Answer All Questions In The Spaces Provided - AQA

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GCSE Mathematics:90 maths problem solving questionsThe new Maths GCSE has an increased focus on problem solving.So that you can help your students practice this type of question,we’ve refreshed our 90 maths problems resource so that it’srelevant to the new GCSE.Visit All About Maths aqa.maths.aqa.org.uk our free mathsresource website to access other resources and guidance.Great supportGreat resourcesAQA Maths GCSE aqamaths.aqa.org.uk

You can get further copies of this Teacher Resource from:The GCSE Mathematics DepartmentAQADevas StreetManchesterM16 6EXOr, you can download a copy from our All About Maths website (http://allaboutmaths.aqa.org.uk/).Copyright 2015 AQA and its licensors. All rights reserved.AQA retains the copyright on all its publications, including the specifications. However, registeredcentres for AQA are permitted to copy material from this specification booklet for their own internaluse.AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales(number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

Contents1Introduction21aOrigin21bPurpose21cProblem solving in GCSE Mathematics32Teaching strategies42aGeneral strategies for problem solving in mathematics42bThe five strategies of the resource52cTeaching using the resource62dOrganisation of teaching62eThe relevance of mathematics learning72fA progression in student problem solving73The Problems84The Commentaries5Answers1546Classification of Problems1617Classification Table by Strategy1628Classification Table by Content Area166991

11aIntroductionOriginThis resource was originally produced by LeedsUniversity’s Assessment and Evaluation Unit tosupport teachers in developing approaches to thetype of problem solving questions that appearedin the pilot GCSE in Additional Mathematics. Thispilot was part of the QCDA funded CurriculumPathways Project. We are grateful to QCDA forgiving their permission to reproduce theseresources to support teachers of GCSEMathematics.1bThe approach taken to problem solving in the pilotqualification formed the basis for AQA’s approachto the assessment of AO3 in the GCSEMathematics specifications, for which firstteaching began in September 2010 (for exams upuntil June 2016), and the problems in thisdocument remain highly relevant in approachingproblem solving in the new AQA GCSEMathematics specification for first examination in2017. Hence, we hope that this resource will beuseful to all teachers of mathematics indeveloping problem solving strategies for lessons.PurposeThis resource is designed to support teaching andpreparation for the problem solving requirement(AO3) of the new AQA GCSE in Mathematics(8300) as well as the previous GCSEs inMathematics (Linear; 4365 and unitised 4360)and the Linked-Pair pilot GCSEs in Methods inMathematics (9365) and Applications ofMathematics (9370)In the remainder of the introductory pages thereare descriptions of: The resource consists of: these introductory pages90 examples of problem solving questionsextended commentaries for 30 of theproblem solving questionsanswers to the 90 questionssummary lists linking questions both toprocess skills and to content areas.The problem solving questions can also be foundas separate problem sheets within the resourcesavailable as part of AQA All About Mathshttp://allaboutmaths.aqa.org.uk2 problem solving in the GCSE AdditionalMathematicsgeneral strategies for problem solving inMathematicsfive strategies that are helpful in solvingthe kinds of problems that are set in theexamination papers for the GCSE inAdditional Mathematicsteaching using the resourcea progression in student problem solvingthe information given in the resourceabout all problemsthe elements of the commentaries on the30 questions.

GCSE MATHEMATICS1cProblem solving in GCSE MathematicsDespite its title, the Pilot GCSE in AdditionalMathematics did not contain any mathematicalcontent additional to that specified in the NationalCurriculum for Key Stage 4. The differencebetween GCSE Mathematics and GCSEAdditional Mathematics was one of emphasis.This is enough to give the examination a differentcharacter, with a commensurate need for differentpreparatory teaching, but without requiring theteaching of new mathematics.Specifically, the GCSE in Additional Mathematicshad proportionately more questions that call onmathematical processes that are not a matter ofprocedural or factual knowledge. There is: more reasoningmore justification of reasoning inexplanationsmore representation of a situationalgebraicallymore manipulation of algebra in previouslyunrehearsed waysmore visualisationmore problem solving - with more‘unstructured’ questions, lacking the stepby-step build up to a solution that is foundin many GCSE questions.Each of these kinds of mathematical process hasbeen explicit in all versions of the MathematicsNational Curriculum since 1989, and has alwaysbeen taught as part of preparation for GCSEMathematics. Nevertheless, the greater emphasison these processes in the new specifications forGCSE Mathematics introduced for firstexamination in 2012 and the further increase inemphasis in the new specifications for firstexamination in 2017 mean there is a need to havea more direct focus on each of them inpreparatory teaching.This will generally just be a matter of a little morepractice, changing the balance in teaching toreflect the different balance in the examination –more practice in reasoning, in representing andmanipulating, in visualising, in explaining andjustifying – and this should be sufficient.However, the new requirements for GCSE maths(exams from June 2017) articulated inassessment objective AO3 clarify:‘Solve problems within mathematics and inother contexts’.25% of all foundation tier and 30% of all highertier assessment in GCSE Mathematics mustaddress this objective. This resource focussesprincipally though not exclusively on problemsolving within mathematics although thestrategies outlined may be more widely applied.For successful problem solving in theexamination, there is a likely need for focusedteaching of strategies. Faced with unstructuredproblems without easy lead-in steps, manystudents do not know how to begin to findsolutions.The main aim of this resource is to offer themeans to equip students with strategies to useon the problem solving questions on theexamination paper: it is a source of problem solving questionssimilar to those in the pilot GCSE inAdditional Mathematics although is not acollection of GCSE questionsit also describes how the problem solvingstrategies that are required for these kindsof questions might be taught.3

22aTeaching StrategiesGeneral strategies for problem solving in mathematicsIn developing problem solving in mathematics,including in GCSE classes, teaching is likely tofocus on general strategies that could be usefulfor any question. For example: thinking of the properties that the answerwill haveformulating and testing hypotheseseliminating options (paths or outcomes)representing the cases, relationships,features or examples symbolically,algebraically or diagrammatically.There is also a ‘meta’-strategy of reviewingprogress and going back and trying again whenthe chosen strategy is not working. Within this,‘trying again’ has a number of different forms: 4trying the strategy again, only morecarefully, presuming that the action hasthe capacity to succeed but was not beingdone well enoughtrying the strategy again, but on a differentbasis, following an amended perception ofthe mathematics in the problemtrying a different strategy altogether.Teaching of these general strategies has alwaysbeen part of Mathematics classes, includingGCSE classes, and remains important.However, the problem solving element of GCSEMathematics is a particular context for problemsolving, in which the problems are constrainedby the fact that they are in a timed examinationsubject to a mark scheme. A range of strategiescan be identified that help with the kinds ofproblems that are set under such conditions, andwhich featured in the pilot GCSE and will featurein the new, national GCSE (exams from June2017). Therefore, to prepare for GCSEmathematics, as well as a continuation of theteaching of general strategies, the classes canalso be directed at the strategies that would behelpful for tackling the kinds of questions foundin the examination. These are the five strategiesspecified in this resource.

GCSE MATHEMATICS2bThe five strategies of the resourceA student facing a new problem is initially likely toexamine it to see if it is like a problem that theyhave done before - and if it is, will try to use theapproach that had worked on the previousoccasion. If the problem is not obviously like onethey have done before, then they could considerthe features of the problem to decide what mightbe done that could be helpful. They may askthemselves questions about the problem, such as:‘are there examples or ‘cases’ that it might behelpful to set out systematically? Is there aprocedural relationship between some elementsthat could be reversed to find others? Are therecriteria to apply to possible solutions? Are therefeatures with relationships between them thatcould be drawn out and expressed in some way(eg, linguistically, diagrammatically, symbolicallyand algebraically)? Is there some recognisablemathematical approach that could be followedthrough, extended or applied?’Progress on the problems in GCSE Mathematicsis likely to be made through one or another ofthose possible approaches. As a result theyrepresent the five strategies with particularrelevance to the examination:12345To set out cases systematically, andidentify how many there are of relevanttypes.To work backwards from a value given inthe problem:(a) where the inverse is familiar, sojust has to be applied but mayhave to be sustained over anumber of steps.(b) where the inverse is unfamiliar, sohas to be worked out ‘from firstprinciples’.To find one or more examples that fit acondition for the answer, and see whetherthose examples fit with the otherconditions in the situation, makingadjustments until they do.To look for and represent relationshipsbetween elements of the situation, andthen act on them to see if any are useful.To find features of the situation that can beacted on mathematically, and see whereusing them takes you (operatingincrementally, yet speculatively).For the purposes of this resource, these fivepossibilities for action, which can be developedas strategies for the individual, will be labelled asfollows:1Set out cases2Work back familiar; work back unfamiliar3Find an example to fit4Find key relationships5Find mathematical featuresHowever, it should be noted that these labels arefor the convenience of reference in the resource,and should not be used in the classroom context.To use labels like this in class is to invitestudents to misunderstand these possibilities foraction as distinct sequences of activity that canbe learned and applied as procedures. Problemsolving is not a matter of identifying a problemas a ‘type 3’ and solving it by applying the ‘findan example to fit’ routine.Even though the problems outlined in thisresource are broadly classified (section 7) bywhich of these five strategies most readily lendsitself, this is not the same as identifying problemtypes, since there are almost invariably differentways of tackling any problem. The purpose ofthe classification is to enable the resource to beused in a systematic way. The five identifiedstrategies are effective approaches to the kind ofproblems found in GCSE Mathematics; and byassociating each problem with a strategy; theresource offers a structure for rehearsing anddeveloping techniques to enable success in theexaminations.* This resource has been developed specificallyin the context of the questions written for theGCSE Additional Mathematics, and is notintended as a structure with applicability to allproblem solving.5

2cTeaching using the resourceIt has long been recognised that problem solvingstrategies (whether the general strategiesoutlined previously, or the five of this resource)cannot be taught directly - that is by describingthem and rehearsing them as procedures.Rather, the teaching is indirect, drawing attentionto strategies as possibilities for action in thecontext of solving problems. This should beremembered in what follows: references to“teaching” should not be taken to imply “telling”.The resource contains a set of problems whichoffer opportunities for students to develop anduse the five processes listed above – processesthat should, if strategically deployed by them,improve their performance on GCSE questionsassessing AO3. A classification is used in whichprocess is matched to problem (section 7) but ofcourse, being problems, there are always otherways to do any of them, and, being strategies,there are no guarantees that they will lead to asolution.2dLessons focusing on problem-solving:Problems are used that can all be solvedby the same approach; then, in laterlessons, another set that suit a secondapproach - and so on.By this approach each of the five strategies couldbe focused on in turn during a sequence ofdedicated lessons, using an appropriate selectionof problems for each strategy (following themapping in section 7). Keep in view that this doesnot mean labelling the lessons by the relatedstrategy, due to the dangers associated withpigeon-holing them into such definitive genericcategories (see section 2b).However, the subtlety involved in learning to solveproblems is such that an indirect approach isnecessary. The more direct approach of describingstrategies and getting students to rehearse themhas been shown to be ineffective.6The resource could be used purely as a set ofexample questions for practice in problemsolving, but it also offers the opportunity to usethe examples in a more structured way. In usingthe resource, therefore (even though answeringthe question is likely to be the first focus for thestudent), the main focus for teaching should beon the strategies - at least at first.Organisation of teachingThe problems in the resource can be organised inat least two different, contrasting ways.(i)By working through the problems in anappropriate way, students will developstrategies and also have experience ofproblems that have been solved using them.Using the resource should therefore enable boththe recognition of similar problems in the future,and the development of confidence in a set ofactions that can lead to problem resolution.(ii)Problem solving within lessons:Problems are used that fit in with thecontent area, as part of teachingmathematics topics.In this approach it is important that themathematics underlying the teaching is ‘usedand applied’ to the problems, and so themapping in section 8 will be a guide as to whento use which problems. Since problems canoften be approached in different ways, and thissometimes means using different mathematics,care would have to be taken to ‘shape’ theapproach of the students so that the desiredapproach is being taken. Whatever is done withthe mathematics, however, the main focus ofthe teaching should still be on the problemsolving strategies, and the progression inteaching outlined below should still be applied.

GCSE MATHEMATICS2eThe relevance of mathematics learningThe main focus of the teaching using this resourceis the development of problem-solving strategies.However, it should be recognised that working withproblems in a social context, in which differentideas that students have are shared across thegroup, helps to develop awareness of interconnectedness in mathematics, the knowledge ofwhat is like what, and what fits with what, and whatoften goes with what, and this makes an importantcontribution to students’ capacity to do problems.2fA progression in student problem solvingFirst StageSecond StageDeveloping the ‘strategies’ as possibleapproachesIn the early phase of the work, as each problem isintroduced to the students and they make theirfirst attempts at engaging with the challenge anddevising an approach, the teacher draws attentionto possibilities for action by making suggestions,by pointing out a feature of the problem, or byasking eliciting questions that bring the student tothat awareness. The classification in section 7 is aguide to which approach each problem is mostsusceptible to, which indicates which possibilitiesthe teacher might highlight. Through these meansthe students’ range of possibilities for action what they consider when looking at a newproblem - is expanded to include those thatcorrespond to the five strategies. They will haveestablished that sometimes it is helpful to set outcases, sometimes to work back, sometimes tofind examples that fit conditions and so on - butwithout thinking of them as techniques oridentifying them through labels.Developing awareness of the approaches asstrategiesIn the second stage, when a new problem isintroduced, teaching should begin with adiscussion with students about what might be ahelpful thing to do on that problem. The mappingin section 7 and / or the commentary on thequestion (section 4) might be used as a guide forwhat would be a good approach. Over a seriesof problems it will be helpful for the teacher toensure that the actions related to all fivestrategies are considered as possibilities, so thatthey remain in the students’ minds as options.Third StageOperating strategicallyThe last stage in the sequence for developingproblem solving using these materials involvesstudents deciding independently what to do - ineffect practising problem solving using examplesfrom the resource. However it should also beaccepted that some students need to bereminded of possibilities as they consider eachproblem. At this stage the measure of success isnot what they have learned or are aware of, butsolving the problem in a reasonable length oftime.7

3The ProblemsThis section contains 90 problem-solving questions. They are not intended to be examination stylequestions but are designed for use in supporting the teaching and learning of mathematics, andparticularly the requirements of AO3 in all new GCSE specifications (exams from June 2017).Apples9Identical Rectangles39Scale Factor69April 1st10Inside Circle40Shares70Bag11Isosceles Grid41Sharing71Boat Hire12Javelin A42Side by Side72Bouncing Ball13Javelin B43Spinners73Box Clever14Line Up44Square Area74Bubble15Loop45Square Root Range75Chart16Maze46Stamps76Chocolate Mousse17Mean47Stationery77Coins18Mean Set48Stretcher78Crate19Mean Street49Sum and Difference79Cubes20Meet50T-Grid80Cuboid Ratio21Middle Sequence51Tape 3N and M53Terms83Double trouble24Number Pairs54Three Different Numbers84Equal Money25Overlap55Three Numbers85Equal Points26P and Q56Three, Four, Five86Expand27Peculiar57Three Squares87Eye r89Five Thousand30Poster60Towers90Five Times31PQR61Trapezium Tiles91Flight cost32Purple Paint62Two Triangles92Flow Chart33Put the Numbers in63Weights93Form34Quadratic Function Graph64Wheels94Gang of Four35Repeater65Whole Numbers95Half Take36Right-Angled l38Rollover68Youth Club Trip988

GCSE MATHEMATICSApplesLottie has a bag of apples.She gives half of them to Fred.Fred eats two and then has four left.How many apples did Lottie have at the start?Answer .9

April 1stExplain why the 1st of April is always on the same day of the week as thefirst of July.10

GCSE MATHEMATICSBagA bag contains only red counters and blue counters.There are 90 red counters in the bag.The probability of choosing a red counter from the bag is 0.3How many blue counters are in the bag?11

Boat HireThe cost of hiring a boat is: 4

AQA Maths GCSE GCSE Mathematics: 90 maths problem solving questions The new Maths GCSE has an increased focus on problem solving. So that you can help your students practice this type of question, we’ve refreshed our 90 maths problems resource so that it’s relevant to the new GCSE.

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