Download Year 11 Pathway A-D Module Plan

HIGHER
Year 11 Pathway B GCSE Programme
MODULE 20:
Pythagoras Theorem
TIME ALLOCATION: 2 sessions
What students already know





Round numbers
Use the square root button on calculator
Identify hypotenuse as the longest side
Simplify squared numbers
To multiply and divide surds (basic)
LEARNING OBJECTIVES
Progression through module
 Derive Pythagoras’ theorem and use it to find the length of
the hypotenuse in any right-angled triangle
 Know and use Pythagoras’ theorem to find any missing length
in a right-angled triangle
 Apply Pythagoras’ theorem to 2D problems
 Link Pythagoras’ theorem to real-life skills for industry
 Use Pythagoras’ theorem to show whether a triangle is rightangled or not
 Understand, recall and use Pythagoras’ Theorem in 2D shapes
Apply Pythagoras’ theorem to 3D problems
Be aware of…..
Fluency in Literacy
Pythagoras theorem: states that in a right-angled
triangle the area of the square on the hypotenuse
is equal to the sum of the areas of the squares of
the other two sides.
Hypotenuse: Side opposite the right angle.
RESOURCES/ACTIVITIES/ICT
MathsWatch clips: 177/178/149/124
ICT: www.mymaths.co.uk for homework
activity or revision
www.teachmathematics.net for software,
resources and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy videos
Geogebra for dynamic geometric illustration
RESOURCES:
Higher Collins
Chapter 4 & 6
Higher Smith
Chapter 18
Recommended curricular assessment materials:
Identifying the hypotenuse correctly
Student may plug in or substitute the wrong values.
Student may forget to square each value
Student may forget to square root their final answer
Student may half their final answer instead of square
rooting it.
..\Year 11 Assessment Resources\Geometry
and Measures\AQA Topic Tests\Pythagoras'
Theorem and basic trigonometry - Topic test 1
H v1.1.doc
..\..\..\..\..\..\..\STAFFS~1\DEPART~1\Maths\1A3BE
~1.SCH\SCHEME~2\YEAR11~1\YEAR11~1\GEOMET~
1\AQATOP~1\Pythagoras' Theorem and basic
trigonometry - Topic test 1 H - Mark Scheme v1.1.doc
HIGHER
Year 11 Fast Track Pathway GCSE Programme
MODULE 25 Trigonometry
TIME ALLOCATION: 6 sessions
What students already know
 How to use sin, cos and tan buttons on calculator.
 Apply Trigonometry in right angles triangles to find missing
angles and sides.
 Apply Pythagoras in right angled and isosceles triangles to
find missing sides and angles.
 Know when to use trigonometry and Pythagoras in multistep questions involving both skills.
LEARNING OBJECTIVES
Fluency in Literacy
Trigonometry – uses the ratios of sides and
angles of a triangle for calculations.
Formulae – an equation used to find quantities
when given certain values
Angle of elevation / depression – when looking
up/down the angle between the line of sight
and the horizontal.
RESOURCES/ACTIVITIES/ICT
Progression through module
 To use the trigonometric ratios given by the
sine, cosine and tangent functions to find
unknown lengths and angles in 2D rightangled triangles
 To know the exact ratios given by sine and
cosine of 0, 30, 45, 60 and 90 degrees and the
exact ratios given by the tangent function for
0, 30, 45 and 60 degrees
 To know the difference between an angle of
depression and an angle of elevation
 To identify when the trigonometric ratios
must be used instead of Pythagoras’ theorem
to solve 2D problems relating to right-angled
triangles, including contextual problems
 Learn exact common trigonometric ratios and
be able to recall these in problems.
 Know and apply 1/2absinC to calculate the
area of triangles.
ICT: www.mymaths.co.uk for homework activity
or revision
www.teachmathematics.net for software,
resources and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy
videos
Geogebra for dynamic geometric
illustration
MathsWatch clips: 201, 202, 203
RESOURCES:
Higher AQA Mathematics for GCSE AQA
Chapter 38
Beware of……..
 Error in identifying the angle under
consideration and the opposite side,
adjacent side etc.
 Error in comprehending the problem
Recommended curricular assessment materials:
..\Year 11 Assessment Resources\Geometry and
Measures\AQA Topic Tests\Pythagoras'
Theorem and basic trigonometry - Topic test 2 H
v1.1.doc
 Your calculator is set on Degree and not Rad
etc.……..
 Incorrect use of the inverse button.
 Not marking angle of depression in the
diagram
 Error in identification of appropriate
trigonometric ratios and the triangle.
 Not mentioning units in the answer
HIGHER
..\..\..\..\..\..\..\STAFFS~1\DEPART~1\Maths\1A3
BE~1.SCH\SCHEME~2\YEAR11~1\YEAR11~1\GE
OMET~1\AQATOP~1\Pythagoras' Theorem and
basic trigonometry - Topic Test 2 H - Mark
Scheme v1.1.doc
..\Year 11 Assessment Resources\Geometry and
Measures\AQA Topic Tests\Trigonometry recap
and extension - Topic test 1 H v1.2.doc
..\..\..\..\..\..\..\STAFFS~1\DEPART~1\Maths\1A3
BE~1.SCH\SCHEME~2\YEAR11~1\YEAR11~1\GE
OMET~1\AQATOP~1\Trigonometry recap and
extension - Topic test 1 H - Mark Scheme
v1.1.doc
Year 11 Pathway B GCSE Programme
MODULE: Direct and indirect proportion
TIME ALLOCATION: 6 sessions
What students already know
●
Solving simple equations
●
Ratios and proportions
●
Recognise direct proportion graphs.
●
Powers and roots
Fluency in Literacy
Constant of proportionality- The constant
value of the ratio of two proportional
quantities
Inversely proportional- when one quantity is in
directly proportion to the reciprocal of other.
LEARNING OBJECTIVES
Progression through module
●
How to work out the constant of proportionality from
straight line graph for direct proportion (use gradient)
●
Finding constant of proportionality by (unitary method)
●
Understand proportion and the equality of ratios
●
Direct proportions involving squares, cubes and square
root
●
Solve problems involving direct and inverse proportion,
using graphs and algebraic techniques
●
Understand that ‘x is inversely proportional to y’ is
proportional to 1/y ●
Interpret equations that describe direct and inverse proportion RESOURCES/ACTIVITIES/ICT
ICT: www.mymaths.co.uk for homework activity or
revision
www.teachmathematics.net for software,
resources and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy
videos
Geogebra for dynamic geometric illustration
RESOURCES:
Higher AQA: Chapter 28,
Recommended curricular assessment materials:
Be aware of…..
●
Students may write a formula with the two variables the
wrong way round (finding the inverse of the constant).
Encourage students to check their formula by
substitution.
..\Year 11 Assessment Resources\Ratio,
Proportion and rates of change\AQA Topic
Tests\Ratio and proportion - Topic test 1 H
v1.1.doc
●
Students may substitute into the formula, or process
incorrectly. Encourage students to write down each step
in their working and check their answers.
●
Students may fail to identify the correct type of
relationship. Encourage them to read the question
carefully, writing down each piece of information as they
go.
..\..\..\..\..\..\..\STAFFS~1\DEPART~1\Maths\1A3
BE~1.SCH\SCHEME~2\YEAR11~1\YEAR11~1\RATI
O_~1\AQATOP~1\Direct and inverse proportion Topic test 1 H v1.1.doc
●
Some students may carry out operations in the wrong
order (such as multiplying before squaring). Remind
students of BIDMAS.
●
Some students may ignore the term ‘inversely’ and treat it
as direct proportion. Encourage students to think about
what ‘inverse proportion’ means (answer: as one variable
increases, the other decreases). Do your answers fit this
pattern?
●
Some students may incorrectly rearrange the equation to
find k. Recap rearranging or solving equations involving a
fraction if appropriate.
HIGHER
Year 11 Pathway B GCSE Programme
MODULE: The Sine, cosine and area rules
TIME ALLOCATION: 6 sessions
What students already know
Use A = 1 bh to find the area of a triangle………
2
Measuring triangles…………..
Constructing triangles………..
Evaluate expressions using the priority of operations….
●
Find the area of a triangle (1/2 ab sin C)
Fluency in Literacy
Trigonometric: relating to or according to
the principles of trigonometry
Sine: ratio of the length of the side opposite
the given angle to the length of the
hypotenuse of a right-angled triangle.
Cosine: ratio of the adjacent side to the
hypotenuse of a right-angled triangle.
Tangent: ratio of the opposite to the adjacent
side of a right-angled triangle
Inverse function: a function obtained by
expressing the dependent variable of one
function as the independent variable of
another; f and g are inverse functions if f(x)=y
and g(y)=x
LEARNING OBJECTIVES
RESOURCES/ACTIVITIES/ICT
Progression through module.
●
Know and apply sine rule to find missing sides and
angles.
●
Know and apply cosine rule to find missing sides and
angles.
●
●
Be able to recognise when to use basic trigonometry,
sine rule, cosine rule or a mixture of all three to solve
multi step problems.
Use the sine rule to solve 2D problems.
●
Use the cosine rule to solve 2D problems.
ICT: www.mymaths.co.uk for homework activity or
revision
www.teachmathematics.net for software,
resources and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy
videos
Geogebra for dynamic geometric illustration
Higher AQA
Chapter: 38
Higher Smith
Chapter: 27
Higher Collins
Chapter: 15
Recommended curricular assessment materials
Be aware of…..
●
Students are expected to know the exact values of sin θ
and cos θ for θ = 0°, 30°, 45°, 60° and 90°, and the exact
values of tan θ for θ = 0°, 30°, 45° and 60°. This is new to
the Higher tier GCSE 2015.
●
Students need to know the formula for the cosine rule and
be able to apply it to work out unknown lengths and
angles. This is new to the Higher tier GCSE 2015.
●
Students may not understand that it takes practice to
recognise the situations where the sine rule may be used
and become frustrated. A brief class discussion about key
indicators may help.
●
The sine rule may be used either to calculate the length of
an unknown side or the size of an unknown angle. It is best
to write the rule so that the unknown value is the
numerator of the fraction as this minimises the amount of
rearranging required. Make sure that students are
:
..\Year 11 Assessment Resources\Geometry
and Measures\AQA Topic Tests\Sine and
Cosine Rules - Topic test 1 H v1.1.doc
..\Year 11 Assessment Resources\Geometry
and Measures\AQA Topic Tests\Sine and
Cosine Rules - Topic test 1 H - Mark Scheme
v1.1.doc
confident of the fact that both representations of the sine
rule are valid, the only difference is whether they want the
missing value to be the numerator or the denominator
HIGHER
Year 11 Pathway B GCSE Programme
MODULE: Changing the subject of the formula, Algebraic fractions
TIME ALLOCATION: 6 sessions
What students already know
Fluency in Literacy
Factorising, squares, cubes and roots
Solve- find the value of unknown
Substitute into a linear formula.
Algebraic expression: is an expression that
Calculating with fractions
contains at least one variable
Simplifying numerical and simple algebraic fractions.
Factorise simple expressions by identifying the common factor.
Factorising, lowest common multiple
Simplify algebraic fractions by cancelling common factors.
LEARNING OBJECTIVES
Algebraic Fractions: They are like normal
fractions with their numerators and denominators
expressed as algebraic fractions.
RESOURCES/ACTIVITIES/ICT
Progression through module
●
Harder linear equations contain many stages
●
Equations and brackets
●
Expanding squares
●
Equations with fractional coefficient
●
Change the subject of a formula where the power of the
subject appears.
●
Change the subject of a formula where the subject appears
twice
●
Change the subject of a formula involving fractions where
all the variables are in the denominators.
●
Add and subtract algebraic fractions.
●
Higher AQA
Chapter: 16, 23,
Multiply and divide algebraic fractions.
Recommended curricular assessment materials:
●
Simplify algebraic fractions.
●
Add and subtract more complex algebraic fractions.
●
Multiply and divide more complex algebraic fractions.
●
Solve equations that involve algebraic fractions.
ICT: www.mymaths.co.uk for homework activity or
revision
www.teachmathematics.net for software,
resources and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy
videos
Geogebra for dynamic geometric illustration
..\Year 11 Assessment Resources\Algebra\AQA
Topic Tests\Algebraic fractions - Topic test 1 H
v1.1.doc
Be aware of…..
●
When squaring, students may only square some of the
terms. Remind them that when they apply an operation
to a formula or an equation, they must do the same to the
whole of both sides.
●
Students may be confused by algebraic fractions and not
know how best to get started on a question.
●
Students may forget to multiply the numerator by a
number, and only multiply the denominator. Remind
students that the fractions must remain equivalent.
●
Students may not factorise quadratic expressions
correctly.
●
Some students may have difficulty finding the common
denominator. Emphasize that the key steps are to
factorise and then look for factors in common.
HIGHER
..\Year 11 Assessment Resources\Algebra\AQA
Topic Tests\Algebraic fractions - Topic test 1 H - Mark
Scheme v1.1.doc
Year 11 Fast Track Pathway GCSE Programme
MODULE 23 Circle Theorems
TIME ALLOCATION: 6 sessions
What students already know
 use ruler, protractor and pair of compasses to accurately
construct angles and shapes
 accurately copy diagrams using rulers and a pair of compasses
only
 Be able to calculate missing angles in triangles and parallel
lines
 Understand the concept of a proof.
 Know the names of parts of a circle
LEARNING OBJECTIVES
Fluency in Literacy
Cyclic Quadrilateral – any quadrilateral with all four
vertices on the circumference of a circle.
Proof – an argument or explanation that establishes
the truth of a preposition.
Theorem – a statement established by proof.
RESOURCES/ACTIVITIES/ICT
Progression through module







ICT: www.mymaths.co.uk for homework activity or
revision
www.teachmathematics.net for software, resources
and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy
videos
Geogebra for dynamic geometric illustration
To construct the perpendicular bisector of a line
To construct the perpendicular at a given point on a
line
To construct a perpendicular from a given point to a
line
To bisect an angle
To use constructions to solve loci problems
Recall the definition of a circle and identify (name) and
draw parts of a circle, including sector, tangent, chord,
MathsWatch clips:
segment;
Prove and use the facts that:
RESOURCES:
Higher AQA Mathematics for GCSE AQA Chapter 31












the angle subtended by an arc at the centre of a circle
is twice the angle subtended at any point on the
circumference;
the angle in a semicircle is a right angle;
the perpendicular from the centre of a circle to a chord
bisects the chord;
angles in the same segment are equal;
alternate segment theorem;
opposite angles of a cyclic quadrilateral sum to 180°;
Understand and use the fact that the tangent at any
point on a circle is perpendicular to the radius at that
point;
Find and give reasons for missing angles on diagrams
using:
circle theorems;
isosceles triangles (radius properties) in circles;
the fact that the angle between a tangent and radius is
90°;
The fact that tangents from an external point are equal
in length.
Beware of…………

Misunderstanding the fact that the tangent at any
point on a circle is perpendicular to the radius at that
point

Misunderstanding the fact that tangents from an
external point area equal in length

Writing the missing angles on the diagram without
showing your working out.

Student should prove and use the facts that:

The angle subtended at any point on the centre of a
circle is twice the angle subtended at any point on
the circumference

Alternate segment theorem

Students must give reasons for angle calculations
involving the use of tangent theorems

Perpendicular from the centre of a circle to a chord
bisect the chord.
Recommended curricular assessment materials:
..\Year 11 Assessment Resources\Geometry and
Measures\AQA Topic Tests\Circle Theorems - Topic test 1
H v1.1.doc
..\Year 11 Assessment Resources\Geometry and
Measures\AQA Topic Tests\Circle Theorems - Topic test 1
H - Mark Scheme v1.1.doc
HIGHER
Year 11 Pathway A GCSE Programme
MODULE 24 Equations and Graphs
TIME ALLOCATION: 6 sessions
What students already know
 Use graphs to find equations o straight lines
 Solve simultaneous equations
 Use compound measures
LEARNING OBJECTIVES
Fluency
Parabola: A special curve, shaped like an arch.
Reciprocal : The reciprocal of a number is: 1 divided by the
number
Intersect: To cross over (have some common point)
Quadratic Equation: An equation contains at least one term that
is squared.
RESOURCES/ACTIVITIES/ICT
ACTIVITIES:









To solve linear equations
To understand that identities are equations for which
there are an infinite number of solutions as they are true
for all values x can take
To form and solve quadratic equations
To understand that different types of equations have a
different possible number of solutions
To solve linear simultaneous equations
To know how to read and interpret graphs in various
contexts
To be able to use graphs to find approximate solutions
to equations
To identify lines that are parallel by considering their
equation
To find the equation of a line parallel to a given line
(perhaps passing through a known point)

Draw and interpret graphs modelling real life situations


Use trial and improvement to find approximate solutions
of equations
Generate points and plot graphs of simple quadratic
functions

Plot graphs of simple cubic reciprocal functions

Find approximate solutions of equations from their
graphs, including one linear and one quadratic and simple
cubic equations

Plots graphs of simple quadratic functions which model
real life situations and formulae
ICT: www.mymaths.co.uk for homework activity or
revision
www.teachmathematics.net for software, resources
and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy
videos
Geogebra for dynamic geometric illustration
MathsWatch clips: 161/170/145/146/116/117
Higher Collins
Chapter 13 & 17
Higher Smith
Chapter 25 & 11
Recommended curricular assessment materials:
..\Year 11 Assessment Resources\Algebra\AQA Topic
Tests\Coordinates and linear graphs - Topic test H
1.1.doc
..\Year 11 Assessment Resources\Algebra\AQA Topic
Tests\Coordinates and linear graphs - Topic test 1 H Mark Scheme 1.1.doc
..\Year 11 Assessment Resources\Algebra\AQA Topic
Tests\Equations - Topic test 1 H 1.1.doc
..\Year 11 Assessment Resources\Algebra\AQA Topic
Tests\Equations - Topic test 1 H - Mark Scheme
1.1.doc
..\Year 11 Assessment Resources\Algebra\AQA Topic
Tests\Further equations and graphs - Topic test 1 H
v1.1.doc
..\Year 11 Assessment Resources\Algebra\AQA Topic
Tests\Further equations and graphs - Topic test 1 H Mark Scheme v1.1.doc
..\Year 11 Assessment Resources\Algebra\AQA Topic
Tests\Linear and quadratic equations and their graphs
- Topic test 1 H v1.1.doc
..\..\..\..\..\..\..\STAFFS~1\DEPART~1\Maths\1A3BE~1.
SCH\SCHEME~2\YEAR11~1\YEAR11~1\Algebra\AQAT
OP~1\Linear and quadratic equations and their
graphs - Topic test 1 H - Mark Scheme v1.1.doc
Beware of……

Errors Related to Order of Operations and Basic Properties: Not distributing to all terms: 6(x+5)
≠6x + 5 instead of the correct form 6(x+5) =6x + 6(5). This mistake occurs often when the
expression is in the form – (3x+2). When the distributive property is applied correctly to this
example you should get -3x – 2 not the -3x+2 which is often given.

Failure to include parentheses when substituting: If f(x) = x 2 - x and you want to find f(x+2) the
correct substitution should look like f(x+2) = (x+2)2 - (x+2) and the understood -1 factor for the
second term would then be distributed over the expression. However many times the substitution
is written as f(x+2) = (x+2)2 - x+2 so that it does not appear that the distributive rule needs to be
applied.

Errors Related to Exponents: Many mistakes result from attempting to apply this rule to additions
or subtractions (x-3)2 ≠ x2+9. Emphasize that (x – 3) (x – 3) = x2 – 6x+9.

Errors Related to Fractions

Errors related to solving equations
MODULE: Transforming Graphs
TIME ALLOCATION: 6 sessions
What students already know
Fluency in Literacy




Coordinates
Use vectors to describe the translation
Reflection
Combined transformation
 Function notation
Column vector- a collection of numbers, as the
components of a vector, written vertically
Transformation- term for four specific ways to
manipulate the shape of a point, a line, or shape.
Translation- Sliding: moving a shape without
rotating or flipping it.
Scale factor- The ratio of any two
corresponding lengths in two similar
geometric figures
Asymptote- is a horizontal, vertical, or slanted
line that a graph approaches but never
touches
LEARNING OBJECTIVES
Progression through module
 To carry out, identify and describe reflections
 To carry out, identify and describe translations using 2D
vectors

To carry out, identify and describe rotations
 To transform Linear graph
 Quadratic functions and parabolas
 Transform a function f(x) to f(x) ± a where a is a constant.
 Transform a function f(x) to f(x ± a) where a is a constant
 Transform a function f(x) to af(x) where a is a Integer.
 Transform a function f(x) to f(ax) where a is a constant
 Transform a function f(x) to af(x) and f(ax) where a is a
rational number.
 f(x) can be a trigonometric function.
Be aware of…..
●
Students may forget the translation vectors for graphs of
the form y = f(x + a) and y = f(x). Discuss ways of
remembering the vectors.
●
Students may confuse the graph of y = f(ax) with y = af(x)
and
y = f( 1 x). Discuss reliable ways of remembering which is
a
which with students.
●
Some students may not plot turning points and intercepts
accurately. Remind students that these are the key
features of a sketch graph and need to be plotted
carefully
●
Emphasis on showing all the working out.
●
Co-ordinates
MODULE 30: Vectors
TIME ALLOCATION: 6 sessions
RESOURCES/ACTIVITIES/ICT
ACTIVITIES
ICT: www.mymaths.co.uk for homework activity or
revision
www.teachmathematics.net for software,
resources and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy
videos
Resources: Mathswatch 167/168/169
Higher AQA: Chapter 41
Recommended curricular assessment materials:
Transformation functions assessment AQA
Transformation functions Mark scheme AQA.
What students already know




Coordinates
Recall properties of basic shapes
Understand translations
Use vectors to describe the translation
LEARNING OBJECTIVES
Progression through module
 Understand and use Vector notation and the associated
vocabulary
 Describe a translation by a vector
 To represent vectors as a diagram or a column vector
 Add and subtract vectors graphically
 Understand and use commutative properties of vector
addition
 Calculate a scalar multiple of a vector and represent it
graphically
 Understand that vectors represented by parallel lines are
multiples of each other.
 Understand how the sign of a vector relates to its
direction
 Solve simple geometric problems in 2D using vector
methods
 Prove that points are collinear in a vector diagram
Fluency in Literacy
Column vector: a collection of numbers, as the
components of a vector, written vertically.
Vectors: a quantity having direction as well as
magnitude, especially as determining the position
of one point in space relative to another.
Magnitude: refers to size or quantity.
Scalar: a physical quantity that only has magnitude
and no other characteristics.
Collinear: Three or more points are said to be
collinear if they lie on a single straight line.
Scale factor: The ratio of any two corresponding
lengths in two similar geometric figures
RESOURCES/ACTIVITIES/ICT
ACTIVITIES
ICT: www.mymaths.co.uk for homework activity or
revision
www.teachmathematics.net for software,
resources and problem
solving episodes
http://mathsapp.pixl.org.uk/PMA.swf for
examination therapy videos
Resources:
MathsWatch clip: 180
RESOURCES:
AQA Higher:
Chapter 32
Higher Collins
Chapter 25
Higher Smith
Chapter 29
Be aware of…..
●
Emphasis on showing all the working out.
●
Co-ordinates
●
Poor arithmetic and showing few working out.
●
Mistakes made by student’s inability to carry out simple
algebraic fraction manipulations efficiently.
Recommended curricular assessment materials :
..\Year 11 Assessment Resources\Geometry and
Measures\AQA Topic Tests\Vectors - Topic test 1 H
v1.1.doc
..\Year 11 Assessment Resources\Geometry and
Measures\AQA Topic Tests\Vectors - Topic test 1 H Mark Scheme v1.1.doc