Mathematics in manufacturing and product design principal learning
The three themes of the manufacturing and product design principal learning are supported by appropriate areas of mathematics, which increase in demand through the levels. The themes are: • business and enterprise • production systems • product design and materials science.The mathematics of finance in manufacturing underpins the theme of business and enterprise; collecting, representing and interpreting data is often used in production systems and product design and materials science; the learner will also have the opportunity to develop the mathematical techniques generally used in materials science.
Level 1At level 1 business and enterprise involves the calculation of costs, involving multi-step operations •(Level 1 unit 1) . Similar skills are required for production systems, where learners need to calculate quantities, estimate required time and explain the implications of production quantity being altered •(Level 1 unit 6) .
In preparing a product specification, learners have to set out quantifiable aspects such as dimensions, weight and cost, and use sector- appropriate units. Some specifications introduce the use of CAD, requiring work with 2D and 3D geometry •(Level 1 unit 4) . Learners apply a range of sector procedures, knowledge and skills, including numeracy, in ‘Manufacturing a product’ •(Level 1 unit 6) .
Learners discover how manufacturers work with properties of materials and what standard procedures are used to test materials. This can involve data handling and the plotting and interpretation of graphs to understand and apply scientific theories, eg Hooke’s law, and the use of word equations such as density = mass/volume •(Level 1 unit 5) .
Teaching implications for level 1 at key stage 4The mathematics in level 1 lies well within the mathematics GCSE specifications at foundation level and it builds from key stage 3. The possible exception is the application of 2D and 3D geometry needed to support working with CAD featured in some specifications (see explanatory note on level 1 unit 4).
Because of this kind of overlap it is vital that manufacturing specialists and mathematics teachers teaching Diploma learners have opportunities to plan together and develop a common terminology. It is also advisable that they have a chance to plan work with science teachers teaching Diploma learners.
Level 2At level 2 business and enterprise introduces finance skills in ‘Running a manufacturing business’ •(Level 2 unit 1) . Here learners become able to interpret basic financial documents, enter data on to a spreadsheet and use graphical methods to calculate break-even figures.
In the unit ‘The global business world’ •(Level 2 unit 2) investigates statistical data, learners research and present information about the manufacturing sector, markets and customer needs and the effects of the global market. There is a further opportunity to use statistical techniques in analysing marketing approaches.
‘Product design and materials science’ •(Level 2 unit 5) requires learners to carry out tests on the properties of materials. This provides the opportunity to bring together data handling with the testing of hypotheses. ‘Designing and developing products’ •(Level 2 unit 4) requires the learner to develop design ideas using CAD and therefore to work with 2D and 3D geometry. In response to a client brief, learners must create a product specification including quantifiable elements.
At level 2 the learner is required to understand the processes involved in manufacturing. ‘Processing systems’ •(Level 2 unit 6) requires numeracy in addressing quality control and other aspects of production complying with legislation. There is the opportunity here to introduce the statistical methods used with quality control graphs (see explanatory notes). In ‘Product manufacture’ •(Level 2 unit 7) learners apply a range of sector procedures, knowledge and skills requiring numeracy, eg interpretation of a product specification, calculation of quantities of materials, accurate calibration of equipment and application of quality control methods, including the possible use of statistical methods.
Teaching implications for level 2 at key stage 4The finance in manufacturing theme draws on principles from key stage 4 higher level mathematics, although learners may not have applied these to business and finance. The mathematics surrounding materials testing and manufacturing processes and procedures lies within key stage 4. However, learners may not have had the opportunities to apply mathematics to CAD •(Level 2 unit 4) . There is the possibility of applying statistical methods to control graphs, which may go beyond key stage 4 (see •(Level 2 unit 6) and •(Level 2 unit 7) .
Because of this kind of overlap it is vital that manufacturing specialists and mathematics teachers teaching Diploma learners have opportunities to plan together and develop a common terminology. It is also advisable that they have a chance to plan work with specialist science teachers teaching Diploma learners.
Level 3Level 3 requires increasing competence in financial techniques. ‘Manufacturing business principles’ •(Level 3 unit 1) presents opportunities to use graphs and algebraic manipulation to perform cost-benefit analyses. Financial planning techniques can entail using spreadsheets, often with algebraic methods.
Level 3 introduces ‘Supply chain management’ •(Level 3 unit 3) requiring multi-step arithmetic calculations and allowing scope for statistical analysis. The unit also provides the chance to work with Gantt charts and critical path analysis. Learners will not have encountered these in key stage 4 mathematics, and they draw on GCE AS or A2 decision (discrete) mathematics modules.
‘Customer needs and marketing requirements’ •(Level 3 unit 2) looks at techniques in the sector and necessitates more advanced data handling, such as selection of appropriate data sets, interpretation of statistics and graphs and possibly collation, analysis and interpretation of questionnaire data.
Data handling drawing on a greater range of statistical and analytical techniques will also be needed in ‘Research, development and the introduction of new products’ •(Level 3 unit 5) .
‘Materials science’ •(Level 3 unit 6) again brings together data handling with the formulation and testing of hypotheses. Learners must follow sector-standard tests for materials properties and collate, interpret and present the results. They must be proficient in number handling, measuring scales, plotting data on to graphs, substitution into formulae and the use of a range of statistical techniques to analyse results.
‘Production and processing systems’ •(Level 3 unit 7) looks at control technology and control methods, offering opportunities to understand statistical elements such as sampling and to work with quality control graphs. The unit provides a further chance to work with Gantt charts and critical path analysis.
‘Quality in manufacturing’ •(Level 3 unit 9) returns to the concept of tolerances and entails proportional reasoning. Again, learners can use quality control graphs; they should understand normal distribution.
‘Management of production and processing operations’ •(Level 3 unit 8) looks at the management of maintenance, involving data handling and problem solving using statistical techniques and again it could involve GCE AS/A level decision mathematics.
Teaching implications for level 3The mathematics in level 3 is largely within the range of GCSE, although there are instances where a learner could explore techniques that fall within areas compatible with GCSE statistics or decision (discrete) mathematics and statistics modules at A level. These will require specific teaching. Where specialist manufacturing teachers and mathematics specialists are both teaching level 3 Diploma learners, it is important that they are able to meet frequently to plan learning opportunities together and to develop a common terminology.
Information, advice and guidance supporting level 3Learners should be advised that manufacturing qualifications beyond level 3 are likely to require grade C GCSE mathematics. Those thinking of progressing to engineering courses need to consider either A level mathematics or the Level 3 Certificate in Mathematics for Engineering as their additional and specialist learning.
Grade C at GCSE is advisable, although not essential, in any case, to support the extent of statistical methods and decision (discrete) mathematics learners are likely to encounter in the level 3 principal learning.
Introduction to manufacturing Level 1 unit 1
Area: Be able to calculate simple costs eg cost of materials and labour within product price, sales per month Mathematical knowledge and understanding: Use of four rules of number with numbers and costs expressed as decimal numbers (may include use of fractions and percentages); Possible multi-step calculations with an appreciation of the order of operations; Basic proportional understandingMathematical skills: Analysing; Interpreting and evaluatingOverlap with key stage 4: Number and algebraSupportive prior learning: Key stage 3 Number
Areas: Be able to, in the following standard manufacturing operating procedures: • calculate quantities of materials for manufacturing operating procedures • comply with health safety procedures in manufacturing operationsMathematical knowledge and understanding: Accurate calculation with addition, subtraction, multiplication and division, decimals and multi-step calculations; Estimation of time needed; Explaining differences if production quantity were altered (requiring basic ideas of proportion)Mathematical skills: Analysing; Interpreting and evaluating Overlap with key stage 4: Number and algebraSupportive prior learning: Key stage 3 number and algebra
Introduction to product design and development Level 1 unit 4
Areas: Be able to: • carry out tests to analyse whether an example product is suitable for a customer/client • draw up an exemplar product design specification • use CAD (some specifications only)Mathematical knowledge and understanding: Tests entailing the use of quantitative data will involve reading scales and assessing whether a measurement lies in a given acceptable range, which could involve graph plotting and data handling; Development of design specifications could involve quantitative factors, such as dimensions and require simple calculations; 2D and 3D visualisation geometry: length, angles, volume and netsMathematical skills: Representing (eg organising data collection, drawing up design specifications); Analysing; Interpreting and evaluating; Communicating Overlap with key stage 4: Number and algebra; Geometry and measures Supportive prior learning: Key stage 3 Number and algebra; Geometry and measures
Areas: Be able to: • carry out basic testing and measurement techniques in a range of different materials within a sub-sector • record, present and store the results of investigations into the main properties, characteristics and uses of materialsMathematical knowledge and understanding: Reading scales and assessing whether a measurement lies in a given acceptable range; Graphical representation of data; (possibly include formulae expressed in words or symbolically eg density = mass/volume or ρ = m/v); (possibly some basic statistical representation)Mathematical skills: Representing (eg deciding on data to collect and its organisation and presentation); Analysing (eg processing data, possibly modelling with formula); Interpreting and evaluating (eg making sense of results of data analysis); CommunicatingOverlap with key stage 4: Number and algebra; Geometry and measuresSupportive prior learning: Key stage 3 Number and algebra; Geometry and measures
Running a manufacturing business Level 2 unit 1
Areas: Be able to: • enter data onto a financial spreadsheet • interpret basic financial documents • calculate and check the accuracy of basic costs and break-even figuresMathematical knowledge and understanding: Basic number techniques including the use of percentages in relation to finance including revenue, fixed cost, variable cost, margins and profit; Interpreting basic financial documents using estimating and checking techniques as well as accurate calculations of and checking break-even figures; Using proportional reasoning to understand how costs vary in relation to the volume of production; (possibly including use of spreadsheets) Mathematical skills: Representing (eg breaking down into multiple steps); Analysis (eg visualising what figures are indicating); Interpreting and evaluating (for errors or discrepancies); Communicating (using appropriate graphs); Analysing (using line graphs in forecasting break-even figures)Overlap with key stage 4: Number and algebraSupportive prior learning: Key stage 3 number
Areas: Be able to: • interpret basic statistical data about manufacturing • present information about market and customer needs analyse marketing approachesMathematical knowledge and understanding: Working with large numbers including standard form; Using ICT to analyse and present findings on statistical information; Interpreting basic statistical diagrams eg bar charts, line graphs and pie charts and measures eg averages and range; May involve collecting and organising sales figures to create a statistical model; Checking statistical model is a good representation of reality; Presenting findings using statistical model to demonstrate and explain sales figuresMathematical skills: Interpreting and evaluating (making inferences from the shapes of distributions and measures of average and spread); Communicating (choosing appropriate graphical representations for the context)Overlap with key stage 4: Number and algebra; Statistics Not usually covered in key stage 4: Learners may not have encountered interpretation of sales figures Supportive prior learning: Key stage 3 Number and algebra; Statistics
Materials science Level 2 unit 5
Areas: Be able to: • use scientific terminology, symbols and units • test, analyse and measure the main chemical, biological and physical properties of exemplar materials within a sub-sector • record observations • analyse results and draw conclusionsMathematical knowledge and understanding: Number (reading scales and assessing whether a measurement lies in a given acceptable range, recording data); Use correct symbols and units; Graphical representation of data; (possibly interpreting data using formulae in words or symbols)Mathematical skills: Representing (identifying data and organising and recording this); Analysing (eg processing data); Interpreting and evaluating (eg understanding results of experimentation in terms of properties of materials); CommunicatingOverlap with key stage 4: Number; Statistics Not usually covered in key stage 4: Depending on the sub-sector studied, learners may not have encountered the scientific terminology, symbols and units Supportive prior learning: Key stage 3 Number; Statistics
Areas: Be able to: • interpret a client brief or product research • develop design ideas for a prototype, using CAD where appropriate • draw up a simple product design specification for a manufactured product to meet a client brief or product research • analyse whether the features and benefits of a designed product meet the client’s briefMathematical knowledge and understanding: Handling of numbers or graphs; Basic statistical research techniques; Measures, accuracy, areas, volume; Size and weight, costs; 2D and 3D geometry, angles, tessellation, nets; The product design specification is likely to include dimensions and costs; Using CAD efficiently will involve mathematicsMathematical skills: Interpreting (comparative tests on products); Representing (breaking down stages of costing); Analysing (visualising the product’s proportions and representing these quantitatively); Checking and justifying estimates and approximations of calculationsOverlap with key stage 4: Number and algebra; Geometry and measures Not usually covered in key stage 4: Learners will not have encountered working with CAD and using mathematics with CAD Supportive prior learning: Key stage 3 Number and algebra; Geometry and measures
Processing systems Level 2 unit 6
Areas: Be able to: • know what processes and systems are used within a manufacturing enterprise • understand how quality control is used to ensure quality and cost effectiveness • use measuring equipment to monitor the quality of a productMathematical knowledge and understanding: Number eg reading scales and assessing whether a measurement lies in a given acceptable range, recording data; Using correct symbols and units; Understanding of proportionality; Measures; Using measuring equipment correctly; (possibly understanding of statistical techniques in relation to quality control)Mathematical skills: Representing (eg identifying data and organising and recording this); Analysing (eg processing data); Interpreting and evaluating (eg understanding results of experimentation) Overlap with key stage 4: Number and algebra; possibly statistics: the data handling cycle Not usually covered in key stage 4: Learners are unlikely to have encountered the particular type of measuring equipmentUse of statistical methods employed with quality control graphs occur only at grades A and B of GCSE statistics. Supportive prior learning: Key stage 3 Number and algebra
Areas: Be able to: • follow a design specification and standard operating procedures • calculate quantities of materials • set up or calibrate equipment • apply quality control methods • check work, check progress and meet production targetsMathematical knowledge and understanding: Measures, area, volume, size and weight, costs, units and symbols; Estimating and checking; In some sectors, understanding of proportion, percentages and ratio; Reading scales; Setting tolerances for control procedures; Statistics: statistical techniques for quality control; Possibly 2D and 3D geometry); Possibly multi-step calculationsMathematical skills: Analysing; Interpreting and evaluating Overlap with key stage 4: Number and algebra; Geometry and measures; Statistics Supportive prior learning: Key stage 3 Number and algebra; Statistics
Manufacturing business principles Level 3 unit 1
Areas: Be able to: • interpret and construct accurate key financial planning documents • carry out financial modelling • evaluate potential financial risksMathematical knowledge and understanding: Basic number techniques including proportion and ratio looking at percentages, simple and compound interest, exchange rates; Algebraic methods, use of formulae in financial contextsMathematical skills: Representing (developing appropriate financial documents); Analysing (cash flow analysis, substitution into formulae and adaptation of existing formulae in word or symbol form to calculate quantities such as profit and perform cost-benefit analyses); Interpreting and evaluating (accounts including cost-benefit analysis Overlap with A level and other qualifications: AS use of mathematics; Advanced level Free Standing Mathematics Qualification (FSMQ) mathematical principles for personal finance; GCSE mathematics; Intermediate level FSMQ financial calculations Supportive prior learning: Key stage 4 Number and algebra
Areas: Be able to: • calculate quantities of materials and supplies (this will probably involve basic number calculations, possibly in conjunction with
word formulae) • perform calculations and make recommendations for effective supply chainMathematical knowledge and understanding: The four rules of number, with decimal and fractional values involving multi-step calculations; Use of proportional reasoning and percentages; Statistics eg interpretation of data presented in different forms; Possibly Gantt charts and critical path analysisMathematical skills: Representing (eg using different formats to make comparisons); Analysing; Interpreting and evaluating (eg selecting appropriate data presented in a range of different forms and interpreting this in relation to the task) Overlap with A level and other qualifications: A mathematical approach to critical path analysis features in AS (discrete) mathematics modules Supportive prior learning: learners will not have encountered critical path analysis or Gantt charts at key stage 4
Customer needs and market requirements Level 3 unit 2
Areas: Be able to: • understand what market research techniques can be used • carry out market researchMathematical knowledge and understanding: Questionnaire construction, ideas of sampling: sample size and reliability/validity, bias; Statistics: the handling data cycle, use of statistical methods of analysis by hand and using technology, to include diagrammatic representation, measures of location and spread; Number: proportion, ratio, percentagesMathematical skills: Representing (eg devising data collection schedules, choosing appropriate diagrammatic representations); Analysing (eg collating, analysing data); Interpreting and evaluating (statistical measures and representations); Communicating Overlap with A level and other qualifications: AS use of mathematics; AS mathematics statistics modules; Advanced level FSMQ data analysis; GCSE statistics; Intermediate level FSMQ data handling Supportive prior learning: Key stage 4 Statistics: the handling data cycle
Research, development and introduction of new products Level 3 unit 5
Areas: Be able to: • use a wide range of research data to assist the development and introduction of a new product • design a new product, test and modify this producing a final product specificationMathematical knowledge and understanding: Data collection (questionnaire design, methods of sampling) and summary analysis of data using key statistics and possibly statistical diagrams/graphs; The four rules of number, with numbers expressed as decimals; Multi-step calculations; Proportional reasoning, percentages and ratio; Reading scales; Estimating and checking; (possibly 2D and 3 D geometry)Mathematical skills: Representing (eg using a range of different representations in product design); Analysing; Interpreting and evaluating (eg data to inform product design) Overlap with A level and other qualifications: AS use of mathematics; Intermediate level FSMQ data handling; Intermediate level FSMQ shape and space Supportive prior learning: Key stage 4 Number and algebra; Geometry and measures; Statistics
Materials science Level 3 unit 6
Areas: Be able to: • calibrate equipment • gather data from tests on materials and calculate, interpret and present the resultsMathematical knowledge and understanding: Reading scales; Real numbers including different representations (for large and small values); Upper and lower bounds; Conversions between measures; Algebra eg substituting into formulae in symbolic form to express the properties of materials graphical representation including scalingMathematical skills: Representing (of data both primary and secondary and processed); Analysing (eg processing data); Interpreting and evaluating (including comparative interpretation ie between different tests); Understanding what data from a particular test is likely to yield; Communicating Overlap with A level and other qualifications: AS use of mathematics; Advanced level FSMQ in working with algebraic and graphical techniques; Intermediate level FSMQ algebra and graphs Supportive prior learning: Key stage 4 Number and algebra; Geometry and measures; Statistics
Areas: Be able to: • know how production processes are planned and improved to maximise efficiency and effectiveness • know how control technology, control methods and safety controls are used to protect a product • monitor quality and tolerances against a product design specificationMathematical knowledge and understanding: Statistical techniques eg sampling techniques or taking a sample and finding a mean; Use of quality control charts – reading and interpreting graphs; Statistical analysis or principles and techniques from decision (discrete) mathematics; (possibly Gantt charts and critical path analysis)Mathematical skills: Representing (using a wide range of diagrams and charts); Analysing; Interpreting and evaluating; Communicating Overlap with A level and other qualifications: GCE AS or A2 decision (discrete) mathematics modules Supportive prior learning: Learners will not have encountered critical path analysis of Gantt charts at key stage 4
Quality in manufacturing Level 3 unit 9
Area: Be able to analyse data, interpret and present results and recommendations about qualityMathematical knowledge and understanding: Number eg reading scales, recording data, tolerances; Using correct symbols and units; Understanding of proportionality; Measures; (could include graphical techniques to summarise and report findings – possibly, quality control charts and normal distribution)Mathematical skills: Representing (eg data in a range of formats); Analysing; Interpreting and evaluatingAnalysing; Interpreting and evaluating; Communicating Overlap with A level and other qualifications: AS statistics modules; Advanced level FSMQ data analysis Supportive prior learning: Key stage 4 Statistics: the handling data cycle
Management of production and processing operations Level 3 unit 8
Areas: Be able to: • monitor and record the inputs and outputs of a production process against the specification • use problem-solving techniques to find ways to improve quality or efficiency, to reduce bottlenecks, improve output, reduce waste • use fault-finding techniques on manufacturing plant and equipment • carry out risk assessmentsMathematical knowledge and understanding: Measures, estimating, checking, tolerances with values expressed using a range of formats including standard form; Conversion between measures; Statistical measures, tables and diagrams; Statistical methods and decision (discrete) mathematics; Probability as applied to risk assessmentMathematical skills: Using ICT where appropriate; Representing; Analysing; Interpreting and evaluating; Communicating Overlap with A level and other qualifications: GCE AS or A2 decision (discrete) mathematics modules; AS statistics modules; AS use of mathematics Supportive prior learning: Data is likely to be presented in industry standard forms that require understanding of mathematical ideas in situations beyond the key stage 4 curriculum experience
