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N5 N5 N5 N5 N5
N5
National
Five
N5
N5 N5 N5 N5 N5
Relationships
Teachers’ Notes
N5 N5 N5
N5
N5 N5
N5
1.1
Applying algebraic skills to linear equations
A1.1 Function Notation
A1.1a
Introduction to f(x) notation.
Build on the pupils’ previous knowledge of number machines.
Teejay Int 2 Book 2
P40
Ex 4.1
Q 1 to 6
A1.1b The common sets and basic set builder notation could be introduced here.
This can be linked to inequalities and reinforced as lesson starters.
Graph drawing packages could be used to draw graphs of functions by
letting y = f(x).
Teejay Int 2 Book 2
P40
Ex 4.1
Q 7 to 9
B1.1 Straight Line
Straight Line is covered earlier in the course. This section should only be a quick
revision. It should also be reinforced through lesson starters.
B1.1a
I can calculate the gradient of a straight line given two points which lie on
the line:
National 5 Bk
Page 41
Ex 6.2
By this point pupils should be familiar with finding the gradient simply as a
process of substitution into a formula.
B1.1b
I have investigated the gradients of horizontal and vertical lines and can
state their equations.
Departmental PowerPoint
Horizontal lines can be explained as all the points where y = k and
similarly for vertical lines.
The PowerPoint ‚BHStraightLineHorizontalAndVertical‛ is available.
B1.1c
I have worked with others to investigate the effect of m and c in the equation
Investigation, Board Examples and Graphing Software
Graphing software is available to illustrate what happens to the
graph of a straight line as m and c are varied. Pupils should attempt
to predict graphs by using a Cartesian show me board. This will be
revision of S3 work.
B1.1d
I can write down the equation of a straight line given the gradient and yintercept and vice versa.
Natioanl 5 Bk
B1.1e
Ex 9.1
I can rearrange the equation of a line into the format
National 5 Bk
B1.1f
Page 62
Page 65
Ex 9.3
I can find the equation of a straight line given the gradient and one point
using
National 5 Bk
.
Page 63
Ex 9.2
C1.1 Linear Equations and Inequalities
This topic has been well covered during the BGE. It should be reinforced by lesson
starters very, very regularly.
C1.1a
I can solve equations that contain brackets.
National 5 Bk
P72
Ex 10.4
National 5 Bk
P73
Ex 10.5
C1.1b I can solve equations which contain fractions.
National 5 Bk
C1.1c
P74
Ex 10.6
I can solve inequalities and understand through investigation that when
then
etc.
National 5 Bk
P77
Ex 10.8
Natioanl 5 Bk
P78
Ex 10.9
C1.1d I can use Equations to make mathematical models.
National 5 Bk
P75
Ex 10.7
D1.1 Simultaneous Equations
Simultaneous Equations are covered earlier in the course. This section should only
be a quick revision. It should also be reinforced through lesson starters.
D1.1a I can solve simultaneous equations (2 equations and 2 unknowns)
graphically.
National 5 Bk
P 82
Ex 11.1
D1.1b I can solve simultaneous equations (2 equations and 2 unknowns) by
elimination.
National 5 Bk
D1.1c
P85
Ex 11.3
Nation al 5 BK P86
Ex 11.4
I can solve simultaneous equations (2 equations and 2 unknowns) by
substitution.
National 5 Bk
P85
Ex 11.5
D1.1h I can model real life problems with simultaneous equations and solve them.
National 5 Bk
P88
Ex 11.6
E1.1 Changing the subject
Pupils will have been solving equations using inverse operations for a number of
years by this point.
Changing the subject should initially be introduced as a series of inverse operations
on the new subject.
e.g.
y = 2x + 3 – x is multiplied by two then 3 is added.
The inverse of this is subtracting 3 followed by division by two.
Apply this to both sides and hey presto!!
There are then 3 special cases where some manipulation prior to finding the inverse
operations is required
a) New subject is part of a negative term.
b) New subject is part of a denominator.
c) New subject appears in more than one term.
Departmental PowerPoint is available
E1.1a
I can substitute into Expressions and Formulae
National 5 Bk
E1.1b
P92
Ex 12.1
I can Change the subject of formulae
Int 2/C2
P22
Ex 2.5
Q1 – 9
Int 2/C2
P24
Ex 2.6
Q1(a – d), 2 – 8
National 5 Bk
P 94
Ex 12.2
National 5 Bk
P 95
Ex 12.3
Nationa 5 Bk
P95
Ex 12.4
Revision and sub skill assessment Relationships 1.1
N5
1.2
Applying algebraic skills to graphs of quadratic
relationships
A1.2 Quadratic functions and Graphs
Pupils will be familiar with the term PARABOLA from the flying car experiment on
projectiles. The National 5 textbook avoids the notation f(x) but this should be
covered by the teacher. It should also be made clear that the function is f(x) and the
graph is y = f(x).
A1.2a
I can make a table of values for a function (f(x)) and draw the graph of y =
f(x).
National 5 Bk
P99
Ex 13.1
A1.2b I can recognise, from its graph, y = x2 and any related graph of the
form y = ax2 + b.
A1.2c
National 5 Bk
P101
Ex 13.2
Int 2/C2
P96
Ex 9.3
Q1 to Q5
I have revised the procedure known as Completing the Square.
National 5 Bk
P30
Ex 4.8
Q3 to Q7
A1.2d I can identify the turning point (and line of symmetry) of a quadratic when
the quadratic is given in the form
and determine the nature
of this turning point. I can also determine the equation of a quadratic in the
form
from its graph.
National 5 Bk
P103
Ex 13.3
Int 2/C2
P94
Ex 9.1
Q1 to Q6
Int 2/C2
P95
Ex 9.2
Q1 to Q4
Revision and sub skill assessment Relationships 1.2
N5
1.3
Applying algebraic skills to quadratic equations
A1.3 Quadratic Equations (Graphically)
Graph drawing packages used with show me boards allow pupils to experience a
large number of examples of finding the points where the graph cuts the x axis and
making the connection between this and the solution to the equation.
Teaching Idea – Have 20 or so graphs of quadratic functions around the classroom or
corridor. Pupils are given some quadratics equations and have to go on a gallery
walk to find the solution. They will first have to rearrange the equations into the
correct form before searching for the correct graph.
Include examples which do not have a solution.
A1.3a
I know the meaning of the term ‘roots of a quadratic equation’ and solve
quadratic equations graphically.
National 5 Bk
P106
Ex 14.1
Int 2/C2
P60
Ex 6.2
Q1 to Q3
B1.3 Quadratic Equations (by factorisation)
Revision of factorisation should be happening regularly as lesson starters for a
number of weeks prior to starting this topic.
B1.3a
B1.3b
I have revised how to factorise
National 5 Bk
P29
Ex 4.6 and Ex 4.7
Int 2/C2 :
P61 :
Ex 6.3 :
Q1 to Q4
I can solve a Quadratic Equation by factorisation.
National 5 Bk
P108
Ex 14.2
Int 2/C2 :
P62 :
Ex 6.4 :
Q1 to Q13
C1.3 Quadratic Formula and the Discriminant
Pupils should ask ‚does the quadratic equation have real roots‛ by calculating the
Discriminant. The Discriminant can then be substituted into the quadratic formula if
it is >=0. It is therefore important to give pupils the opportunity to experience
quadratic equations with no real roots.
A graph drawing package could be used to illustrate the three conditions graphically
while checking answers. This is well suited to pupils working in active pairs with a
show me board between them.
This will reinforce and deepen our pupils’ understanding of the discriminant.
Negative – No real roots. Can’t solve the equation (for now!)
Zero – Repeated roots, equal roots producing one solution to the equation.
Positive – Two real and distinct roots producing two solutions to the equation.
C1.3a
I know how to determine nature of the roots of a quadratic equation using
the discriminant.
National 5 Bk
C1.3b
C1.3c
P110
Ex 14.4
Q1 and Q2
I can solve a quadratic equation using the Quadratic Formula
National 5 Bk
P109
Ex 14.3
National 5 Bk
P110
Ex 14,4
Q10
Int 2/C2
P98
Ex 9.4
Q1 to 10
I can identify problems where the discriminant can be used as a strategy to
solve a problem.
National 5 Bk
P110
Ex 14.4
Q3 to Q9
D1.3 Problem Solving Using Quadratic Equations
D1.3a
I can find a Quadratic Equation which models a real life situation and by
solving the Quadratics Equation I can find and communicate the solution to
a problem.
National 5 Bk
P112
Ex 14.5
Revision and sub skill assessment Relationships 1.3
N5
1.4
Applying geometric skills to lengths, angles and
similarity
A1.4 Pythagoras (converse and 3D)
Converse has previously been covered in the BGE.
3D Pythagoras is new at this point and could be perfect for some model building.
Build a skeleton model. Use Pythagoras to calculate the length of a space diagonal.
Cut out a length and check that it fits.
What types of 3D shape could be used. Are there packs available commercially?
Could we just use straws?
A1.4a
I have revised the use of Pythagoras Theorem including the Converse of
Pythagoras.
National 5 Bk
P115
Ex 15.1
A1.4b I can use Pythagoras to solve problems in 3 Dimensions.
National 5 Bk
P118
Ex 15.3
B1.4 Properties of Shape (Triangles, Quadrilaterals, Polygons, Angles in a Circle)
Angles In The Circle has been covered in the BGE.
This could be new having this type of plane geometry so prominent at this level.
Development required.
Some form of extension to the quadrilateral project might be good only for regular
polygons.
B1.4a I have spent time revising properties of 3D shape
National 5 Bk
P123 Ex 16.1
Q3 to Q6
National 5 Bk
P125 Ex 16.2
Q5 to Q7
National 5 Bk
P127 Ex 16.3
B1.4b I have revised using angle facts in a circle.
National 5 Bk
P130 Ex 16.5
C1.4 Similarity
C1.4a
I know how to check if two shapes are Similar and can use the fact that two
shapes are Similar to calculate unknown sides.
C1.4b
National 5 Bk
P135
Ex 17.1
Int 2 C2
P48
Ex 5.1
I can show when triangles are Similar and can use Similarity in Triangles to
find the lengths of unknown sides.
C1.4c
National 5 Bk
P137
Ex 17.2
Int 2 C2
P50
Ex 5.2
Int 2 C2
P52
Ex 5.3
I know the relationships between Length Scale Factor, Area Scale Factor and
Volume Scale Factor and can apply Scale Factor to solve problems.
National 5 Bk
P140
Ex 17.3
Int 2 C2
P54
Ex 5.4
Int 2 C2
P56
Ex 5.5
Revision and sub skill assessment Applications 1.4
N5
1.5
Applying trigonometric skills to graphs and identities
A1.5 Related Angles in the range 0 - 360˚
Pupils should use graphs of sin, cos and tanx to identify related angles which satisfy
sin, cos, tan x = ±k.
Pupils should then be guided towards an understanding that every angle 0 - 360 has
a related angle in each quadrant (with the exception of the border angles 0, 90, 180,
270, 360) and that sometimes it is positive and sometimes it is negative.
This may seem a bit back to front but it lets pupils see where they are going before
introducing the new definition of sinx, cosx and tanx as ratio of coordinates
produced by a rotating arm rotating counter clockwise by an angle x.
This cognitive conflict caused by rejecting the nice opp/hyp etc can cause some
distress, particularly to the fixed mindset pupils, as they will not like finding out that
something they though was easy can in fact be quite challenging. Care must be
taken !!
A1.5a
I understand the new definitions of
,
and
Insert Required
A1.5b Given an angle θ where
where
, I can state a related base angle A
and all of its related angles (both positive and negative) in
the other three quadrants.
Insert Required
National 5 Bk
P149
Ex 18.2
B1.5 Trig Graphs (basic, amplitude, vertical translation, horizontal translation
(phase angle), multiple angles
This topic will build upon the translation work in the blue course and the quadratic
graph section of this unit.
Pupils will create their own graphs of
,
and
.
Then they will investigate the input transformations.
i.e. Given
there are two transformations taking place. Both affect
the graph in the x-direction.

is first translated by –b in the x-direction according to the
transformation matix

.
The new period is found using the formula
.
Finally they will investigate the output transformations.
i.e. Given
there are two transformations taking place. Both affect the
graph in the y-direction.

The range of values of y is changed from
amplitude changes from 1 to p.

to
so the
is translated by q in the y-direction according to the transformation
matix
.
Pupils could be asked to match a list of Trig Graphs to the Equations using a gallery
walk similar to that in section A1.3 of this unit.
B1.5a
I have used a table of values to plot the graphs of
.
,
and
Class activity
B1.5b
I know the meaning of the terms Transformation; Translation; Period and
Amplitude.
Class activity
B1.5c
Sketch and identify trigonometric functions which involve transformations
to the basic graphs of
,
and
.
National 5 Bk
P146
Ex 18.1
Int 2/C2
P75
Ex 7.4
Q1 to Q6
Int 2/C2
P77
Ex 7.5
Q1 to Q5
Int 2/C2
P82
Ex 7.7
Q1 to Q2
Int 2/C2
P83
RR
Q1 to Q6
C1.5 Trig Equations in the range 0 - 360˚
Pupils were introduced to the concept of a base angle in the first quadrant and
related angles in the other three. This will be developed into a formal procedure for
solving trig equations as below.
this is the base angle in the first quadrant
From the diagram below we can see that Cosine is negative in the 2nd and 3rd
quadrant
Sine

180 - A
All

A
180 + A

Tan
360 - A

Cos
Leading to our two solutions
Solution 1
C1.5a
Solution 2
I can solve Trig Equations Algebraically
National 5 Bk
P150
Ex 18.3
Int 2/C2 :
P116
Ex 11.1
Q1 - 11
D1.5 Trig Identities
D1.5a
I can use the Trig Identities
and
National 5 Bk
P152
Ex 18.4
Int 2/C2
P120
Ex 11.3
Revision and sub skill assessment Applications 1.5
.
Q1 to Q4, Q6(a – e), Q7(a, b)