qcaa general maths assignment example

General Mathematics 2019 v1.2 - IA1: High-level annotated sample response September 2021 - Queensland ...

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Bachelors of Nursing (675484)

James cook university.

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General mathematics 2019 v1., ia1 high-level annotated sample response, problem-solving and modelling task (20%), this sample has been compiled by the qcaa to assist and support teachers to match evidence, in student responses to the characteristics described in the instrument-specific marking guide, this resource contains content that may be of a sensitive nature for students. teachers should, consult with school leaders and consider the suitability of the investigation in the school’s, assessment objectives, this assessment instrument is used to determine student achievement in the following, objectives:, 1. select, recall and use facts, rules, definitions and procedures drawn from unit 3 topics 1, 2, 2. comprehend mathematical concepts and techniques drawn from unit 3 topics 1, 2 and/or 3, 3. communicate using mathematical, statistical and everyday language and conventions, 4. evaluate the reasonableness of solutions, 5. justify procedures and decisions by explaining mathematical reasoning, 6. solve problems by applying mathematical concepts and techniques drawn from unit 3, topics 1, 2 and/or 3., queensland curriculum & assessment authority.

Instrument-specific marking guide (ISMG) Criterion: Formulate Assessment objectives 1. select, recall and use facts, rules definitions and procedures drawn from Unit 3 Topics 1, 2 and/or 3 2. comprehend mathematical concepts and techniques drawn from Unit 3 Topics 1, 2 and/or 3 5. justify procedures and decisions by explaining mathematical reasoning The student work has the following characteristics: Marks

 documentation of appropriate assumptions

 accurate documentation of relevant observations,  accurate translation of all aspects of the problem by identifying mathematical concepts and, techniques.,  statement of some assumptions,  statement of some observations,  translation of simple aspects of the problem by identifying mathematical concepts and,  does not satisfy any of the descriptors above. 0.

Criterion: Solve Assessment objectives 1. select, recall and use facts, rules, definitions and procedures drawn from Unit 3 Topics 1, 2 and/or 3 6. solve problems by applying mathematical concepts and techniques drawn from Unit 3 Topics 1, 2 and/or 3 The student work has the following characteristics: Marks

 accurate use of complex procedures to reach a valid solution

 discerning application of mathematical concepts and techniques relevant to the task,  accurate and appropriate use of technology.,  use of complex procedures to reach a reasonable solution,  application of mathematical concepts and techniques relevant to the task,  use of technology.,  use of simple procedures to make some progress towards a solution,  simplistic application of mathematical concepts and techniques relevant to the task,  superficial use of technology.,  inappropriate use of technology or procedures. 1.

Task Investigate the phenomenon of ancestral heredity by focusing on the height of a parent and their biological child of the same sex, using data from students at your school. The investigation should explore the dependence of a male’s height on his father’s height, or the dependence of a female’s height on her mother’s height. Can a person’s height be reliably predicted from their relative’s height? To complete this task, you must:  respond with a range of understanding and skills, such as using mathematical language, appropriate calculations, tables of data, graphs and diagrams  provide a response to the context that highlights the real-life application of mathematics  respond using a written report format that can be read and interpreted independently of the problem-solving and modelling task sheet  develop a unique response  use both analytic procedures and technology. See IA1 sample assessment instrument: Problem-solving and modelling task (20%) (available on the QCAA Portal). Sample response Criterion Marks allocated Provisional marks Formulate Assessment objective/s 1, 2, 3 4 4 Solve Assessment objective/s 1, 6 7 6 Evaluate and verify Assessment objective/s 4, 5 5 5 Communicate Assessment objective/s 3 4 4 Total 20 19

The annotations show the match to the instrument-specific marking guide (ISMG) performance- level descriptors. Table of contents 1 Introduction 2 Considerations 2 Observations and assumptions 2 Mathematical concepts and techniques 2 Use of technology 3 Developing a solution 4 Evaluation to verify results 4 Improving the model 4 Strengths and limitations 5 Conclusion 6 Appendixes 7 Reference list 1 Introduction Sir Francis Galton (1822–1911) created the statistical concept of correlation. In 1903, assisted by Alice Lee, Pearson decided to supplement Galton’s study on the inheritance of physical characteristics. The results presented in this report are a product of the investigation into the ancestral heredity of stature with a focus on a son’s height compared to their father’s height. A sample of 100 male students from the current Year 12 cohort was selected for the study. 2 Considerations 2 Observations and assumptions While fathers and sons represent a very large population, a sample of Year 12 peers will be used to represent the sons. Year 12 boys will be sampled because most boys reach their full height by the age of 16. A sample of 100 was randomly selected from various form classes as an appropriate size to investigate the stature of sons and their fathers. Since the Year 12 cohort has 213 male students, the decision to include approximately half of the cohort was deemed sufficient to allow for a variety of height data. Therefore, a valid comparison between father and son heights could be made. As the father’s height will be used to predict the

Communicate [3–4]

Coherent and concise, organisation of the, response, appropriate, genre, including a, suitable introduction, formulate [3–4], documentation of, relevant observations, accurate translation of, problem by identifying, mathematical concepts, and techniques, suitable body, solve [6–7], accurate and, appropriate use of, formulate [3-4], accurate use of, complex procedures to, reach a valid solution,, application of, relevant to the task.

2 Use of technology A spreadsheet program was used extensively during the investigation process to organise the father and son data, prepare graphs, confirm the regression equation and coefficient of determination, and calculate residuals. The program was also used to calculate the statistical measures of mean, standard deviation and the correlation coefficient, which are required to develop the least-squares regression equation analytically. 3 Developing a solution The height data appears in Appendix 1. The data is presented below in Graph 1, using a scatterplot to identify a possible association between father and son heights. Graph 1: Scatterplot of son height against father height On first inspection there did not appear to be a strong association between the two variables. However, a weak positive linear relationship was identified as plausible. Based on this conclusion, a linear regression equation was developed using the least-squares method of regression. The general form of the least-square regression line is given by: 𝑦 = 𝑎 + 𝑏𝑥 where 𝑏 = 𝑟 × ௦೤ ௦ೣ and 𝑎 = 𝑦ത − 𝑏𝑥̅, given 𝑟 is Pearson’s correlation coefficient, 𝑠௫ and 𝑠௬ are the sample standard deviations, and 𝑥̅ and 𝑦ത are the sample means. As determined using the spreadsheet function CORREL: 𝑟 = 0. As determined using the spreadsheet function AVERAGE: 𝑥̅ = 168 and 𝑦ത = 169. As determined using the spreadsheet function STDEV: 𝑠௫ = 5 .74252876, and 𝑠௬ = 4. Refer to Appendixes 2 and 3 for spreadsheet functions used.

145 150 155 160 165 170 175 180 185 190

Son height (cm), father height (cm), all aspects of the, correct use of, appropriate technical, vocabulary, procedural, vocabulary and, conventions to develop, the response.

𝑏 = 𝑟 × ௦೤ ௦ೣ = 0 × ସ.ଶଶଽଷଵ଺଺଼଼ ହ.଻ସଶହଶ଼଻଺ = 0. 𝑎 = 𝑦ത − 𝑏𝑥̅ = 169 − 0 × 168. = 131. ∴ The least-squares regression line equation for the data is given by: 𝑦 = 131 + 0𝑥 𝑦 = 131 + 0𝑥 (correct to two decimal places). The regression line was added to the scatterplot using the trendline function of the spreadsheet program and the distribution of data points did not follow the trendline very closely (see Graph 2). The calculated correlation coefficient (𝑟) value of 0 .302 was confirmed using the coefficient of determination (𝑅ଶ) value of 0 generated by the spreadsheet program. The equation of the line, as determined by the spreadsheet trendline function, confirmed the regression equation constants calculated using formulas. Graph 2: Scatterplot of son height vs father height with regression line The result of 0 is close to zero, which indicates a (positive) weak correlation. The 𝑅ଶ value of 0 means that only 9% of the variation can be explained by the relationship between the heights of fathers and sons. It is worth noting that if we expected a son to be the same height as his father, we would expect the slope (gradient) of the least-squares regression line to be exactly 1. The determined slope of 0 indicates that, in some instances, sons are taller than their fathers.

y = 0 + 131.

Son height,, father height, x (cm), evaluate and verify, evaluation of the, reasonableness of, solutions by, considering the results,, assumptions and, observations, documentation of the, strengths and, limitations of the, correct use of technical, and procedural, vocabulary to develop.

relationship is most likely linear. It can be seen in Graph 3, the residual plot, that the points are randomly scattered across the 𝑥-axis. Therefore, it can be concluded that a linear regression model was appropriate for this study. In trying to explain the unexpected low correlation between son and father heights, the initial assumptions and observations were revisited. The following could explain the weak association:  Errors may have occurred during the collection of data.  Self-reported data (father height) lacks reliability.  Some students may not have reached their full adult height.  Personal information about health and family circumstances is unknown and could be affecting the data. 4 Improving the model To improve the study, data collection of son heights and father heights could result in a stronger association. This would eliminate the assumption of growth. Directly measuring father height rather than using self-reported data could also improve reliability. Further, it must not be ignored that a son has two genetic pools — mother and father. Only the father has been studied in this investigation, with no acknowledgement of the influence the mother may have on the son’s height. It would be an interesting study to also include the mothers’ heights against their sons’ heights. As an additional presentation of father–son height analysis, Pearson’s data was accessed (The University of Alabama in Huntsville n.). Randomly selecting 50 pairs (out of a total of 1078) of father and son heights from Pearson’s 1903 dataset produced the scatterplot and regression equation shown in Graph 4. Graph 4: Pearson’s data, random sample n = 50, scatterplot and regression line Unexpectedly, this sample produced a negative moderate linear association with a stronger relationship between the variables than the father–son height in this study. However, when graphing Pearson’s entire dataset, a positive linear association was found. This leads to the conclusion that the

y = -0 + 195.

155 160 165 170 175 180 185 190 195.

Son height, y (cm) Father height, x (cm)

solution and/or model

Appropriate genre,, including a suitable, conclusion; coherent, and concise, response, which can, be read independently, of the task sheet..

analysis is very dependent on the sample selected, and a sufficient sample size is required. A larger sample size with greater diversity, or the entire Year 12 male cohort, may result in a stronger association. 4 Strengths and limitations Some strengths of the model are that:  the method is generalisable and will work for other datasets or other hereditary characteristics,  it can tell us how much the father’s height explains the son’s height since the coefficient of determination measures how much of the variation is explained by the explanatory variable  the model’s gradient can also tell us about the dependence of inherited characteristics on the value of the characteristic (e. if the gradient is 1, we expect father and son to be the same height; less than 1 and we might expect shorter fathers to have taller sons and vice-versa). Some limitations of the model include:  the sample size may be insufficient to make general claims about the predictions of heights; however, there is some evidence from the initial data analysis that is worth pursuing with additional data samples to produce a more reliable solution  there are other factors that affect height, e. mothers’ and grandparents’ heights, nutrition, general health, exercise (Danish 2017) The accuracy of the recorded data (father’s height) may have been unreliable. 5 Conclusion The relationship between a father’s height and his son’s height is not very strong, according to this study. While there was a positive linear association, as demonstrated by the linear regression and correlation coefficient, the coefficient of determination (𝑅ଶ) value of 0 means this association is very weak. Given limitations such as the relatively small dataset, the self-reported heights of fathers, and the assumption that the students had reached their full adult height, it is not possible to draw a strong conclusion about whether a son’s height is dependent on his father’s height, from this study. Further research could be undertaken to investigate female students’ heights compared to their mothers’ heights. Word count Word count excluding, contents page, reference list, appendixes, data and tables is approximately 1743.

Statistical functions used in a spreadsheet program r = =CORREL(A2:A101,B2:B101) mean (x) = =AVERAGE(A2:A101) mean (y) = =AVERAGE(B2:B101) - General Mathematics 2019 v1. - April Queensland Curriculum & Assessment Authority - 173 168. - 176 168. - 182 168. - 158 170. - 163 171. - 163 170. - 162 169. - 166 170. - 165 169. - 166 176. - 174 175. - 168 156. - 164 166. - 176 175. - 164 171. - 165 170. - 167 170. - 167 170. - 166 171. - 167 171. - 167 169. - 170 171. - 169 170. - 171 169. - 170 170. - 169 171. - 173 171. - 172 170. - 173 171. - 172 171. - 175 171. - 175 170. - 176 169. - 178 169. - 177 169. - 184 171. - 159 172. - 159 173. - 162 172. - 163 173. - 163 173. - 167 173. - 167 173. - General Mathematics 2019 v1. - April Queensland Curriculum & Assessment Authority - 167 173. - 166 172. - 166 173. - 169 172. - 169 174. - 170 172. - 169 172. - 169 173. - 170 172. - 171 173. - 173 172. - 171 173. - 173 172. - 171 174. - 174 172. - 176 173. - 174 172. - 175 174. - 176 173. - 174 174. - 176 172. - Appendix - r = 0. Statistical measures calculated using a spreadsheet program - mean (x) = 168 mean (y) = 169. - std dev (x) = 5 std dev (y) = 4. - Appendix

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