Grade 9 Assessment Of Mathematics 2012 Answers

EXPLORE PISA 2012 MATHEMATICS, PROBLEM SOLVING AND FINANCIAL LITERACY TEST QUESTIONS

The OECD's Programme for International Student Assessment (PISA) evaluates education systems worldwide by testing 15-year-olds in key subjects. The focus of PISA 2012 was mathematics. Some countries chose to assess other subjects too such as problem-solving and financial literacy. To understand more about the PISA 2012 mathematics, problem-solving and financial literacy tests, click below to answer sample questions. You can also explore the concepts and skills being tested and learn what 15-year-olds students at different proficiency levels can do.

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Test Levels

The PISA test contains questions representing 6 levels of proficiency. Learn more about the levels for each subject.

Question Categories

The PISA test contains question which provide different contexts and test different skills. Click on the subject below to learn more about the subject categories and skills being tested.

At Level 1, students can explore a problem scenario only in a limited way, but tend to do so only when they have encountered very similar situations before. Based on their observations of familiar scenarios, these students are able only to partially describe the behaviour of a simple, everyday device. In general, students at Level 1 can solve straightforward problems provided there is only a simple condition to be satisfied and there are only one or two steps to be performed to reach the goal. Level 1 students tend not to be able to plan ahead or set sub-goals.

At Level 2, students can explore an unfamiliar problem scenario and understand a small part of it. They try, but only partially succeed, to understand and control digital devices with unfamiliar controls, such as home appliances and vending machines. Level 2 problem-solvers can test a simple hypothesis that is given to them and can solve a problem that has a single, specific constraint. They can plan and carry out one step at a time to achieve a sub-goal, and have some capacity to monitor overall progress towards a solution.

At Level 3, students can handle information presented in several different formats. They can explore a problem scenario and infer simple relationships among its components. They can control simple digital devices, but have trouble with more complex devices. Problem-solvers at Level 3 can fully deal with one condition, for example, by generating several solutions and checking to see whether these satisfy the condition. When there are multiple conditions or inter-related features, they can hold one variable constant to see the effect of change on the other variables. They can devise and execute tests to confirm or refute a given hypothesis. They understand the need to plan ahead and monitor progress, and are able to try a different option if necessary.

At Level 4, students can explore a moderately complex problem scenario in a focused way. They grasp the links among the components of the scenario that are required to solve the problem. They can control moderately complex digital devices, such as unfamiliar vending machines or home appliances, but they don't always do so efficiently. These students can plan a few steps ahead and monitor the progress of their plans. They are usually able to adjust these plans or reformulate a goal in light of feedback. They can systematically try out different possibilities and check whether multiple conditions have been satisfied. They can form an hypothesis about why a system is malfunctioning, and describe how to test it.

At Level 5, students can systematically explore a complex problem scenario to gain an understanding of how relevant information is structured. When faced with unfamiliar, moderately complex devices, such as vending machines or home appliances, they respond quickly to feedback in order to control the device. In order to reach a solution, Level 5 problem-solvers think ahead to find the best strategy that addresses all the given constraints. They can immediately adjust their plans or backtrack when they detect unexpected difficulties or when they make mistakes that take them off course.

At Level 6, students can develop complete, coherent mental models of diverse problem scenarios, enabling them to solve complex problems efficiently. They can explore a scenario in a highly strategic manner to understand all information pertaining to the problem. The information may be presented in different formats, requiring interpretation and integration of related parts. When confronted with very complex devices, such as home appliances that work in an unusual or unexpected manner, they quickly learn how to control the devices to achieve a goal in an optimal way. Level 6 problem-solvers can set up general hypotheses about a system and thoroughly test them. They can follow a premise through to a logical conclusion or recognise when there is not enough information available to reach one. In order to reach a solution, these highly proficient problem-solvers can create complex, flexible, multi-step plans that they continually monitor during execution. Where necessary, they modify their strategies, taking all constraints into account, both explicit and implicit.

At Level 1 students can answer questions involving familiar contexts where all relevant information is present and the questions are clearly defined. They are able to identify information and to carry out routine procedures according to direct instructions in explicit situations. They can perform actions that are obvious and follow immediately from the given stimuli.

At Level 2 students can interpret and recognise situations in contexts that require no more than direct inference. They can extract relevant information from a single source and make use of a single representational mode. Students at this level can employ basic algorithms, formulae, procedures, or conventions. They are capable of direct reasoning and making literal interpretations of the results.

At Level 3 students can execute clearly described procedures, including those that require sequential decisions. They can select and apply simple problem solving strategies. Students at this level can interpret and use representations based on different information sources and reason directly from them. They can develop short communications reporting their interpretations, results and reasoning.

At Level 4 students can work effectively with explicit models for complex concrete situations that may involve constraints or call for making assumptions. They can select and integrate different representations, including symbolic, linking them directly to aspects of real-world situations. Students at this level can utilise well-developed skills and reason flexibly, with some insight, in these contexts. They can construct and communicate explanations and arguments based on their interpretations, arguments, and actions.

At Level 5 students can develop and work with models for complex situations, identifying constraints and specifying assumptions. They can select, compare, and evaluate appropriate problem solving strategies for dealing with complex problems related to these models. Students at this level can work strategically using broad, well-developed thinking and reasoning skills, appropriate linked representations, symbolic and formal characterisations, and insight pertaining to these situations. They can reflect on their actions and formulate and communicate their interpretations and reasoning.

At Level 6 students can conceptualise, generalise, and utilise information based on their investigations and modelling of complex problem situations. They can link different information sources and representations and flexibly translate among them. Students at this level are capable of advanced mathematical thinking and reasoning. These students can apply this insight and understandings along with a mastery of symbolic and formal mathematical operations and relationships to develop new approaches and strategies for attacking novel situations. Student at this level can formulate and precisely communicate their actions and reflections regarding their findings, interpretations, arguments, and the appropriateness of these to the original situations.

Students can identify common financial products and terms and interpret information relating to basic financial concepts. They can recognise the difference between needs and wants and can make simple decisions on everyday spending. They can recognise the purpose of everyday financial documents such as an invoice and apply single and basic numerical operations (addition, subtraction or multiplication) in financial contexts that they are likely to have experienced personally.

Students begin to apply their knowledge of common financial products and commonly used financial terms and concepts. They can use given information to make financial decisions in contexts that are immediately relevant to them. They can recognise the value of a simple budget and can interpret prominent features of everyday financial documents. They can apply single basic numerical operations, including division, to answer financial questions. They show an understanding of the relationships between different financial elements, such as the amount of use and the costs incurred.

Students can apply their understanding of commonly used financial concepts, terms and products to situations that are relevant to them. They begin to consider the consequences of financial decisions and they can make simple financial plans in familiar contexts. They can make straightforward interpretations of a range of financial documents and can apply a range of basic numerical operations, including calculating percentages. They can choose the numerical operations needed to solve routine problems in relatively common financial literacy contexts, such as budget calculations

At level 4 students can apply their understanding of less common financial concepts and terms to contexts that will be relevant to them as they move towards adulthood, such as bank account management and compound interest in saving products. They can interpret and evaluate a range of detailed financial documents, such as bank statements, and explain the functions of less commonly used financial products. They can make financial decisions taking into account longer-term consequences, such as understanding the overall cost implication of paying back a loan over a longer period, and they can solve routine problems in less common financial contexts.

At level 5 students can apply their understanding of a wide range of financial terms and concepts to contexts that may only become relevant to their lives in the long term. They can analyse complex financial products and can take into account features of financial documents that are significant but unstated or not immediately evident, such as transaction costs. They can work with a high level of accuracy and solve non-routine financial problems, and they can describe the potential outcomes of financial decisions, showing an understanding of the wider financial landscape, such as income tax.

Space and Shape encompasses a wide range of phenomena that are encountered everywhere in our visual and physical world: patterns, properties of objects, positions and orientations, representations of objects, decoding and encoding of visual information, navigation and dynamic interaction with real shapes as well as with representations. Geometry serves as an essential foundation for space and shape, but the category extends beyond traditional geometry in content, meaning and method, drawing on elements of other mathematical areas such as spatial visualisation, measurement and algebra.

Change & Relationships involves understanding fundamental types of change and recognising when they occur in order to use suitable mathematical models to describe and predict change. Mathematically this means modelling the change and the relationships with appropriate functions and equations, as well as creating, interpreting, and translating among symbolic and graphical representations of relationships.

Quantity may be the most pervasive and essential mathematical aspect of engaging with, and functioning in, our world. Engaging with the quantification of the world involves understanding measurements, counts, magnitudes, units, indicators, relative size, and numerical trends and patterns. Aspects of quantitative reasoning – such as number sense, multiple representations of numbers, elegance in computation, mental calculation, estimation and assessment of reasonableness of results – are the essence of mathematical literacy relative to quantity.

Uncertainty is a phenomenon at the heart of the mathematical analysis of many problem situations, and the theory of probability and statistics as well as techniques of data representation and description have been established to deal with it. The uncertainty and data content category includes recognising the place of variation in processes, having a sense of the quantification of that variation, acknowledging uncertainty and error in measurement, and knowing about chance.

Personal questions include those focusing on activities of one's self, one's family or one's peer group e.g. food preparation, shopping, games, personal health, personal transportation, sports, travel, personal scheduling and personal finance.

Occupational questions are centred on the world of work e.g. measuring, costing and ordering materials for building, payroll/accounting, quality control, scheduling/inventory, design/architecture and job-related decision making.

Societal questions focus on one's community (whether local, national or global) e.g. voting systems, public transport, government, public policies, demographics, advertising, national statistics and economics. Although individuals are involved in all of these things in a personal way, in the societal context category the focus of problems is on the community perspective.

Scientific questions relate to the application of mathematics to the natural world and issues and topics related to science and technology e.g. weather or climate, ecology, medicine, space science, genetics, measurement and the world of mathematics itself.

Formulating situations mathematically means being able to recognise and identify opportunities to use mathematics and then provide mathematical structure to a problem presented in some contextualised form.

Employing mathematical concepts, facts, procedures and reasoning means being able to apply mathematical concepts, facts, procedures, and reasoning to solve mathematically-formulated problems to obtain mathematical conclusions.

Interpreting, applying and evaluating mathematical outcomes means the ability to reflect upon mathematical solutions, results, or conclusions and interpret them in the context of real-life problems.

The problem is interactive when not all information is disclosed at the outset and some information has to be uncovered by exploring the problem situation. The problem is static when all relevant information for solving the problem is disclosed at the outset.

The problem belongs to the technology category when it involves a technological device. It belongs to the non-technology category when there is no technological device.

The problem belongs to the personal category when the focus is on one's self, family or close peers. It belongs to the social category when the focus is the community or society in general.

Exploring and understanding involves building mental representations of each of the pieces of information presented in the problem. This includes exploring the problem situation: observing it, interacting with it, searching for information and finding limitations or obstacles; and understanding given information and information discovered while interacting with the problem situation; demonstrating understanding of relevant concepts.

Representing and formulating involves building a coherent mental representation of the problem situation. To do this, relevant information must be selected, mentally organised and integrated with relevant prior knowledge. This may involve: representing the problem by constructing tabular, graphical, symbolic or verbal representations, and shifting between representational formats; and formulating hypotheses by identifying the relevant factors in the problem and their interrelationships; organising and critically evaluating information.

Planning and executing means planning by setting goals, including clarifying the overall goal, and setting sub-goals, where necessary; and devising a plan or strategy to reach the goal state, including the steps to be undertaken; as well as executing by carrying out a plan.

Monitoring and reflecting involves monitoring progress towards the goal at each stage, including checking intermediate and final results, detecting unexpected events, and taking remedial action when required. Reflecting involves reflecting on solutions from different perspectives, critically evaluating assumptions and alternative solutions, identifying the need for additional information or clarification and communicating progress in a suitable manner.

This content area includes the awareness of the different forms and purposes of money and handling simple monetary transactions such as everyday payments, spending, value for money, bank cards, cheques, bank accounts and currencies.

Income and wealth need planning and managing over both the short term and long term. This content area includes knowledge and ability to monitor income and expenses as well as knowledge and ability to make use of income and other available resources in the short and long terms to enhance financial well-being:

Risk and reward is a key area of financial literacy, incorporating the ability to identify ways of managing, balancing and covering risks and an understanding of the potential for financial gains or losses across a range of financial contexts.

There are two types of risk of particular importance in this domain. The first relates to financial losses that an individual cannot bear, such as those caused by catastrophic or repeated costs. The second is the risk inherent in financial products, such as credit agreements with variable interest rates, or investment products.

This content area relates to the character and features of the financial world. It covers knowing the rights and responsibilities of consumers in the financial marketplace and within the general financial environment, and the main implications of financial contracts. Information resources and legal regulation are also topics relevant to this content area. In its broadest sense, financial landscape also incorporates an understanding of the consequences of changes in economic conditions and public policies, such as changes in interest rates, inflation, taxation or welfare benefits.

Education and work questions include understanding payslips, planning to save for higher education, investigating the benefists and risks of taking out a student loan, and participating in workplace savings schemes.

Home and family questions include financial issues relating to the costs involved in running a household such as buying household items or family groceries, keeping records of family spending and making plans for family events. Decisions about budgeting and prioritising spending are also included in this context.

Individual questions include contractual issues around events such as opening a bank account, purchasing consumer goods, paying for recreational activities and dealing with relevant financial services that are often associated with larger consumption items, such as credit and insurance.

Societal questions cover matters such as consumer rights and responsibilities, taxes and local government charges, business interests, and consumer purchasing power. Financial choices such as donating to non-profit organisations and charitites are also included in this context.

This process is engaged when the individual searches and accesses sources of financial information, and identifies or recognises its relevance. The information is in the form of printed texts such as contracts, advertisements, charts, tables, forms and instructions. A typical task might ask students to identify the features of a purchase invoice, or recognise the balance on a bank statement.

Analysing information in a financial context includes interpreting, comparing and contrasting, synthesising, and extrapolating from information that is provided. It involves recognising something that is not explicit: identifying the underlying assumptions or implications of an issue in a financial context e.g. a task may involve comparing the terms offered by different mobile phone contracts, or working aout whether an advertisement for a loan in likely to include unstated conditions.

Evaluating financial issues involves recognising or constructing financial justifications and explanations, drawing on financial knowledge and understanding applied in specified contexts. It involves explaining, assessing and generalising. It also involves critical thinking such as drawing on knowledge, logic and plausible reasoning to make sense of and form a view about a finance-related problem.

Applying financial knowledge and understanding focuses on taking effective action in a financial setting by using knowledge of financial products and contexts, and understanding of financial concepts. This process is reflected in tasks that involve performing calculations and solving problems, often taking into account multiple conditions. Examples of these kinds of tasks are calculating the interest on a loan over two years or working out whether purchasing power will decline or increase over time when prices are changing at a given rate.