Demonstrating knowledge and understanding of:

Content guidance:

The influence the following social factors have on learning and developing number skills:

1.      Socio-economic status

2.      Ethnicity / culture

3.      Gender

Understanding the social factors that can influence an individual’s current and prior learning experiences is an important concern. The section is concerned, in part, with the lack of achievement in numeracy both from the point of view of the individual concerned and from a wider viewpoint.

When considering the individual the focus should be on the consequences of having limited or restricted number attainment in today’s society. Consider the financial, economic and social consequences of limited number attainment.

Learners should also consider the following types of wider issues:

·          Roles and responsibilities

·          Political, social and economic ramifications

·          Moral  issues

In particular, time should be spent in critical analysis of education issues. For example:

·          The nature of current curriculum provision

·          Curriculum approaches

·          Validity in assessment

·          Key theories related to numeracy

How limited number attainment or the presence of a number disorder/difficulty can disadvantage individuals.

The wider education and socio-economic issues related to levels of numeracy attainment.

Demonstrating knowledge and understanding of:

Content guidance:

Number learning disorders and disabilities like developmental arithmetic disorder and problems some individuals face with processing deficits.

This will involve:

1.      Processing problems

2.      Attention deficits

3.      Visual-spatial deficits

4.      Auditory-processing difficulties

5.      Memory and sequence difficulties

6.      Motor disabilities, and

7.      Unusually high anxiety.

Learners need to be encouraged to develop a firm understanding of the types of disorders and difficulties that exist relating to numbers and symbols recognition, memorising facts, aligning numbers, and understanding abstract concepts like place value and fractions. In particular learners should know about:

·          The main developmental arithmetic disorders (e.g. dyscalculia), and

·          Ways to recognise and test for number disorders like dyscalculia.

Difficulties in processing numerical information:

·          Attention deficits: For example, difficulty maintaining attention to steps in algorithms or problem solving or difficulty sustaining attention to critical instruction.

·          Visual-spatial deficits:

1.  Student loses place on the worksheet.

2.  Student has difficulty differentiating between numbers (e.g. 6 and 9; 2 and 5; or 17 and 71), coins, the operation symbols, and clock hands.

3.  Student has difficulty writing across the paper in a straight line.

4.  Student has difficulty relating to directional aspects of math, for example, in problems involving up-down (e.g. addition), left-right (regrouping), and aligning of numbers.

5.  Student has difficulty using a number line.

·          Auditory-processing difficulties: For example, student has difficulty doing oral drills or student is unable to count on from within a sequence.

·          Memory problems:

1.  Student is unable to retain number facts or new information.

2.  Student forgets steps in an algorithm.

3.  Student performs poorly on review lessons or mixed probes.

4.  Student has difficulty telling time.

5.  Student has difficulty solving multi-step word problems.

·          Motor disabilities: For example, student writes numbers illegibly, slowly, and inaccurately or student has difficulty writing numbers in small spaces (i.e., writes large).

Recognition of strategies used by individuals to compensate for lack of number skills.

Signs of learning disorders in the work of others and strategies to help learners with learning disorders and disabilities.

The influence of the following factors on number learning and development:

1.      Age

2.      Personal circumstance

3.      Prior learning

Demonstrating knowledge and understanding of:

Content guidance:

Number systems including denary, binary and octal.

Explain the nature of denary, binary and octal number systems including the advantages of different systems for specific applications.

Perform manual conversion of positive and negative integers between bases 10, 2 and 8.

Using floating point arithmetic in base 10, expressing numbers in scientific notation.

Perform repeated multiplying of a factor (base, exponent and power) and understanding and applying the rules of exponents.

Appreciation of the types and significance of absolute and relative errors, including those for numbers expressed in scientific notation.

Solving financial problems using formulas associated with arithmetic and geometric progressions and using financial tables and spreadsheets.

Using numbers and number models to formulate hypotheses and make deductions and being able to interpret and create suitable graphic techniques to represent data..

Understanding and applying proportion, variation and scaling including inverse proportion and other non-linear variation.

Explain the application of the communicative, associative and distributive laws for carrying out arithmetic and algebraic operations.

How to represent numbers by letters making firm connections between number theory and algebraic symbolism.

Algebraic symbolism used to express relationships between entities and data.

Laws of arithmetic operations and rules of precedence in simplifying arithmetic expressions.

Demonstrating knowledge and understanding of:

Content guidance:

The concepts and properties of geometry as a means of describing the physical world.

Classifying and comparing geometric figures and applying the use of appropriate technologies to the study of geometry and spatial sense.

Relating geometric ideas to number and measurement ideas, including the concepts of perimeter, area, volume, angle measure, capacity, weight and mass.

Exploring the various transformations of geometric figures.

An understanding of the nature and properties of polygons, quadrilaterals, circles, triangles and solids. Work with regular polygons, circles (chord, arc and sectors), triangles (Scalene: obtuse and acute) and quadrilaterals (parallelogram, rhombus, and trapezoids).

Co-ordinate geometry:

Combining elementary algebra and Euclidean geometry to establish equations and to solve problems involving points, lines, planes and circles using co-ordinates and vectors.

Applying midpoint and distance formulas to find the midpoint and find the distance between two points.

Determining the equation of a straight line given two points on the line and one point and the gradient.

Calculating the distance between two points and the midpoint of a line segment.

Finding the gradient of a straight line using m = tan è.

Finding the equation of a line parallel to and a line perpendicular to a given line.

Finding areas using co-ordinates.

Understand the link between gradient of a curve at a given point and the concept of differentiation in calculus.

Understand the link between the area of strips beneath a curve and the concept of integration in calculus.

The basis and effects of non-linear scales such as logarithmic and exponential scales.

How the techniques of calculus can be built from concepts of geometry.

How to apply geometric properties and relationships to real life and mathematical problems to make deductions about spatial reasoning. This will involve:

1.      Working with geometric inequalities,

2.      Understanding and applying the appropriate postulates and theorems

3.      Demonstrating an understanding of co-ordinate geometry of the straight line.

Demonstrating knowledge and understanding of:

Content guidance:

Concepts and calculations associated with discrete probability distributions and the probability of events.

The appropriate notation and terminology of probability.

The laws of probability and probability rules and the concepts of classical probability, relative frequency and the law of large numbers.

Properties of estimation; confidence limits.

Numerical measures of variability for ungrouped univariate data involving numerical values that measure the spread or variability of a numerical data set. How to compute these measures and investigate some of their properties.

Concepts and calculations associated with dispersion include techniques like standard deviation, range and confidence limits, making appropriate inferences from the data.

Range, inter-quartile range, mean absolute deviation, variance or standard deviation and coefficient of variation as measures of variation for numeric data.

Numeric values to measure the location or position of a data value in numerical data sets. Understanding how to compute these measures and investigate some of their properties. Address measures of position as standard score (z score) and percentiles.

How to interpret and create graphical information in the form of more sophisticated graphic methods e.g. frequency polygons, box plots, scatter graphs.

Significance tests: Chi-squared, t-test, Z-test etc.

Using calculators to find: mean, standard deviation, sum of values, correlation coefficient, linear regression coefficients.

Using spreadsheets to record data in tables; calculate values from data; plot graphs; use the ‘fill down’ facility; draw statistical diagrams; calculate statistical measures.

Estimation of means, variances and proportions.

Understanding and applying concepts and calculations associated with dispersion.

Quantitative skills including data analysis, interpretation and extrapolation.

How to analyse and draw reasoned conclusions concerning structured and unstructured problems from data.

Numerical measures of position for ungrouped univariate data.

Significance tests: their role and application.

Graphical information associated with this level of probability and statistics.

Demonstrating knowledge and understanding of:

Content guidance:

How relationships between number operations underpin the techniques used in manipulating algebraic expressions and how algebra gives precise form to mathematical relationships and calculations.

The generalised rules of arithmetic to manipulate algebraic expressions (common factors, like terms, multiplying out, factorising).

How to convert practical situations and statements from words into appropriate algebraic symbols and expressions.

Strategies for solving quadratic expressions, and for solving cubic and quartic expressions with one or two linear factors.

Algebraic methods to solve and represent mathematical problems. Working with quadratic expressions and evaluating and graphing polynomials.

The algebra of polynomials and rational functions.

Complex numbers in Cartesian and polar form.

The value of lower-order polynomial functions (quadratic, cubic, quartic) for given values of the polynomial.

Graphical representations of lower order polynomial functions and understand the links between parts of the expression and parts of the graph.

How to use the memory and function facilities on a standard scientific calculator and use the appropriate graph plotting software.

How to manipulate and solve algebraic expressions and interpret results as practical outcomes.

Graphic calculators / graph plotting software to:

·          Plot graphs of data pairs

·          Plot graphs of functions

·          Use function facilities

·          Use trace graphs and zoom facilities

How to representing practical situations symbolically and using formulas to produce useful outcomes.

How to apply algebraic methods to solve and represent a variety of real world and mathematical problems.

How to represent arithmetic patterns and real-world situations using tables, graphs and equations, exploring the interrelationships of these presentations.