LESMAHAGOW HIGH SCHOOL
Mathematics Department
New Higher
Scheme of Work
This document is a guide to be used when studying the new
higher mathematics course using the Heinemann higher
mathematics textbook.
Contained is a list of ‘I can …’ statements and each outlines
what you should be able to do after completing the set
exercise(s). It will also allow you to follow the course should
you be absent for a period of time. The number of noted hours
should be used as a rough guide only.
There are three units and after each there is an assessment
standard that must be met. The order of these will be changed
and assessed in groups to suit the order in which the topics
are taught at Lesmahagow High School. You are allowed only
one reassessment for each assessment standard as per SQA
guidelines.
There will also be A/B level assessments and a prelim exam
during the year. These are a much better indication as to how
you are coping with the course as the aforementioned
assessment standards are merely a check of minimum
competency.
It is important that you keep up with all work in class and
revise at home on a regular basis. A minimum of five hours
per week (excluding study periods) is recommended.
Homework is an essential part of the higher mathematics
course and any homework that is issued should be done to
the best of your ability and handed in on time. Should you
experience difficulty with your homework then speak to your
teacher at the earliest opportunity.
The topics that were completed during the national 5
mathematics course are essential skills and will not
necessarily be covered in the new higher course. These are
areas that you should look to revise yourself.
Applications 1.1
Applying algebraic skills to rectilinear shapes
Properties of a Straight Line
Revision of straight line
Time
SPW
1 period
Gradients

Gradient formula/Oblique lines
HHM 1
1A
(1 – 3), 7, 8
HHM 3
1B
3-5, 8
Completed N5

Horizontal lines
HHM 1
1A
4, 5
Completed N5

Vertical lines
HHM 1
1A
6
Completed N5

m = tan Ø
HHM 1
1A
9, 10
HHM 3
1B
6, 7, 10
1B
1, 2, 9
1 period
 I can calculate the gradient of a line using m = tan𝜃
Collinearity
HHM 3
Distance formula

1 period
HHM 205
12A
1
HHM 206
12B
(1 – 5)
 I can find the Distance between 2 points- using the Distance Formula
Perpendicular lines

Introduction
2 periods
HHM 6
1D
(1 – 10)
 I can calculate the gradient of perpendicular lines using 𝑚1 × 𝑚2 = −1
Equation of a straight line
Note
Pupils should be made aware of the meaning of the term
‘locus’

y = mx + c revision
HHM 8
1E
(1 – 8)
Completed N5

y – b = m(x – a)
HHM 11
1G
(1 – 4)
Completed N5

Ax + By + C = 0
HHM 10
1F
(1 – 3)
1 period
 I can rearrange any straight line into the general equation form 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
Geometrical constructions
Note

Pupils should be made aware of the meaning of the term
‘concurrency’
Perpendicular bisectors
HHM 13
1I
1,2
 I know the properties of perpendicular bisectors and can find their equations
1 period

Altitudes
HHM 15
1K
(1 – 5)
2 periods
(1 – 3)
2 periods
 I know the properties of altitudes and can find their equations

Medians
HHM 16
1M
 I know the properties of medians and can find their equations
Lines in a triangle
1 period

HHM
18
1N
(1 – 7)
 I can find the point of intersection of straight lines
TO BE COVERED DURING REVISION FOR EXAMS

Circumcentre, Orthocentre
and concurrency
HHM 12
1H
1,2
HHM 14
1J
1,2
HHM 16
1L
1,2
1 period
 I have knowledge of the terms; circumcentre, orthocentre and concurrency
13 periods
Applications 1.2
Applying algebraic skills to circles
The Equation of a Circle

The equation of a circle, centre
Time
HHM 207 12D
(1 – 10)
period
the origin and radius r
 I have used Pythagoras to develop the equation of a circle with centre
the origin and radius r using 𝑥 2 + 𝑦 2 = 𝑟 2

The equation of a circle centre (a,b)
HHM 210 12F
(1 – 10)
period
and radius r
 I can determine the centre and radius of a circle given its equation using
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
 I can determine the equation of a circle given its centre and radius using
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2
 I can use (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2 in mixed problems
 I can determine if a point lies inside, outside or on the circle

The general equation of a circle
HHM 213 12H
(1 – 16)
2 periods
 I can determine the centre and radius of a circle given its equation
using 𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 (Qu 1, 2, 3, 6, 11)
 I can identify when an equation in the form x2 + y2 + 2fx + 2fy + c = 0
represents a circle and then determine its radius (Qu 4, 5)

Intersection of a line and a circle
HHM 217 12J
(1 – 5)
1 period
 I can determine the point of intersection of a line and a circle or two circles

Tangents to circles
HHM 218 12K
(1 – 7)
1 period
(1 – 6)
1 period
 I can determine the point of intersection of a tangent to a circle

Equations of tangents
HHM 220 12L
 I can determine the equation of a tangent to a circle
TOTAL 7 periods
Applications 1.3
Applying algebraic skills to sequences
Recurrence relations

Recurring problems
Time
HHM 69
5A
(1 – 5)
1 period
 I have revised percentage appreciation and depreciation questions
using real life examples

Using recurrence relations to
HHM 70
5B
(1 – 4)
1 period
HHM 72
5C
( 1 – 11)
1 period
(1 – 5)
1 period
(1 – 3)
1 period
solve problems

More complex recurrence relations
 I can set up a recurrence relation from given information

Linear recurrence relations
HHM 73
5D
 I can calculate a required term in a recurrence relation

Investigating long-term effects
HHM 74
5E
 I have investigated the long term effects of a recurrence relation

Investigating 𝑢𝑛+1 = 𝑎𝑢𝑛 + 𝑏
HHM 76
5G
(1 – 3)
1 period
 I have investigated convergent and divergent sequences
Note Mention Convergent and Divergent sequences using example
diagrams from pages 75 & 76

The limit of a recurrence relation
HHM 77
5H
(1 – 10)
1 period
5I
(1 – 4)
1 period
 I can calculate the limit of a recurrence relation
 I can explain and interpret a limit where it exists

Solving recurrence relations to
HHM 79
find a and b
 I have experience of questions solving recurrence relations to find
missing values
8 periods
Applications 1.4
Applying calculus skills to optimisation and area
Optimisation, Area under a curve
 These topics are to be taught at the same time as differentiation
– R&C 1.3 and integration – R&C 1.4.
Optimisation
Time

Optimisation : Maxima and minima
HHM 112
6Q
(1 – 8)
1 period

Further applications
HHM 114
6R
(1 – 5)
1 period

I can solve optimisation problems in context using differentiation
Area under a curve

Definite integrals
HHM 167
9K
(1 – 3)
1 period

Fundamental theorem of calculus
HHM 169
9L
(1 – 4)
1 period

Areas above and below the x –axis
HHM 172
9N
(1 – 4)
1 period
(1,2)
1 period
(1 – 6)
1 period

I can find the area enclosed between a curve and the x-axis

Area between two graphs

Calculating the area between two graphsHHM174 9P

HHM 173
9O
I can evaluate the area enclosed between two functions
TOTAL 7 Periods
Expressions and Functions 1.1
Applying algebraic skills to logarithms and
exponentials
Logarithmic and Exponential Functions
15C
(1 – 10)
1 period
15D
(2 – 5)
1 period
HHM 286 15E
(1 – 3)
1 period
HHM 30
2G
1
HHM 31
2H
1
HHM

Time
 I have investigated exponential growth and decay
HHM

 I can evaluate 𝑒 𝑥 and for equations solve for x

Logarithms
 I can identify (and sketch) the inverse function of an exponential or a logarithmic
function

Laws of logarithms
HHM 287 15F
(1 – 8)
1 period
HHM 288 15G
(1 – 4)
2 periods
(1 – 7)
2 periods
1 (a, b), 2
1 period
1 (a, b), 2
1 period
 I can use the laws of logarithms

Logarithmic equations
 I can solve logarithmic equations using the laws of logarithms

Natural logarithms
HHM 289 15H
 I can solve exponential equations using natural logarithms
 I can solve logarithmic equations in real life contexts

y = kxn
HHM 291 15I
 I understand formulae from experimental data

y = abx
HHM 293 15J
 I understand further formulae from experimental data
TOTAL 10
periods
Expressions and Functions 1.2
Applying trigonometric skills to manipulate
expressions
Trigonometric Formulae
Addition formulae and Double angle formulae

Compound angles
HHM 187 11A
Time
1
period
 I understand concept of compound angles

Addition formulae
3 periods
sin (  +  )
HHM 189 11B
(1 – 7)
sin (  -  )
HHM 190 11C
(1 – 10)
cos (  +  ) and cos (  -  )
HHM 191 11D
(1 – 9)
HHM 193 11E
(1 – 8)
period
(1 – 6)
period
 I can use addition formulae

Trigonometric identities
 I can use addition formula to prove trigonometric identities

Applications of addition formulae
HHM 194 11F
 I can apply Addition Formula Formulae to problem solving context

Formulae involving 2
HHM 196 11G
(1 – 14)
period
 I can use the double angle formulae, including expanding sin4x, cos6x etc

Trigonometric equations
HHM 198 11H
(1 – 6)
 I can solve trigonometric equations that require the substitution of a
trigonometric identity
2 periods
Wave function

Waves and graphs


Time
HHM 301 16A
1 period
I can recall deriving equations, max and min values from graphs
Adding two waves
HHM 302 16B
1 period
 Introduction to the Wave Function through adding 2 waves

Expressing a cos x + b sin x
HHM 304 16C
(Odd nos.)
1 period
in the form k cos (x - )
 I can convert acosx + bsinx to kcos(x - α), with angle α in the first
quadrant and k > 0
|

The difference of two waves
HHM 304 16D
1 (a, d, g), 2 (a, c, e) 1 period

Expressing a cos x + b sin x in
HHM 305 16E
(1 – 5)
other forms
1 period
Parts a, c and f
 I can convert acosx + bsinx to kcos(x ± α) or ksin(x ± α), with angle α
as a value in 1 of 4 quadrants and k > 0

Multiple angles
HHM 306 16F
(1 – 3)
1 period
1, 2, 6, 8, 9, 10
1 period
 I can apply the wave function formula to multiple angles

Maxima and minimum values
HHM 307 16G
 I can find maximum and minimum values of a function by expressing
the function acosx + bsinx as a single trigonometric function

Solving equations
HHM 309 16H
1, 2
1 period
 I can solve equations involving acosx + bsinx using the wave function

Solving equations and sketching
graphs
HHM
16I
1 period
 I can solve mathematical applications in the form acosx + bsinx and
sketch associated graph
TOTAL 18 periods
Expressions and Functions 1.3
Applying algebraic and trigonometric skills to
functions
Composite and Inverse Functions

Set notation
Time
HHM 23
2A
(1 – 3)
1 period
2B
(1 – 8)
1 period
 I can understand and use basic set notation

Functions and mappings
HHM 24
 I have investigated the domain and the range of a function

Composition of functions
HHM 26
2C
(1 – 10)
1 period
HHM 28
2D
(1 – 5)
1 period
HHM 29
2E
(1 – 2)
1 period
HHM 29
2F
(1 – 2)
1 period
 I can determine a composite function

Inverse of a function
 I can determine the inverse of a linear function

Graphs of inverses
 I understand that f(g(x)) = x implies that f(x) and g(x) are inverses
 I can sketch and draw the inverse of each function

Exponential function
HHM 30
2G
1

Logarithmic function
HHM 31
2H
1
HHM
1 period
3B-3M
3 periods
 I can identify and sketch a function after a transformation of the form
f(x) + k, f(x + k), kf(x), f(kx), -f(x), f(-x), or a combination of these
HHM
3N
1, 3
HHM
3O
1, 3
 I can evaluate unknowns from exponential and logarithmic graphs

Graphs of associated functions
HHM 47

Related graphs
HHM 296 15K
3P
(1 – 12)
1 period
(1 – 7)
1 period
 I can use my knowledge of Graphs and Functions and graphs of y= 𝑒 𝑥
𝑎nd y= ln x to sketch related graphs
TOTAL 12 periods
Expressions and Functions 1.4
Applying geometric skills to vectors
Vectors in Three Dimensions
Note
Time
Pupils are expected to have an understanding of both
two and three dimensional vectors
Two dimensional vectors

Magnitude
HHM 230 13A
(1 – 5)
Completed N5

Equal vectors
HHM 232 13B
(1 – 4)
Completed N5

Addition of vectors
HHM 233 13C
(1 – 7)
Completed N5

Zero vector

Subtraction of vectors
HHM 236 13D
(1– 6)
Completed N5

Multiplication by a scalar
HHM 237 13E
(1 – 4)
Completed N5

Unit vectors
HHM 238 13F
1, 2
period
HHM 239 13G
1, 2
period
HHM 240 13I
(1 – 5)
HHM 247 13N
(1 – 4)
Pupils should be made aware of th
 I can find 2D unit vectors

Position vectors
 I can find 2D position vectors

Collinearity
 I can determine whether or not coordinates are collinear, using the appropriate
language

Section formula
HHM 241 13J
(1 – 3)
HHM 242 13K
(1 – 4)
HHM 247 13N
(19 – 24)
 I can apply the section formula to both 2D and 3D vectors
1 period

Three dimensional coordinates
HHM 243 13L
(1 – 4)
Completed N5

Three dimensional vectors
HHM 245 13M
(1 – 4)
Completed N5
Scalar Product
2periods

Vector form
HHM 250 13O
(1 – 3)

Component form
HHM 252 13P
(1 – 4)
 I can calculate the scalar product of two vectors

Angle between two vectors
HHM 253 13Q
(1 – 6)
1 period
(1 – 8)
1 period
 I can calculate the angle between two vectors

Perpendicular vectors
HHM 254 13R
 I can use the properties of perpendicular vectors to solve problems

Properties of the scalar product
HHM 257 13U
(1 – 9)
1 period

Further questions??
HHM
13S
(1 – 7)
1 period
HHM
13T
(1,2)
HHM
13V
9 periods
Relationships and Calculus 1.1
Applying algebraic skills to solve equations
Quadratic Theory

Sketching quadratic functions
Time
HHM 144
8C
1 (a, b, f)
Completed N5
 I can sketch a Quadratic
 I can determine the equation of a Quadratic

Completing the square
(1 – 4)
HHM 336
5
1 period
HHM 146
8D
(1-6)
HHM 148
8F
(1-3)
1 period
8F
(3)
1 period
 I can complete the square

Solving quadratic equations
 I can find the roots of a Quadratic by factorising

Quadratic inequations
HHM 148
 I can solve Quadratic inequalities using a sketch of the graph

The quadratic formula
HHM 150
8G
(1, 2)
Completed N5
(1, 2)
Completed N5
 I can find the roots of a Quadratic using the quadratic formula

The discriminant
HHM 151
8H
 I can find the nature of the roots of a Quadratic using the discriminant

Using the discriminant
HHM 152
8I
(1 – 8)
Completed N5
8J
(1 – 6)
1 period
7H
(1 – 4)
1 period
 I can use the discriminant to find an unknown value

Condition for tangency
HHM 155
 I understand the condition for tangency

Determine quadratic equation
HHM 135
with given coefficients
 I can find an expression for a quadratic from its graph
5 periods
Factor Theorem and Remainder Theorem

What is a polynomial?
HHM 126
Time
7A
(1 – 8)
period
 I can identify a polynomial expression (Qu 1,2)
 I can identify the coefficients of each term in a polynomial (Qu 3)
 I have had experience of substituting and evaluating expressions (Qu 6 – 8)

Nested form of a polynomial
HHM 127
7B
1, 2
period
 I can use the nested form to evaluate polynomials

Division by (x – a)
HHM 129
7C
(1 – 4)
1 period

Remainder Theorem
HHM 130
7D
(1 – 8)
1 period
7E
( 1 – 7)
1 period
 I can find the remainder using the remainder theorem

Factor Theorem
HHM 131
 I can factorise a polynomial expression using the factor theorem

Finding a polynomial’s coefficients
HHM 132
7F
( 1 – 5)
1 period
 I can evaluate an unknown coefficient of a polynomial given (x – a)

Solving polynomial equations
HHM 134
7G
(1 – 6)
1 period
HHM
7H
(5-15)
1 period
(1-7)
1 period
 I can solve polynomial equations

Functions from graphs
 I can find coefficients of polynomials from their graphs

Approximate roots
HHM
7J
 I can find approximate roots
9 periods
Total 14 periods
Relationships and Calculus 1.2
Applying trigonometric skills to solve equations
Time

Revision of y = Asin (Bx + C) + D
Completed N5
and y = Acos (Bx + C) + D

Period and amplitude
HHM 53
4A
1,3
1 period
 I can state the period and amplitude for a given trigonometric graph

Graphs of trigonometric functions
HHM 55
4B
3
1 period
 I can sketch and annotate a basic trig graph under the following transformations
kf(x), f(x) + k, f(kx), f(x + k), -f(x), f(-x) – including combination of these

Radians
HHM 56
4C
(1 – 5)
1 period
 I can convert Degrees to Radians (and vice versa) and sketch and interpret
Trigonometric Graphs using radians

Special angles and triangles
HHM 57
4D
(1 – 3)
1 period
 I have revised how to find exact values and the origins of the CAST diagram

Using exact values
HHM 59
4E
(1 & 2)
1 period
 I can find exact values for any given angle using the CAST diagram

Graphical solution of equations
HHM 59
4G
(1 & 2)
1 period
HHM 56
4H
(1 – 5)
2 periods
 I can use graphs to solve equations

Algebraic solution of equations
 I have revised solving basic trigonometric equations in degrees and
radians – (Qu 1a, b, 2 and 3)
 I can solve a trigonometric equation including those involving multiple
angles – (Qu 1c, d, e, f and 4)
 I can solve trigonometric equations that require factorisation – (Qu 5)

Algebraic solution of compound
angle equations
HHM 56
4I
(1 – 4)
1 period
 I can solve trigonometric equations that involve phase angles
TOTAL 9 Periods
Relationships and Calculus 1.3
Applying calculus skills of differentiation
Time
1. Derivative of xn
HHM 91
6D
(1 – 40)
1 period
I can differentiate algebraic functions which have positive and
negative powers of x (Qu 1 – 20)
I can differentiate algebraic functions which can be simplified
to an expression in powers of x from a surd (Qu 21 – 27)
I can differentiate algebraic functions which can be simplified
to an expression in powers of x originally as the denominator (Qu 28 – 40)
2. Derivative of axn
HHM 94
6F
(1 – 27)
1 period
I can differentiate algebraic functions which have a multiplier (Qu 1 – 18)
I can differentiate algebraic functions which have more than
3. Derivatives of products and quotients HHM 95
one term (Qu 19
6G
(1 – 27)
1 period
I can differentiate algebraic functions containing products (Qu 1 – 16)
I can differentiate algebraic functions containing quotients (Qu 17 – 27)
4. Rate of change
HHM 92
6E
1, 3, 5, 6
period
HHM 96
6H
(6 – 14)
1 period
6I
(2,3,4)
1 period
6J
(1 – 8)
1 period
I can find the rate of change of a function
5. Applications of derivatives
I can use differentiation to solve real-life problems
6. Leibniz notation
HHM 100
I can use Leibniz Notation as an alternative to f(x)
7. Equations of tangents
HHM 101
I can determine the equation of a tangent to a curve, at a given
8. Stationary points
HHM 106 6M
point by differenti
(1-9)
I can determine use the stationary points of a curve and state
9. The shape of curves
HHM 104
6L
1 period
their nature
(2-6)
1 period
(1 – 9)
3 periods
I can determine where a function is strictly increasing/decreasing
10. Curve sketching
HHM 107
6N
I can sketch the graph of an algebraic function by determining
stationary points a
(Extra practice
HHM 137
7I
(1-9) )
11. Closed interval
HHM 109
60
1, 2
1 period
I can determine where the maximum/minimum values lie in
12. Graphs of derived functions
HHM 110
a closed interv
6P
(1 – 9)
1 period
I can find the graph of a derived function
(The following two learning intentions are assessed in Applying calculus
skills to optimisation and area – Applications 1.4. Teach together but
assess after completing the teaching of both differentiation and
integration – R&C 1.3 & 1.4.)
13. Optimisation : Maxima and minima HHM 112
6Q
(1 – 8)
1 period
14. Further applications
6R
(1 – 5)
1 period
(1 - 6)
1 period
HHM 267 14E
1, 2
1 period
17. Derivative of (ax+ b)
HHM 269 14G
1
1 period
18. The chain rule
HHM 271 14H
(1 – 5)
1 period
HHM 114
I can solve optimisation problems in context using differentiation
15. Differentiation of sin x and cos x
HHM 264 14B
I have experience of differentiating sinx and cosx
16. Derivative of (a + x)n
n
I can differentiate a composite function using the chain rule
20 periods
Relationships and Calculus 1.4
Applying calculus skills of integration
Time

Anti-differentiation
HHM 162
9E
1
1-2 periods
 I understand that integration is the reverse process of differentiation

Indefinite integrals
HHM 164
9G
1
1 period
 I can find an indefinite integral remembering to include C – the
constant of integration

Rules of integration
HHM 164
9H
(1 – 3)
1 period

Further integrals
HHM 165
9I
1
1 period
 I can integrate an algebraic function which can be simplified to
an expression of powers of x

Definite integrals
HHM 167
9K
( 1 –3)
1 period
 I can evaluate the area enclosed between a function and the
x – axis and can explain why an area is positive or negative

Fundamental Theorem of Calculus
HHM 169
9L
(1 – 4)
2 periods
 I can evaluate the definite integral of a polynomial functions
with integer limits (Qu 1 – 3)
 I can evaluate one of the limits of a definite integral given the
value of the definite integral. (Qu 4)

Differential equations

HHM 176
9Q
(1 – 6)
1 period
I can use differential equations to find the equation of a line
(The following five learning intentions are assessed in Applying calculus skills to
optimisation and area – Applications 1.4. Teach together but assess after
completing the teaching of both differentiation and integration – R&C 1.3 &
1.4.)

Definite integrals
HHM 167
9K
(1 – 3)
1 period

Fundamental theorem of calculus
HHM 169
9L
(1 – 4)
1 period

Areas above and below the x –axis
HHM 172
9N
(1 – 4)
1 period

I can find the area enclosed between a curve and the x-axis

Area between two graphs

Calculating the area between two graphsHHM174 9P


HHM 173
9O
(1,2)
1 period
(1 – 6)
1 period
(1 - 7)
1 period
(1 – 7)
1 period
(1 - 8)
1 period
I can evaluate the area enclosed between two functions
Integration of sin x and cos x
HHM 265 14C
 I can integrate standard trigonometric functions

Integrating (ax + b)n
HHM 274 14J
 I can apply a standard integral of the form
𝑓(𝑥) = (𝑝𝑥 + 𝑞)𝑛 with n≠-1

Integrating sin (ax + b) and
HHM 275 14K
cos (ax + b)
 I can apply a standard integral of the form 𝑓(𝑥) = 𝑝𝑐𝑜𝑠(𝑞𝑥 + 𝑟)
and 𝑓(𝑥) = 𝑝𝑠𝑖𝑛(𝑞𝑥 + 𝑟)
16-17 periods