The KING’S Medium Term Plan – Mathematics

Year 11 senior programme – Learning cycle one

Module Developing Algebra

Overarching Challenging question

‘How can we apply algebra and graphs when solving problems and proving it works?’

Lines of Enquiry

Week 1 (B): Why does having X2 in an equation create a curve in a graph? (Deeper thinking: is there a relationship between and hyperbola and the function 1/x?) Week 2 (A): How do we graph inequalities? Week 3 (B): How can we solve simultaneous equations using graphs? Week 4 (A): How can we prove or disprove using mathematics? (What use are counter examples?) Week 5 (B): How do I understand which skills to apply when problem solving? Week 6 (A)-7 (B): Assessment followed by gap teaching – from assessment analysis.

Progress Objectives

By the end of LC1 in Mathematics SWBAT achieve these AQA objectives: (Objectives underlined form the HIGHER curriculum at GP5+) Algebra AQA objectives (Week 1) In this number unit pupils will master the following;

A17 Ø Solve linear equations in one unknown algebraically including those with the unknown on both sides of the equation Ø Find approximate solutions using a graph

A18 Ø Solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula

Ø Find approximate solutions using a graph

A12 Ø Recognise, sketch and interpret graphs of linear and quadratic functions

A11 Ø Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square (including the symmetrical property of a quadratic)

A21 Ø Translate simple situations or procedures into algebraic expressions or formulae; derive an equation and the solve the equation and interpret the solution

Algebra AQA objectives (Week 2) In this number unit pupils will master the following;

A22 Ø Solve linear inequalities in one variable Ø Represent the solution set on a number line - know the conventions of an open circle on a number line for a strict

inequality and a closed circle for an included boundary Ø Graph the regions of inequalities, solve quadratic inequalities

Algebra AQA objectives (Week 3) In this number unit pupils will master the following;

A19 Ø Solve two simultaneous equations in two variables (linear / linear or quadratic/linear) algebraically Ø Find approximate solutions using a graph including the approximate solution of a quadratic equation by drawing a

straight line to intersect with another quadratic equation

A21 Ø Translate simple situations or procedures into algebraic expressions or formulae; Derive two simultaneous equations Ø Solve the equations and interpret the solution (including the solution of geometrical problems and problems set in

context)

AQA objectives – algebraic and geometric proof (Week 4-5) During this week the pupils will do a variety of problem solving tasks using proofs from a range of areas; G

6 Ø Apply angle facts, triangle congruence, Pythagoras’ Theorem, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides including the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs

A6 Ø Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

Assessment at the end of week 6 will be against the above AQA objectives following on from 2 lessons of REACH and revision. Gap teaching from analysis of assessments will take place in week 7.

AQA objectives – problem solving (Week 5-6) During this time pupils will apply a range of skills to worded problems, real life applications and specific exam problems.

IMPORTANT INFORMATION AND WEEKLY NEEDS

Personalised Learning and Reach work and Mastery

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.

Maths in real life

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.

Planning for Feedback

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.

REACH and Support

Each week there will opportunities for support with in class intervention, group intervention and after school catch-up.

MEDIUM TERM PLAN OVERVIEW

Week 1

4 1hr lessons plus 1hr homework

Line of enquiry: Why does having X2 in an equation create a curve in a graph? (Deeper thinking: is there a relationship between and hyperbola and the function 1/x?) Hypothesis 1 – X has a positive value when the coefficients a, b and c are positive Learning intention: Solve linear and quadratic equations using a range of methods GP 5 = Recall how to solve linear 2 step equations and equations with an unknown on both sides. GP 6 = Understand how to solve quadratic equations by factorising. GP 7 = Evaluate how to apply the quadratic formula to solve equations. Hypothesis 2 – A negative coefficient of X at the start of a quadratic greatly effects the graph of that equation Learning intention: Recognise functions, sketch the graphs of quadratic and cubic functions and find simple solutions (interpret) GP 4 – 5 = Recall how to complete co-ordinate grids for quadratic and cubic graphs then plot onto a set of axes. GP 6 – 7 = Understand how to plot functions without a co-ordinate grid. Solve quadratic equations by plotting the graph for solutions of X. Explain the effect of altering the coefficients on the curve. Hypothesis 3 – All graphs pass through the origin on a set of axes Learning intention: Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square (including the symmetrical property of a quadratic). Transform functions and plot inverse functions. GP 7 = Understand how to plot inverse functions and parabolas. GP 8 = Evaluate what happens and why in the graph y=1/x. Explain the effect of negative value of x in quadratic and cubic functions. (This lesson, those at a LGP or sitting Foundation will continue work on the above outcomes.)

Week 2

4 1hr lessons plus 1hr homework

Line of Enquiry: How do we graph inequalities? (Deeper thinking: Why is it important to know our limits?) Lesson1: REACH (Review & Recap, Evaluate & Endeavour, Attainment & Achievement, Challenge yourself, Hone your skills) Pupils will use this lesson to engage with and respond to work marked and done the previous week. They will read through teacher comments and respond by following a given set of criteria. This will allow them to make improvements on their work, carry out corrections, seek help and make further progress before moving to the next unit of work. Hypothesis 1 – An empty circle should be used for numbers that are not inclusive Learning intention: Solve linear inequalities and represent them on a number line.

GP 4 = Recall how to solve simple one step inequalities and how to draw simple inequalities (solutions) on a number line. GP 5 = Solve 2 step inequalities. Draw inequalities on a line within 2 parameters (e.g. x>3 and x<10, -3< x < 4). Hypothesis 2 – It is impossible to graph inequalities that have no link e.g. x<4 and x>7 Learning intention: Represent inequalities on a graph and locate regions that satisfy 2 or more inequalities. GP 5 – 6 = Recall sketching straight line graphs in order to draw inequalities on a set of axes. GP 7 – 8 = Graph inequalities in order to define a region. GP 8+ = Solve and graph quadratic inequalities. Lesson 3 – GCSE questions, worded problems and real life applications of inequalities. Learning intention: Convert worded problems into inequalities and represent areas on graphs ALL GP = Pupils need to be able to apply their knowledge at different levels in relation to the exam specification. Questions from AQA specimen papers will be worked through on this topic. LGP = e.g. Vanessa has at most $5, represent this as an inequality on a number line. MGP = e.g. Kelly works for Quick Oil Change. If customers have to wait longer than 20 minutes for the oil change, the company does not charge for the service. The fastest oil change that Kelly has ever done took 6 minutes. Show the possible customer wait times in which the company charges the customer. HGP = e.g. Write an inequality to represent each situation, then graph the solution:

Such as; 1. Blayton is at most 2 meters above sea level. 2. Edith must read for a minimum of 20 minutes.

LEARNING CYCLE ONE MIDTERM TO ALSO BE DONE THIS WEEK (in lesson 4)

Week 3

4 1hr lessons plus 1hr homework

Line of Enquiry: How can we solve simultaneous equations using graphs? Lesson1: REACH (Review & Recap, Evaluate & Endeavour, Attainment & Achievement, Challenge yourself, Hone your skills) Pupils will use this lesson to engage with and respond to work marked and done the previous week. They will read through teacher comments and respond by following a given set of criteria. This will allow them to make improvements on their work, carry out corrections, seek help and make further progress before moving to the next unit of work. Pupils will use this lesson to learn from the mistakes made in their LC1 midterm. They will be given tasks that support further practice and development to fill in their knowledge gaps. Hypothesis 1 – Substitution and numeracy skills are the basis of all simultaneous equations Learning intention: Solve sets of simultaneous equations GP 4 – 5 = Understand how to find solutions for X and Y methodically by solving linear simultaneous equations with integer coefficients. GP 6 = Evaluate how to solve more complex simultaneous equations with fractional coefficients or difficult solution values. GP 7 = Create sets of simultaneous equations from worded problems and solve. GP 8 = Understand how to solve sets of quadratic simultaneous equations. Hypothesis 2 – (This will last for 2 lessons and will involve solving simultaneous equations with graphs and real life applications) You can only solve simultaneous equations by graphing them if they are linear. Learning intention: Find the values of X and Y in sets of simultaneous equations through graphs. GP 4 – 5 = Pupils will continue with the outcomes above. GP 6 = Recall how to plot the graphs of equations. Apply this in order to plot the graphs of simultaneous linear equations to find the solutions of X and Y. GP 7 – 8 = Evaluate how to graph quadratic simultaneous equations to find the solutions of X and Y. Home learning: Given Tuesday of each week and due in Tuesday the following week.

This week pupils will complete a worksheet graphing and interpreting functions dependent upon their KGP.

Week 4 Line of Enquiry: How can we prove or disprove using mathematics? (What use are counter examples?) Lesson1: REACH (Review & Recap, Evaluate & Endeavour, Attainment & Achievement, Challenge yourself, Hone your skills)

4 1hr lessons plus 1hr homework

Pupils will use this lesson to engage with and respond to work marked and done the previous week. They will read through teacher comments and respond by following a given set of criteria. This will allow them to make improvements on their work, carry out corrections, seek help and make further progress before moving to the next unit of work. To extend their knowledge of the content from week 2 pupils will be given examination type problems from the new GCSE specification at and above their targeted grade point to stretch their comprehension. Real life application and problem solving will be provided as REACH work linked to their feedback. Hypothesis 1 – (This will take 2 lessons to cover the content within the AQA objective) You can prove that triangles are congruent without mathematical reasoning. Learning intention: Apply angle facts, triangle congruence, Pythagoras’ Theorem, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides including the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs. GP 3 – 4 = Recall shape properties important to quadrilaterals and triangles. Describe with examples the term congruent and similar. Prove algebraically that angles in quadrilaterals add up to 360 degrees. GP 5 = Understand how to apply Pythagoras’ Theorem to prove that a triangle is right-angled. Evaluate by applying algebra if 2 triangles are similar/congruent. GP 6 – 7 = Apply trigonometry to evaluate similarity or congruence in triangles.

Hypothesis 2 – (This will take 2 lessons to cover the content within the AQA objective, which will take this into week 5 lessons) Algebra can be used to prove what type of number we have. Learning intention: Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs GP 3 = Recall types of number, particularly prime, odd and even. Examples;

1. Amy says that for every whole number, 6n – 1 is always a prime number.

Give a counter example to show that she is wrong.

2. Prove that the answer to every line of the pattern below is 6

3 × 4 - 1 × 6

4 × 5 - 2 × 7

5 × 6 - 3 × 8

6 × 7 - 4 × 9

3. Below is Joe’s attempt at forming a proof that 1 = 2

Let a = b

Þ ab = b 2 (multiply by b)

Þ ab - a 2 = b 2 - a 2 (subtract a 2)

Þ a(b - a) = (b - a)(b + a) (factorise)

Þ a = b + a (divide by b - a)

Þ a = a + a (a = b)

Þ a = 2a (a + a = 2a)

Þ 1 = 2 (divide by a)

Which step in Joe’s argument is not allowed?

Home learning: Given Tuesday each week and due in by Tuesday the following week.

This week pupils will have a graded revision booklet. This will be utilised in GAP week as well as help prepare them for the exam.

Week 5

4 1hr lessons including end of term exam

Line of enquiry: How do I understand which skills to apply when problem solving? (4 hours) http://www.greatmathsteachingideas.com/wp-content/uploads/2015/06/AQA-90-Problem-solving-questions.pdf Over 3 lessons (week 4 proof work will be continued into this week as well) pupils will work through problem solving questions linked to the graphs, shapes and algebra work seen this LC and within the proof content. Example of tasks are shown below under each hypothesis. Lesson1: REACH (Review & Recap, Evaluate & Endeavour, Attainment & Achievement, Challenge yourself, Hone your skills) Pupils will use this lesson to engage with and respond to work marked and done the previous week. They will read through teacher comments and respond by following a given set of criteria. This will allow them to make improvements on their work, carry out corrections, seek help and make further progress before moving to the next unit of work. Hypothesis 1 – (algebra problem solving) To utilise algebra is to abbreviate long sentences

Learning intention: Use smaller numbers and algebraic expressions to solve problems such as:

Hypothesis 2 – (geometry problem solving) A pair of axes does not help when solving geometrical problems Learning intention: Apply geometric sketching to solve problems such as:

Hypothesis 3 – (graphs and functions) Graphing situations is paramount to the financial sector Learning intention: To apply a variety of functions to model real life situations such as:

Week 6

4 1hr lessons including end of term exam.

Line of enquiry: Continued from week 5 (problem solving Hypothesis – continued from week 5 (Proof work) Lesson 2, 3 and 4 will be for the end of LC exam mock and beginning of GAP (Calc and non-calc broken into 3 smaller papers)

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

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