Learning Area: Mathematics
Phase: Intermediate (Grades 4-6)

Overview of the Learning Outcomes

Learning Outcome 1: Numbers, Operations and Relationships

The learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems.

Learning Outcome focus

Learning Outcome 1 builds the learner’s number sense, which is the foundation of further study in Mathematics.

It also develops the learner’s understanding of:
  • what different kinds of numbers mean;
  • how different kinds of numbers relate to one another;
  • the relative size of different numbers;
  • how different numbers can be thought about and represented in various ways; and
  • the effect of operating with numbers.

Essential to the development of number sense is knowledge of basic number facts, the use of efficient and accurate methods for calculation and measurement, and a range of strategies for estimating and checking results.

Learning Outcome 1 also provides opportunities for the learner to use appropriate technology and to engage with the historical and cultural developments of numerical counting and writing systems. Learners with a good sense of number and operations have the mathematical confidence to make sense of problems and results in various contexts.

Contexts should be selected in which the learner has to count, estimate and calculate in a way that builds awareness of other Learning Areas, as well as human rights, social, economic, cultural, political and environmental issues. For example, the learner should be able to:
  • compare counting in different African languages and relate this to the geographical locations of the language groups;
  • count animals in the environment with an awareness of animals at risk of becoming extinct;
  • compare national health statistics with an awareness of how learners’ own regions are affected;
  • calculate and compare the ratios of elements in a chemical compound;
  • interpret climatic conditions (e.g. read temperature and rainfall information); and
  • calculate within financial contexts found in the Economic and Management Sciences Learning Area.

Intermediate Phase focus

The range of numbers developed by the end of Grade 6 is extended to at least 9-digit whole numbers, decimal fractions to at least 2 decimal places, common fractions and fractions written in percentage form.

In this phase, the learner is expected to move from counting reliably to calculating fluently with all four operations. The learner should be encouraged to:
  • memorise multiplication fluently to at least 12 x 12;
  • sharpen mental calculation skills; and
  • use calculators confidently.

Since the range of numbers that the learner works with is continually expanding, much attention needs to be focused on understanding the concept of place value so that the learner develops a sense of large numbers and decimal fractions.

Fraction concepts should be expanded through the use of fraction walls and number-lines to compare and order fractions. Measurement is a useful context in which the learner can practice calculations with fractions, including decimal fractions.

Through the study of a variety of number patterns, the learner should recognise and describe properties of numbers and operations, including identity properties, factors, multiples, and commutative, associative and distributive properties. The purpose should be for the learner to recognise:
  • what the properties are; and
  • how they can be used to solve problems and simplify calculations.

Learning Outcome 2: Patterns, Functions and Algebra

The learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills.

Learning Outcome focus

Algebra is the language for investigating and communicating most of Mathematics. Algebra can be seen as generalised arithmetic, and can be extended to the study of functions and other relationships between variables. A central part of this outcome is for the learner to achieve efficient manipulative skills in the use of algebra.

Learning Outcome 2 focuses on:
  • describing patterns and relationships through the use of symbolic expressions, graphs and tables; and
  • identifying and analysing regularities and change in patterns and relationships that enable learners to make predictions and solve problems.

Investigating patterns and relationships allows the learner to develop an appreciation of the aesthetic and creative qualities of Mathematics. These investigations develop mathematical thinking skills such as generalising, explaining, describing, observing, inferring, specialising, creating, justifying, representing, refuting and predicting.

Mathematical skills are developed over time. The learner should be given opportunities at every grade level to develop these skills to greater levels of sophistication so that they can be used with greater competence and confidence.

Contexts should be selected in which the learner can use algebraic language and skills to describe patterns and relationships in a way that builds awareness of other Learning Areas, as well as human rights, social, economic, cultural, political and environmental issues. For example, the learner should be able to:
  • investigate geometric patterns in art and architecture;
  • study symmetrical patterns that occur in nature;
  • understand formulae used to calculate pensions and medical aid rates;
  • understand and use formulae for calculating quantities encountered in Natural Sciences (e.g. air pressure, resistance, voltage);
  • consider how graphs in the media can be manipulated to misrepresent trends and patterns; and
  • use mathematical models to represent relationships within an ecosystem.

Intermediate Phase focus

In the Intermediate Phase, the study of numeric and geometric patterns is extended with a special focus on the relationships:

  • between terms in a sequence; and
  • between the number of the term (its place in the sequence) and the term itself.

The study of numeric and geometric patterns develops the concepts of variable, relationship and function. The understanding of these relationships by the learner will allow her or him to describe the rules generating the patterns.

This phase has a particular focus on the use of different, yet equivalent, representations to describe problems or relationships by means of flow diagrams, tables, number sentences or verbally.

Graphs are not dealt with in this Learning Outcome in the Intermediate Phase. However, the learner is given opportunities to read, interpret and draw graphs within data contexts (see Learning Outcome 5).

Learning Outcome 3: Space and Shape (Geometry)

The learner will be able to describe and represent characteristics and relationships between two-dimensional shapes and three-dimensional objects in a variety of orientations and positions.

Learning Outcome focus

The study of space and shape improves understanding and appreciation of the pattern, precision, achievement and beauty in natural and cultural forms. It focuses on the properties, relationships, orientations, positions and transformations of two-dimensional shapes and three-dimensional objects.

The study of space and shape enables the learner to:
  • develop the ability to visualise, interpret, calculate relevant values, reason and justify; and
  • interpret, understand, classify, appreciate and describe the world through two-dimensional shapes and three-dimensional objects, their location, movement and relationships.

The learner should gain these skills from experiences with concrete objects, through drawing and construction, and in the abstract justification of spatial relationships. It is important that the study of two-dimensional shapes and three-dimensional objects be contextualised to include the study of natural and cultural forms and artefacts.

Contexts should be selected in which the learner can study space and shape in a way that can be used to build awareness of other Learning Areas, as well as human rights, social, economic, cultural, political and environmental issues. For example, the learner should be able to:
  • use national flags to demonstrate transformations and symmetry in designs;
  • investigate and recognise the geometrical properties and patterns existing in traditional and modern architecture;
  • use maps in Geography as specific forms of grids; and
  • investigate geometric patterns in art.

Intermediate Phase focus

The learner’s experience of space and shape in this phase moves from recognition and simple description to classification and more detailed description of features and properties of two-dimensional shapes and three-dimensional objects.

Learners should be given opportunities to:

  • draw two-dimensional shapes and make models of three-dimensional objects; and
  • describe location, transformations and symmetry.

Learning Outcome 4: Measurement

The learner will be able to use appropriate measuring units, instruments and formulae in a variety of contexts.

Learning Outcome focus

The ability to measure appropriately has been developed by humans over time and through various cultures. Measurement focuses on the selection and use of appropriate units, instruments and formulae to quantify characteristics of events, shapes, objects and the environment. Measuring relates directly to the scientific, technological and economic worlds of the learner, enabling the learner to:

  • make sensible estimates; and
  • be alert to the reasonableness of measurements and results.
Contexts should be selected in which the learner can measure in a way that builds awareness of other Learning Areas, as well as human rights, social, political, economic, cultural and environmental issues. For example, the learner should be able to:
  • measure and compare distances and times taken by learners from home to school;
  • compare the capacity of dams and the volume of water available through taps in a particular community;
  • measure working hours and their relation to income earned;
  • compare the distribution and allocation of land areas to population size; and
  • use measuring units from Technology, Natural Sciences and Social Sciences.

Intermediate Phase focus

In this phase, the learner is introduced to the use of standardised units of measurement and appropriate instruments for measuring. The learner should be able to estimate and verify results through accurate measurement.

A useful teaching and learning strategy is to expose the learner to a variety of measurement activities that will make it possible to select and convert between appropriate units of measurement. Measurement is an opportunity to use common fractions and decimals in context.

Measurement in this phase should also enable the learner to:

  • informally measure angles, area, perimeter and capacity/volume; and
  • discuss and describe the historical development of measuring instruments and tools.

Learning Outcome 5: Data Handling

The learner will be able to collect, summarise, display and critically analyse data in order to draw conclusions and make predictions, and to interpret and determine chance variation.

Learning Outcome focus

Data – meaning information in statements, graphs and tables – bombard our senses through television, newspapers and other media. For example, crime rates, rainfall, sports results, election polls, government spending, population and economic growth are all reported in graphical or summarised form as statistics.

Through the study of data handling, the learner develops the skills to collect, organise, display, analyse and interpret this information. This enables the learner to participate meaningfully in political, social and economic activities.

Making sense of data involves collecting, organising, analysing, summarising and interpreting it, as well as drawing conclusions and making predictions.

Through the study of chance, the learner will also develop skills and techniques for making informed choices, and coping with randomness and uncertainty.

In this Learning Outcome, the learner will develop a sense of how Mathematics can be used to manipulate data to represent or misrepresent trends and patterns. The learner will also develop a sense of how Mathematics can provide solutions that sustain or destroy the environment, and promote or harm the health of others. The learner is thereby able to use Mathematics effectively and critically, showing responsibility towards the environment and health of others. For example, the learner should be able to understand:
  • the distribution of resources according to class, race or gender; and
  • economic trends and patterns between developing countries and developed countries.

Intermediate Phase focus

The focus in the teaching and learning of data handling in the Intermediate Phase is on gaining the skills to gather and summarise data so that they can be interpreted and predictions made from them.

The learner should become aware that:

  • different questions reveal different features of a situation, and that this will affect the ability to understand the situation; and that
  • different forms of representation highlight some aspects of the data while hiding others, and that this, too, has a role in limiting interpretations of the data.

The learner should begin to develop sensitivity to how the data-gathering context limits interpretation and prediction (e.g. interviewing only boys on the role of peer pressure in deciding whether or not to start smoking may give different results compared to interviewing only girls or interviewing both boys and girls).

Contexts should be selected to read, interpret and represent data that build awareness of human rights and other social, economic and environmental issues. Such contexts should focus on discrete data involving whole numbers only.

The learner should develop the capacity to analyse critically interpretations and predictions from data. The study of chance (probability) develops awareness that:

  • different situations have different probabilities of occurring; and
  • for many situations there are a finite number of different possible outcomes.
In this phase, the learner is not expected to calculate the probability of events occurring.