 5th year
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in participating in investigating and understanding:

Revision of statistics concepts from 5th year
Lesson Idea LCOL.53
Title
Bivariate data, scatter plots, correlation
45 SeniorCycleOrdinaryLevel
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:

Bivariate data versus univariate data

Different types of bivariate data

The use of scatter plots to determine the relationship between numeric variables

That correlation always has a value from -1 to +1 inclusive, and that it measures the extent of the
linear relationship between two variables

How to match correlation coefficients values to appropriate scatter plots

That correlation does not imply causality
Lesson Idea LCHL.54
Title
Drawing inferences from data, margin of error, the concept of a hypothesis test,
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:
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How sampling variability influences the use of sample information to make statements about the
population
How to use appropriate tools to describe variability drawing inferences about the population from the
sample
How to interpret the analysis and relate the interpretation to the original question
How to interpret a histogram in terms of the distribution of data
The use of the empirical rule
1
How to calculate the margin of error for a population proportion (
)
n
The concept of a hypothesis test
The distinction between a null and an alternative hypothesis
46 SeniorCycleOrdinaryLevel
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How to conduct a hypothesis test on a population proportion using the margin of error
Note: The margin of error referred to here is the maximum value of the 95% confidence interval.
47 SeniorCycleOrdinaryLevel
Section 9: Synthetic Geometry 2
Lesson Idea LCOL.55
Title
Revision of fifth year synthetic geometry
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:

The synthetic geometry from 5th year
Lesson Idea LCOL.56
Title
Corollaries 3 and 4
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:
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The term “corollary”
Corollary 3: Each angle in a semi-circle is a right angle.
Corollary 4:If the angle standing on a chord [BC] at some point of the circle is a right-angle, then
[BC] is a diameter.
Lesson Idea LCOL.57
Title
Theorem 20, Corollary 6 and Construction 19
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
48 SeniorCycleOrdinaryLevel
Content
These lessons will involve the students in investigating and understanding:

Theorem 20: (i) Each tangent to a circle is perpendicular to the radius that goes to the point of
contact.
(ii) If P lies on the circle , and a line through P is perpendicular to the radius to P,
then is a tangent to .
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Corollary 6: If two circles share a common tangent line at one point, then the two centres and that
point are collinear.
Construction 19: Tangent to a given circle at a given point on it.
Lesson Idea LCOL.58
Title
Construction 18
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
This lesson will involve the students in investigating and understanding:
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Construction 18: Angle of 60⁰ without using a protractor or set square
Lesson Idea LCOL.59
Title
Theorem 21
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:

Theorem 21: (i) The perpendicular from the centre of a circle to a chord bisects the chord.
(ii) The perpendicular bisector of a chord passes through the centre of a circle.
49 SeniorCycleOrdinaryLevel
Lesson Idea LCOL.60
Title
Construction 21
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:
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Definition 45:
medians and centroid
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Construction 21:
Centroid of a triangle
Suggested class activities
Students might engage in the following activities:
Draw the medians and centroid for an acute-angled triangle, a right-angled triangle and an obtuse-angled
triangle.
In which instances is the centroid inside the triangle?
Lesson Idea LCOL.61
Title
Constructions 16 and 17
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:

Construction 16: Circumcentre and circumcircle of a given triangle, using only straight edge and
compass.
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Construction 17: Incentre and incircle of a given triangle, using only straight edge and compass.
Suggested class activities
Students might engage in the following activities:
Draw the circumcentre and incentre of an acute-angled triangle, a right-angled triangle, an obtuse-angled
triangle.
50 SeniorCycleOrdinaryLevel
They should then answer the following questions explaining their reasoning in each case:
In which instances is the circumcentre inside the triangle?
Where is the circumcentre in a right-angled triangle? (see Theorem 19, corollary 3)
In which instances is the incentre inside the triangle?
51 SeniorCycleOrdinaryLevel
Section 10: Co-ordinate geometry of the circle
Lesson Idea LCOL.62
Title
Co-ordinate geometry of the circle
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:

2
2
2
That x  y  r represents the equation of a circle centre (0,0) and radius of length r (Link to
Pythagoras’ Theorem - distance from any point p  x, y  on the circle to the centre of the circle is
equal to the length of the radius of the circle.)
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That
 x  h   y  k 
2
2
 r 2 represents the relationship between the and
co-ordinates of points
on a circle wit centre (h, k) and radius r
(Link to Pythagoras’ Theorem - distance from any point p  x, y  on the circle to the centre of the
circle is equal to the length of the radius of the circle.)
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How to solve problems involving a line and a circle with centre (0,0)
52 SeniorCycleOrdinaryLevel
Section11: Trigonometry
Lesson Idea LCOL.63
Title
Trigonometry 1
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:
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The use of Pythagoras’ Theorem to solve problems (2D only)
Trigonometric ratios in a right-angled triangle
The use of the trigonometric ratios to solve problems involving right-angled triangles
The use of similar triangles to find unknowns in right-angled triangles
The use of the clinometer
How to use trigonometry to calculate the area of a triangle
How to solve problems using the sine and cosine rules (2D)
How to define sin  and cos for all values of 
1
How to use the unit circle to solve equations such as sin   , cos  0 etc.
2
How to define tan 
How to solve problems involving the area of a sector of a circle and the length of an arc
Work with trigonometric ratios in surd form
53 SeniorCycleOrdinaryLevel
Section 12: Complex numbers
Lesson Idea LCOL.64
Title
Complex numbers
Resources
Online resources on the Project Maths website
Dynamic software package
A mathematical instruments set
Content
These lessons will involve the students in investigating and understanding:
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The origin and need for complex numbers
The use of complex numbers to model two dimensional systems as in computer games,
alternating current and voltage etc.
How to interpret multiplication by i as a rotation of 900 anticlockwise
How to express complex numbers in rectangular form a  ib and illustrate them on the
Argand diagram
How to investigate the operations of addition and subtraction of complex numbers in the
rectangular form  a  ib  using the Argand diagram
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How to interpret the modulus as distance from the origin on an Argand diagram
How to multiply a complex number by another complex number
Interpret the results of the above multiplication on the Argand diagram
How to interpret the complex conjugate as a reflection in the real axis
Division of complex numbers in the  a  ib  rectangular form and representation on the

Argand diagram
How to solve quadratic equations having complex roots and how to interpret the solutions
54 SeniorCycleOrdinaryLevel
Appendix A
Geometry: Thinking at Different Levels
The Van Hiele Theory
The Van Hiele model describes how students learn geometry. Pierre van Hiele and Dina van Hiele-Geldof,
mathematics teachers from the Netherlands, observed their geometry students in the 1950's.The following is
a brief summary of the Van Hiele theory. According to this theory, students progress through five levels of
thinking starting from merely recognising a shape to being able to write a formal proof. The levels are as
follows:
*Visualisation (Level 0)
The objects of thought are shapes and what they look like.
Students have an overall impression of a shape. The appearance of a shape is what is important. They may
think that a rotated square is a “diamond” and not a square because it is different from their visual image of
a square. They will be able to distinguish shapes like triangles, squares, rectangles etc but will not be able to
explain, for example, what makes a rectangle a rectangle. Vocabulary: Students use visual words like
“pointy”, “curvy”, “corner” as well as correct language like angle, rectangle and parallelogram.
*Analysis (Level 1)
The objects of thought are “classes” of shapes rather than individual shapes.
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Students think about what makes a rectangle a rectangle and can separate the defining characteristics
of a rectangle from irrelevant information like size and orientation. They recognize its parts (sides,
diagonals and angles) and compare their properties (similar, congruent)
They understand that if a shape belongs to a class like “rectangle”, then it has all the properties of
that class (2 pairs of equal sides, right angles, 2 equal diagonals, 2 axes of symmetry).
Vocabulary: words like parallel, perpendicular and congruent relating to properties within a figure
and the words all, always, sometimes, never, alike, different.
A concise definition of a figure, using a sufficient rather than an exhaustive list of properties is not
possible at this level.
They do not deal with questions like “Is a square a parallelogram?” but just look at the properties of
each class of shape, without comparing the classes.
*Some visualisation and analysis is covered in Primary School.
Relational/ Ordering/Informal Deduction (Level 2)
The objects of thought are the properties of shapes.
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Students are ready to understand interrelationships of properties within figures and between figures.
Opposite sides of a parallelogram are parallel and opposite angles are equal.
A rectangle is a parallelogram since it has all the properties of a parallelogram as well as having all
900 angles.
55 SeniorCycleOrdinaryLevel
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Students can recognise the difference between a statement and its converse. All squares are
rectangles (true) is different from all rectangles are squares (not true).
Capable of “if –then” thinking – if a shape is a rectangle then all the angles in it are right angles. If
|<A |= |<B |and |<B| = |<C| then |< A| =|<C|
They can select one or two properties to define a figure rather than an exhaustive list. If a
quadrilateral has 4 equal sides and one right angle it must be a square.
Students can discover new properties by simple deduction. The two acute angles in a right angled
triangle add to 900 because all the angles in a triangle add up to 1800. They can explain logically
without having to measure everything.
Formal deduction (Level 3)
Students learn how to use an axiomatic system to establish geometric theory. This is the level at which proof
of Theorems is learned. The sequence of theorems given in the appendix is arranged in such a manner that
each theorem builds on the previous theorem(s).
Rigor (Level 4)
Comparing different axiomatic systems – not done at secondary level
Characteristics of these levels: Students cannot function at any particular level unless they are
competent at all previous levels. The teacher’s role is crucial in structuring activities to bring students from
one level to the next.
How does the teacher bring students from any one level to the next?
5 phases of learning:
1. In an informal discussion of the topic, students are asked to give their initial observations.
2. The teacher provides structured activities such as drawing, making and measuring.
3. The students then verbalise and write down what they have learned and report back in groups to the
class, which leads to a class discussion.
4. The teacher then provides an activity which will require students to apply what they have discovered
5. In the last stage students are required to summarise all they have learned and should be able to
remember it as they have discovered it through guidance.
A PowerPoint presentation of the Van Hiele theory can be got at www.projectmaths.ie
2 examples are given on the PowerPoint slides
(1) Using similar triangles to show advancement between levels and
(2) Using an investigation of the rhombus to show how to progress from level 0 to level 1 with this figure
using the 5 teaching phases.
A mind map of Van Hiele can be found at
http://agutie.homestead.com/files/mindmap/van_hiele_geometry_level.html
56 SeniorCycleOrdinaryLevel
Appendix B
Guide to Theorems, Axioms and Constructions at all Levels*
This is intended as a quick guide to the various axioms, theorems and constructions as set out in the
Geometry Course for Post-Primary School Mathematics. You can get this from the project maths website:
www.projectmaths.ie
It is not intended as a replacement for this document, merely as an aid to reading at a glance which material
is required to be studied at various levels. The sequence of theorems as given must be followed.
As stated in the heading, these theorems and constructions are underpinned by 46 definitions and 20
propositions which are all set out in the Geometry Course for Post-Primary School Mathematics, along
with many undefined terms and definable terms used without explicit definition.
*An axiom is a statement accepted without proof, as a basis for argument
*A theorem is a statement deduced from the axioms by logical argument. Theorems can also be deduced
from previously established theorems.
* A proposition is a useful or interesting statement that could be proved at this point, but whose proof is not
stipulated as an essential part of the programme. Teachers are free to deal with them as they see fit, but they
should be mentioned, at least (Appendix p. 20, footnote).
*The instruments that may be used for constructions are listed and described on page 38 of the Appendix
and are a straight edge, compass, ruler, protractor and set-square.
Terms
Students at Junior Certificate Higher level and Leaving Certificate Ordinary level will be
expected to understand the meanings of the following terms related to logic and deductive
reasoning:
Theorem, proof, axiom, corollary, converse, implies.
In addition, students at Leaving Certificate Higher level will be expected to understand the
meanings of the following terms related to logic and deductive reasoning:
Is equivalent to, if and only if, proof by contradiction.
57 SeniorCycleOrdinaryLevel
Synthetic Geometry
Guide to Axioms, Theorems and Constructions for all Levels
Interactive files are available in the Student Area on the Project Maths website.
Axioms and Theorems
(supported by 46 definitions, 20 propositions)
*proof required for JCHL only
** proof required for LCHL only
 These results are required as background knowledge for
constructions and/or applications of trigonometry.
Axiom 1: There is exactly one line through any two given
points
Axiom 2: [Ruler Axiom]: The properties of the distance
between points.
Axiom 3: Protractor Axiom (The properties of the degree
measure of an angle).
1
2
3
4*
5
6*
7
8
9*
10
Vertically opposite angles are equal in measure.
Axiom 4: Congruent triangles conditions (SSS, SAS,
ASA)
In an isosceles triangle the angles opposite the equal sides
are equal. Conversely, if two angles are equal, then the
triangle is isosceles.
Axiom 5: Given any line l and a point P, there is exactly
one line through P that is parallel to l.
If a transversal makes equal alternate angles on two lines
then the lines are parallel. Conversely, if two lines are
parallel, then any transversal will make equal alternate
angles with them.
The angles in any triangle add to 180⁰.
Two lines are parallel if, and only if, for any transversal, the
corresponding angles are equal.
Each exterior angle of a triangle is equal to the sum of the
interior opposite angles.
The angle opposite the greater of two sides is greater than
the angles opposite the lesser. Conversely, the side
opposite the greater of two angles is greater than the side
opposite the lesser angle.
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Two sides of a triangle are together greater than the third.
In a parallelogram, opposite sides are equal, and opposite
angles are equal. Conversely, (1) if the opposite angles of a
convex quadrilateral are equal, then it is a parallelogram;
(2) if the opposite sides of a convex quadrilateral are equal,
then it is a parallelogram.
Corollary 1. A diagonal divides a parallelogram into two
congruent triangles.
The diagonals of a parallelogram bisect each other.
Conversely, if the diagonals of a quadrilateral bisect one
another, then the quadrilateral is a parallelogram.
58 CMN
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11**
12**
13**
14*
15
16
Axioms and Theorems
(supported by 46 definitions, 20 propositions)
*proof required for JCHL only
** proof required for LCHL only
 These results are required as background knowledge for
constructions and/or applications of trigonometry.
If three parallel lines cut off equal segments on some
transversal line, then they will cut off equal segments on
any other transversal.
Let ABC be a triangle. If a line l is parallel to BC and cuts
[AB] in the ratio m:n, then it also cuts [AC] in the same
ratio.
Conversely, if the sides of two triangles are in proportion,
then the two triangles are similar.
If two triangles are similar, then their sides are
proportional, in order (and converse)
[Theorem of Pythagoras]In a right-angled triangle the
square of the hypotenuse is the sum of the squares of the
other two sides.
[Converse to Pythagoras]. If the square of one side of a
triangle is the sum of the squares of the other two, then the
angle opposite the first side is a right angle.
Proposition 9: (RHS). If two right-angled triangles have
hypotenuse and another side equal in length respectively,
then they are congruent.
For a triangle, base x height does not depend on the choice
of base.
Definition 38: The area of a triangle is half the base by
the height.
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A diagonal of a parallelogram bisects the area.
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18
The area of a parallelogram is the base x height.
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19*
The angle at the centre of a circle standing on a given arc
is twice the angle at any point of the circle standing on the
same arc.
Corollary 2†: All angles at points of a circle, standing on
the same arc are equal (and converse).
Corollary 3: Each angle in a semi-circle is a right angle.
20
21
Corollary 4: If the angle standing on a chord [BC] at
some point of the circle is a right-angle, then [BC] is a
diameter.
Corollary 5: If ABCD is a cyclic quadrilateral, then
opposite angles sum to 180⁰(and converse).
(i)
Each tangent is perpendicular to the radius that
goes to the point of contact.
(ii)
If P lies on the circle S, and a line l is
perpendicular to the radius to P, then l is a
tangent to S.
Corollary 6: If two circles intersect at one point only,
then the two centres and the point of contact are collinear.
(i)
The perpendicular from the centre to a chord
bisects the chord.
(ii)
The perpendicular bisector of a chord passes
through the centre.
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† The corollaries are numbered as in the Geometry for Post-primary School Mathematics; corollary 2 is the first one relating to
theorem 19
♦ These results are required as background knowledge for constructions and/or applications of trigonometry
59 SeniorCycleOrdinaryLevel
Constructions
(Supported by 46 definitions, 20 propositions, 5 axioms and 21
theorems)
1
2
3
4
5
6
7
Bisector of an angle, using only compass and straight edge.
Perpendicular bisector of a segment, using only compass and
straight edge.
Line perpendicular to a given line l, passing through a given point
not on l.
Line perpendicular to a given line l, passing through a given point
on l.
Line parallel to given line, through a given point.
Division of a line segment into 2 or 3 equal segments without
measuring it.
Division of a line segment into any number of equal segments,
without measuring it.
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Introd.
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Triangle, given lengths of 3 sides.
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Triangle, given SAS data.
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Triangle, given ASA data
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Right-angled triangle, given length of hypotenuse and one other side
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Line segment of a given length on a given ray.
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17
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Circumcentre and circumcircle of a given triangle, using only
straight edge and compass.
Incentre and incircle of a triangle of a given triangle, using only
straight edge and compass.
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Angle of 60⁰ without using a protractor or set square.
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Tangent to a given circle at a given point on it.
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Parallelogram, given the length of the sides and the measure of the
angles.
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Centroid of a triangle.
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Orthocentre of a triangle.
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60 SeniorCycleOrdinaryLevel
Appendix C
Investigations of quadrilaterals and triangles
Investigating Quadrilaterals
Quadrilaterals
Square
Rhombus
Rectangle
Parallelogram
Trapezium
(not a square)
(not a rectangle
or a rhombus)
(not a
parallelogram)
(not a square)
and not an isosceles
trapezium which has the
non parallel sides equal in
length)
Describe it in
words.
Draw three
examples in
different
orientations.
How many
axes of
symmetry does
it have? Show
on a diagram.
Does it have a
centre of
symmetry?
Show on a
diagram.
Which sides
are equal?
What is the
sum of all the
angles?
Are all angles
equal?
Which angles
are equal?
What is the
sum of two
61 SeniorCycleOrdinaryLevel
Quadrilaterals
Square
Rhombus
Rectangle
Parallelogram
Trapezium
(not a square)
(not a rectangle
or a rhombus)
(not a
parallelogram)
(not a square)
and not an isosceles
trapezium which has the
non parallel sides equal in
length)
adjacent
angles?
Does a
diagonal bisect
the angles it
passes
through?
Does a
diagonal divide
it into two
congruent
triangles?
Given the
length of its
sides, can you
calculate the
length of a
diagonal?
Are the two
diagonal s
equal in
length?
Do the
diagonals
divide it into
four congruent
triangles?
Do the
diagonals
divide it into
four triangles
of equal area?
Are the
diagonals
perpendicular?
62 SeniorCycleOrdinaryLevel
Quadrilaterals
Square
Rhombus
Rectangle
Parallelogram
Trapezium
(not a square)
(not a rectangle
or a rhombus)
(not a
parallelogram)
(not a square)
and not an isosceles
trapezium which has the
non parallel sides equal in
length)
Do the two
diagonals
bisect each
other?
What
information do
you need to
calculate its
area?
How do you
calculate it?
Does a
diagonal bisect
its area?
Investigating triangles
Triangles
Equilateral
Isosceles
Describe it in words.
Draw three examples
63 Right angled
Obtuse Angled
SeniorCycleOrdinaryLevel
Triangles
Equilateral
Isosceles
in different
orientations.
How many axes of
symmetry does it
have? Show on a
diagram.
Does it have a centre
of symmetry? Show
on a diagram.
What is the sum of
the three angles?
Are all angles equal?
Are there any equal
angles? Where?
Can you say for
certain what size the
angles are?
Apart from the
isosceles triangles
themselves which of
the others could also
be isosceles?
What information do
you need to calculate
its area?
How do you calculate
it?
Draw 3 diagrams for
each type of triangle
showing each side as
a base and the
corresponding
perpendicular
height?
How do you calculate
the area?
Is the centroid inside
64 Right angled
Obtuse Angled
SeniorCycleOrdinaryLevel
Triangles
Equilateral
Right angled
Isosceles
Obtuse Angled
the triangle always?
Is the circumcentre
inside the triangle
always?
Is the incentre inside
the triangle always?
Quadrilaterals
Square
Rhombus
Rectangle
Parallelogram
Trapezium
(not a square)
(not a rectangle
or a rhombus)
(not a
parallelogram)
(not a square)
and not an isosceles
trapezium which has the
non parallel sides equal in
length)
Describe it in
words.
A square is a
quadrilateral
in which all
sides are
equal in
length and all
angles are
900. (need
only say that
one angle is
900)
A rhombus is
a quadrilateral
with all sides
equal and
opposite
angles equal.
(a
parallelogram
with all sides
equal in
length.)
A rectangle is
a
quadrilateral
with opposite
sides equal
and parallel
and all
interior
angles equal
to 900.
A
parallelogram
is a
quadrilateral
with opposite
sides equal and
parallel and
opposite
angles equal.
A trapezium is
a quadrilateral
which has 1
pair of parallel
sides.
Draw three
examples in
different
orientations.
How many
axes of
symmetry does
it have? Show
on a diagram.
4
2
2
65 None
None if the
non parallel
sides are not
equal in
length.
SeniorCycleOrdinaryLevel
Quadrilaterals
Square
Rhombus
Rectangle
Parallelogram
Trapezium
(not a square)
(not a rectangle
or a rhombus)
(not a
parallelogram)
(not a square)
and not an isosceles
trapezium which has the
non parallel sides equal in
length)
Does it have a
centre of
symmetry?
Show on a
diagram.




No
Which sides
are equal?
All
All
Opposite
Opposite
none
What is the
sum of all the
angles?
3600
3600
3600
3600
3600
Are all angles
equal?

x

x
x
Which angles
are equal?
What is the
sum of two
adjacent
angles?
All angles
Opposite
angles
All angles
Opposite
angles
1800
1800
1800
1800
1800
Does a
diagonal bisect
the angles it
passes
through?


x
x
x
Does a
diagonal divide
it into two
congruent
triangles?




x
No. Need to
know an angle.
Need to know
the lengths of
two adjacent
sides and the
angle between
them.
Given the
length of its
sides, can you
calculate the
length of a
diagonal?

No. Need to
know an
angle.
Investigate
using
geostrips.

66 SeniorCycleOrdinaryLevel
Quadrilaterals
Square
Rhombus
Rectangle
Parallelogram
Trapezium
(not a square)
(not a rectangle
or a rhombus)
(not a
parallelogram)
(not a square)
and not an isosceles
trapezium which has the
non parallel sides equal in
length)
Are the two
diagonal s
equal in
length?

x

x
x
Do the
diagonals
divide it into
four congruent
triangles?


x
x
x
Do the
diagonals
divide it into
four triangles
of equal area?




x
Are the
diagonals
perpendicular?


x
x
x
Do the two
diagonals
bisect each
other?




x
Base ( b) and
perpendicular
height (h)from
a vertex to that
base
Lengths of 2
adjacent sides
l and b.
Area = l x b.
Base ( b) and
perpendicular
height (h)from
a vertex to that
base
Area = b x h
(Base ( b) and
Area = b x h
The lengths of
its parallel
sides ( a and
b)and the
perpendicular
distance
between them.
What
information do
you need to
calculate its
area?
How do you
calculate it?
One side
length x.
Area = x2
(Base ( b) and
perpendicular
height (h)from
a vertex to that
base
Area = b x h)
Does a
diagonal bisect
its area?

If you know
the lengths of
the diagonals x
and y
Area = ½ x y.

perpendicular
height (h)from
a vertex to that
base
Area = b x h)

67 Area =
½ (a+b) h

Appendix D
How to register for CensusAtSchool, complete the online questionnaire and retrieve class data for analysis and interpretation
Part 1: Registration
www.CensusAtSchool.ie Click “Teacher Registration” under “Take Part” in the main menu This is the Irish CensusAtSchool homepage. Click “Take Part” under the main menu. 1 2 Fill out and click on “Submit” Click “ONLINE REGISTRATION FORM” Once your registration form has been
received and approved an email with
a personal username and password
will be dispatched to you. Check your
email spam folder if you do not
See “Forgotten Log In
Details”
Queries:
[email protected]
receive these details
within 48 hours.
3 4 SeniorCycleOrdinaryLevel
Part 2: Accessing the questionnaire
Click “Take Part”, “Questionnaires” and then click on the latest questionnaire which will be at the top of the list. Click “Take the CensusAtSchool* online questionnaire” 6 5 Part 3: Filling out the questionnaire
Each student must fill in accurately; the school roll number and the teacher’s username (sent by email) Click on “Next”
When the questionnaire is completed, the student clicks “Submit”.
7 69 8 SeniorCycleOrdinaryLevel
Logging in to retrieve data
Part 4 Now that my students have filled in the data – what next?
Fill in your username and password. (If you have forgotten them, click on the “forgot your password or username” link.) Under “Get Data” click on “Request your class data” Click “Login”. 9 10 Your data will look similar to the spreadsheet below.
Once logged in you will be asked for your school roll number and username. Select whichever questionnaire (phase) your students have completed. Click on “Submit” and your data will be returned to you in spreadsheet form. 12 11 70 
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