Download JC HL scheme 2015/2016

St. Vincent’s Secondary School Glasnevin
Maths Department
Scheme of work
Junior Cycle Higher Level (2nd and 3rd Year)
396 classes
Dat
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Topic
Time
Frame
Algebra
Sept
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Simplifying expressions: Students will be able to identify
expressions, terms, variables, coefficients and constants. They
will also realise that only like terms may be added or subtracted
Removing brackets: Students will be able to simplify algebraic
expressions involving brackets by multiplying terms together.
They will learn how signs affect the answer and how indices
work.
Evaluating expressions: Students will be able to find out the
value of expressions by substituting real numbers for the letters
or the variables in the expression.
Solving linear equations: Students will be able to find the value
of the variable that makes the equation true. This will be done
algebraically by balancing the equation repeatedly until we
isolate the variable and hence, calculate its value.
Solving problems using linear equations: Students will be able
to change a problem expressed in words into a mathematical
equation and then solve it.
Plotting numbers on the number line/Inequalities: Students
will be able to identify the different number systems; Natural
Numbers, Integers, Real Numbers. They will also be able to solve
inequalities by treating them like equations.
REVISION
TEST
Factors
Factorising with common factors: Students will use the highest
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Assessment
Oct.
common factor to simplify and factorise algebraic expressions
Factorising by grouping terms: Students will be able to factorise
a four-termed expression by pairing the four terms and finding
common factors.
Difference of two squares: Students will identify numbers that
are perfect squares and will be able to simplify and factorise
expression involving perfect squares.
Factorising quadratic expressions: Students will be able to
factorise a quadratic trinomial in the form 𝑎𝑥 2 + bx + c
Using factors to simplify algebraic fractions: By finding common
factors, students will be able to simplify the numerator and
denominator.
REVISION
TEST
Late
Oct.
Sets
Revision of sets terminology: Students will revisit the areas of
equal sets, union, intersection, subsets, universal set,
complement, cardinal number of a set.
Set difference: Students will be able to list and identify from a
set A all the elements which are NOT in the set B. They will also
learn that intersection and union of sets is commutative, but set
difference is not.
Venn diagrams involving three sets: Students will be able to
interpret and draw Venn diagrams involving three sets. Students
will see that the union and intersection of sets are associative
but set difference is not. They will also learn that union of sets is
distributive over intersection and that intersection of sets is
distributive over union.
Solving problems using three sets: Students will use Venn
diagrams to solve problems involving three sets. They will use x
to represent the number of elements in a region that is not
given directly.
REVISION
TEST
Nov
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Applied Arithmetic:
V.A.T. – Profit and Loss: Students will be able to convert a
percentage to a decimal, and use this knowledge, with a
calculator, in order to work out given percentages. Students will
also be able to work out percentage profit and loss on a number
of quantities.
Income tax: Students will become familiar with the standard
rate and higher rate of tax. They will know how to work out the
gross tax and the tax payable by deducting the tax-credit, and
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hence, the take-home pay. Students will also learn how to use
and work out the Universal Social Charge and Pay-Related Social
Insurance.
Currency exchange: Students will be able to covert amounts of
currency to other international currencies.
Compound interest: Students will be able to identify the
principal, rate and final amount. They will be able to work out
the compound interest of a sum of money up to a maximum of
four years by using. They will also learn how to work out the
rate and the principal given the other respective values.
REVISION
TEST
Probability
Listing Outcomes: Students will learn to list all of the possible
outcomes of a trial or experiment. We will also discuss the
“fundamental principle of counting”. Particular attention will be
given to the term “fundamental principle of counting” as
students generally understand the meaning but forget the
terminology.
Chance and the probability scale: Students will recap previous
knowledge of chance and use terms already familiar to them
such as likely, unlikely and so on.
Probability and equally likely events: Students will learn to
quantify the probability of an event happening. They will discuss
terms such as “favourable outcomes”. They will learn that a
sample space is a list of all possible outcomes- recapping their
listing of outcomes.
Estimating probability from experiments: The students will
learn to predict the chances of an event happening based on
statistics from the past. They will work out the probabilities by
calculating the experimental probability of an event happening.
Probability using Venn diagrams: When information is
presented in the form of a Venn diagram it is easy to write down
the probability of different events happening. These will include
both two and three set diagrams.
Tree diagrams: By using a tree diagram, students will be able to
show the possible outcomes of two or more events.
REVISION
TEST
Statistics 1 – Collecting Data
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Types of data: Students will learn about the different types of
data; categorical, numerical, discrete, continuous, nominal data,
ordinal, discrete, continuous data.
Collecting data: Students will learn how to gather data through
surveys and questionnaires. They will use both primary and
secondary data and present this information in a suitable
fashion. They will learn how to interpret and produce frequency
table. They will learn about bias and how to look out for it.
Sampling: Students will learn how to collect information from a
group within a population, a Sample. They will learn to avoid
bias by using a simple random sample.
Statistics 2 - Averages and Spread
Averages – Mode and Median: Students will learn that there
are different types of averages, but that the most commonlyused averages are the mode, the median and the mean.
Students will learn how to calculate these and also how extreme
values work. Students will learn which average to use in a given
context.
Frequency Tables: Students will learn how to construct and
interpret frequency distribution tables. They will deal with
group frequency distributions and mid-interval values and will
learn how to calculate the mean and mode of such tables.
Range and variability: Students will learn that the range of a set
of given values is the highest value minus the lowest value. They
will also learn through comparing data that values can be
divided into the interquartile range.
REVISION FOR CHRISTMAS TEST
CHRISTMAS TEST
Perimeter – area – volume
Review of Perimeter and Area: Students will revisit area and
perimeter of squares, rectangles and triangles.
Area of a parallelogram: Students will learn how to work out
the area of a parallelogram, namely
Area and circumference of a circle: Students will learn several
new terms in relation to circles, such as radius, diameter, chord,
semicircle, segment, sector, quadrant and tangent. They will
learn how to approximate a value for 𝜋. They will learn
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They will also learn to calculate the perimeter of a sector of a
circle:
They will learn to calculate the area of a circle and the area of a
sector of a circle:
Rectangular solids: They will learn that the space taken up by a
3-D object is called its volume. They will learn how to calculate
the volume = length x breadth x height. They will learn that the
surface area of a rectangular solid is 2lb + 2lh + 2bh. They will
also see that the volume of a cube is l x l x l and the surface area
is 6𝑙 2 .
Nets of solids – Capacity: Students will learn hoe to interpret
the nets of solids (i.e. folded out into 2D. From this they will be
able to work out total surface area (inside and out) of a solid.
They will also learn that 1000 cubic centimetres is equivalent to
1 litre of capacity, and that to convert cubic centimetres to
litres, you must divide by 1000.
Prisms: They will learn that a solid figure with the same crosssection along its length is called a prism. They will also learn that
the volume of a prism is:
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They will also learn that the surface area of a prism is the sum of
the area of its faces.
Scale Drawing: Students will learn how to read and understand
scales in relation to drawings and models. They will be able to
work out actual sizes and model sizes based on information
given.
REVISION
TEST
Feb.
Quadratic Equations
Solving quadratic equations using factors: Students will learn
how to identify equations of the form a𝑥 2 + bx + c = 0. They will
learn that the solutions or roots of an equation are the values of
x that satisfy the equation. They will learn that if the product of
two numbers is zero, then at least one of them must be zero, i.e.
if ab = 0, then a = 0 or b = 0. They will learn to factorise a
quadratic expression and let those factors equal zero in order to
find the roots of a quadratic equation.
Using the quadratic formula: Students will learn that if a
quadratic expression cannot be factorised, then the quadratic
formula can be used:
Problems leading to quadratic equations: When a problem
expressed in words is changed to a mathematical sentence, it
can often result in a quadratic equation and can be solved.
Students will learn how to form these equations and solve them.
Forming a quadratic equation from its roots: Students will learn
how to use the reverse method of solving a quadratic equation
by factorisation in order to form an equation when given its
roots.
REVISION
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Late
Feb.
MID-TERM BREAK
Geometry 1 – Triangles and Quadrilaterals
Revision of lines and angles: Students will revisit topics covered
in the Common Introductory Course such as lines, angles and
parallel lines.
Angles of a triangle: Students will look at the following diagrams
and learn the information in relation to triangles.
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They will learn that an isosceles triangle is a triangle where
opposite sides and angles are equal.
Quadrilaterals: Students will become familiar with the different
quadrilaterals ( four-sided figures ) such as a square, a rectangle,
a parallelogram and a rhombus. They will learn that the interior
angles of a quadrilateral add up to 360⁰. They will also learn that
the diagonals of a parallelogram bisect each other.
Congruent triangles: Students will learn that figures which are
exactly the same size and shape are said to be congruent.
Students will learn how to try and identify conditions for
congruence; Side, side side (SSS); side, angle, side (SAS); angle,
side, angle (ASA); right-angle, hypotenuse, side (RHS). They will
learn that if two triangles are congruent, then corresponding
sides and angles are equal.
The Theorem of Pythagoras: Students will explore the Theorem
of Pytahgoras and learn that in a right-angled triangle, the
square on the hypotenuse is equal to the sum of the squares on
the other two sides.
REVISION
TEST
Co-ordinate geometry – the line
Revision of formulae: Students will revisit topics such as the
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distance between two points and the midpoint of a line segment
The slope of a line: By looking at the relationship between the
vertical change (rise) and the horizontal change (run), students
will be able to work out the slope of a given line. They will learn
about positive and negative slopes, and how parallel lines have
equal slopes and the product of the slopes of perpendicular
lines is always – 1.
Equation of a line: By observing the relationship between the x
and y values of each point and looking at the slope of a given
line, students will be able to form the equation of a line by: y –
y1 = m(x – x1). They will learn that if given two points, they must
first find out the slope and then use the above formula.
The equation y = mx + c: Students will study the equation of a
line in the form y = mx + c , and see that the slope is m and the
line intersects the y-axis at the point (0, c)
Parallel and Perpendicular Lines: Students will learn that
parallel lines have equal slopes and perpendicular lines have
slopes that multiply to give -1. They will be asked to work out
equations of lines that are parallel or perpendicular to a given
equation.
Graphing lines: Students will learn how to graph lines parallel to
either axis, plot lines that contain the origin. They will also be
able to verify a point is on a given line by proving the coordinates of the point satisfy the given equation of the line.
Intersection of two lines: By using simultaneous equations,
students will be able to locate the point of intersection of two
lines.
Interpreting slope: Students will learn how to interpret graphs
and extract useful information by using the slope of a given
graph.
REVISION
TEST
Late
Mar
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Ratio – Time – Speed
Ratio and proportion: Students will learn how to use ratios to
compare quantities to each other, and to use proportions to
compare an amount to a total. They will also learn how inverse
proportion works.
Time and timetables: Students will learn how the 24-hour clock
works, and how to use a.m. and p.m. They will also learn how to
extract relevant information from given timetables.
Speed – Distance – Time: Students will discover the relationship
between distance, speed and time i.e. Distance = Speed x Time,
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Late
Apr.
and work out the unknown when given the other two.
REVISION
TEST
Easter Holidays
Statistics 3 – Presenting Data
Line plots and bar charts revisited: Students will revisit how to
compile and interpret line plots and bar charts. They will revise
how to work out the mean, mode, range and locate outliers and
clustered information.
Pie charts: Students will learn how to interpret and construct
pie charts. By using the information that a full circle is 360⁰,
students will be able to work out values of given sectors of a pie
chart.
Histograms: Students will understand the difference between
bar charts and histograms and will be able to construct
histograms and extract relevant information from given
histograms.
Stem and leaf plots: Students will learn how to interpret and
construct stem and leaf plots, including the interquartile range.
Misleading graphs: Students will learn how to find common
misleading aspects to statistical information.
REVISION
TEST
May
May
Simultaneous Equations
Solving simultaneous equations: By eliminating one of the
unknowns in both equations, students will be able to solve for
the value of the other variable, and hence, work out the other
variable also.
Solving simultaneous equations graphically: Students will
discover that two simultaneous equations may be solved by
drawing graphs of the two equations (lines) and then reading
the x-value and y-value of their point of intersection.
REVISION
TEST
Indices – Scientific Notation – Surds
The laws of indices: Students will learn the various laws
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Fractional indices: Students will learn how to compute and
understand indices that involve fractional powers, i.e.:
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Equations involving indices: By expressing given equations in
terms of the same base, students will be able to solve given
equations involving indices.
Irrational numbers – Surds: Students will learn more about the
characteristics of rational and irrational numbers,. They will
learn what a surd is and how to add, subtract and multiply
surds.
Numbers in standard form: Students will learn how to put
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numbers in scientific notation or standard form, i.e.
Students will learn how to add, subtract, multiply and divide in
standard form.
Significant figures – Approximation: There are many times
when having an approximation of an answer will be invaluable,
in this section, students will learn how to interpret decimal
places and significant figures when making an estimate.
Reciprocals and use of calculator: Students will learn that the
reciprocal of a number is 1/number, they will learn how to
perform operations using powers and roots on the calculator.
REVISION FOR SUMMER TEST
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SUMMER TEST
Sep.
YEAR 2:
Geometry 2: Similar Trianlges – Circles – Theorems
Similar trinagles: Students will learn that similar triangles have
the same shapes but different sizes. They will also learn that
equiangular means all the corresponding angles are equal. They
will learn that if two triangles are similar, then their sides are
proportional, in order.
Transversals and triangles: Students will see that a line that
intersects three parallel lines. They will learn that if three
parallel lines cut off equal segments on some transversal line,
then they will cut off equal segments on any other transversal.
They will learn a line drawn parallel to one side of a triangle
divides the other two sides in the same ratio.
Angles and circles: Students will revise the meanings of the
terms frequently used when dealing with circles, such as centre,
circumference, diameter, radius, semicircle, chord, tangent,
sector and segment. They will learn that the angle in a
semicircle is a right angle and that the angle subtended at the
centre of a circle is twice the angle at the circumference. They
will see that angles in the same segment are equal. They will
prove that the opposite angles of a cyclic quadrilateral add to
180⁰. They will learn that if the angle standing on a chord [BC] at
some point of the circle is a right angle, then [BC] is a diameter.
Proofs of theorems 4, 6, 9, 14, 19: Students will learn what the
terms axiom, theorem, corollary, converse and implies. They will
learn the angles in any triangle add to 180⁰, each exterior angle
of a triangle is equal to the sum of the interior opposite angles.
They will learn that in a parallelogram, opposite sides are equal
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and opposite angles are equal. They will learn that in a rightangled triangle, the square of the hypotenuse is equal to the
sum of the squares of the other two sides. They will learn that
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.
REVISION
TEST
Late
Sep.
Cylinder – Sphere – Cone
The cylinder: Students will learn how to calculate the volume,
curved surface area and the total surface area of a cylinder
The sphere and hemisphere: Students will learn how to
calculate the volume and surface area of a solid sphere and
hemisphere.
The cone: Students will work out the volume, curved surface
area and total surface area of a cone.
REVISION
TEST
Oct.
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Patterns and sequences
Sequences: Students will observe patterns and see how they
can repeat themselves, and that these types of patterns are
called sequences. They will learn what a term is, the significance
of its position, and the term-to-term rule.
Repeating patterns: Students will look at repeating patterns and
can see how to work out the value of any term in any position.
Linear sequences: Students will see that when the term-to-term
rule involves adding or subtracting a constant, it is known as a
linear equation or an arithmetic sequence. They will also learn
how to find any term (the nth term) by discovering the rule of a
given sequence.
Sequences formed from shapes: Students will deal with
geometric figures and the patterns they form.
Quadratic sequences: Students will look at non-linear patterns,
and see that sequences that have an nth term containing 𝑛2 as
the highest power are called quadratic sequences. They will
learn that in a quadratic sequence, the coefficient of 𝑛2 in the
nth term is half the second difference. They will also learn how
to work out the nth term of a quadratic sequence.
Graphing sequences: Students will learn how to graph terms
values in relation to position and see how linear, quadratic and
exponential graphs compare to each other.
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REVISION
TEST
Nov
Late
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Functions
Functions: Students will learn about rules, and how an input
leads to an output. They will work with function machines and
flow charts.
Mapping diagrams: Students will learn that the input numbers
are called the domain, and the output numbers are called the
range. They will also learn to use the word function for any rule
that produces one output value only for each input value.
Students will learn how to use the appropriate notation for
functions, and that the set of possible outcomes is called the
codomain. They will learn how to identify functions and sets of
ordered pairs.
Notation for functions: Students will learn how to use the
appropriate notations, i.e. f(x), f : x , y =
Finding coefficients of functions: Students will learn how to
draw a graph of a function (a parabola) within a given domain,
and from this, will be able to find the coefficient of a linear
function and the roots of a quadratic function.
REVISION
TEST
Drawing and interpreting real-life graphs
Distance - time graphs: Students will learn how to interpret
change in a graph to extract information, and also use D = S X T
Directly proportional graphs: Students will see that directly
proportional graphs are straight lines through the origin.
Real-life graphs: Students will see the different rates of change
in graphs in relational to real-life graphs.
Dec.
Algebraic Fractions – Formulae:
Adding algebraic fractions: Students will learn how to use a
common denominator to simplify algebraic terms and fractions.
Solving equations involving fractions: Students will use the
same strategies as above to solve algebraic equations involving
fractions.
Solving problems involving fractions: Students will learn to
form equations from worded problems and hence, using
strategies as above, solve those problems.
Algebraic division: Students will learn how to divide a linear
expression such as x + 4 into a quadratic expression such as 𝑥 2 +
x - 12
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Rearranging formulae: By using the fact that an equation
remains unchanged if the same operation is performed on both
sides, students will be able to rearrange a formula or equation
in order to isolate any variable.
Evaluating and writing formulae: Students will learn how
important it is to be able to derive a formula from information
that you are given.
REVISION FOR XMAS TEST
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XMAS TEST
Jan.
Trigonometry
The Theorem of Pythagoras: Students will revisit the Theorem
of Pythagoras and work out unknown sides in given problems.
Sine, Cosine and Tangent ratios: Students will learn about sine,
cosine and tangent ratios, and from that, work out unknown
sides.
Using a calculator to find ratios and angles: Students will learn
how to compute the sin, cos and tan of angles on their
calculators. They will learn that an angle is divided into 60
minutes and how to change decimal angles into D,M,S. They will
also learn how to use the inverse function on the calculator to
work out an angle when given the sin, cos or tan ratio.
Solving right-angled triangles: Students will use the sine, cosine
and tangent ratios to find an unknown side or an unknown angle
in a right-angled triangle.
Using trigonometry to solve problems: Students will learn
about angles of elevation and depression, when dealing with
trigonometric problems.
The angles 30⁰, 45⁰ and 60⁰: Students will see the above angles
are used very frequently and triangles will be used to express
the sine, cosine and tangent ratios of these angles as fractions
or surds.
REVISION
TEST
Late
Jan.
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Graphing functions
Graphing linear and quadratic functions: Students will learn
how to draw a graph of a function (a parabola) within a given
domain, and from this, will be able to find the coefficient of a
linear function and the roots of a quadratic function.
Using quadratic graphs: Students will learn how to solve the
equation f(x) = 0, how to solve the equation f(x) = k when k is a
real number. They will learn when a function is negative, i.e.
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under the x-axis. They will learn how to draw intersecting graphs
to find points of intersection and when f(x) is less than g(x) or
vice versa. Students will learn how to locate maximum and
minimum values of functions by inspection. Students will learn
how to find f(k) by inspection from the graph and when a
function is increasing or decreasing.
Quadratic graphs and real-life problems: Students will learn
how to use quadratic graphs to solve certain types of real-life
problems.
Graphs of exponential functions: Students will learn how to
draw and interpret graphs in which the x appears as a power.
REVISION FOR MOCKS
MID-TERM BREAK
Mock Examinations
Mar
Late
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Geometry 3: Transformations – Constructions
Transformational geometry and symmetries: Students will
learn how to identify and perform translations and symmetries
of shapes and points.
Constructions 1: Students will learn how to bisect an angle, how
to construct the perpendicular bisector of a line segment, how
to construct a line perpendicular to a given line l passing
through a given point on l, how to construct a line perpendicular
to a given line passing through a given point not on l, how to
draw a line parallel to a given line, through a given point, how to
divide a line segment into three equal parts, how to divide a
line segment into any number of equal segments without
measuring it, how to draw a line segment of a given length on a
given ray, and how to draw an angle of a given number of
degrees with a given ray as one arm.
Constructing triangles and rectangles: Student’s will learn how
to construct a triangle given the lengths of three sides, how to
construct a triangle given a side, angle and side measurements,
how to construct a triangle given angle, side, angle
measurements, how to construct a right-angled triangle, given
the length of the hypotenuse and one other side, how to
construct a right-angled triangle given one side and one of the
acute angles, and how to construct a rectangle.
Easter Holidays
REVISION FOR STATE EXAMINATIONS
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