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STAGE 2 MATHEMATICAL APPLICATIONS FOLIO TASK Body Parts and Height Predictors Topic: Statistics and Working with Data Subtopics: 7.1 – Sampling from Populations 7.2 – Analysis and Representation of Sets of Data 7.4 – Linear Correlation A completed investigation should include: an introduction that outlines the problem to be explored, including it significance, its features, and the context the method required to find a solution, in terms of the mathematical model or strategy to be used the appropriate application of the mathematical model or strategy, including - the generation or collection of relevant data and/or information, with details of the process of collection - mathematical calculations and results, and appropriate representations - the analysis and interpretation of results - reference to the limitations of the original problem a statement of the results and conclusions in the context of the original problem appendices and a bibliography, as appropriate. Learning Requirements Assessment Design Criteria Capabilities 1. Understand fundamental mathematical concepts and relationships. Mathematical Knowledge and Skills and Their Application Communication Identify, collect, and organise mathematical information relevant to investigating and finding solutions to questions/problems taken from social, scientific, economic, or historical contexts. MKSA1 Knowledge of content and understanding of mathematical concepts and relationships. MKSA2 Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find solutions to routine and complex questions. 2. 3. Recognise and apply the mathematical techniques needed when analysing and finding a solution to a question/problem in context. 4. Make informed use of electronic technology to provide numerical results and graphical representations. 5. Interpret results, draw conclusions, and reflect on the reasonableness of these in the context of the question/problem. 6. Communicate mathematical ideas and reasoning using appropriate language and representations. 7. Work both independently and cooperatively in planning, organising, and carrying out mathematical activities. Page 1 of 3 The specific features are as follows: Citizenship Personal Development Work Learning MKSA3 Application of knowledge and skills to answer questions in applied contexts. Mathematical Modelling and Problem-solving The specific features are as follows: MMP1 Application of mathematical models. MMP2 Development of mathematical results for problems set in applied contexts. MMP3 Interpretation of the mathematical results in the context of the problem. MMP4 Understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. Communication of Mathematical Information The specific features are as follows: CMI1 Communication of mathematical ideas and reasoning to develop logical arguments. CMI2 Use of appropriate mathematical notation, representations, and terminology. Stage 2 Mathematical Applications task Ref: A185579 (revised January 2013) © SACE Board of South Australia 2010 PERFORMANCE STANDARDS FOR STAGE 2 MATHEMATICAL APPLICATIONS Mathematical Knowledge and Skills and Their Application A Comprehensive knowledge of content and understanding of concepts and relationships. Appropriate selection and use of mathematical algorithms and techniques (implemented electronically where appropriate) to find efficient solutions to complex questions. Highly effective and accurate application of knowledge and skills to answer questions set in applied contexts. B Some depth of knowledge of content and understanding of concepts and relationships. Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find some correct solutions to complex questions. Accurate application of knowledge and skills to answer questions set in applied contexts. C Generally competent knowledge of content and understanding of concepts and relationships. Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find mostly correct solutions to routine questions. Generally accurate application of knowledge and skills to answer questions set in applied contexts. D Basic knowledge of content and some understanding of concepts and relationships. Some use of mathematical algorithms and techniques (implemented electronically where appropriate) to find some correct solutions to routine questions. Sometimes accurate application of knowledge and skills to answer questions set in applied contexts. E Mathematical Modelling and Problemsolving Development and effective application of mathematical models. Complete, concise, and accurate solutions to mathematical problems set in applied contexts. Concise interpretation of the mathematical results in the context of the problem. In-depth understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. Attempted development and appropriate application of mathematical models. Mostly accurate and complete solutions to mathematical problems set in applied contexts. Complete interpretation of the mathematical results in the context of the problem. Some depth of understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made. Appropriate application of mathematical models. Some accurate and generally complete solutions to mathematical problems set in applied contexts. Generally appropriate interpretation of the mathematical results in the context of the problem. Some understanding of the reasonableness and possible limitations of the interpreted results, and some recognition of assumptions made. Application of a mathematical model, with partial effectiveness. Partly accurate and generally incomplete solutions to mathematical problems set in applied contexts. Attempted interpretation of the mathematical results in the context of the problem. Some awareness of the reasonableness and possible limitations of the interpreted results. Limited knowledge of content. Attempted application of a basic mathematical model. Attempted use of mathematical algorithms and techniques (implemented electronically where appropriate) to find limited correct solutions to routine questions. Limited accuracy in solutions to one or more mathematical problems set in applied contexts. Attempted application of knowledge and skills to answer questions set in applied contexts, with limited effectiveness. Page 2 of 3 Limited attempt at interpretation of the mathematical results in the context of the problem. Limited awareness of the reasonableness and possible limitations of the results. Communication of Mathematical Information Highly effective communication of mathematical ideas and reasoning to develop logical arguments. Proficient and accurate use of appropriate notation, representations, and terminology. Effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Mostly accurate use of appropriate notation, representations, and terminology. Appropriate communication of mathematical ideas and reasoning to develop some logical arguments. Use of generally appropriate notation, representations, and terminology, with some inaccuracies. Some appropriate communication of mathematical ideas and reasoning. Some attempt to use appropriate notation, representations, and terminology, with occasional accuracy. Attempted communication of emerging mathematical ideas and reasoning. Limited attempt to use appropriate notation, representations, or terminology, and with limited accuracy. Stage 2 Mathematical Applications task Ref: A185579 (revised January 2013) © SACE Board of South Australia 2010 STAGE 2 MATHEMATICAL APPLICATIONS FOLIO TASK Body Parts and Height Predictors Introduction Forensic scientists and Anthropologists use bones to determine as much information from human remains in their efforts to identify a victim, or to compile information about the past. No two human bodies are identical in body dimensions, not even identical twins. But there are rules of thumb that can allow us to predict a reasonably accurate body measurement based entirely on the relationship of one part of the body to another body part. It is believed that you can predict a person’s height from the length of their tibia (the bone running from the foot to the kneecap), or from the length of the humerus (the bone running from the elbow to the shoulder). Skeletal remains have been discovered that are not complete, however a tibia and humerus bone were collected. Mathematical Investigations Investigate the correlation between the length of the tibia bone and the height of a person, and also the correlation between the length of the humerus bone and the height of a person. Determine if the skeletal remains could be used to accurately determine the height of the deceased. You may choose to consider: Different age groups Effect of sample size used Gender differences Outliers and their effect on the result Hints on making measurements on living specimens To measure the humerus, bend the subject’s arm at the elbow, and feel for the “knot” on the side of the elbow (called the tuberosity). This is one end of the humerus. Now feel for a similar “knot” at the shoulder, this should be the top of the humerus. Carefully measure the entire length of this bone. To determine the length of the tibia you need to measure the distance between the middle of the kneecap and the top of the foot. Analysis/Discussion Critically analyse your results in regard to the implications for forensic scientists and anthropologists to predict height of humans from their remains. You should consider: the appropriateness of the size of the samples used the information the data has provided the validity of any height predictions made using either of these methods any assumptions and limitations of the investigation. Teacher’s Note This task would lend itself to working with another student or small group in the collection of data and the investigation of the effect of sample size on the correlation between the sets of data. Each student could collect data pairs related to a particular age group or gender group and then investigate the correlation of their sample and compare to the correlation of a larger sample once the like samples are combined. Website resources: http://library.thinkquest.org/4116/Science/body_size.htm www.nsbri.org/HumanPhysSpace/focus6/student1.html www.surveysystem.com/correlation.htm Page 3 of 3 Stage 2 Mathematical Applications task Ref: A185579 (revised January 2013) © SACE Board of South Australia 2010