Essential Mathematics 2019 v1.1

Transcription

1 Problem-solving and modelling task 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 standards. 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 all Unit 3 topics 2. comprehend mathematical concepts and techniques drawn from all Unit 3 topics 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 all Unit 3 topics

2 Instrument-specific standards Formulate Solve Evaluate and verify Communicate Grade The student work has the following characteristics: documentation of appropriate assumptions accurate documentation of relevant observations accurate translation of all simple and complex aspects of the problem by identifying mathematical concepts and techniques. accurate use of complex procedures to reach a valid solution discerning application of simple and complex mathematical concepts and techniques relevant to the task accurate and appropriate use of technology. evaluation of the reasonableness of solutions by considering the results, assumptions and observations documentation of relevant strengths and limitations of the solution and/or model justification of decisions made using mathematical reasoning. correct use of appropriate technical vocabulary, procedural vocabulary and conventions to develop the response coherent and concise organisation of the response, appropriate to the genre, including a suitable introduction, body and conclusion. A statements of appropriate assumptions statements of relevant observations translation of simple and complex aspects of the problem by identifying mathematical concepts and techniques. use of complex procedures to reach a reasonable solution application of simple and complex mathematical concepts and techniques relevant to the task appropriate use of technology. statements about the reasonableness of solutions by considering the context of the task statements about relevant strengths and limitations of the solution and/or model statements about decisions made relevant to the context of the task. use of technical vocabulary, procedural vocabulary and conventions to develop the response organisation of the response, including a suitable introduction, body and conclusion. B statement of assumptions statement of observations translation of simple aspects of the problem by identifying mathematical concepts and techniques. use of simple procedures to make some progress towards a solution application of simple mathematical concepts and techniques relevant to the task use of technology. statement about the reasonableness of solutions statement about strengths and/or limitations of the solution and/or model statement about decisions made. use some of appropriate language and conventions to develop the response adequate organisation of the response. C statement of an assumption or an observation translation of some simple aspects of the problem by identifying mathematical concepts and techniques. application of some simple procedures, mathematical concepts or techniques superficial use of technology. statement about a decision and/or the reasonableness of a solution. use of everyday language to develop a response basic organisation of the response. D statement of an assumption, observation or translation of an aspect of the problem. inappropriate use of technology or procedures. inappropriate statement about a decision or the reasonableness of a solution. unclear and disjointed organisation of the response. E Page 2 of 11

3 Task Context Whether we live in a city, in a small town, on a farm or in a remote community, water availability affects us all. With uncertain rainfall during periods of drought, many householders try to minimise water wastage and supplement their council s water supply with rainwater. The Queensland Government encourages this through its Using water wisely campaign, which shows how to save water and gives advice on using rainwater tanks. Tanks are available in shapes and sizes to fit even the smallest spaces. It is becoming more common to have a rainwater tank in the backyard of a suburban house, but, as space is often limited, certain restrictions must be placed on the tank size and design. Task You are to develop a report for Alyssa and Preston, who want to install a rainwater tank to supplement their family s water needs. Their single-storey home has gutters 2.4 m above the ground. You will: 1. Analyse past rainfall data and use it to predict rainfall patterns. 2. Consider how rainfall is measured and analysed to decide which statistical measure should be used. 3. Calculate the amount of rainwater collected from a roof each month using local rainfall data. 4. Create an appropriate rainwater-use plan based on their current and predicted monthly water consumption. 5. Make a recommendation for a suitable rainwater tank. Your teacher will give you information about the couple and the Careful water use table, which shows how much water certain activities use. Your response will be in the form of a report. It should outline options and considerations for the couple s goals, so they can decide on an appropriate rainwater tank. Page 3 of 11

4 Sample response Criterion Grade awarded Formulate Assessment objectives 1, 2, 5 Solve Assessment objectives 1, 6 Evaluate and verify Assessment objectives 4, 5 A Communicate Assessment objective 3 The annotations show the match to the standard descriptors of the instrument-specific standards. coherent and concise organisation of the response The introduction clearly describes what the task is about and concisely outlines the intent of the task. Formulate [A] documentation of appropriate assumptions and accurate documentation of relevant observations Introduction Based on the available local information, a variety of models of rainwater tanks and house plans, developed with assistance from the internet and phone apps, our company has provided a recommendation for a rainwater tank to meet your family s estimated needs. Assumptions and observations Assume a 4-bedroom home would comfortably sleep a family of five and the number of people in the household will determine the total amount of water usage. There is no loss or reduction of rainwater in the tank due to evaporation or leaking, as this will greatly affect the ongoing costs of maintenance and require a larger tank to be installed. The rainwater tank is positioned at ground level to minimise the installation costs. Rainfall data from Tully for only the last 10 years was collected and analysed, as longer-term data may be unrepresentative due to the effects of climate change. The Careful water use table provides information about a person s typical water use. It has been assumed that children and adults both use the same amount of water and, if used for drinking and cooking, would need to be filtered to be safe to consume. Page 4 of 11

5 correct use of appropriate technical vocabulary, procedural vocabulary and conventions to develop the response Analysis and discussion The following mean and median rainfall data for Tully was obtained from the Bureau of Meteorology (BOM) website. The past 90 years of data has been used to create the bar graphs below Comparison of 2017 rainfall to the long-term averages at the Tully Sugar Formulate [A] accurate documentation relevant observations Source: Bureau of Meteorology, 2017 Rainfall, Tully Sugar Mill, av?p_display_type=datasgraph&p_stn_num=032042&p_nccobscode=136&p_month=13&p_startyear=2017 In all months, it was noted that the mean values are higher than the median. Examining the data of one of the months over the last ten years February (see table below) shows why this is the case. February annual rainfall for Tully Sugar Mill 2007 to 2016 Year Solve [A] discerning application of simple and complex mathematical concept relevant to the task Rainfall (mm) As can be seen, there is a significant amount of variation from year to year. The reason why the median is lower than the mean (from the graph, in February the median is less than 700 mm and the mean is more than 700 mm), is that on occasion there are years with significantly higher rainfall in the month, which skews the average higher, e.g. in 2007 and This could have been due to certain heavy downpours of rain (or floods) that occurred in those years. As such, the median is a more useful and robust predictor of typical rainfall than the mean. Page 5 of 11

6 Calculating the rain collected The dimension ratios from the floor plan have been calculated as follows: Length: 11.3 cm:21.44 m 1 cm:1.897 m Formulate [A] documentation of appropriate assumptions coherent and concise organisation of the response use of table to clearly summarise information and data Solve [A] discerning application of mathematical concepts and techniques relevant to the task Width: 6.9 cm:13.1 m 1 cm:1.899 m Assuming for the purpose of mathematical accuracy, a scale factor of 1.9 (correct to 1 d.p.) has been used. The above floor plan was divided into four composite rectangles to approximate the total roof area of the house as indicated by the different coloured rectangular outlines. Summary table of area calculations (rounded to 1 d.p.): Area (m 2 ) Working out 1 AA 1 = AA m 2 2 AA 2 = AA m 2 3 AA 3 = AA m 2 4 AA 4 = AA m 2 Formulate [A] accurate translation of all simple and complex aspects of the problem by identifying mathematical concepts and techniques Total roof area Area = AA 1 + AA 2 + AA 3 + AA 4 Area m 2 It is reasonable to approximate the total roof area to be 240 m 2 (more realistic if rounded to the nearest square metre). The volume of water it can collect is the total roof area depth of rainfall. For example, in 2013, based on 281 mm of rainfall, the amount of rainwater (rounded to 1 d.p.) collected by the roof is: Page 6 of 11

7 VV = AA h VV = VV 67.4 m 3 Evaluate and verify [A] evaluation of the reasonableness of solutions by considering the results, assumptions and observations coherent and concise organisation of the response use of table to clearly summarise information and data The approximate potential volume of rainwater collected was L, i.e ). However, the rainwater tank can t realistically collect 100% of the rain that hits the roof. Given the flow-off rate of 75%, the approximate potential amount of rainwater was L, i.e Typical monthly rain collection A spreadsheet can calculate the expected monthly median rainfall collection, as summarised in the table below (to the nearest millimetre): Month Median rainfall (mm) Total rainwater (kl) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average Here s a snippet of the spreadsheet values: Page 7 of 11

8 Solve [A] accurate and appropriate use of technology use of spreadsheet software to produce entries for easier data manipulations Here s a snippet of the spreadsheet entries with the formulas showing: Typical monthly water usage This report has used the Careful water use information to calculate your family s needs. Purpose Amount per person per month (L) Total amount consumed by the family per month (L) Solve [A] accurate use of complex procedures to reach a valid solution The solution consists of an involved combination of parts that are interconnected. correct use of appropriate technical vocabulary, procedural vocabulary and conventions to develop the response Solve [A] accurate use of complex procedures to reach a valid solution The process of connecting the total roof area to then determine a volume of rainwater collected and the total amount of water that it consumed and comparing these values to recommend a rainwater tank consists of an involved combination of parts that are interconnected. Evaluate and verify [A] documentation of relevant strengths and limitations of the solution and/or model Drinking and cooking = 1500 Dishwashing = 750 Bathroom and toilet = Washing clothes = 1000 Watering garden (30 mins session) 350 L (per half hour) 3 (times a week) 52 (weeks in a year) 12 (months in a year) = 4550 L Total amount of rainwater used by the family (per month) = L The median monthly rainwater was found to be L (see Typical monthly rain collection table). Using this as a basis, the average excess amount of water remaining in the tank per month is L, i.e Recommendation Based on the potential rainwater that could be collected, it is recommended that you opt for a rainwater tank larger than L (your average water consumption). This should be more than sufficient to cater for most rainfall periods in that area. An even larger tank (e.g L) would likely allow you to store extra rainwater from the heavy rain months (January to May) for use in the lighter rain months (August to October). Page 8 of 11

9 Evaluate and verify [A] justification of decisions made using mathematical reasoning Evaluate and verify [A] justification of decisions made using mathematical reasoning The above recommendation considers the gutter height of 2.4 m (the ceiling height of the house) meaning that a tank needs a height less than this to allow for water to drain via the pipes. A number of rainwater tanks have been considered and a L round tank is recommended. It has a diameter of 3.95 m and a height of 2.45 m, with the inlet height at 2.05 m. Because it has an overall height of 2.45 m it will need to be located a few metres from the side of the house, which is only 2.4 m high. The tank could be placed immediately beside the house but would need to be dug into the ground, which would drastically increase the installation costs, but provide no additional benefits. coherent and concise organisation of the response, appropriate to the genre, including a suitable conclusion Evaluate and verify [A] documentation of relevant strengths and limitations of the solution and/or model Conclusion Based on a house roof area of 240 m 2, historical data, and the family s predicted water usage, it was determined that a rainwater tank of L capacity would be ideal. A number of strengths and limitations for choosing this sized tank are: One tank, not directly beside the house, would be cheaper to install than two tanks beside the house. In typical weather conditions, this tank would provide sufficient water for all of your predicted needs. It assumes average (and not below average) rainfall. It doesn t account for the very dry months, so you will need to supplement your water supply with town water. Page 9 of 11

10 coherent and concise organisation of the response, appropriate to the genre, including a suitable conclusion Taking all this into account, it is strongly recommended that you purchase a L round tank with a diameter of 3.95 m and a height of 2.45 m. Page 10 of 11

11 Appendix Page 11 of 11