Download Developing Geometry `How can we apply geometric rules when

The KING’S Medium Term Plan – Mathematics
Year 11 senior programme – Learning cycle 2
Module
Developing Geometry
‘How can we apply geometric rules when solving problems and proving it works?’
Lines of
Enquiry
Progress
Objectives
Week 1 (A): Why are circles so important?
Week 2 (B): Why is proof necessary when we are told the rules?
Week 3 (A): What is the link between vectors and transformations?
Week 4 (B): What are the key differences between SOH CAH TOA and the Sine and Cosine rule?
Week 5 (A): Why are some values of Sin, Cos and Tan equal to zero?
Week 6 (B)-7 (A): Assessment followed by gap teaching – from assessment analysis.
By the end of LC1 in Mathematics SWBAT achieve these AQA objectives: (Objectives underlined form the HIGHER curriculum at GP5+)
Geometry AQA objectives (Weeks 1-2) In this number unit pupils will master the following;
G9

Identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector
and segment (review of Year 9)
G10
Apply and prove the standard circle theorems concerning angles, radii, tangents and chords and use them to prove related results:

angle at centre is equal to twice angle at circumference;

angle in a semi-circle is 90°;

angles in the same segment are equal;

opposite angles in a cyclic quadrilateral sum to 180°;

tangent at any point on a circle is perpendicular to the radius at that point

tangents from an external point are equal in length;

the perpendicular from the centre to a chord bisects the chord;

alternate segment theorem
Geometry AQA objectives (Week 3)
In this number unit pupils will master the following;
G12

Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representation
of vectors
Geometry AQA objectives (Week 4)
In this number unit pupils will master the following;
R12 G22
Compare lengths using ratio notation (Review of Year 10 - 3 year route); Make links to trigonometric ratios

Know and apply the Sine rule
and Cosine rule
to find unknown lengths and
angles
G23
Know and apply
to calculate the area, sides or angles of any triangle
AQA objectives – algebraic representation through graphing (Week 5)
During this week the pupils will do a variety of problem solving tasks using proofs from a range of areas;
G21

Know the exact values of:
0°, 30° 45°, 60° and 90°

Know the exact value of:
0°, 30°, 45° and 60°
A12

Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions and the reciprocal
function
degrees)
with
, exponential functions
for positive values of
for angles of any size.
, and the trigonometric functions (with arguments in
IMPORTANT INFORMATION AND WEEKLY NEEDS
Personalised
Learning and
Reach work
and Mastery
Maths in real
life
Planning for
Feedback
REACH and
Support
The AQA objectives above cover a wide range of mathematical skills and applications at varying levels of difficulty.
Each practitioner has access to sets of exam based questions and activities that are aimed at these different levels of
application and will ensure that all pupils are provided with work that will both challenge and support them at their targeted
Grade Point as well as pushing them towards the next. All pupils will meet the progress objectives outlined above at a pace
that suits them and will be delivered in a way that is personalised to how they learn. The use of iPads will be planned for
carefully so that they can maximise learning.
Each week, there will be discussion and slides planned in so that pupils can value the relevance of what they are learning,
which areas of life or careers that skill may be useful to and lessons will, as much as possible, contain resources where maths
has to be applied to real world problems in order to find solutions. Percentages for instance, will be applied to calculating
interest, value comparisons and rate of change.
Pupils will receive written feedback each week in the form of teacher marking, peer/self-assessment and small quizzes to check
key knowledge. Mark schemes will be provided where appropriate for pupil self-assessment and development. REACH lessons
each week will allow time for acting on feedback and making improvements to their work in order to develop further and fill in
GAPs. To extend mastery in learning REACH lessons will now also include extension problem solving tasks.
Each week there will opportunities for support with in class intervention, group intervention and after school catch-up.
MEDIUM TERM PLAN
Week 1
Line of enquiry: Why are circles so important?
4 1hr lessons
plus 1hr
homework
Hypothesis 1 – You cannot make a right angle using a circle
Learning intention: Apply the parts of a circle to the first 3 circle theorems
Foundation:
Identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and
segment; define vocabulary such as subtended.
Identify the double angle theorem to solve simple problems
Identify the right angle in a semi-circle theorem
Identify angles that are equal by locating ones subtended between two points on the circumference
Higher:
As Foundation plus
Algebraic proof and application questions of the first 3 theorems. Pupils will experience more complex diagrams using the 3 theorems.
Hypothesis 2 – The properties of triangles and quadrilaterals are useful when working with circle theorems (This will last 2
lessons to include mastery and development time)
Learning intention: Apply the parts of a circle to circle theorems 4, 5 and 6
Foundation:
Recall circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment; define
vocabulary such as subtended.
Identify and measure the single tangent theorem
Identify and understand the alternate segment theorem
Calculate angles using the quadrilateral cyclic theorem
Begin to find missing angles in a variety of situations and explain using technical language why angles are the size they are (justification
of their calculations)
Higher:
As Foundation plus
Algebraic proof and application questions of theorems 4, 5 and 6. Pupils will experience more complex diagrams using the 3 theorems.
Hypothesis 3 – Tangents are equal in length when on the same circle
Learning intention: Apply the parts of a circle to circle theorems 7 and 8
Foundation:
Identify and use the 2 tangent theorem
Identify and understand the chord bisector theorem
Continue to find missing angles in a variety of situations and explain using technical language why angles are the size they are
(justification of their calculations)
Higher:
As Foundation plus
Algebraic proof and application questions of theorems 7 and 8. Pupils will experience more complex diagrams using the 2 theorems.
Home learning: Given Tuesday of each week and due in by Tuesday the following week.
Learn how to recognise each theorem ready for a knowledge check test on MathsWatch next week.
Week 2
Line of Enquiry: Why are proofs necessary when we are told the rules?
Lesson 1 – Pupils will work through GCSE questions on circle theorems for mastery in the topic.
Pupils will use their revision guides in this lesson as we want to develop revision skills and exam technique within this LC.
4 1hr lessons
plus 1hr
homework
Lesson 2 – Pupils take the circle theorems test (MathsWatch) then improve their knowledge through mini gap resources.
Lessons 3 and 4 – mastery and further development
Those at Foundation and lower end of Higher will practice more exam style applications of the 8 theorems with support.
Those at Higher will use these 2 lessons to develop their understanding of Proofs.
Weekly line of enquiry and hypotheses discussions will take place in detail in lesson 4 to pull the unit together.
Home learning: Given Tuesday of each week and due in by Tuesday of the following week.
Week 3
4 1hr lessons
plus 1hr
homework
Personalised exam questions using ‘MathsWatch’ and the revision guides and workbooks.
Line of Enquiry: What is the link between vectors and transformations?
Hypothesis 1 – We can travel in a diagonal direction
Learning intention: Apply Pythagoras to calculate the magnitude of vectors
Foundation/Low Higher:
Define and understand the key terms ‘magnitude’ and ‘direction’
Describe a vector and the rules applied to them
Determine the column pair for vectors given
Understand what vectors can be used to represent (e.g. force, velocity and acceleration)
Find the magnitude of a vector by applying Pythagors’ theorem
Higher:
As Foundation plus
Real life application of vectors using Pythagoras
Exam technique development of vector questions
Hypothesis 2 – Two vectors result in a diagonal direction
Learning intention: Apply addition and subtraction of vectors
Foundation/Low Higher:
Recall the key terms ‘magnitude’ and ‘direction’
Determine the column pair for vectors given
Understand and use the nose-to-tail method to find a resultant vector and adding the column pairs
Higher:
As Foundation plus
Real life application of vectors with addition and subtraction
Understand and use the effect of having a negative sign in front of a vector
Draw and use vectors
Exam technique development of vector questions
Hypothesis 3 – Scalar quantities do not alter direction
Learning intention: Apply multiplication of vectors by a scalar, and diagrammatic and column representation of vectors
Top Foundation/Low Higher:
Recall the key terms ‘magnitude’ and ‘direction’ and ’scalar’
Determine the column pair for vectors given
Understand that applying a scalar alters magnitude of a vector but not direction
Higher:
As Foundation plus
Real life application of vectors with scalars
Draw and use vectors
Exam technique development of vector questions
Lesson 4: MIDTERM to test key knowledge so far on geometry modules covered.
Home learning: Given Tuesday of each week and due in Tuesday the following week.
Week 4
4 1hr lessons
plus 1hr
homework
Personalised exam questions using ‘MathsWatch’ and revision guides.
Line of Enquiry: What are the key differences between SOH CAH TOA and the Sine and Cosine rule?
Hypothesis 1 – Higher: Only one ratio will work for any given trigonometry problem (This will need 2 lessons)
Foundation/Higher: The position of an angle is more useful than the position of the hypotenuse to solve problems
Learning intention: Compare lengths using ratio notation and calculate the missing side of a right angled triangle
Foundation
Pupils who are working at the lower end of Foundation will use this week to practice and master applications of Pythagoras’ Theorem.
Foundation/Higher
Pupils will be required to learn/revise the 3 ratios.
Pupils will understand how to label the sides of right angled triangles based on the position of an interior angle. They will learn by heart a
rhyme to help them remember the 3 ratios.
Pupils will begin to use the ratios to calculate the opposite and adjacent sides of right angled triangles when given an angle and the
hypotenuse.
Higher
Pupils will recall how to apply the 3 ratios to calculate any missing side of right angled triangles given an angle and a side length. They
will recall how to calculate a missing angle in right angled triangles given 2 of the sides.
Pupils will apply trigonometric ratios to a variety of problems, including those with 2 triangles placed together or real life situations.
Hypothesis 2 – Higher: Trigonometry is used for cases involving right angled triangles
Foundation/Higher: Only one ratio can be used at a time to solve problems with trigonometry
Learning intention: Know and apply the Sine rule
lengths
and Cosine rule
to find unknown
Foundation/Higher
Pupils will continue to work on the 3 ratios with right angled triangles. They will continue calculating the opposite and adjacent sides in
real life situations.
Pupils will then apply the ratios to calculate the hypotenuse of right angled triangles in different situations.
Higher
Pupils will calculate the missing sides of non-right-angled triangles using the sine and cosine rules. They will apply this to real life
problems and exam questions.
Hypothesis 3 – Higher: The position of 2 angles will determine the rule we use when calculating a missing angle
Foundation/Higher: We always need the position of the non-right-angle angle and 2 sides when calculating the missing angle
Learning intention: Know and apply the Sine rule
angles
and Cosine rule
to find unknown
Foundation/Higher
Pupils will continue to work on the 3 ratios with right angled triangles. They will apply the ratios to calculate the missing interior angles of
right angled triangles in different situations, given 2 of the side lengths.
Higher
Pupils will calculate the missing interior angles of non-right-angled triangles using the sine and cosine rules. They will apply this to real
life problems and exam questions.
Home learning: Given Tuesday each week and due in by Tuesday the following week.
Week 5
Personalised exam questions using ‘MathsWatch’ and revision guides. REVISION.
Line of enquiry: Why are some values of Sin, Cos and Tan equal to zero?
4 1hr lessons
including end
of term exam
Pupils doing the Higher paper with targets of 7 or more, will participate in the lessons below. Pupils who are Foundation/Higher
that are sitting the Higher paper with targets of 5 and 6 will continue with week 4 learning intentions to gain further mastery in
trigonometric applications.
Hypothesis 1 – All the values of sin, cos and tan are between 0 and 1

Learning intention: Know the exact values of:
0°, 30° 45°, 60° and 90° and
0°, 30°, 45° and 60°
Higher only
Pupils will calculate the values above using a calculator and write in decimal, integer and surd form.
Pupils will begin to understand why they have these particular values. For example sin30o is always ½ and Pythagoras can be used to
demonstrate why for 30o triangles.
Hypothesis 2 – Graphs of sin, cos and tan always pass through the origin
Learning intention: Sketch and recognise key graphs of sin, cos and tan and other reciprocal graphs.
Higher only
Pupils will plot the graphs of y=sinx, y=cosx and y=tanx using co=ordinate grids.
Pupils will learn to recognise each graph by looking at their shape and the points on the x-axis they pass through in cycle.
At REACH level pupils will learn how to manipulate and transform graphs of sin, cos and tan.
The second half of the week will be saved for beginning the Mock exams.
Week 6
Lessons will be for the end of LC mock exams and beginning of GAP (Calc and non-calc Higher or Foundation will be sat)
Gap Analysis Reinforcement
Gap
As seen in the lesson activities each week, gap teaching will not just be at the end of the LC after exam analysis has taken place. Gap
Reinforcement teaching is an integral part to each unit of work and will consist of summary sheets, mini-tests and tasks where gaps can be filled and
in week 7
REACH activities can be delivered.
Extended Learning and useful websites
Extended learning will in a variety of forms. During home learning pupils may be asked to use the following sites where they complete
quick quizzes, CIMT tasks, GCSE style questions and more open ended problem solving tasks.
1) Levelled quizzes http://www.educationquizzes.com/ks3/maths/
2) Lots of maths online help and activities – as well as mini tests http://www.bbc.co.uk/schools/websites/11_16/site/maths.shtml
3) http://uk.ixl.com/math/year-7
This link is useful for additional revision and practice on all areas of maths. For LC1 pupils should click on the Geometry areas for
practice questions.
4) www.onlinemathlearning.com
5) https://corbettmaths.com/more/gcse_practice_papers/
6) www.studymaths.co.uk
7) Corbettmaths.com
8) Vle.mathswatch.com
9) http://www.transum.org/
10) www.mathsisfun.com
11) http://www.mathsgenie.co.uk/gcse.html