Numeracy Stage 6 CEC Teaching Guide - Module 4 NSW Education Standards Authority - NSW ...
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NSW Education Standards Authority Numeracy Stage 6 CEC Teaching Guide Module 4 Effective from Term 4, 2022 Year 12 Publication date July 2021 Updated NA
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Contents Introduction to the Teaching Guides .................................................................................. 7 What are the Teaching Guides? ......................................................................................................... 7 What types of resources are in the Teaching Guides? ...................................................................... 7 How do the Teaching Guides connect to other Numeracy CEC resources? .................................... 7 What are the Teaching and Learning Programs? .............................................................................. 8 What types of resources are in the Teaching and Learning Programs? ........................................... 8 Using the Numeracy CEC resources .................................................................................................. 9 Module 4 .............................................................................................................................. 10 Outcomes ........................................................................................................................................... 10 Content ............................................................................................................................................... 10 4.1 Rates and ratios............................................................................................................ 11 4.1.1 Rates ........................................................................................................................... 11 Contexts for rates ............................................................................................................................... 11 Stimulus questions ............................................................................................................................. 11 Language and literacy ....................................................................................................................... 11 Common misconceptions .................................................................................................................. 12 Introductory tasks ............................................................................................................................... 12 Task 1: Speed limit ....................................................................................................................... 12 Task 2: Expensive tastes ............................................................................................................. 13 Task 3: If this then ........................................................................................................................ 13 What if my students find rates difficult?............................................................................................. 13 Suggested activities ........................................................................................................................... 14 Activity 1: Would you like fries with that? ..................................................................................... 14 Activity 2: The inflation rate .......................................................................................................... 14 Activity 3: Rate bloopers ............................................................................................................... 15 4.1.2 Ratios.......................................................................................................................... 16 Contexts for ratios .............................................................................................................................. 16 Stimulus questions ............................................................................................................................. 16 Language and literacy ....................................................................................................................... 16 Common misconceptions .................................................................................................................. 16
Introductory tasks ............................................................................................................................... 17 Task 1: Sharing chocolates .......................................................................................................... 17 Task 2: Enlarging Felix ................................................................................................................. 17 Task 3: How long will it take? ....................................................................................................... 18 Suggested activities ........................................................................................................................... 18 Activity 1: Bad date ....................................................................................................................... 18 Activity 2: Netflix or Stan? ............................................................................................................ 19 Activity 3: Which device is better? ............................................................................................... 19 Activity 4: Mocktails ...................................................................................................................... 20 Activity 5: Mixing paint .................................................................................................................. 20 Activity 6: Baseball ....................................................................................................................... 21 Activity 7: Landscape plans .......................................................................................................... 21 4.1.3 Rates and ratios......................................................................................................... 21 Suggested activities ........................................................................................................................... 21 Activity 1: Ramp it up .................................................................................................................... 21 Activity 2: Screen-time habits ....................................................................................................... 22 Activity 3: Shower vs bath ............................................................................................................ 23 4.2 Statistics and probability ............................................................................................. 24 4.2.1 Statistics..................................................................................................................... 24 Contexts for statistics ......................................................................................................................... 24 Stimulus questions ............................................................................................................................. 24 Language and literacy ....................................................................................................................... 24 Common misconceptions .................................................................................................................. 24 Introductory task................................................................................................................................. 25 Task: Source and context ............................................................................................................ 25 Suggested activities ........................................................................................................................... 25 Activity 1: Screen time .................................................................................................................. 25 Activity 2: Misleading graphs........................................................................................................ 26 Activity 3: Mean, mode and median............................................................................................. 26 Activity 4: Graphing in Excel ........................................................................................................ 26 4.2.2 Probability .................................................................................................................. 27 Contexts for probability ...................................................................................................................... 27
Stimulus questions ............................................................................................................................. 27 Language and literacy ....................................................................................................................... 27 Common misconceptions .................................................................................................................. 28 Introductory tasks ............................................................................................................................... 28 Task 1: That’s random.................................................................................................................. 28 Task 2: Best chance ..................................................................................................................... 29 Suggested activities ........................................................................................................................... 29 Activity 1: There’s still a chance ................................................................................................... 29 Activity 2: Reasonable risk ........................................................................................................... 30 4.3 Exploring with NRMT ................................................................................................... 32 The NRMT process ............................................................................................................................ 32 The Numeracy journey so far … ....................................................................................................... 32 Contexts for NRMT ............................................................................................................................ 33 Language and literacy ....................................................................................................................... 33 Exploring the elements ...................................................................................................................... 33 Interpreting ......................................................................................................................................... 33 The challenge of interpreting........................................................................................................ 33 Interpreting task: Coffee shop confusion ..................................................................................... 33 Designing activities that focus on interpreting ............................................................................. 34 Choosing ............................................................................................................................................ 34 The challenge of choosing ........................................................................................................... 34 Choosing task 1: Discount fuel .................................................................................................... 35 Choosing task 2: Kitchen renovation ........................................................................................... 35 Designing activities that focus on choosing ................................................................................. 35 Applying .............................................................................................................................................. 36 The challenge of applying ............................................................................................................ 36 Applying task: Stopping distances ............................................................................................... 36 Designing activities that focus on applying .................................................................................. 36 Reflecting ........................................................................................................................................... 37 The challenge of reflecting ........................................................................................................... 37 Reflecting task 1: Seven-day forecast ......................................................................................... 37 Reflecting task 2: Garden lights ................................................................................................... 38 Designing activities that focus on reflecting................................................................................. 38
Communicating .................................................................................................................................. 38 The challenge of communicating ................................................................................................. 38 Communicating task: Unfair games ............................................................................................. 39 Designing activities that focus on communicating....................................................................... 39 Putting it all together .......................................................................................................................... 39 The purpose of a project .............................................................................................................. 40 Choosing an area of interest ........................................................................................................ 40 Presenting the personal numeracy project .................................................................................. 40 Selecting knowledge and skills .................................................................................................... 41 Possible scenarios ........................................................................................................................ 42 Embedded objects ............................................................................................................. 43 Web links ............................................................................................................................. 43
Introduction to the Teaching Guides What are the Teaching Guides? The Teaching Guides illustrate ways to engage with the content and skills associated with the Numeracy Stage 6 Syllabus (2021). A Teaching Guide has been created for each module. Key resources from each Teaching Guide are referenced within the associated Teaching and Learning Program. What types of resources are in the Teaching Guides? Materials provided within the Teaching Guides are organised according to the following categories: Contexts connect the content to age-appropriate contexts and establish the place of numeracy in the real world. Stimulus questions are age-appropriate questions that aim to ignite student curiosity. They help students identify with the usefulness and importance of learning the content and skills. Language and literacy highlights the content-specific literacy needs of students and provides some teaching ideas for how terms, ideas and concepts can be addressed. Common misconceptions identify assumptions or learned errors, which may affect student understanding or readiness to progress into new learning. Explicit teaching may be required to address the misconception. Introductory tasks are intended to contribute to teachers’ understanding of their students’ numeracy needs through informal identification of common misconceptions or numeracy ‘gaps’. Designed to be short and non-threatening, they provide immediate feedback for teachers and students by encouraging discussion or actions aimed at revealing student thinking. Activities engage students with everyday situations that require them to identify and apply numeracy skills in meaningful contexts. Specifically, NRMT activities provide students with opportunities to combine skills and understanding from multiple topics and apply the Numerical Reasoning and Mathematical Thinking process to interpret and resolve a situation. How do the Teaching Guides connect to other Numeracy CEC resources? Figure 1 summarises the connection between the various Numeracy CEC resources. The Teaching Guides have been created in partnership with the Teaching and Learning Programs. Each Teaching Guide provides content-specific advice for teachers for each of the content areas listed in the associated Teaching and Learning Program. Teachers are encouraged to adapt, refine and personalise the activities to create resources that are appropriate to the age, interests and aspirations of the students in their class. The Teaching Guides are not an exhaustive list of possible learning activities but serve as a starting point that may seed further investigation and fuel teacher creativity. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 7 of 44
The Teaching Guides and Teaching and Learning Programs have been created to model best practice in the teaching and assessment of the Numeracy Stage 6 Syllabus (2021). Numeracy Stage 6 Syllabus Teaching & Learning Programs − Scope, sequence and assessment schedule Teaching Guides − Week-by-week anticipated content − Introductory tasks − Links to: − Contexts, stimulus questions, language and literacy, and common o National Numeracy Learning Progression misconceptions o Teaching Guide resources o Other useful online materials − Activities and resources Figure 1: Connecting Numeracy CEC resources What are the Teaching and Learning Programs? The Teaching and Learning (T&L) Programs illustrate ways to scope, sequence and program the Numeracy Stage 6 Syllabus (2021). A T&L Program has been created for each module. What types of resources are in the Teaching and Learning Programs? Materials provided within the T&L Programs are organised according to the following categories: Suggested course structure includes a sample scope, sequence and assessment schedule, a week-by-week breakdown of anticipated content, links to the National Numeracy Learning Progression and identification of Teaching Guide resources appropriate to the content. Recall, revise, relearn indicates the skills and content that may need to be revisited to ensure that students are prepared to meet new concepts. Review and consolidation provides links to content and skills met in the previous week(s) that may require additional attention. Anticipated content indicates the syllabus content that could be addressed during the week. Professional reading includes published articles or research related to relevant aspects of numeracy, pedagogical approaches, or Teaching Guide activities. Reference materials are materials that contain information to learn from or to use while supporting the learning activities of others. Online interactive materials are materials to learn with or to use while supporting the learning activities of others. Learning objects can be defined in a number of ways such as: ‘any entity, digital or non- digital, that may be used for learning, education or training’. Such objects are self-contained, reusable, and applicable in multiple contexts and small chunks of learning. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 8 of 44
Using the Numeracy CEC resources When using the Teaching Guides, the T&L Programs and other associated materials, teachers should ensure that teaching and learning materials are accessible, age appropriate, contextually relevant and suitable amendments have been made to meet the needs of their students. Students with disability may require adjustments and/or additional support in order to engage in the teaching, learning and assessment activities. This could include alternate modes of assessment, including resources in a range of formats, and ensuring images, graphics and other resources are accessible. Decisions regarding curriculum options, including adjustments, should be made in the context of collaborative curriculum planning with the student, parent/carer and other significant individuals to ensure that decisions are appropriate for the learning needs and priorities of individual students. Successful learning in numeracy for Aboriginal students requires teaching and learning experiences that are culturally relevant, academically rigorous, and explicitly linked to students’ social contexts. The Numeracy Stage 6 CEC provides the opportunity to do this by presenting numeracy concepts and skills through age-appropriate activities that are contextually relevant to the everyday experiences of students. Effectively incorporating Aboriginal perspectives as part of this context allows Aboriginal students to see themselves in their learning and makes numeracy a powerful and purposeful learning experience. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 9 of 44
Module 4 Outcomes A student: N6-1.1 recognises and applies functional numeracy concepts in practical situations, including personal and community, workplace and employment, and education and training contexts N6-1.2 applies numerical reasoning and mathematical thinking to clarify, efficiently solve and communicate solutions to problems N6-1.3 determines whether an estimate or an answer is reasonable in the context of a problem, evaluates results and communicates conclusions N6-2.1 chooses and applies appropriate operations with whole numbers, familiar fractions and decimals, percentages, rates and ratios to analyse and solve everyday problems N6-2.2 chooses and applies efficient strategies to analyse and solve everyday problems involving metric relationships, distance and length, area, volume, time, mass, capacity and temperature N6-2.3 chooses and applies efficient strategies to analyse and solve everyday problems involving data, graphs, tables, statistics and probability N6-2.4 chooses and applies efficient strategies to analyse and solve everyday problems involving money and finance N6-2.5 chooses and applies efficient strategies to analyse and solve everyday problems involving location, space and design N6-2.6 chooses and applies appropriate numeracy operations and techniques to analyse and resolve everyday situations N6-3.1 chooses and uses appropriate technology to access, organise and interpret information in a range of practical personal and community, workplace and employment, and education and training contexts N6-3.2 chooses and uses appropriate technology to analyse and solve problems, represent information and communicate solutions in a range of practical contexts Content The content for this module is drawn primarily from the content areas listed in Module 4. From time to time it may be necessary to include aspects from the content listed in Modules 1, 2 or 3 because activities that require the application of numeracy skills do not fall neatly into compartmentalised content areas or contexts. Teachers are encouraged to select from across the range of content areas rather than teaching through a single content area. This facilitates presentation of activities within real-life contexts relevant to the students who are undertaking the course. Teachers are encouraged to address areas of specific need or extension as they arise. In these documents, such opportunities will be referred to as ‘learning-ready’. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 10 of 44
In Module 4, students resolve situations independently using the NRMT process by: ▪ interpreting the situation ▪ choosing information, strategies and skills relevant to the situation ▪ applying information, strategies and skills to resolve the situation ▪ reflecting on the situation as it is being resolved ▪ communicating throughout the resolution of the situation. 4.1 Rates and ratios 4.1.1 Rates Contexts for rates The following situations provide meaningful contexts for learning activities that involve rates: ▪ driving a car, for example speed limits, blood alcohol limits, fuel costs ▪ comparing pay rates and hiring costs ▪ interpreting quantities that are used together in descriptions and comparisons, for example price and weight ▪ estimating cooking times ▪ calculating best buys to check that price tags have been changed correctly. Stimulus questions ▪ What does it mean if an aircraft breaks the sound barrier? ▪ What is the world record for the 100-m sprint? How fast was that person running? ▪ How much does the price of fuel affect a young adult in NSW? ▪ How did the 2019/2020 NSW bushfires affect air quality? How is air quality measured? Language and literacy Numeracy and literacy are interrelated and teachers will need to address the context-specific literacy needs of their students during this course. This section includes some terms, ideas and concepts that may need to be addressed. For example: ▪ the symbol / should be read as ‘per’ ▪ the use of the word per, meaning for every, in describing rates should be made explicit to students ▪ the abbreviations for common units use a mix of capital and lower-case letters, and each is specifically assigned in The International System of Units, for example: cm, °C, g and L. When describing distance/time graphs (travel graphs), teachers are advised to model a story and graph, or jointly construct a story with students before setting independent work. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 11 of 44
Common misconceptions 1. Students may be ‘overfamiliar’ with common rates such as km/hr and not recognise that these represent two measurements in two different units – distance in kilometres and time in hours. The common display of speed restrictions on roads removes the units and this may have led some students to think of speed only as the number, for example 60. This can affect a student’s thinking about questions such as: How far would you travel in 15 minutes at this speed? How long would it take to travel 90 km at this speed? 2. As rates are typically presented as unit rates, students can miss the significance of the ‘second’ measurement. For example, a price of $6.90/kg can be perceived as $6.90. This can impede the ability to make well-reasoned comparisons between different products. Students can miss the meaningfulness of the kilogram measurement and forget to consider what 1 kg of the different products actually yield when making comparisons. 3. Misconceptions about the conversion of units of measurement will influence the ability to apply proportional reasoning to problems. For example, when considering km/hr and m/sec, students can be unsure whether division or multiplication is required and hence fail to recognise an unreasonable answer should they arrive at one. 4. Misconceptions about the application of fractions, decimals and percentages will affect the ability to interpret and apply rates. These are identified in Modules 2 and 3. Introductory tasks The following tasks have been designed to provide opportunities for teachers to determine students’ ways of thinking about proportion and the fixed relationships between quantities measured in different units. Students may use repeated addition or multiplication to work out simple rates before they develop proportional reasoning. The development of proportional reasoning follows the development of additive and then multiplicative thinking. It requires the formation of complex connections and an understanding of the relationship between quantities. Task 1: Speed limit This task can help the teacher learn more about how a student interprets rates as a relationship between two different types of quantities (CoU2). Students interpret a 50 km/hr speed limit sign. Students could respond on post-it notes or mini whiteboards or enter their answers into a digital response collecting tool such as Mentimeter or Wooclap. Speed limit Understanding student responses Student responses to the first two instructions can reveal their understanding of ordering and comparing rates and the concept of a limit as a boundary between greater than and less than values. An error in the response to Item 3 may indicate a range of misconceptions and these should be explored through discussion. Student responses to Item 4 could assist teachers in differentiating appropriately for their readiness in this topic. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 12 of 44
Task 2: Expensive tastes This task can help the teacher learn more about how a student interprets rates as a relationship between two different types of quantities (CoU2). Students are shown three expensive foods and their price tags: avocados @ $9.00/kg, lobster @ $14.90/100 g and saffron @ $13.80/100 mg. Students may be interested to know that saffron is a spice used to colour and season food. It is known as the world’s most expensive spice by weight. Expensive tastes Questions 1. List these three food items in order of most to least expensive. 2. Is the most expensive item a lot more, a little more or about the same price as the least expensive? Explain your answer. Understanding student responses 1. The response, lobster, saffron, avocado, suggests that students have not considered the rate and have only read the dollar amount. 2. This question does not ask students to convert to a common unit, but some students may be aware of the need to do this and, of them, some may know the calculation required. Teachers may observe errors in the decision about whether to multiply or divide the dollar amount. Responses to this question may also reveal misconceptions about the relative size of kg, g and mg and how to convert between these. Task 3: If this then This task can help the teacher learn more about how a student applies direct proportion (CoU3). Students compare pairs of images to estimate values inferable from the information provided. Students reflect on and describe the method they used to decide on their answers. If this then If this then What if my students find rates difficult? The part-part comparison that is ratio becomes a rate when it implies a constant, indicated by ‘per’. For example, speed in kilometres per hour describes a ratio between a measure of distance and a measure of time. Students could establish confidence in rates as a constant relationship between the two quantities being measured by first using an additive strategy. In the context of a long stretch of road and a speed limit of 100 km/hr, students could set out a table or diagram showing how far a car would travel in one hour, two hours, etc. This can then be developed for shorter lengths of time such as half an hour, a quarter of an hour, both also Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 13 of 44
being expressed as minutes to allow students to improve their understanding of the relationship between time and the distance travelled. Maintaining this context, students could complete a table or diagram from the perspective of distance travelled, answering questions about how long it will take to travel 100 km, 200 km, etc. This can then be developed to shorter distances such as 50 km and 25 km. Students should be encouraged to use ‘per’ in their discussions. Some students may have experience in measuring their own heart rate. After defining ‘resting heart rate’, ask the class what the most efficient way would be for each person to measure their own. Practising the common technique of counting the pulse for 15 seconds and multiplying by four can assist students in recognising the multiplicative nature of rates. A discussion drawing out the difference between the two things being measured, a beating heart and the passing of time, can emphasise that ‘resting heart rate’ is neither a heartbeat nor a minute, but the constant relationship between these two different things. Suggested activities Activity 1: Would you like fries with that? This activity invites students to think critically about the relative value of small and large servings by observing a reporter in the field calculating and comparing unit rates. Discussion ▪ If students buy McDonald’s fries, do they buy the large or small serving? ▪ What are they considering when they make this decision? ▪ Do students believe the large serving is better, worse or equal value to the small? ▪ On what do they base this opinion? Students view the video, McDonald’s fries – this is how much you ACTUALLY get, and complete the response table. Students then choose another product to test and use an online tool such as the unit rate calculator to convert advertised prices to unit rates for meaningful comparisons. The linked video does not contain closed captions. It is recommended that students who are deaf or hard of hearing be provided with the written article that accompanies the video. Would you like fries with that Activity 2: The inflation rate This activity focuses on understanding the meaning and effect of the inflation rate. It will also engage students in reviewing their knowledge and skills associated with percentages. Students are provided with an opportunity to make informed financial decisions and increase their ability to interpret credit and savings interest rates, pay rises and freezes, and the changing cost of living. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 14 of 44
Discuss ▪ In general, what does the word ‘inflation’ mean? What does it mean in a financial context? ▪ Have students heard any stories from older people about what things used to cost? ▪ Do students have any personal examples of things that have become more or less expensive in the last ten years? Explore Students use the Reserve Bank of Australia inflation explorer to explore how average prices have changed in recent years. A printable set of instructions for exploring inflation can be downloaded from Reserve Bank of Australia learning activities. Students view the video, Petrol prices: How to save at the pump, embedded in the October 2019 news article ‘Potential time bomb’: Graph shows how high your bills will go. Teachers then guide the class to interpret the information about inflation and its impact on the cost of living as presented in the article. Teachers should point out to students that prices usually fluctuate within an overall trend, so while the price of a product may be increasing with inflation, it may still fall at particular times due to mini-cycles within the trend. The Petrol prices: How to save at the pump video does not contain closed captions. It is recommended that students who are deaf or hard of hearing be provided with the written article How to save money on fuel. Research and compare Students look for current news articles or video clips to see whether the inflation rate and the cost of living has changed since the October 2019 article. Apply In small groups, students consider how to decide when the time is right to make a purchase. Buy now or wait? ▪ Does this depend on what the item is? ▪ Does it depend on whether the money is available or a loan is needed? ▪ Do interest rates play a role in the decision? ▪ Will everything be more expensive in the future or can it be expected that the price of some things will come down? Activity 3: Rate bloopers Students explore examples where rates have been applied incorrectly. In pairs or small groups they identify the error, decide what was intended and explain the correction needed. Rate bloopers Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 15 of 44
4.1.2 Ratios Contexts for ratios The following situations provide meaningful contexts for learning activities that involve ratios: ▪ making comparisons, for example aspect ratios in television, screens and photography or population density ▪ altering recipes ▪ using map scales, floor plans and landscape plans ▪ working with fuel, concrete or paint mixes ▪ calculating odds. Stimulus questions ▪ What does changing gears on a bike or in a car actually do? ▪ If the odds of winning are 1: 5 would you buy a ticket? Would you buy 5 tickets? ▪ If the aim of training is to prepare players for their real matches, is there an optimal training-to-match ratio? How could you find out? Language and literacy Numeracy and literacy are interrelated and teachers will need to address the context-specific literacy needs of their students during this course. This section includes some terms, ideas and concepts that may need to be addressed. For example: ▪ the symbol : in a ratio is read as 'to' ▪ when comparing two quantities, order of language is important, eg 'if I have 4 ties for every shirt, the ratio of ties to shirts is 4: 1 and the ratio of shirts to ties is 1: 4'. Common misconceptions 1. Students may confuse ratios with fractions, for example thinking 2: 3 is the same as 2/3 or 2 2 3 . Students who make this error are likely to describe 3 as 2 parts out of 3, rather than two 1 lots of . 3 2. Students may misunderstand the difference between multiplicative and additive comparisons. Equivalent ratios have a multiplicative relationship, so students need to understand the concept of multiplicative comparison. Students can mistakenly assume 5: 8 is equivalent to 7: 10 because each part of the original ratio has been increased by 2. 3. Students may misunderstand direct versus proportional division. For example, if it takes two people four hours to do a certain task, students may mistakenly think that it would take one person two hours rather than eight hours. 4. When writing ratios in the form 1: students incorrectly assume that has to be a whole number or must be greater than 1. 5. Misconceptions about the application of fractions, decimals and percentages will affect a student’s ability to interpret and apply ratios. These misconceptions are identified in Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 16 of 44
Modules 2 and 3. Introductory tasks The following tasks have been designed to provide opportunities to determine your students’ ways of thinking about proportion and the maintenance of consistent proportions between quantities measured in the same units. Task 1: Sharing chocolates This task can help the teacher learn more about a student’s interpretation of ratio as a comparison of quantities with the same unit of measure (CoU2). Students are presented with a sharing situation and respond individually. Responses can be recorded on slips of paper or submitted using a digital tool. The situation can then be opened up to class discussion. In an alternate method of responding, after considering Question 1, students can be invited to move to one side of the class or another depending on their answer to Question 2. Students can then explain the reason for their choice. Students who wish to change sides after hearing these reasons are invited to do so. Sharing chocolates Sharing chocolates Understanding student responses Students who think Hunter is correct will have focused on the ‘three for you, four for me’ relationship, rather than the relationship between what Brody has and the total chocolates in the packet. A table showing each person’s chocolates and the total chocolates as they are distributed can help students visualise the difference. Task 2: Enlarging Felix This task can help the teacher learn more about a student’s interpretation of ratio as a comparison of quantities with the same unit of measure (CoU2). Students are presented with a situation involving scale and respond individually. Responses can be recorded on slips of paper or submitted using a digital tool. Students are then invited to discuss their responses as a class. Enlarging Felix Understanding student responses Students may misunderstand the question and suggest that it all depends on the quality of the original picture. Students who have some understanding of equivalent ratios may correctly identify that the photograph can be enlarged without needing to cut or distort the picture, but may be unable to explain mathematically why this is the case. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 17 of 44
Development Students could be asked to bring in their own 4 by 6 photo, or one could be provided. If they needed to fit the photo into a square, how would they do it? Could they do it? Tactile photographs or cards can be used for students with vision impairment. Task 3: How long will it take? This can help the teacher learn more about a student's understanding and application of indirect proportion (CoU3). Students are presented with a problem-solving situation and respond individually. Responses can be recorded on the embedded Word document or submitted using a digital tool. Students are then invited to discuss their responses as a class. How long will it How long will it take take Understanding student responses Student responses can reveal their understanding of proportion. For example, students may mistakenly halve the time to do the job, since there are now half the people, indicating confusion between direct and indirect proportion. Suggested activities Activity 1: Bad date In this activity, students will consider ratio in the context of a conversation between two people on their first date. Teachers guide students’ viewing of the video Bad date, pausing at specified breaks to invite student responses to a set of prompts. At the conclusion of the video, students work in pairs and respond to four questions. First pause: After the first date, when the ratio is 1: 7. ▪ Students state what they think the video is about. ▪ Students share everyday situations that are in a 1: 7 ratio. Second pause: After the second date, when the ratio is 6: 1. ▪ Students comment on how well the second date went. ▪ Students predict what Isabella may be looking for in a date. Bad date Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 18 of 44
Activity 2: Netflix or Stan? In this activity, students are presented with three summary statements about the results of a survey testing customer preferences. Students will need to employ numerical reasoning to decide what can and can’t be determined from each statement. This activity will contribute to their ability to think critically about statistical statements in the media. The questions could be completed individually by students and then students could share the numerical reasoning they used to make their decisions in a class discussion. Netflix or Stan Activity 3: Which device is better? This activity invites students to think mathematically about one of their most regular, day-to- day activities – looking at screens. Students will consider the strengths and weaknesses of the different, common screen ratios and explore the origins and evolution of these. Stimulus Students watch the TED video: Why the shape of your screen matters. Discussion Student preferences for watching a movie on their home television: ▪ the black bars top and bottom ▪ the stretched screen where everything is slightly distorted ▪ changing the aspect ratio depending on what they are watching. Research Students then use the information in the stimulus video and conduct further research to answer the question: Which device is best when watching a movie on a plane? ▪ The screen on the back of the seat in front of you ▪ Your own mobile phone ▪ Your own laptop or tablet. Which device is better Scaffolding The development of student autonomy in applying NRMT is a priority of Module 4 and so teachers give students the opportunity to develop their response on their own or in small groups. Where students do require scaffolding to support their work towards an independent answer, the following is suggested: Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 19 of 44
▪ Revisit the part of the TED video showing that movies are in an aspect ratio of 2.35: 1. ▪ Prompt students to search for the size and aspect ratio of different screens online: ˗ television screens ˗ inbuilt economy class screens on planes, eg the size of the screens on Virgin economy seats is currently 11.6 inches which is an aspect ratio of 16: 9 ˗ phones, eg ones that students have ˗ laptops or tablets, eg ones that students use or own. ▪ Prompt research of the aspect ratios of smartphones (most Smartphones manufactured since 2010 have an aspect ratio of 16: 9 but later Smartphones such as the iPhone X have an aspect ratio of 19.5: 9). ▪ Ask how students would decide whether the screen on the seat in a plane or the phone screen is better. Which has better definition? Depending on how new students' phones are, they may find they are better off watching a movie on their phone if they prefer the movie to look more like the original! Or, they may prefer to watch the movie on their seat screen, where the movie will need to be downsized to fit, but at least the screen will be bigger. Activity 4: Mocktails This activity introduces ratios with three terms in the context of mixing a mocktail. The initial information includes a fraction as one of the terms. In Stage 4 Mathematics, students are asked to simplify such ratios; however, the Numeracy Stage 6 CEC focus is on using and interpreting ratios in everyday contexts. This activity provides a useful platform to emphasise the importance of and assumption of a common unit of measurement when proportions are presented as ratios. Students could be asked to consider the resulting flavour if the sour ingredient was measured in cups and the sweet ingredient in teaspoons. Where resources allow, this activity could be conducted as a practical experiment where students perform taste tests on correctly and incorrectly interpreted proportions. Mocktails Activity 5: Mixing paint Students who have developed fluency with interpreting simple ratios may be ready to apply their skills and understanding to more complex problems. As this activity is in the context of mixing paint colours, students should be encouraged to visualise the resulting shade of the initial and subsequent ratios. Where resources allow, this activity could include a practical experiment where students mix teaspoons of paint to see the contrast between the two resulting shades. This would provide further opportunity to discuss proportion in terms of the colour being the same when measuring in teaspoons, cups or litres. As long as the ratio is maintained, the quantity will change but not the ‘quality’. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 20 of 44
Mixing paint This activity is adapted from a PBS Learning media video activity. Activity 6: Baseball This activity asks students to apply prior learning about fractions, decimals and percentages. Students are given information in the form of a ratio and encouraged to practise their fluency with these different forms of expressing a value. Baseball Activity 7: Landscape plans In this activity, students consider ratio in the context of scale on a site plan. Landscape plans As an alternative to the site plan provided in the student resource, students could choose their own ratio and create a landscape plan. This could be loosely based on where they live, a part of the school, eg bushfoods garden, or be a creation of their own. If students are creating their own landscapes, they should explain how they are determining the actual size of the features in their plan in order to justify the scale they have used. 4.1.3 Rates and ratios Suggested activities Activity 1: Ramp it up One of the rates that students encounter every day is the rate of ascent or descent, determined by gradient or steepness. This activity encourages students to recognise that steepness is the relationship between the distance moved forward and the distance moved up or down with each step. This activity also considers speed, another everyday rate in student experience. Stimulus Students watch a video of the Cooper’s Hill Cheese-Rolling and Wake: UK: Daredevils launch themselves downhill in cheese rolling race. Discussion ▪ Would it be possible for a person to beat the cheese down the hill? How fast is the cheese Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 21 of 44
going? (Students should estimate this before looking it up.) ▪ What is a comfortable rate of ascent or descent: ˗ when walking ˗ when running ˗ when cycling ˗ when driving? Investigation Students watch a second video, World’s steepest street: Baldwin Street in New Zealand gains internet fame, and note the description of steepness as a rise of 1 m for every 2.86 m horizontal distance. This is presented as the ratio 1: 2.86 on the Baldwin Street page. Students determine a way to measure steepness and investigate whether the different steps, staircases and ramps around the school are built to a consistent ratio. Students who are wheelchair users could contribute valuable insight based on their experience using ramps. Students working from remote locations could conduct a similar investigation within their homes or immediate environment. If students suggest that the steepness should be measured as an angle, teachers could invite the class to divide into two groups: one that is going to test steepness as a ratio; and one that will test angles. Preparation for this could include a discussion about the implication of an error of 10 degrees compared to 10 mm. These values are chosen as 10 degrees ‘looks’ like about 10 mm around a common protractor. Activity 2: Screen-time habits This activity relates to Activity 1: Screen time in 4.2.1 Statistics. In this activity, students use numerical reasoning in their consideration of a diagram from the article Australians are spending more than one-third of their day in front of a screen. Discussion for slide 1 ▪ Do these numbers seem reasonable? ▪ Does this match how much time students think they themselves spend using screens and sleeping? ▪ If people in Australia are spending 143 days per year on average in front of a screen, what percentage of their year are they spending on their screens? ▪ What other information would you like to know? Investigation Students consider the second slide in the stimulus PowerPoint. The information comes from an article exploring the screen time to reading ratio for children during their summer holidays. Students should look at the survey results and consider the questions on the slide. Students could also consider how the hours spent outdoors compare to their own. A good discussion point could also be whether the time that students spend reading on their phones is screen time or reading time. Does it depend on what they are reading? Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 22 of 44
Screen-time habits Activity 3: Shower vs bath This activity invites students to apply their understanding about rates and ratios in a context that incorporates prior learning about estimation, time, measurement and money. Students are asked to consider whether a shower or a bath is more expensive. Alternatively, this activity could be used to investigate the water used when washing up by hand compared to using a dishwasher. Discussion Ask students to estimate how much water they think they use in a typical shower or bath. ▪ How did they come up with that estimate? ▪ What information would they want to gather in order to make a more informed guess? ▪ How long do they spend in the shower? In order to decide whether a shower or a bath is cheaper, students need to work out what a shower or a bath actually costs. ▪ What information would they need to resolve this situation? Shower vs bath Shower vs Bath Investigation Students could use the Hunter Water website to estimate the water rate of the average showerhead in Australia and the water rate of the average tap. The average water flow rate for a showerhead could be anywhere from 15 to 20 litres per minute, or 9 L/min if a newer water- saving showerhead was installed. The water flow rate for an average bathroom tap is likely to be 12 L/min, again depending on how old it is. If there is a tap/sink available, the teacher or students could test the water rate from the tap using a bucket and a phone timer. A good point of discussion could be whether you can just run the tap for 30 seconds and then just double your answer to get the rate per minute. Students will need to be able to convert from litres to kilolitres to estimate the cost from a water bill. Students could also consider other ways they use water in the home. They could consider how much water they use when brushing their teeth and how much water they would save if they didn't leave the tap running while they did this. Numeracy Stage 6 CEC: Teaching Guide Module 4, published July 2021 Page 23 of 44
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