How can real life problems be solved using chance and an awareness of shape and space?

Transcription

1 The KING S Medium Term Plan Mathematics Y9 LC4 Programme Module Building on prior learning Overarching subject Challenge question Lines of Enquiry Area, perimeter and volume. Probability. Pupils will cover all objectives, but the depth at which they will study the objectives and each hypothesis is dependent on their end of year target grade point. Key objectives are taken from the AQA GCSE Specification for the new curriculum. In LC1, 2 and 3 the pupils developed skills working with number, algebra, shape, angles, FDP and ratio, shapes. This LC pupils will need to recall knowledge of shape properties to apply it to area, perimeter and volume and at a higher level apply algebraic expressions for higher level problem solving. Pupils will need to recall angles and shapes to develop Pythagoras skills. Pupils will need to recall knowledge of FDP in order to find probabilities of events. Number skills, worded problems and methods of working will support pupil progress in these areas. How can real life problems be solved using chance and an awareness of shape and space? Week 1: How can we use our knowledge of shapes in the real world? Week 2: How can I apply Pi and area to calculate volume of objects in the real world? Week 3: Is there a real life purpose for using and applying Pythagoras Theorem? Week 4: How can maths be used to predict chance? Week 5: How can fractions be used to calculate probabilities? Week 6-7: Revision and assessment followed by gap teaching from assessment analysis. Progress Topic Statement By the end of LC4 in Mathematics SWBAT: Geometry Shape and reasoning (AQA objectives G9, 12, 16, 17, 20 supported by A1, A3, A4) Weeks 1-3 In these unit pupils will study the properties of 2D and 3D shapes and apply them to problems. They will develop their understanding of area, perimeter and volume from basic shapes revision up to more difficult compound shapes. They will apply some algebra at varying levels to area and perimeter through writing expressions or creating simple equations to solve problems.

2 Pupils will develop a better understanding of how to work with circles, particularly in real life contexts and calculate volumes of a variety of shapes and objects. Pupils will also develop more applications of Pythagoras Theorem, using it to calculate the length of any side of a right angles triangle and see where this is useful in the real world. Proofs will be learned with regards to Pythagoras and certain shape properties. Probability (AQA objectives P1-8) Weeks 4-5 In this unit pupils will recall how to use a probability scale, how mutually exclusive events are calculated and the use of key vocabulary to describe the likeliness of events. Pupils will then develop their understanding further, using sample space diagrams and probability trees to identify outcomes and calculate probabilities of a variety of events and scenarios. Pupils will analyse the differences between theoretical probability and experimental probability and begin to use relative frequency to predict future probabilities. The application of fractions from LC3 will be very useful to pupils. In week 3 (lesson 3/4) there will be a mid LC assessment to check current progress. Assessment at the end of week 6 will be against the above AQA objectives. Gap teaching from analysis of assessments will take place in week 7. Week 1 4 hours of lessons plus 1 hour of homework each week Line of Enquiry: How can we use our knowledge of shapes in the real world? How each week looks; Ø Hypotheses for the week s lessons; These will act as the title for the lessons, in which the work done will be reflected upon to either prove or disprove each hypothesis. It may be that 1 hypothesis can last more than 1 lesson yet others are achieved quickly. This depends upon how far the pupils move on from the knowledge section and get through the different success criteria within the main body of the lesson. All hypotheses should be answered to some degree over the course of the week. Ø Learning Intentions: These are the key objectives laid out by the exam board. Ø Weekly success criteria for completion across 4 lessons; This is where after teaching the knowledge necessary the pupils will work at their grade point on exam questions in order to achieve the learning intention.

3 Hypothesis 1 2D properties are the same as 3D properties of shape Ø Identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres : GP 2 = Delivery to recall names of 2D and 3D shapes GP 3 = Pupils will use the properties of shapes to identify symmetry GP 4 = Pupils will apply their knowledge of symmetry to identify rotational symmetry, NETs and planes of symmetry in 3D shapes GP 5 = Pupils will apply knowledge of shape properties to visualise 2D representations of 3D shapes GP 2 =Quick questions where pupils will match up names to the images of shapes. Care will be taken with spelling. GP 3 = Pupils will both identify and draw lines of symmetry and explain why some shapes do not have them. They will compare regular and irregular shapes through investigating symmetry. GP 4 = Pupils will demonstrate how to find rotational symmetry of both regular and irregular polygons. They will match up a 3D object with its NET. Pupils will have to be able to draw and shade in the plane of symmetry on 3D shapes and explain why we cannot have a line of symmetry like we do for 2D shapes. GP 5 = Pupils will recall NETS and apply this to help them identify viewpoints of a variety of 3D shapes/objects. They will draw the 3 key viewpoints from drawings of 3D images using 2D shapes on squared paper. Additional challenge will be to apply scales and draw them accurately to the correct size. Hypothesis 2 Lengths and widths are only used for area of rectangles Ø Calculate the perimeter of a 2D shape and composite shapes : GP 3 = Pupils will recall how to find perimeters of simple shapes GP 4 = Pupils will find the perimeters of compound shapes GP 5 = Pupils will recall how to simplify algebraic terms and apply it to perimeter

4 GP 3 = Pupils will complete sets of questions finding the perimeters of shapes, being sure to write down the measurement of each side. GP 4 = Pupils will complete sets of questions and real life problems finding the perimeter of irregular and compound shapes, but must spot missing sides and work out their lengths first. GP 5 = Pupils will solve problems finding the perimeter by first writing an expression or equation for the perimeter then getting the solution. Pupils may need to apply skills using rules of algebra and inverses. Hypothesis 3 There are no formulas for finding the area of irregular shapes Ø Calculate the area of composite shapes : GP 3 = Pupils will recall how to find the area of shapes using known formulae GP 4 = Pupils will understand how to distinguish between area and perimeter. Pupils will calculate the areas of compound shapes made from rectangles GP 5 = Pupils will understand how to find the area of parallelograms, trapeziums and varieties of triangles. Pupils will calculate areas of more complex compound shapes GP 3 = Quick questions findings areas of rectangles, squares and right triangles using both the formulae and by counting squares. GP 4 = Pupils will apply knowledge of area of rectangles to find the area of compound shapes. GP 4 = Pupils will work through questions calculating the area of a variety of quadrilaterals by applying formulae. GP 5 = Pupils will learn how to prove and deduce the formula for area of trapeziums, parallelograms, kites and triangles. GP 5 = Pupils will apply knowledge to calculate more complex shapes made from different quadrilaterals such as compound shapes made from 3 or more shapes, and calculate shaded areas by applying subtraction, for instance, when a section of a shape has been cut away, what area is left? Hypothesis 4 Area is useful when calculating volume of 3D shapes Ø Know and apply formulae to calculate the volume of cuboids and other right prisms : GP 2 = The class will recall and discuss the meaning of volume, dimensions of shapes and finding volum3 by counting squares

5 GP 3 = Understand how and why we use lxwxh for volume of cubes and cuboids GP 4 = Demonstrate how to find the volume of a triangular prism GP 5 = Demonstrate formulae needed to calculate volume of prisms GP 2 = Pupils will find the volume of a variety of 3D shapes by counting cubes. Pupils will use multilink to make shapes with certain volumes to develop understanding. GP 3 = Pupils will calculate the volume of cubes and cuboids and then reverse the method and apply inverses to find missing lengths given the volume. E.g. if the volume of a cube is 8cm cubed, what is its length? GP 4 = Pupils will answer questions applying knowledge of cuboids and the correct formula to calculate the volume of different triangular prisms (both right angles and non-right angled triangular cross-sections). GP 5 = Pupils will work through a set of problems where they will need to deduce why the formula works for volume of prisms. They will need be able to explain why shapes are prisms. They will work through a variety of problems calculating volume of prisms. GP 6 = Pupils will apply all their knowledge of area and volume to calculate the volume of compound 3D shapes/real life objects. Home learning: Given each Thursday, due in by the following Thursday each week. Week 2 4 1hr lessons plus 1hr homework For the first home learning tasks, pupils will do an online assignment using the Stuck for schools website. Questions will be based around all work done this week and will last for 1 hour. Videos and hints are available to support development. Pupils will get instant feedback on their progress and have the opportunity to improve scores. This work can be set up to suit their target GP but also challenge them to go beyond. Line of enquiry; How can I apply Pi and area to calculate volume of objects in the real world? Lesson 1: R.E.A.C.H Session and COGNITIVE ACCELERATION TASK Pupils will use this lesson to engage with and respond to work marked and done in week one. 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 1 pupils will be given examination type problems from the new GCSE specification at and above their targeted grade point to stretch their comprehension. Stuck for schools will also be used as REACH activities and can be personalised in the level of difficulty. Prior learning will be ascertained in this lesson on the key names and parts of a circle.

6 COGNITIVE ACCELERATION TASK Pupils will also take part today in a CA task to develop their deep thinking skills. From a square sheet of paper 20 cm by 20 cm, we can make a box without a lid. We do this by cutting a square from each corner and folding up the flaps. Will you get the same volume irrespective of the size of the squares that are cut out? Investigate what volumes are possible for different sizes of cut-out squares. What is the maximum possible volume and what size cut produces it? Try different sized square sheets of paper. Can you find a relationship between the size of paper and the size of cut that produces the maximum volume?

7 Hypothesis 1: I need to use the radius to calculate area and circumference Learning Intention: Ø Identify and apply circle definitions and properties, including centre, radius, chord, diameter, circumference, tangent, arc, sector and segment Ø Know the formulae needed to find the circumference and area of circles GP 2 = A quick recall of naming parts of a circle GP 3 = Demonstrate the use of the formula for calculating the circumference of circles GP 4 = Demonstrate the use of the formula for calculating the area of circles GP 2 = Pupils will be able to write down the parts of a circle on a diagram and define them. GP 3-4 = Pupils will calculate the circumference of circles using both the radius and diameter. GP 4 = Pupils will calculate the area of circles using both the radius and diameter. GP 5-6 = Some pupils may now be able to apply the methods from GP3 4 to calculate the area and perimeter of compound shapes involving circles. Hypothesis 2: There is no relationship between area of circles and volume of cylinders Ø Calculate areas of circles and composite shapes Ø Calculate the volume of a cylinder GP 4 = Recall methods of finding the area of circles GP 5 = Recall how to find the area of prisms GP 4 = Pupils will practice quick area of circle questions. GP 5 = Pupils will apply their knowledge of prisms and circles to evaluate how we find the volume of a cylinder, then complete a variety of problems calculating the volume knowing either the radius or the diameter of the circle. Hypothesis 3: Pi is not useful for compound objects

8 Learning intentions: Ø Calculate the volume of compound shapes involving circular sections For all GP = Recall with pupils how we find the volume of key 3D shapes. (This lesson may not be suitable for certain groups, so more development of previous lessons may be useful). GP 5-6 = Pupils will apply knowledge of area, volume and 2D shape to calculate the volume of compound 3D shapes/real life objects. Pupils may calculate the volume of buildings, towers, boxes etc. They will need to apply previous skills of scale drawings, measuring and shape properties. Home learning: Given every Monday, due in by Friday each week. Home learning this week will be a targeted GP booklet on all work done over the last 2 weeks in preparation for their mid LC mini test in week 3. Week 3 Line of enquiry; Is there a real life purpose for using and applying Pythagoras Theorem? Lesson 1: R.E.A.C.H Session 4 1hr lessons plus 1hr homework Pupils will use this lesson to engage with and respond to work marked and done in week one. 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 1 pupils will be given examination type problems from the new GCSE specification at and above their targeted grade point to stretch their comprehension. Stuck for schools will also be used as REACH activities and can be personalised in the level of difficulty. Prior learning will be ascertained in this lesson about types of triangles and area of squares. Hypothesis 1: The 3 dimensions of a right angled triangle have a relationship Ø Know the formula for Pythagoras' Theorem and apply it to calculate the length of the hypotenuse GP 3 = Recall the area of squares and parts of a right angled triangle GP 4 = Demonstrate how Pythagoras Theorem works (proof) GP 5 = Discuss and use the formula for Pythagoras Theorem

9 GP 3 = Pupils will discuss and do a quick investigation regarding the area of squares on the sides of right angled triangles. They will be able to describe how the area of the squares on the 2 shorter sides add up to the area of the larger square on the hypotenuse. Pupils need to be able to identify and label the sides of the triangle with a, b and c (shorter sides and hypotenuse). GP 4 = Pupils solve a variety of problems using the theorem to calculate the hypotenuse of right-angled triangles. GP 5+ = Can pupils apply the theorem to identify if a triangle is right angled or not? Hypothesis 2: Inverses are not useful when applying Pythagoras Theorem to real life Learning intentions: Ø Apply Pythagoras Theorem to find any length in right angled triangles in two dimensional figures GP 4 = Recall the formula for Pythagoras Theorem and how to find the hypotenuse GP5 = Demonstrate how to calculate the shorter side of right angled triangles using the theorem GP 6 = Demonstrate how to apply the theorem to 3D shapes to find the length of a plane of symmetry in a cube for instance. GP 4 = Pupils will discuss and do a quick task to calculate the hypotenuse of right-angled triangles. GP 5+ = Pupils will look at a set of triangles to determine if they need to use the theorem to find the hypotenuse or one of the shorter sides. Pupils will then solve problems finding the missing sides. GP4-6 = A range of problems where pupils need to apply Pythagoras Theorem to real life problems from simple ladder problems to harder ones with 3D shapes such as 3D Pythagoras problems and bearings (LC3). Lesson 4: Mid-term/LC mini test Pupils will end the week by completing their Mid LC assessment. The purpose of this short test is to check current understanding and progress in the Geometry units. Home learning: Given every Monday, due in by Friday each week. Week 4 4 1hr lessons plus 1hr homework Home learning this week will be to create a revision keynote on their ipad showing applications and questions on a topic from the last 3 weeks priority to be put on questions they did not understand in their Mid LC test. Line of enquiry; How can maths be used to predict chance? Lesson 1: R.E.A.C.H Session and COGNITIVE ACCELERATION TASK During this time today, pupils will reflect on their mini test and work in groups to go through corrections, practice questions that will help them to improve their score and develop a better understanding of questions they have struggled with. Pupils will also use their strengths to support and teach their peers. Prior learning of the vocabulary used in describing chance will be assessed.

10 COGNITIVE ACCELERATION TASK Pupils will also take part today in a CA task to develop their deep thinking skills. It's a Tie One Friday morning, Kaia said, "Do you like that tie? You usually wear it at least twice a week." "I do not," replied her father, "I have eight ties and I only work five days a week. I certainly don't wear any tie more than once a week." "You have worn that tie twice a week in at least five of the ten weeks you have been taking me to school," Kaia insisted. "Impossible! Prove it!" challenged her father. Hypothesis 1: Probability only goes up to 100% Learning intentions: Ø Understand that probabilities should be written as fractions, decimals or percentages Ø Apply the property that the probabilities of an exhaustive set of outcomes always sum up to one GP 2 = Recall the vocabulary for probability and how the probability scale is used GP 3 = Recall how to list outcomes of a variety of events GP 3-4 = Recall how find the probability of events NOT happening GP 2 = Pupils answer quick questions on naming events that are certain, impossible etc, and place them on the probability scale. Pupils need to be able to create and describe events as well as then define these key words. GP 3 = Pupils will realise the outcomes of events and apply this to calculate the probability of different single events using fractions. Pupils will explain the difference between equally likely and even chance. GP 3-4 = Pupils apply the knowledge that all probabilities add up to 1 in order to calculate the probabilities of different events NOT happening, using fractions, decimals and percentages (recall LC3). Hypothesis 2: Not all events can happen at the same time Ø Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one

11 GP 3 = Provide the definition of mutually exclusive events with an example GP 4 = Demonstrate the OR Rule for mutually exclusive events GP 3 = Pupils will identify sets of events that are/are not mutually exclusive and be able to explain why, as well as create their own examples. GP 4 = Pupils will move onto more difficult tasks applying the OR Rule to calculate probabilities of mutually exclusive events ( P(A or B) = P(A) + P(B) ). Hypothesis 3: Two-way tables can be used to find outcomes Learning intentions: Ø Construct theoretical possibility spaces for single and combined events GP 3 = Recall writing outcomes of events GP 4 = Recall how outcomes can be written in and calculated from a sample space diagram GP 5 = Demonstrate how data can be represented in 2-way tables GP 3 = Quick questions to write lists of outcomes. Discussion as to why this is not always a suitable method (time). GP 4 = Pupils will complete or draw from scratch sample space diagrams for 2 combined events such as rolling dice, using spinners or flipping coins. Pupils then need to calculate the probability of certain events using fractions. GP 5 = Pupils will calculate the probability of events from 2-way tables, being careful of the language and therefore which totals need to be used in the answer (fraction). Pupils should be able to complete a 2-way table from the data/information given. GP 6 = Pupils can apply previous knowledge to problem type probability questions. These questions involve the use of converting FDP and applying ratio along with probability. Challenge question - A discussion on what we have done so far this LC and how it helps to answer the LC Challenge Question. Pupils will also reflect on the weekly lines of enquiry in books. Home learning: Given every Monday, due in by Friday each week. Home learning this week will be targeted tasks placed onto Google Classroom pulling together work on probability in real life in order to answer the line of enquiry for the week.

12 Week 5 4 1hr lessons plus 1hr homework Line of enquiry; How can fractions be used to calculate probabilities? Lesson 1: R.E.A.C.H Session Pupils will use this session to respond to comments in their books. They will be given instructions as to how they can improve work, how to correct work and how to carry out further REACH work to stretch them beyond target grades. Today will see a much fuller and deeper reflection as to their progress, improvements and confidence levels in certain topics. Prior learning will be ascertained in the lesson recalling how probability works from historical data as well as theory. Hypothesis 1: Theory is the same as real life Learning intentions: Ø Record, describe and analyse the frequency of outcomes of probability experiments using tables GP 3 = Recall calculating simple probabilities from tables of experimental data GP 4 = Demonstrate how we can use frequency to calculate expectations GP 5-6 = Demonstrate and recall how algebra is used to write expressions to help solve problems GP 3 = Pupils will calculate probabilities from tables. They will plan and carry out experiments using spinners, dice and coins. They will record their data in tables similar to the ones they have practiced and apply their knowledge of 2-way tables or sample space diagrams to write and calculate probabilities of a variety of events. Pupils will discuss what they expect to happen based upon theoretical probability, e.g. if I roll a dice 12 times theory says I should get.. GP 4 = Pupils will apply numeracy skills and understanding of frequency to complete worded questions such as A coin is flipped 100 times, how many times would you expect it to land on heads?. The pupils will make their predictions for an experiment and create a set of simple hypotheses. Pupils will conduct the experiment then respond to their hypotheses explaining why they were true/false. The next step is for pupils to calculate what they would expect to happen in a number of scenarios. GP 5-6 = Pupils will use algebraic expressions to help solve probability/theoretical situations. All pupils should be able to write up why theory, and what we expect to happen, is often different to experimental results (real life) using examples from their experiments. They will repeat the experiment and observe what happens to the results the more and more it is repeated. They will develop an understanding that the more data you collect, the more reliable the results are. Hypothesis 2: Probability can predict future events

13 Ø Understand how to calculate relative frequency from experimental results GP 3-4 = Recall the way we calculated what we expected to happen from previous lesson GP 5-6 = Discuss how the experimental results could be applied to proportion/fractions in order to make further predictions GP 3-4 = Pupils will recap quick questions from the GP 4 success criteria above to develop their understanding further. GP 5-6 = Pupils will work on a variety of situations with historical data, experimental data or theoretical probabilities in order to calculate the relative frequency of events. For instance, they may need to explain the reliability of results, calculate what they expect from a table of results, and predict what should happen if an experiment was repeated again, through the application of relative frequency calculations. Hypothesis 3: Knowing how to multiply and add fractions is not necessary for probability Learning intentions: Ø Record, describe and analyse outcomes using probability trees with and without replacement GP 4+ = Recall how and when the OR Rule is needed to calculate probabilities GP 5 = Demonstrate how to use and complete a probability tree diagram using both OR and AND Rules GP 6 = Demonstrate how to complete tree diagrams for events with non-replacement GP 4 = Pupils will develop their understanding of the OR Rule and multiplying fractions by solving probability questions recalling mutually exclusive events. They can then move on to using/completing simple replacement tree diagrams to apply the OR Rule to calculate the probabilities and outcomes. GP 5 = Pupils will complete and draw tree diagrams with replacement and calculate the different probabilities. GP 6 = Pupils will develop their understanding of using tree diagrams in a variety of scenarios using all fractions, decimals and percentages, in order to calculate the probability using the AND/OR Rule. Pupils will also have worded problems and be expected to draw their own tree in order to simplify and solve the problem. Answer the overarching challenge question: How can real life problems be solved using chance and an awareness of shape and space?

14 Home learning: Given every Monday, due in by Friday each week. Revision of unit for exams next week. Pupils will be provided with online resources, videos and booklets where requested. Week 6 Lesson 1: R.E.A.C.H Session Pupils will use this session to respond to comments in their books. They will be given instructions as to how they can improve work, how to correct work and how to carry out further REACH work to stretch them beyond target grades. Today they will use all their personal reflection to begin revision prioritising the topics they need to work on. Resources will be provided to allow for personalised, independent and structured revision. Lesson 2 and 3: Lesson 2 may be used to develop any final areas of work from week 5, leading in to revision using levelled booklets followed by assessments. This time will also be used to complete revision from year 8 work done on distance time graphs. Lesson 4: End of LC4 exam Gap Analysis Reinforcement Week 7 Gap Reinforcement As seen in the lesson activities each week, gap teaching will not just be at the end of the LC2 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 This will be in the form of REACH questions to challenge pupils in lessons beyond their target grade. They will be required to apply their knowledge from each week in order to solve these problems. Extended learning will also come in the form of access to websites, revision videos and additional independent assignments set at the right level, yet will challenge them to reach further.

15 Examples of GCSE questions These are an example of how the level of difficulty can vary between Foundation and Higher. Pupils will be given work to suit their target level, those studying the Higher curriculum will work towards these more difficult types of problems in year 10 but will have chance to try them in REACH.