How can we apply geometric rules when solving problems and proving it works?

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Maurice Paul
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1 Year 11 senior programme Learning cycle 2 The KING S Medium Term Plan Mathematics 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

2 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 with, exponential functions for positive values of, and the trigonometric functions (with arguments in degrees) for angles of any size.

3 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 plus 1hr homework Line of enquiry: Why are circles so important? 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 ü Identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment; define vocabulary such as subtended. ü Identify the double angle theorem to solve simple problems ü Identify the right angle in a semi-circle theorem ü Identify angles that are equal by locating ones subtended between two points on the circumference ü As Foundation plus: ü Understand how to complete algebraic proof questions. ü Apply the first 3 theorems to more complex diagrams.

4 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 ü Recall circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment; define vocabulary such as subtended. ü Identify and measure the single tangent theorem ü Identify and understand the alternate segment theorem ü Calculate angles using the quadrilateral cyclic theorem ü 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) ü As Foundation plus: ü Understand how to complete GCSE algebraic proof questions. ü Apply the first all theorems to more complex diagrams. Hypothesis 3 and 4 Tangents are equal in length when on the same circle Learning intention: Apply the parts of a circle to circle theorems 7 and 8 ü Recall circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment; define vocabulary such as subtended. ü Identify and use the 2 tangent theorem ü Identify and understand the chord bisector theorem ü 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) ü As Foundation plus: ü Understand how to complete GCSE algebraic proof questions. ü Apply the first all theorems to more complex diagrams. ü Apply their knowledge of circle theorems to GCSE questions. Home learning: Given Monday of each week and due in by Monday the following week.

5 Week 2 - Higher plus 1hr homework Line of enquiry: Why is proof necessary when we are told the rules? the baseline assessment. Hypothesis 1 and 2 Even numbers are always a multiple of 6. Learning intention: Use your knowledge of algebra to ratify a statement. ü Understand how to express certain characteristics of numbers in algebraic language (E.g. odd numbers, even numbers, square ). ü Analyse how to find key words in the questions to determine what we need to demonstrate. ü Evaluate how to apply your knowledge of algebra to solve GCSE questions. Hypothesis 3 and 4 Factorising quadratics does not help to prove that a number is negative. Learning intention: Being able to demonstrate difficult relationships between algebraic expressions. ü Recall how to represent simple expressions in algebraic language. ü Understand how to solve more difficult proof questions. ü Evaluate how to use your knowledge of algebra to determine if a question is right or not. Week 2 - Foundation plus 1hr homework Home learning: Given Monday of each week and due in by Monday the following week. Line of enquiry: Is there any link between standard form and percentages? the baseline assessment. Hypothesis 1 and 2 Standard form only uses powers of 10 Learning intention: Being able to solve standard form problems.

6 ü Recall how to convert ordinary numbers to standard form and vice versa. ü Understand how to do calculations (+, -, x, /) with numbers in standard form. ü Analyse how to solve GCSE worded problems involving standard form. Hypothesis 3 To calculate a 28% increase, we multiply by Learning intention: Being able to calculate with percentages. ü Recall how to calculate percentages with and without a calculator. ü Analyse how to calculate percentage increase and decrease worded questions. ü Analyse how to calculate percentage change worded questions. ü Analyse how to calculate reverse percentage worded questions. Hypothesis 4 For the same investment, compound interest gives you less money than simple interest. Learning intention: Being able to apply your knowledge of percentages to calculate interest. ü Recall how to calculate percentage increase questions. ü Analyse what the difference are between simple and compound interest. ü Evaluate how to solve GCSE worded questions involving interest. Week 3 plus 1hr homework Home learning: Given Monday of each week and due in by Monday the following week. Line of Enquiry: What is the link between vectors and transformations? the baseline assessment. Hypothesis 1 We can travel in a diagonal direction Learning intention: Apply Pythagoras to calculate the magnitude of vectors ü Define and understand the key terms magnitude and direction ü Describe a vector and the rules applied to them ü Determine the column pair for vectors given

7 ü Understand what vectors can be used to represent (e.g. force, velocity and acceleration) ü Find the magnitude of a vector by applying Pythagors theorem ü As Foundation plus ü Real life application of vectors using Pythagoras ü Exam technique development of vector questions Hypothesis 2 Two vectors result in a diagonal direction Learning intention: Apply the knowledge of directions to do bearings. ü Recall the three rules of bearings. ü Understand how to link the direction of vectors to bearings. ü Analyse how to calculate with bearings. Learning intention: Apply addition and subtraction of vectors ü Real life application of vectors with addition and subtraction ü Understand and use the effect of having a negative sign in front of a vector ü Draw and use vectors ü Exam technique development of vector questions Hypothesis 3 Scalar quantities do not alter direction Learning intention: Apply your knowledge of directions and bearings to problem solving GCSE questions. ü Recall the three rules of bearings. ü Understand how to calculate reverse bearings. ü Evaluate how to solve worded questions without a sketch. Learning intention: Apply multiplication of vectors by a scalar, and diagrammatic and column representation of vectors ü Real life application of vectors with scalars ü Draw and use vectors ü Exam technique development of vector questions

8 MIDTERM to test key knowledge so far on modules covered will be done this week. Home learning: Given Monday of each week and due in Monday the following week. Week 4 plus 1hr homework Line of Enquiry: What are the key differences between SOH CAH TOA and the Sine and Cosine rule? the baseline assessment. This week, improvements will be based on the midterm. 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: Being able to apply Pythagoras Theorem to all type of questions involving algebra and different 2D/3D shapes. ü Recall Pythagoras Theorem. ü Understand how to apply Pythagoras Theorem to problem solving questions. ü Analyse how to apply Pythagoras Theorem to questions that don t have triangles. ü Evaluate how to apply Pythagoras Theorem to 3D questions. Learning intention: Being able to apply SOHCAHTOA to 2D and 3D questions. ü Recall how to use SOHCAHTOA in 2D triangles and other shapes. ü Understand how to get a right angle triangle from a 3D shape. ü Analyse how to use SOHCAHTOA to solve GCSE problem solving questions involving ratio and algebra. Hypothesis 2 and 3 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: Being able to apply SOHCAHTOA to 2D and 3D questions. ü Recall how to use SOHCAHTOA in 2D triangles to find missing sides and angles. ü Understand how to get a right angle triangle from different shapes such as isosceles triangles or rectangles. ü Apply your knowledge of SOHCAHTOA to GCSE questions.

9 ü Evaluate how to use trigonometry to calculate the area of a triangle. Learning intention: : Know and apply the Sine rule and Cosine rule to find unknown lengths ü Recall the sin and cos rules for all triangles (not right angled ones). ü Understand how to use the sin and cos rules to find missing sides and angles. ü Analyse how to use the sin and cos rules in GCSE questions and problem solving. ü Evaluate how to calculate the area of a triangle using trigonometry. Home learning: Given Monday of each week and due in Monday the following week. Week 5 including end of term exam Line of enquiry: Why are some values of Sin, Cos and Tan equal to zero? the baseline assessment. 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 ü Learn and understand the exact trigonometry values above. ü Analyse how the graphs for the cos, sin and tan look like depending on those values. ü Identify each of the trigonometric graphs amongst other graphs. Learning intention: 0, 30 45, 60 and 90 and 0, 30, 45 and 60 ü Understand the values above and using a calculator write them in decimal, integer and surd form and memorise them. ü Understand why they have these particular values using the circumference of radius 1. ü Analyse how to plot the graphs for each of the trigonometric ratios.

10 Hypothesis 2 and 3 Graphs of sin, cos and tan always pass through the origin Learning intention: There are no right angled triangles in a cuboid. ü (This two lessons will be dedicated to revise all trigonometric content from the past two weeks) ü Recall how to use Pythagoras Theorem in 2D shapes and problem solving questions. ü Analyse how to use SOHCAHTOA in 2D shapes. ü Evaluate how to use trigonometry to find the area of a triangle. Learning intention: Sketch and recognise key graphs of sin, cos and tan and other reciprocal graphs. ü Understand the values above and using a calculator write them in decimal, integer and surd form and memorise them. ü Understand why they have these particular values using the circumference of radius 1. ü Analyse how to plot the graphs for each of the trigonometric ratios. ü Evaluate how to translate graphs along the x and y axis. ü Applky their knowledge to GCSE questions. Week 6 Week 7 Home learning: Given Monday of each week and due in Monday the following week. Mock GCSE papers the baseline assessment. This week the results from the Mocks in Week 6 will be analysed and individual work will be produced for students to do in lessons. Gap Analysis Reinforcement Gap Reinforcement in week 7 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 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 REACH activities can be delivered. Extended Learning and useful websites

11 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 2) Lots of maths online help and activities as well as mini tests 3) 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) 5) 6) 7) Corbettmaths.com 8) Vle.mathswatch.com 9) 10) 11)

AGS THE GREAT REVIEW GAME FOR PRE-ALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS

AGS THE GREAT REVIEW GAME FOR PRE-ALGEBRA (CD) CORRELATED TO CALIFORNIA CONTENT STANDARDS 1 CALIFORNIA CONTENT STANDARDS: Chapter 1 ALGEBRA AND WHOLE NUMBERS Algebra and Functions 1.4 Students use algebraic