directorate of national examinationل...grade 12 problem solving test specification 2014 date...

Directorate of National Examination

Problem Solving - Grade 12

Test Specification

2014

Kingdom of Bahrain

Grade 12 Problem Solving Test Specification 2014

Date published: October 2013 2

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Contents

1 PURPOSE 3

1.1 Aims of the Grade 12 Problem Solving examinations 3

2 DEFINITIONS 4

2.1 Skills 4

2.2 Questions and items 4

3 TEST CONSTRUCT 5

3.1 Description of papers 5

3.2 Additional materials 5

3.3 Balance of marks 5

4 PAPER 1: PROBLEM SOLVING 6

5 PAPER 2: PROBLEM ANALYSIS AND SOLUTION 8

6 PERFORMANCE DESCRIPTORS 11

7 EXAMPLE MATERIALS 12

7.1 Paper 1 12

7.2 Paper 2 27

7.3 Mark scheme for Paper 2 examples 36

All details in this document are correct at the time of publishing, but may be subject to change as a result of ongoing efforts to ensure the quality of the examination product.

Please ensure you refer only to the most current version.

Issue 4 of the test specification applies from the examination in March 2014.



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1 PURPOSE

The main purpose of the examinations is to provide information about the performance of students in Problem Solving in the Kingdom of Bahrain. Their performance on subject content aims to identify what students should be able to do after 12 years of formal schooling and to establish whether they have the skills and the knowledge to be able to find work within the global marketplace of the 21st century using problem solving and higher thinking skills. Additional purposes of the National Examinations are to:

to be of use in the selection processes for Further Education, Higher Education or employment;

provide a beacon of good assessment practice;

provide robust information for the monitoring of standards over time.

1.1 Aims of the Grade 12 Problem Solving examinations

The aims of the grade 12 Problem Solving examinations are:

to assess a student’s ability to analyse numerical and graphical information, which is based in real life situations, and apply the right numerical techniques to find new information or derive solutions;

to develop the use of a range of skills such as data handling, reading, modelling, logic and reasoning. Students will be assessed on these various sub-skills, which are the building blocks to successfully solving wider and more complex problems;

to encourage the application of simple mathematics to new situations to show that students can manipulate numerical and graphical data. They need to extract and use relevant data, and find ways of drawing conclusions from information. Students need to be able to present that same data in different forms. They are expected to think critically about information, find possible reasons for unexpected variations and be able to use information for informed decision-making.



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2 DEFINITIONS

2.1 Skills

In Grade 12 examinations a number of skills will be assessed:

Extracting relevant data (PS1)

Processing data (PS2)

Finding procedures for solving problems (PS3)

Searching for solutions (PS4)

Identifying similar data (PS5)

Suggesting hypotheses for variations (PS6)

Using spatial reasoning (PS7)

Identifying necessary and sufficient data (PS8)

Choosing and working with models (PS9)

Making choices and decisions (PS10)

Developing a model (PS11)

Analysing complex data and drawing conclusions (PS12)

For paper 1, students need to be familiar with basic techniques of mathematical manipulation covered during primary education (Grades 1-6).

For paper 2, students need to be familiar with basic techniques of mathematical manipulation covered during intermediate education (Grades 7-9).

Problem Solving is about using logical methods of handling numerical, graphical and pictorial data. Problem solving skills are not only desirable but essential to lawyers, sociologists, geographers, historians and those in other professions. They have to understand and use numerical information, to analyse it and to draw conclusions from it. For example, a lawyer may need to know about and understand the probabilities of a DNA test being conclusive. Sociologists frequently employ statistics and demographic data.

2.2 Questions and items

A task (or question) is a collection of items which is linked by a common written or spoken text. An item is the smallest separately identified testing point within an assessment to which a mark or marks are allocated. A question may consist of several items or just one item.



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3 TEST CONSTRUCT

3.1 Description of papers There will be 2 question papers, 1 assessing problem solving skills and 1 assessing problem analysis and solution skills. For paper 1 students answer on the question paper and for paper 2 students answer on the provided answer booklet.

3.1.1 Paper 1: Problem Solving Students have 90 minutes to complete 30 multiple choice questions, each made up of a stimulus, a question and 4 options – only one of which is the correct answer. Each question will have a skill focus (for example PS 10), however, frequently, the question will involve one or more other skills, but to a lesser degree. Paper 1 tests all of the skills PS1–PS10 inclusive. Each question is worth 1 mark giving a maximum mark of 30. 3.1.2 Paper 2: Problem Analysis and Solution

Students have 120 minutes (2 hours) to complete 3 structured answer questions worth a total of 50 marks. Paper 2 tests skills PS1-PS12 according to the following split:

One question will require students to analyse given complex data and draw conclusions (PS12). A subset of skills PS1-PS10 will also be required;

One question will require students to develop a model and explore possible solutions (PS11). A subset of skills PS1-PS10 will also be required;

The third question will require students to apply a range of skills from PS1–PS10 to solve complex problems.

3.2 Additional materials

Students are permitted to use calculators in both examinations.

3.3 Balance of marks

The 2 papers are equally weighted to produce a final total for each student.



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4 PAPER 1: PROBLEM SOLVING

The questions examine the student’s abilities to use the following skills: PS1: Extracting relevant data PS2: Processing data PS3: Finding procedures for solving problems PS4: Searching for solutions PS5: Identifying similar data PS6: Suggesting hypotheses for variations PS7: Using spatial reasoning PS8: Identifying necessary and sufficient data PS9: Choosing and working with models PS10: Making choices and decisions The questions are multiple choice with a short stimulus passage. The students select the most appropriate response from four options A to D. Information may be presented in five categories: tables, graphs, words, pictures or diagrams. The stimulus may include information in more than one category. The stimulus material is based on a realistic scenario. In general, students should be able to answer the question from the stimulus, rather than having to work backwards from the list of possible answers. However, there will be some questions where the options in effect form part of the stimulus. Students do not have to have knowledge specific to any subject, for example knowledge of physics or statistics; they do have to have knowledge of the basic mathematics required. Each of the 10 sub-skill categories for Problem Solving is explained below. An example of a question that tests the sub-skill is also provided in section 7. PS1 Extracting relevant data

Questions may give information in words, tables or graphs. Most is quantitative but some qualitative elements may be included (e.g. colour or shape).

Students are given criteria for choosing one item or a range of items from the data presented. The skill is to select the correct item or range of items from a set which includes some unnecessary data.

This type of question may also include a small amount of simple processing, e.g. to find a journey time from a timetable giving departure and arrival criteria; the correct departure and arrival time should be found from the timetable and the journey time found by subtraction.

PS2 Processing data

Students are given information in words, tables or graphs.

This data needs to be used to answer a question.



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One or more items from the data should be used correctly to give an answer. The method of using the data will be clear and will not need to be clarified. The skill is to choose and use the data correctly.

This category of question may rely on some data extraction as well as processing, but this will not be the main emphasis of the question.

PS3 Finding procedures for solving problems

Questions may give information in words, tables, graphs, pictures or diagrams.

Students are given a problem to solve (this will in most cases have a numerical solution but it may also be in other forms e.g. a spatial manipulation).

The method of solution is not immediately obvious and the student has to choose a suitable method of using the data to come up with the answer. The skill is to work out how to use the data in a suitable way to solve the problem, and then produce a solution.

PS4 Searching for solutions

Questions may give information in words, tables, graphs, pictures or diagrams.

Students have to solve a problem by searching for a solution.

Searching can be in two forms. It may be necessary to search the data given to find the correct pieces of information to use. In general, it will also be necessary to consider several possible scenarios which may solve the problem, then to decide which, for example, is the best according to a given factor.

The required skills are to identify a suitable method of searching and to carry it out correctly.

PS5 Identifying similar data

Students will be asked to match information in one form to the same information in another form (e.g. table to graph), or in a different representation of the same form (e.g. matching a time-distance graph with the same data presented as a time-velocity graph).

The information given and derived may be in words, tables, graphs, pictures or diagrams.

The skill required from the student is to recognise the correct data item or data set from several others which are presented.

PS6 Suggesting hypotheses for variations

Students are given information, as graphs, tables or text. This will typically describe the relationship between two variables.

Students select from a list, a statement or an additional piece of data which would explain the nature of relationship in the data.

The skill is to make a reasoned choice of a possible cause of relationship. PS7 Using spatial reasoning

Students are given information, usually as pictures (although other forms such as a written description may be used).

They have to identify the result of a spatial operation such as deformation, reflection or rotation or, for example, to identify a view from a different direction.

The skill is in being able to visualise shapes and views, and how they may change, in two or three dimensions.

PS8 Identifying necessary and sufficient data

Students must decide what data is either necessary to contribute to the solution of a problem or sufficient, along with other data, to solve the problem. For example, a



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question could be of the form ‘Which of the following additional pieces of data would be sufficient to answer this question?’

The student is not usually asked to solve the problem. PS9 Choosing and working with models

A model of a set of data may be in the form of a graph, a diagram or a written description.

Students are given a scenario which includes information on the way in which a parameter or a process varies.

They are then asked to choose a representation of this variation in another form which models the data presented.

A more difficult question might expect the student to derive a mathematical model and use it to produce an answer.

The skill is to recognise how one set of information may model either another set or a real situation.

PS10 Making choices and decisions

Questions give data in words, tables, graphs, pictures or diagrams and a set of criteria.

Students choose from a set of options based on the given data and criteria. This involves a search of some sort.

In the simplest sense, this would be just extracting an item from the data set (and would be equivalent to type PS1) or the search may involve an amount of data handling. The skill is to be able to make an informed decision by combining a set of information and given criteria for selection.

5 PAPER 2: PROBLEM ANALYSIS AND SOLUTION This paper tests the same sorts of skills identified in Paper 1, i.e. PS1–PS10, or a subset of these skills. It also requires the additional skills of (a) developing a model (PS11) and (b) analysing complex data and drawing conclusions (PS12). The mathematical content for paper 2 requires, in addition to primary education (Grade 1– 6) mathematics, some knowledge of:

expressing problems as linear algebraic equations and the solution of these equations;

expressing problems as a pair of linear simultaneous equations with integral coefficients and the solution of these equations;

use and manipulation of integer powers (e.g. knowing that 33 × 32 = 35) and square roots;

the calculation and use of averages (mean);

the use of diagrams for solution of categorisation problems (e.g. Venn diagram) PS11 Developing a model Modelling requires the student to develop a set of equations, a table or some other set of data which represents the mathematics of a real situation. As a very simple example, a telephone bill which consists of a standing charge plus a fixed rate per time of calls could be modelled as: b = s + at, where b is the bill, s is the standing charge, a the cost per minute and t the minutes used.



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The student might be required to take a real physical situation and extend the data and parameters given to draw conclusions about other related and possibly more complex situations. PS12 Analysing complex data and drawing conclusions Analysis requires students to combine raw data in appropriate ways so they are able to draw conclusions. The data may be presented as words, tables and/or graphs. The students are expected to:

select the pieces of data which are relevant to the questions asked;

look for patterns in the data;

analyse the data mathematically to lead to the conclusions required. Question 1 or 2: Analysing Complex Data and Drawing Conclusions (15 marks) Students are presented with various data that relate to a single problem. The data may be spread across more than one table and/or graph. Data may also be included in the introductory description of the problem. Some of the data may be irrelevant to the problem to be solved. Faced with relatively complex data in disparate forms, the student will need to recognise how the data can be used to solve the problem posed. This might include identifying what is the relevant data and/or combining data from different sources to generate a solution or intermediate solution. The question will make clear what form of solution is required. But the form of what is required will vary between questions. For instance, the student might be asked to:

identify the relevant datum;

find an optimal solution (maximum/minimum) or an extreme case;

verify a ‘claim’ that has been stated (is it true or false?);

identify an assumption that must be made to reach a solution with a particular attribute;

rank the given options according to some criteria;

select from a number of options;

make a recommendation;

reach some other form of conclusion from the data given.

The question will consist of several parts. Question 1 or 2: Developing a model (15 marks) A realistic ‘task’ is described to the student. This task might be of a physical nature (e.g. the amount of netting needed to cover the sides of a birdcage) or non-physical (e.g. a business challenge such as maximising profit). Some tasks may include a mixture of physical and non-physical aspects – such as maximising sound output from a speaker system while working within a financial budget.

The task description might include:

an overall project ‘aim’;

one or more essential objectives (requirements);

one or more additional objectives that are desirable but non-essential;

some constraints which the solution must satisfy.



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The student will be given a simple model to apply while solving problems relating to the scenario. The final parts of the question will require the student to use the information given in the task to develop the model further.

For example, the application of the model might be used to identify the range of solutions that are possible within the constraints, or the most desirable outcome that can be achieved, or which constraint would need to be relaxed in order to obtain a better outcome. The development of the model might then include one further constraint to be accommodated by the revised model.

The question will consist of several parts. [NB There will always be one question on Analysing Complex Data, and always one question on Developing a Model – but their order may be reversed in different years.] Question 3: Solving Complex Problems (20 marks) The question will have some stimulus source material based around a single topic. Information may be presented in five categories: tables, graphs, words, pictures or diagrams. The stimulus will probably include more than one of these categories. The stimulus will be based on a realistic scenario.

The question will require the student to demonstrate several of the skills PS1–PS10. The complexity of the problem will be greater than on paper 1, requiring students to demonstrate a more sophisticated level of application of skills PS1–PS10. To answer some parts of the question, students may need to select and apply more than one of these problem solving skills. The question will have several parts. Students will need to give a written response to each part.



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6 PERFORMANCE DESCRIPTORS

Grades are awarded on the basis of ‘best-fit’, i.e. where all or most of the descriptors apply.

The Grade A candidate can: The Grade E candidate can:

understand information when presented in a wide variety of ways and can reliably recognise different representations of both simple and complex sets of information including in situations where the data needs to undergo some processing before the representation.

understand simple representations of data and can recognise representations of sets of information where little or no processing of the data is required.

extract and process the information relevant to a given problem from simple and complex sets of data.

identify information relevant to a given problem from sets of data containing little or no distracting information.

develop procedures for solving (possibly complex) problems where a method is not immediately obvious, showing initiative, precision and attention to detail.

apply appropriate procedures for solving problems where the solution only involves one or two steps from the information given.

create appropriate systems to work through a series of possibilities when solving problems.

identify some of the possible cases that exist when solving problems.

understand the relevant aspects of the structure of a (possibly complex) problem and can identify information that enables a solution to be found.

identify information that enables a solution for a simple problem (one which does not involve a large number of variables) to be found.

provide clear reasoning to explain a wide range of scenarios.

identify suitable explanations for familiar scenarios containing little or no irrelevant information.

develop models for simple and complex situations.

apply a given model in situations involving substitution of values to the variables of the model.

make informed decisions based on a given set of parameters for completing complex tasks.

make decisions that satisfy some of the parameters for a task.



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7 EXAMPLE MATERIALS 7.1 Paper 1

PS1 sample question (Extracting relevant data) The cost of sending letters from the United Kingdom to continental Europe is shown below.

Not over £ p

20 g 0.22

60 g 0.37

100 g 0.53

150 g 0.70

200 g 0.88

250 g 1.06

300 g 1.25

350 g 1.44

400 g 1.64

450 g 1.83

500 g 2.02

750 g 2.77

1000 g 3.52

1250 g 4.07

1500 g 4.62

A firm in London wishes to send two letters to separate clients in continental Europe. The letters weigh 75 g and 215 g. What is the total cost of sending the two letters?

A £1.25

B £1.41

C £1.43

D £1.59

Key D Stimulus Type Table Justification



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The 75 g letter will cost 53p to post (over 60 g but under 100 g) and the 215 g letter will cost £1.06p (over 200 g but under 250 g). The total cost is 53p + £1.06p = £1.59p. Candidates are expected to select the two correct values from the table, given the weights of the parcels, and add these together. The primary skill is extraction with a small amount of processing.



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PS2 sample question (Processing data)

We had 76 people wanting orange juice at a conference. The caterers provided enough orange juice for each of 80 people to have a 400 ml cup three-quarters full. We carefully filled each cup to exactly three-quarters full as we handed them out, but failed to notice that they had given us 500 ml cups. How many people went without?

A 0

B 4

C 8

D 12

Key D Stimulus Type Words Justification The amount of orange juice provided by the caterers was 80 (people) × 400

(ml cups) × 4

3 (full) = 24000 ml. The amount of orange juice in one of the

larger cups is 500 (ml) × 4

3 (full) = 375 ml.

The number of people that can be catered for is 24000 (the amount of orange juice provided) 375 (in each cup) = 64.

There are 76 people so 76 – 64 = 12 people will go without. The candidate must use the data correctly (all the data is relevant so the only extraction skill is to use the correct numbers at the correct time). The skill is processing. Finding a method is a minor part of the answer, as the method of solution is straightforward.



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PS3 sample question (Finding procedures for solving problems)

Each of two identical cars can carry enough fuel to travel 100 km only. To make a longer journey over a deserted area, they set out together and then at some stage the first car transfers fuel to the other and returns home. The second car travels on. What, approximately, is the furthest distance from home that the second car can travel?

A 125 km

B 133 km

C 150 km

D 167 km

Key B Stimulus Type Words Justification A method must be developed to solve this problem – it cannot be done just using extraction or processing. One way is by trial and error. For example, if the two cars travel 50 km together, it would be possible to transfer 50 km worth of fuel from one to another but then the car donating the fuel would be empty so this is clearly too far. Similarly, at 25 km only 25 km worth could be transferred so the donor car would have 25 km left when it reaches home.

It becomes clear that the donor car must divide its fuel into 3: 3

1 each for the

outward journey, to transfer to the other car and to return. At 33 km the car which is continuing is completely refilled and would have enough for another 100 km, making 133 km in total.



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PS3 sample question 2 (Finding procedures for solving problems)

To make cake Ahmed needs 125 g of butter and 2 eggs. He has 1000 g of butter and 24 eggs. Ahmed wants to make cake using all of the eggs. How much extra butter will he need? A 500 g B 1000 g C 1500 g D 2000 g

Key A Stimulus type Words Justification

24 2 = 12 12 × 125 = 1500 g 1500 – 1000 = 500 g With 24 eggs, 12 batches of cakes can be made. This will require 1500 g of butter, so another 500 g is needed. Distractors B The 1000 g of butter will make 8 batches, which leaves 8 eggs. 8 × 125 = 1000 g C Does not take off the 1000 g of butter that is already there. D 24 batches would require 3000 g of butter, meaning that 2000 g extra would be needed.



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PS4 sample question (Searching for solutions) A private mail delivery company makes the following charges for delivering letters and packages:

Weight up to 60 g 250 fils Each extra 10 g or part thereof 50 fils

A woman wishes to use this company to send a manuscript either as a single package of weight 138 g or two or more packages with a total weight of 138 g. What is the lowest cost of postage with this company?

A 590 fils B 600 fils C 640 fils D 650 fils

Key B Stimulus Type Words/Table Justification The skill is in performing a search of the options on splitting up the package to find how many pieces, and of what weights, is most effective. As a single package it would cost 250 fils for the first 60 g and 8 × 50 fils for the remaining 78 g, or 650 fils in total. If sent as two packages, it is most economical to have both at 60 g or more, as the first 60 g is pro-rata cheaper than the remaining weight. Thus 60 g + 78 g is as effective as any other split. This costs: 250 fils + 250 fils + (2 × 50 fils) = 600 fils. If sent as three packages, once again it is best to keep as many as possible at 60 g or more. They would then divide as 60 g + 60 g + 18 g at 3 × 250 fils = 750 fils. The two package option is the best at 600 fils.



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PS5 sample question (Identifying similar data) The table shows the age distribution of the inhabitants of a small town:

Age group 0–15 16–25 26–45 46–64 65 +

Number of inhabitants

2997 4432 3761 5980 7440

Which of the following charts, if suitably labelled, could represent the data above?

Key A Stimulus Type Table Justification The second bar should be approximately 1.5 × the height of the first, the third should be in between the first and second, the fourth approximately twice the height of the first and the fifth the largest. Only chart A shows this.



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PS6 sample question (Suggesting hypotheses for variations)

Fatima is a part-time teacher in Dubai. She teaches her lessons during the day and, when she has finished her last lesson, she leaves the school and goes to the bus stop to get a bus home. All lessons start on the hour, or at quarter past, half past or quarter to the hour and each lasts 45 minutes. It takes her 15 minutes from the end of a lesson to reach the bus stop. She doesn’t know what the bus timetable is, but she notices that she always has to wait either 5 minutes or 20 minutes for a bus. Which of the following explains the times she has to wait?

A The buses run every 15 minutes at 5, 20, 35 and 50 minutes past the hour

B The buses run every half hour at 5 and 35 minutes past the hour C The buses run every 20 minutes at 0, 20 and 40 minutes past the hour D In the morning the buses run every hour at 5 minutes past the hour; in the

afternoon they run every hour at 20 minutes past the hour

Key B Stimulus Type Words Justification Fatima always arrives at the bus stop on an exact quarter hour – on the hour or quarter past, half past or quarter to the hour (the lessons start on a

quarter hour, take 4

3 hour and she takes 15 min to get to the bus stop).

If she arrives at the bus stop at the hour or half past, she waits five minutes. If she arrives at quarter past or quarter to, she waits 20 minutes. The buses are at 5 and 35 minutes past the hour. B explains this. The skill is in matching the verbal stimulus to one of the verbal explanations.



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PS 6 sample question 2 (Suggesting hypotheses for variations)

I sent my son to buy 4 bottles of lemonade. I gave him BD 2.400 as I know that one bottle of lemonade costs 600 fils. I was surprised when he came back with 600 fils change. Apparently there was a special offer, though he could not remember the details of the offer. Which one of the following special offers could NOT explain the 600 fils change? A Buy 1 bottle, get a second bottle for 300 fils B Buy 2 bottles, get a third bottle free C Buy 3 bottles, get a fourth bottle for 100 fils D All bottles reduced to 450 fils

Key C Stimulus Type Words Justification The options need to be checked. Only C does not give a saving of 600 fils. Distractors A Two bottles will be reduced by 300 fils, giving a total of 600 fils. B The third bottle being free gives a saving of 600 fils. D There is a saving of 150 fils on each bottle, giving a total of 600 fils.



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PS7 sample question (Using spatial reasoning)

The diagram shows a factory. There is a square chimney in the south-eastern corner of the factory. There is an entrance in the western end which can be seen in the diagram. There are two windows in the southern wall. There is also a door in the northern wall of the factory and a window in the eastern wall of the factory, which are hidden in the diagram.

Which of these is a possible view of the eastern end of the factory?

Key D Stimulus Type Picture and Words Justification The view from the eastern end of the factory should show the chimney on the left and a window in the wall. The roofline should appear behind the chimney. Only figure D shows this.



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PS7 sample question 2 (Using spatial reasoning)

The plan shows three buildings: one building is black, one building is grey and one building is white.

Which one of these is a view of the three buildings?

Key A Stimulus Type Words and diagram Justification The black building is behind the grey building and the white building is on the right. Distractors B not correct as the white building would need to be on the left. C not correct as the black building would need to be on the right. D not correct as the grey building would need to be on the right.



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PS8 sample question (Identifying necessary and sufficient data)

Bader and Salah are twin brothers. They plan to meet at a hotel between their houses to exchange birthday presents. Bader’s journey is on motorways and he travels at an average 120 km/hr. Salah’s journey is on minor roads and he travels at 80 km/hr. Bader leaves at 10 am. They expect to arrive at the hotel at the same time. Which one of the following further pieces of information would be sufficient to determine at what time they will meet?

A Salah’s journey time B Salah’s travel distance C Salah’s departure time D Bader’s travel distance

Key D Stimulus Type Words Justification In order to find the meeting time, from information on either journey, the following calculation has to be carried out:

Meeting time = departure time + travel distance average speed.

For Bader, we have two of the values on the right hand side (departure time and average speed), for Salah we have only one (average speed). Thus, the only single piece of information which would allow us to calculate the meeting time, would be Bader’s travel distance, so D is correct.



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PS9 sample question (Choosing and working with models)

The diagram below shows a bottle used in a laboratory. In order to determine the volume of liquid in the bottle for various heights, water is allowed to flow into the bottle at a constant rate. An observer makes frequent notes of the height the water has reached and of the time. He draws a graph to illustrate his observations.

Which of the following graphs could be the one drawn by the observer?

Key D Stimulus Type Words and Picture Justification The height will rise quickly at first (as the bottle is narrow at the bottom), it will then rise more slowly as the level reaches the wider section and more quickly again at the narrower section towards the top. All these segments will show curves as the area changes continuously. The rate of rise will then become constant in the cylindrical section at the top, giving a straight line segment. D is the only graph to show this.



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PS9 sample question 2 (Choosing and working with models)

Every month Kareem puts some of his salary into a savings account. He calculates the proportion of his salary that he will save using the following method:

He does not save any money from the first BD 500.

From the next BD 1000 he saves 10%.

He saves 25% of any further money he earns. Last month Kareem earned BD 1200. How much money did he put into the savings account? A BD 70 B BD 100 C BD 120 D BD 150

Key A Stimulus Type Words and list Justification 1200 – 500 = 700 10% of 700 is BD 70 Distractors B 10% of BD 500 (1000 – 500), plus 25% of BD 200 (1200 – 1000) C 10% of BD 1200 D 10% of BD 1000 (ignores first rule), plus 25% of BD 200 (1200 – 1000)



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PS10 sample question (Making choices and decisions)

I have none of my nephew’s favourite biscuits left. My nephew visits me at least 3 times a week, though never more than five times. On each visit I know he will eat at least 6 biscuits but I won’t let him have more than 8. Packets of biscuits can contain as few as 10 biscuits, or as many as 12. How many packets of biscuits must I buy to make sure I do not run out within the next two weeks?

A 4

B 5

C 6 D 8

Key D Stimulus Type Words Justification A decision on how many to buy is made by dividing my nephew’s maximum requirement by the minimum number of biscuits in a pack. This is the only way of ensuring there are enough. The most biscuits my nephew will eat in two weeks is 2 (weeks) × 5 (maximum visits per week) × 8 (maximum number of biscuits per visit) = 80. The minimum number of biscuits in a pack is 10, so 8 packets will be required. D is correct.



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7.2 Paper 2 1 Study the information below and answer the questions that follow.

Show your working. The Island of Ferrin is linked to the mainland by a causeway (road). The island becomes cut off from the mainland twice a day as the incoming tide covers the road. It is possible to make accurate predictions of when it is possible to cross the causeway. Crossing times are published well in advance for the benefit of residents and tourists. The table below shows an extract from the crossing times for next July. For example, on 1 July the causeway is open from 10:15 to 16:35 and then from 22:05 on the same day until 05:00 the following day.

CAUSEWAY OPEN CAUSEWAY OPEN

July From To From To

1 10:15 16:35 22:05 05:00

2 10:35 17:40 22:30 06:15

3 10:55 18:55 23:05 07:50

4 11:45 20:25

5 00:10 09:20 13:05 21:45

6 01:40 10:35 14:45 22:50

7 03:15 11:30 16:10 23:35

8 04:40 12:10 17:15 00:15

9 05:45 12:45 18:05 00:45

10 06:35 13:15 18:40 01:15

11 07:10 13:40 19:10 01:50

12 07:40 14:05 19:40 02:25

13 08:05 14:40 20:05 03:00

14 08:25 15:15 20:30 03:40

15 08:45 15:55 20:55 04:25

16 09:05 16:35 21:15 05:10

17 09:25 17:25 21:40 06:05

18 09:55 18:20 22:10 07:10

19 10:35 19:25 22:55 08:20

20 11:40 20:35

21 00:10 09:30 13:15 21:45

22 01:40 10:30 14:55 22:35

23 03:20 11:15 16:25 23:15

24 04:45 11:50 17:40 23:50

25 06:05 12:20 18:45 00:20



© QQA Issue: 4

Bus times The mainland end of the causeway is in the village of Linkham. A local bus service runs from Barwell to Linkham. Upon arrival at Linkham the bus continues on to Ferrin. The causeway must be open for the bus’s outbound time and the inbound time at Linkham. For instance, for the first bus of the day to make the crossing, the causeway must be open for the whole of the time between 09:26 and 10:19 (see table below). Here is the Bus Timetable (which operates daily):

Depart Depart Arrive Depart Arrive Arrive

Barwell Linkham Ferrin Castle Ferrin Castle

Linkham Barwell

09.05 09:26 09:40 10:05 10:19 10:40

10:10 10:31 10:45 11:05 11:19 11:40

12:10 12:31 12:45 13:05 13:19 13:40

14:15 14:36 14:50 15:05 15:19 15:40

15:05 15:26 15:40 15:45 15:59 16:20

16:15 16:36 16:50 17:05 17:19 17:40

18:10 18:31 18:45 18:45 18:59 19:20

Castle opening times Ferrin Castle is the Island’s main tourist attraction.



© QQA Issue: 4

It opens 30 minutes after each time the causeway opens or at 09:00 if the causeway opens before 08:30.

It closes 30 minutes before each time the causeway closes or at 18:00 if the causeway closes after 18:30.

If the castle cannot open for three hours or more, it remains closed.

(a) State the arrival times at the Castle bus stop of all the buses that will

continue from Linkham to Ferrin on 24 July. (The bus makes the journey whether or not the Castle is open). [3]

(b) At what time will the Castle open, and at what time will it close: (b1) On 17 July? [2] (b2) On 20 July? [2] (c) The first date in July when the Castle will be closed all day is 6 July.

On what date will the Castle next open? Show clearly how you reach your conclusion. [2]

(d) Sara and Dana will be staying in Barwell from 2 July to 12 July. During

their holiday they would like to take the bus to Ferrin. However, they will only do so if they will be able to spend at least five hours exploring the Island, including a visit to the Castle.

On which date(s) would this be possible? Show all your working, and for each date that is possible state the

earliest arrival time at, and the latest departure time from, the Castle bus stop. [6]



© QQA Issue: 4

2 Study the information below and answer the questions that follow. Show your working. A cage is to be built to house some exotic birds. It is to be built against a wall, as shown in the diagram, and thus needs 12 struts altogether and netting on four sides (i.e. the three vertical faces and the roof). The structure must be in the shape of a cuboid.

Physical constraints Throughout this question all your answers must abide by the following four constraints.

The wall is 15 metres high and 20 metres long.

There is 600 m2 of netting available.

The smallest strut which can be used for the framework is 9 metres long, so none of the dimensions of the cage can be less than this.

It must have dimensions which are whole numbers of metres. An example of the dimensions of a cage which could be made from the netting is given below. Note that the dimension referred to as ‘length’ is that which is parallel to the wall. Example: Length: 11 m, Width: 12 m, Height: 13 m Area of netting required: roof = 12 × 11 = 132 m2

front = 13 × 11 = 143 m2

sides = 2 × (13 × 12) = 312 m2

TOTAL = 587 m2 (which is within specified limit of netting area)



© QQA Issue: 4

Volume = 13 × 12 × 11 = 1716 m3 (a) If a cage with dimensions 10 m, 11 m and 15 m was to be constructed,

what is the smallest total area of netting that would be needed? [2] (b) Consider possible cages which are of the same height as the wall. What

is the maximum length that the cage can be? Justify your answer. [3] (c) What is the largest volume cage that can be built with a height of 14

metres? Justify your answer. [3]

For parts (d) and (e) of this question you will also need to consider the following additional outcome: For animal welfare reasons the cage should have a volume of 1900 m3 or more.

(d) Show that it is possible to fulfil the outcome of a minimum volume of

1900 m3. Give the length, width and height of the cage which does this. [3] (e) It is decided that the wall is bad for the birds’ welfare, and is to be

demolished. Plans are drawn up for the new cage [which must have 5 netted sides, now the wall has gone].

Four 10 metre struts are found, and it is decided that these should be used for the cage [i.e. at least one of the dimensions should be 10 metres]. The other struts are subject to the same restrictions as before.

What is the minimum extra amount of netting needed if the cage is still to fulfil the 1900 m3 requirement? [4]



© QQA Issue: 4

2 (Second example) Study the information below and answer the questions that follow. Show your working. A local restaurant supplies plates of sandwiches for business meetings. The different plates of sandwiches available are shown below.

Plate type Price per plate of 20

sandwiches (BD) Price per plate of 50

sandwiches (BD)

Salad 12 20

Meat 18 40

Mixed* 17 31

*On a mixed plate, half of the sandwiches are meat and half are salad.

(a) What is the total price for a plate of 50 meat sandwiches and a plate of 20 salad sandwiches? [1]

(b) If it costs 400 fils to make each salad sandwich, what is the percentage profit made when a plate of 20 salad sandwiches is sold? [2]

(c) Explain how it is possible to buy exactly 30 salad sandwiches and 50 meat sandwiches. [2]

(d) A meeting requires exactly 70 sandwiches, of which at least 20 must be

meat sandwiches. What is the lowest cost that could be paid for the order? [2]

(e) A businessman, Zeid orders two plates of sandwiches every week.

This week he has made a change to his order and the total price has gone up by BD 9. Suggest how Zeid’s order might have been changed. [2]

(f) The restaurant manager decides to have a special offer: when you order

any three plates of sandwiches, the cheapest one will be half price. For orders of more than three plates it will always be the cheapest plates that



© QQA Issue: 4

are half price (for example, if the order is for 8 plates then the 2 cheapest plates will be half price).

Zeid wants to place a large order next week and realises that by splitting

the order into multiple smaller orders he can save some money.

Zeid wants to order 3 plates of 50 salad sandwiches, and 7 plates of 50 meat sandwiches. What is the smallest amount that he could have to pay in total for the sandwiches? [3]

(g) A new manager Mohammed, comes to run the restaurant and wants to

have a completely different way of pricing the sandwiches. In the new system there will be a standard price per individual sandwich and a reduced price for every sandwich bought beyond a certain number.

Mohammed decided to set the prices for salad sandwiches as below.

Price per sandwich

Reduced price

Number after which reduced price applies

600 fils 100 fils 30

This means that 20 salad sandwiches will still cost BD 12 and 50 salad sandwiches will still cost BD 20.

What will be the price of 100 salad sandwiches in Mohammed’s system? [1]

(h) Develop one possible new pricing structure for the meat sandwiches. 20 meat sandwiches should still cost BD 18 and 50 meat sandwiches should still cost BD 40. [2]



© QQA Issue: 4

3 Study the information below and answer the questions. Show your working. In the world of taxi drivers who work in a city with a regular grid, a straight line is defined as a shortest path between two points. Sometimes there can be more than one, as demonstrated in the example below. In this case, between the corner of 21st (Avenue) and 5th (Street) and 23rd and 7th, the minimum path length (distance) is 4. We write these as (21,5) and (23,7). Two possible paths of length 4 are:

We could describe the paths as a series of points:

We could also describe the paths as a sequence of directions:

(a) There are four other paths between points (21,5) and (23,7) of length 4. List these four paths. (You can write your answer by using either of the

two ways above) [2]

North

(21,5); (22,5); (23,5); (23,6); (23,7)

(21,5); (21,6); (22,6); (22,7); (23,7)

East, East, North, North

North, East, North, East



© QQA Issue: 4

(b) The following journeys all have a shortest path of 5. What is the total number of paths of length 5 in each case?

(b1) From (3,4) to (8,4). (b2) From (4,5) to (8,4). (b3) From (6,7) to (8,4). [5] (c) The road between (4,4) and (5,4) is closed for roadworks. For each case

below, identify the length of the shortest path and the new total number of shortest paths:

(c1) From (3,4) to (8,4). (c2) From (4,5) to (8,4). (c3) From (6,7) to (8,4). [7]

Automatic equipment for storage has a similar way to measure distance when items are moved from one cube to another in three dimensions. In this case we need to use 3 numbers (coordinates) to specify where an item is. Of course, the coordinates can only be whole numbers. In this world of automatic storage, a sphere is a set of points which are all the same distance from the centre. (d) List all the points on the sphere of distance 1 from (3,4,5). [2] (e) How many shortest paths are there in each case: (e1) (4,3,3) to (8,3,3)? (e2) (4,3,3) to (3,4,5)? [4]



© QQA Issue: 4

7.3 Mark scheme for Paper 2 examples

1 (a) State the arrival times at the Castle bus stop of all the buses that

will continue from Linkham to Ferrin on 24 July. (The bus makes the journey whether or not the Castle is open). [3]

09:40, 10:45 and 18:45 – award 3 marks Deduct 1 mark for each extra time (minimum 0 marks). If a candidate has selected the three correct buses, but consistently given the times for one of the other bus stops (e.g. 09:05, 10:10 and 18:10) award 1 mark. Possible 1 mark combinations [any row of this table]

First Second Last

09:05 10:10 18:10

09:10 10:15 18:15

09:24 10:29 18:15

09:26 10:31 18:31

09:40 10:45 18:45

10:05 11:05 18:45

10:19 11:19 18:59

10:21 11:21 19:01

10:35 11:35 19:15

10:40 11:40 19:30

(b) At what time will the Castle open, and at what time will it close: (b1) On 17 July? [2] Castle will open at 09:55 – award 1 mark Castle will close at 16:55 – award 1 mark (b2) On 20 July? [2] Castle will open at 12:10 – award 1 mark Castle will close at 18:00 – award 1 mark Skill: Extracting relevant data. (c) The first date in July when the Castle will be closed all day is 6 July.

On what date will the Castle next open? Show clearly how you reach your conclusion. [2]

9 July – award 2 marks



© QQA Issue: 4

If a candidate does not state the correct answer, but shows evidence of having calculated how long the castle is open on 7, 8, 9 or 10 July, then award 1 mark: Working:

Date Time which the castle could open

(given causeway openings)

7 2 hours

8 2 hours 40 minutes



(d) Sara and Dana will be staying in Barwell from 2 July to 12 July. During their holiday they would like to take the bus to Ferrin. However, they will only do so if they will be able to spend at least five hours exploring the Island, including a visit to the Castle.

On which date(s) would this be possible? Show all your working, and for each date that is possible state the earliest arrival time at, and the latest departure time from, the Castle bus stop. [6]

4 July – award 1 mark Earliest arrival 12:45 – award 1 mark Latest departure 18:45 – award 1 mark 9 July – award 1 mark Earliest arrival 09:40 – award 1 mark Latest departure 18:45 – award 1 mark Deduct 1 mark for each extra date given in addition to 4 July and 9 July (minimum 0 marks). If only one correct date is given do not deduct mark unless two or more extra dates are given. Each of the marks for bus times can only be awarded if linked to a correct date. If a candidate gives the answer 7 July or 8 July to question (c), then do not deduct marks for these dates appearing in question (d). Just ignore them Skill: Analysing complex data and drawing conclusions.



© QQA Issue: 4

2 (a) If a cage with dimensions 10 m, 11 m and 15 m was to be

constructed, what is the smallest total area of netting that would be needed? [2]

535 metres2 needed (if L = 15) if length is 10 metres, 590 metres2 of netting is needed if length is 11 metres, 575 metres2 of netting is needed if length is 15 metres, 535 metres2 of netting is needed 1 mark for a correct calculation of surface area that is not optimal [590 or 575] Also 1 mark available for candidates who state that the sides must be as small as possible (or that the width and height must be as small as possible), if they have not managed to get any of the actual areas correct. (b) Consider possible cages which are of the same height as the wall.

What is the maximum length that the cage can be? Justify your answer [3]

13 metres - award 1 mark Justification: 13 metres gives 582 m2 - award 1 mark 14 metres gives 606 m2 - award 1 mark If none of the above information is given, award 1 mark for considering any case involving a height of 15 m, a width of 9 m and any length: any attempt to calculate the area of netting. Full marks for an algebraic justification: equation + length = 13.75 m seen (c) What is the largest volume cage that can be built, with a height of

14 metres? Justify your answer. [3] 1890 metres3 3 marks if EITHER the candidates gives this answer and consider the next solution up [14 × 16 × 9] OR Algebraic justification is given: 252 + 23x < 600 [1 mark if this is sighted (regardless of a correct answer or not)] 23x < 348 X = 15.13… If any of these lines of an algebraic solution are sighted, give a mark for justification. 2 marks if the candidate offers 1890 metres3 as a solution without support. 1 mark if the candidate considers frameworks with dimensions 14 × 9 ×?



© QQA Issue: 4

(d) Show that it is possible to fulfil the outcome of a minimum volume of 1900 m3. Give the length, width and height of the cage which does this. [3]

Any of the shaded and emboldened solutions offered in the table below

Height length width volume netting

9 14 14 1764 574

9 15 14 1890 597

9 16 13 1872 586

9 17 12 1836 573

9 18 12 1944 594

9 19 11 1881 578

9 20 11 1980 598

10 14 13 1820 582

10 15 12 1800 570

10 16 12 1920 592

10 17 11 1870 577

10 18 11 1980 598

10 19 10 1900 580

10 20 9 1800 560

11 13 13 1859 598

11 14 12 1848 586

11 15 11 1815 572

11 16 11 1936 594

12 13 12 1872 600

2 marks for a correct solution, in which either the dimensions, the area of netting or the volume is omitted. OR 2 marks for one of the suboptimal solutions given in the table, as long as it does not have height or width 14 m [since these solutions are prompted by the question in part (c)]. 1 mark for a suboptimal solution, in which either the dimensions, the area of netting or the volume is omitted. (e) It is decided that the wall is bad for the birds’ welfare, and is to be

demolished. Plans are drawn up for the new cage [which must have 5 netted sides, now the wall has gone].



© QQA Issue: 4

Four 10 metre struts are found, and it is decided that these should be used for the cage [i.e. at least one of the dimensions should be 10 metres]. The other struts are subject to the same restrictions as before. What is the minimum extra amount of netting needed if the cage is still to fulfil the 1900 m3 requirement? [4]

152 m2 of netting 4 marks for this answer If candidate does not reach this answer award as following: 1 mark for a solution in which the height measurement is the smallest of the three. AND 1 mark for subtracting 600 m2 from the amount of netting they find, in order to calculate the extra netting needed. AND 1 mark for calculating any of the suboptimal solutions given below (not the shaded and emboldened, optimal solution).

Length width height volume netting

22 10 9 1980 796

19 10 10 1900 770

18 11 10 1980 778

16 12 10 1920 752

15 13 10 1950 755

14 14 10 1960 756



© QQA Issue: 4

2 (Second example) (a) What is the total price for a plate of 50 meat sandwiches and a plate

of 20 salad sandwiches? [1] A plate of 50 meat sandwiches costs BD 40. A plate of 20 salad sandwiches costs BD 12. The total cost is BD 52. 1 mark for the answer of BD 52 Skill tested: PS1 (b) If it costs 400 fils to make each salad sandwich, what is the

percentage profit made when a plate of 20 salad sandwiches is sold? [2]

20 salad sandwiches would cost BD 8 to make So the profit is BD 4. This is a 50% profit Award 2 marks for the answer 50% If 2 marks cannot be awarded, award 1 mark for identifying the profit of BD 4. Skills tested: PS2, PS9.

(c) Explain how it is possible to buy exactly 30 salad sandwiches and

50 meat sandwiches. [2] There are two possible answers: 1 plate of 20 salad, 2 plates of 20 meat and 1 mixed plate of 20. or 1 plate of 20 meat and 3 mixed plates of 20. 2 marks for either solution. If 2 marks cannot be awarded, award 1 mark for any solution that includes a mixed plate of 20 sandwiches. Skill tested: PS4. (d) A meeting requires exactly 70 sandwiches, of which at least 20

must be meat sandwiches. What is the lowest cost that could be paid for the order? [2]

1 plate of 20 meat and 1 plate of 50 salad. The cost is BD 38. 2 marks for BD 38. If 2 marks cannot be awarded, award 1 mark for identifying one of the following combinations of plates: 20 meat and 50 salad. 20 salad and 50 meat.



© QQA Issue: 4

20 salad and 50 mixed. Skills tested: PS4, PS9. (e) A businessman, Zeid orders two plates of sandwiches every week.

This week he has made a change to his order and the total price has gone up by BD 9.

Suggest how Zeid’s order might have been changed. [2] Award 2 marks for any of the following solutions: A mixed plate of 50 has been changed to a plate of 50 meat sandwiches. A plate of 20 salad has been changed to a plate of 50 salad and a mixed plate of 20 has been changed to a plate of 20 meat. If 2 marks cannot be awarded, award 1 mark for a solution that achieves an increase of BD 9, but involves an original order containing more than 2 plates of sandwiches. Skills tested: PS1 and PS6. (f) Zeid wants to place a large order next week and realises that by

splitting the single order into multiple smaller orders he can save some money. Zeid wants to order 3 plates of 50 salad sandwiches, and 7 plates of 50 meat sandwiches. What is the smallest amount that he could have to pay in total for the sandwiches? [3]

A plate of 50 salad sandwiches costs BD 20. A plate of 50 meat sandwiches costs BD 40. If all 3 salad plates are in one order then the discount can be applied to one plate of salad sandwiches and two plates of meat sandwiches. The total cost would be:

2 at BD 20 and 1 at BD 10 for the salad sandwiches 5 at BD 40 and 2 at BD 20 for the meat sandwiches

The total cost is BD 290 Award 3 marks for BD 290 If 3 marks cannot be awarded: Award 1 mark for applying the discount to three of the plates. and/or Award 1 mark for identifying that at least one plate of salad sandwiches must be reduced (can be implied by the discount being applied to one of the plates of salad sandwiches). Skills tested: PS3, PS4 and PS9.



© QQA Issue: 4

(g) Mohammed decided to set the prices for salad sandwiches as below.

Price per sandwich

Reduced price



This means that 20 salad sandwiches will still cost BD 12 and 50

salad sandwiches will still cost BD 20. What will be the price of 100 salad sandwiches in Mohammed’s

system? [1] [1] 30 sandwiches at 600 fils will cost BD 18. 70 sandwiches at 100 fils will cost BD 7. The total price is BD 25 Award 1 mark for BD 25 Skill tested: PS2, PS9. (h) Develop one possible new pricing structure for the meat

sandwiches. 20 meat sandwiches should still cost BD 18 and 50 meat sandwiches should still cost BD 40. [2] Award 2 marks for any of the possibilities shown below:

Price per sandwich

Reduced price






If 2 marks cannot be awarded, award 1 mark for a price per sandwich of 900 fils. Skills tested: PS3, PS11. 3 (a) There are four other paths between points (21,5) and (23,7) of length 4. List these four paths. (You can write your answer by using either of the two ways above) [2] Award 1 mark for correctly identifying two of the paths. Award 2 marks for correctly identifying all four paths. Using the points to refer to the path: (21,5); (21,6); (21,7); (22,7); (23,7) (21,5); (21,6); (22,6); (23,6); (23,7)



© QQA Issue: 4

(21,5); (22,5); (22,6); (22,7); (23,7) (21,5); (22,5); (22,6); (23,6); (23,7) Using directions to refer to the path: North, North, East, East North, East, East, North East, North, North, East East, North, East, North Note: The other two paths of length 4 are provided in the question. [Total 2 marks] (b) The following journeys all have a shortest path of 5. What is the

total number of paths of length 5 in each case? [5] (b1) From (3,4) to (8,4) Award 1 mark for identifying that there is only one path: (4,4); (5,4); (6,4); (7,4) or East, East, East... (a description of the path is not required). [Total 1 mark] (b2) From (4,5) to (8,4) Using points: The jump from (*,5) to (*,4) can occur in 5 places, so five paths. Using directions: There must be one South within the 5 steps, so five paths. Award 2 marks for identifying that there are five paths. Award 1 mark for identifying two correct paths. [Total 2 marks] (b3) From (6,7) to (8,4) We can construct a table of the number of paths to each point along the way: 0 1 1 1 2 3 1 3 6 1 4 10 There are ten paths.



© QQA Issue: 4

Alternatively, the movement must require 2 East and 3 South steps. Any order is acceptable, so a list can be found systematically. Award 2 marks for identifying that there are ten paths. Award 1 mark for identifying two correct paths. [Total 2 marks] (c) The road between (4,4) and (5,4) is closed for roadworks. For each

case below, identify the length of the shortest path and the new total number of shortest paths: [7]

(c1) From (3,4) to (8,4) Award 1 mark for identifying that the shortest length is now 7. The route must travel either North or South at the first or second step. The route must also travel back South or North (i.e. the opposite direction to the one in the first two steps) in the fourth, fifth, sixth or seventh steps. All other steps must be East, so there are 2 × 2 × 4 = 16 paths. We can construct a table of the number of paths to each point along the way:

1 2 2 2 2 2 0 1R 4 8 12 16 1 2 2 2 2 2

Where R indicates the position of the roadwork. Award 1 mark for identifying a route of length 7. Award 1 mark for identifying that there are 16 paths. [Total 3 marks] (c2) From (4,5) to (8,4) Award 1 mark for identifying that the shortest length is still 5. The shortest path must make an East step first, but can then take the South step in any of the other 4 steps. the other 4 steps.



© QQA Issue: 4

Award 1 mark for identifying that there are 4 paths. [Total 2 marks] (c3) From (6,7) to (8,4) Award 1 mark for identifying that the shortest length is still 5. The closed road is not on any of the shortest paths, so the number is still 10 paths. Award 1 mark for identifying that there are 10 paths. Note: If an incorrect answer for the number of paths is given and it is the same as the answer to (b3) then award the mark (Error carried forward). [Total 2 marks] (d) List all the points on the sphere of distance 1 from (3,4,5). [2] By adding or subtracting one from each coordinate in turn: (2,4,5) (4,4,5) (3,3,5) (3,5,5) (3,4,4) (3,4,6) [1 mark for any three correct] (e) How many shortest paths are there in each case: [4] (e1) (4,3,3) to (8,3,3)? There is only one path as there is only a need to move in one direction. Award 1 mark for identifying that there is one path. [Total 1 mark] (e2) (4,3,3) to (3,4,5)? The permutations of (a move of 1 unit in) x, y, z and z can be listed: xyzz yxzz zxyz zyzx xzyz yzxz zyxz zzxy



© QQA Issue: 4

xzzy yzzx zxzy zzyx Where x, y and z may be seen as West, North and Up respectively as an extension of the notation provided in the first section of the question. 1 mark for evidence of a systematic approach. or 2 marks for obtaining 6 paths to (3,4,4), or listing 6 correct permutations. Max 3 marks for an answer of twelve paths. [Total 3 marks] [Total for question: 20 marks]

directorate of national examinationل...grade 12 problem solving test specification 2014 date...

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