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NATIONAL QUALIFICATIONS CURRICULUM SUPPORT Mathematics Advice and Guidance for Staff [NATIONAL 4] This advice and guidance has been produced to support the profession with the delivery of courses which are either new or which have aspects of significant change within the new national qualifications (NQ) framework. The advice and guidance provides suggestions on approaches to learning and teaching. staff are encouraged to draw on the materials for their own part of their continuing professional development in introducing new national qualifications in ways that match the needs of learners. Staff should also refer to the course and unit specifications and support notes which have been issued by the Scottish Qualifications Authority. http://www.sqa.org.uk/sqa/34714.html Acknowledgement © Crown copyright 2014. You may re-use this information (excluding logos) free of charge in any format or medium, under the terms of the Open Government Licence. To view this licence, visit http://www.nationalarchives.gov.uk/doc/open-government-licence/ or e-mail: [email protected] Where we have identified any third party copyright information you will need to obtain permission from the copyright holders concerned. Any enquiries regarding this document/publication should be sent to us at [email protected] This document is also available from our website at www.educationscotland.gov.uk. 2 MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 Contents Introduction 4 Resources 10 MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 3 INTRODUCTION Introduction Applying statistical skills to representing data and interpreting situations involving probability to solve real-life problems Mathematics is important in everyday life, allowing us to make sense of the world around us and to manage our lives. This advice and guidance suggests an approach to the learning and teaching of probability, scattergraphs and line of best fit within the context of Mathematics National 4. The learning and teaching approaches incorpora te opportunities for collaborative and independent learning using a range of real -life situations. This will enable learners to draw conclusions, assess risk and gain the ability to respond to mathematical situations that could arise in everyday life. Learners will be able to use mathematical reasoning skills to make and explain their decisions within a given relevant context. The knowledge, skills and understanding within third level experiences and outcomes, as detailed below, provide a firm foundation for further learning within SCQF level 4. I can work collaboratively making appropriate use of technology to source information presented in a range of ways, interpret what it conveys and discuss whether I believe the information to be robust, vague or misleading. MNU 3-20a I can display data in a clear way using a suitable scale, by choosing appropriately from an extended range of tables, charts, diagrams and graphs, making effective use of technology. MTH 3-21a I can find the probability of a simple eve nt happening and explain why the consequences of the event, as well as its probability, should be considered when making choices. MNU 3-22a New national courses have been designed to draw on and build on Curriculum for Excellence experiences and outcomes as appropriate. SCQF level 4 and fourth level experiences and outcomes are broadly equivalent in terms of level of demand, although qualifications at SCQF level 4 will be more specific to 4 MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 INTRODUCTION allow for more specialist study of subjects. The following fourth le vel experiences and outcomes may have been experienced by learners: I can evaluate and interpret raw and graphical data using a variety of methods, comment on relationships I observe within the data and communicate my findings to others. MNU 4-20a I can select appropriately, from a wide range of tables, charts, diagrams and graphs when displaying discrete, continuous or grouped data, clearly communicating the significant features of the data . MNU 4-21a By applying my understanding of probability, I can d etermine how many times I expect an event to occur, and use this information to make predictions, risk assessment, informed choices and decisions . MNU 422a MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 5 INTRODUCTION Area of mathematics Constructing a scattergraph: Given a set of data 6 Learning and teaching approaches Exemplification Learners should be given the opportunity to collect their own data for constructing scattergraphs. Contexts should reflect the age and stage of the learner, for example using charity fundraising data to plot money raised against the distance participants ran or swam in a sponsored event. Alternatively, links with social subjects and travel and tourism can be made by considering, for example, the price changes of a convenience item along an environmental gradient near a major tourist attraction. The hypothesis tested is that prices should decrease with distance from the key area surrounding the major tourist attraction. Straightforward examples can be adapted to be age specific. Learners should be able to use graphs to answer related questions. Staff should reflect on how they will incorporate some ‘thinking questions’ into their teaching. Sequences of questions can gradually increase the level of challenge, for example ‘What is wrong with the statement…?’ , ‘How would you describe…?’, ‘What is the same and what is different about these…?’ Peer- and self-assessment are encouraged. Straightforward examples of contexts for constructing a scattergraph can be found using the following links: MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 http://www.tes.co.uk/teaching-resource/Line-ofBest-Fit-6017667/ http://www.cimt.plymouth.ac.uk/projects/mepres/bo ok7/bk7i3/bk7_3i1.htm INTRODUCTION Drawing and applying a line of best fit: The line should have roughly the same number of data points on either side. Use the line of best fit to estimate one variable given the other. Making and explaining decisions based on interpretation of data from straightforward graphical forms: Make decisions based on observations of patterns and trends in data. Staff are encouraged to use a wide range of learning and teaching approaches to enhance the understanding of learners. Real-life contexts, for example health statistics against environmental (http://www.metoffice.gov.uk/climate/uk/stationda ta/), social or economic factors, can enhance learning. Staff are encouraged to promote learning and understanding through interactive activities using a variety of real-life contexts. Using data that is fit for purpose will support learners’ understanding of this graphical form. For example, discuss graphs of data related to global warming, such as historic mean global temperature and CO 2 level in the atmosphere. Opportunities exist for staff and learners to explore, research and investigate data and its representation. Higher-order thinking can be developed through high-level questioning and Staff may use activities that link to other areas of learning within mathematics. For example, investigating the connection between finger and arm proportions will lead to learners discovering the golden ratio and there are possible extensions to its real-life applications such the design of the Parthenon, the ‘perfect’ body and the works of da Vinci. Learners should be encouraged to understand the importance of their findings. http://www.goldennumber.net/body.htm is a website that gives an opportunity for further research by individuals or for extension activities. See the ‘Finger and arm proportions’ PowerPoint. Maths case studies will promote deeper understanding and reasoning skills. Examples of contexts for learning can be found at http://www.bowlandmaths.org.uk/index.html. Staff may use a matching activity to promote understanding and interpretation of correlation. MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 7 INTRODUCTION Make decisions based on calculations involving data. Make decisions based on reading scales in straightforward graphical forms. Offer reasons for decisions made based on interpretation of data. Making and explaining decisions based on probability Recognise patterns and trends and use these to state the probability of an event happening. 8 creating open-ended questions. Who may want to know about the correlation between these quantities, and why? Discuss high positive or negative correlation, indicating the connection between the variables. Discuss correlation, interpolation and extrapolation, and the real possibilities of errors in interpreting graphs. Learners should be encouraged to use correct mathematical terminology within their response. Staff have the opportunity to adopt a range of resources, particularly ICT and other media. A range of activities using a variety of resources will enable learners to acquire an understanding of probability and risk. Resources should include activities that are interactive and provide opportunities for collaborative learning, for example working collaboratively to arrange in order on a probability scale the possibilities of an event occurring, as detailed on probability card s. Use activities promoting an awareness of, and creating opportunities for discussion on, factors that affect the likelihood of an event occurring. MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 See the ‘Misleading graphs leading to misleading probabilities’ and ‘Stock market game’ Word documents. Learners could obtain a prediction for weather for the coming week and compare it with actual INTRODUCTION Make predictions and use these predictions to make decisions. Promote discussion through use of scenarios to enable learners to deepen their understanding of how to make and explain decisions based on probabilities. Staff can encourage learners to understand risk within the financial sector by linking activities to aspects of financial education , eg examining electricity and gas tariffs. Alternatively, staff may encourage learners to explore aspects of health and wellbeing, eg the probability of sustaining attendance at a local gym or examining health issues linked to smoking. Enhance learners’ understanding through experiments and contingency tables. Using real life contexts can illustrate how making predictions is based on probability data, eg weather predictions based on the previous year’s weather. It is important to promote an awareness of how data, information and statistics are portrayed in the media and use this to make informed decisions and justify those decisions. Learners could use probability as a measure of chance and uncertainty. This could include reference to the likelihood of events happening in familiar contexts, such as selecting a holiday destination from seasonal tables of average rainfall, sunshine and temperatures. weather. They could discuss how accurate such forecasts are likely to be. http://theweatheroutlook.com/forecast/uk/Glasgow. MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 9 RESOURCES Resources http://www.tes.co.uk/teaching-resource/Provoking-discussion-about-correlation6087038/ Staff notes plus an interactive Excel activity using the ‘hide and reveal' technique to explore the interpretation of scattergraphs with respect to correlation, including an option for line of best fit, and to encourage learners to write explanations of the relationships shown. It encourages learners to develop the quality of their written explanations. http://www.tes.co.uk/teaching-resource/Estimating-from-Scatter-GraphsTreasure-Hunt-6124066/ Treasure hunt activity on estimating from scattergraphs using the line of best fit. http://www.bbc.co.uk/schools/ks3bitesize/maths/handling_data/representing_dat a/revise9.shtml Straightforward revision on drawing scattergraphs, line of best fit, correlation and interpolation. For learners to use and test themselves. http://www.scotlandscensus.gov.uk Census from 27 March 2011. http://www.socialbakers.com/facebook-statistics/ Daily updated Facebook statistics from over 200 different countries. Clicking on a country gives Facebook statistics together with demography and a graph in time. http://data.gov.uk/dataset/annual_abstract_of_statistics The national abstract of statistics published annually . http://www.nationwide.co.uk/hpi/datadownload/data_download.htm UK house prices data. 10 MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 RESOURCES http://www.statistics.gov.uk/hub/index.html UK National Statistics. http://www.scotland.gov.uk/Topics/Statistics Scottish Government statistics. http://www.ons.gov.uk Office for National Statistics. http://world.bymap.org/ World by Map presents several world statistics and charts by tables, maps and diagrams. MATHEMATICS (NAT 4, MATHEMATICS) © Crown copyright 2014 11