# Numeracy Stage 6 CEC Teaching Guide - Module 1 NSW Education Standards Authority - NESA

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NSW Education Standards Authority Numeracy Stage 6 CEC Teaching Guide Module 1 Effective from Year 11, 2022 Publication date July 2021 Updated NA

© 2021 NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales. The NESA website holds the ONLY official and up-to-date versions of these documents available on the internet. ANY other copies of these documents, or parts of these documents, that may be found elsewhere on the internet might not be current and are NOT authorised. You CANNOT rely on copies from any other source. The documents on this website contain material prepared by NESA for and on behalf of the Crown in right of the State of New South Wales. The material is protected by Crown copyright. All rights reserved. No part of the Material may be reproduced in Australia or in any other country by any process, electronic or otherwise, in any material form, or transmitted to any other person or stored electronically in any form without the prior written permission of NESA, except as permitted by the Copyright Act 1968. When you access the material you agree: ▪ to use the material for information purposes only ▪ to reproduce a single copy for personal bona fide study use only and not to reproduce any major extract or the entire material without the prior permission of NESA ▪ to acknowledge that the material is provided by NESA ▪ to include this copyright notice in any copy made ▪ not to modify the material or any part of the material without the express prior written permission of NESA. The material may contain third-party copyright materials such as photos, diagrams, quotations, cartoons and artworks. These materials are protected by Australian and international copyright laws and may not be reproduced or transmitted in any format without the copyright owner’s specific permission. Unauthorised reproduction, transmission or commercial use of such copyright materials may result in prosecution. NESA has made all reasonable attempts to locate owners of third-party copyright material and invites anyone from whom permission has not been sought to contact the Copyright Officer. Phone: (02) 9367 8289 Fax: (02) 9279 1482 Email: copyright@nesa.nsw.edu.au Published by NSW Education Standards Authority GPO Box 5300 Sydney NSW 2001 Australia www.educationstandards.nsw.edu.au D2020/149431

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 1 .............................................................................................................................. 10 Outcomes ........................................................................................................................................... 10 Content ............................................................................................................................................... 10 Introduction to the course ................................................................................................. 11 Getting to know your students ........................................................................................................... 11 The numeracy learning landscape .................................................................................................... 11 Introductory tasks ............................................................................................................................... 12 Assessing multiplicative reasoning .............................................................................................. 12 Using the Introductory tasks ......................................................................................................... 13 Task 1: Growth ............................................................................................................................. 13 Task 2: How many times bigger? ................................................................................................ 14 Task 3: A number table ................................................................................................................ 14 Task 4: Big brother ....................................................................................................................... 15 Task 5: Mr Tall and Mr Short........................................................................................................ 15 Where to next? ................................................................................................................................... 16 1.1 Whole numbers ............................................................................................................ 17 Contexts for whole numbers .............................................................................................................. 17 Stimulus questions ............................................................................................................................. 17 Language and literacy ....................................................................................................................... 18 Common misconceptions .................................................................................................................. 18 Suggested activities ........................................................................................................................... 18 Activity 1: What is the largest number you know you can say? .................................................. 18 Activity 2: Which number can you read? ..................................................................................... 19

Activity 3: Creating large numbers ............................................................................................... 19 Activity 4: People count ................................................................................................................ 20 Activity 5: Explaining estimations ................................................................................................. 20 Activity 6: It’s hot (or cold) outside ............................................................................................... 21 1.2 Operations with whole numbers ................................................................................. 22 Contexts for operations with whole numbers .................................................................................... 22 Stimulus questions ............................................................................................................................. 22 Language and literacy ....................................................................................................................... 22 Common misconceptions .................................................................................................................. 23 Additive strategies .............................................................................................................................. 23 Counting-on strategies ................................................................................................................. 23 Using doubles and near doubles ................................................................................................. 23 Compensation strategy ................................................................................................................ 23 Changing the order of addends to form multiples of 10 .............................................................. 24 Jump strategy on a number line................................................................................................... 24 Split strategy ................................................................................................................................. 24 Inverse strategy ............................................................................................................................ 24 Multiplicative strategies ...................................................................................................................... 25 The commutative and associative properties of multiplication.................................................... 25 Doubling and repeated doubling .................................................................................................. 25 Halving and repeated halving ...................................................................................................... 25 Factorising one number................................................................................................................ 25 Inverse operations ........................................................................................................................ 26 Multiplication and division facts .................................................................................................... 26 Introductory tasks ............................................................................................................................... 26 Task 1: Sum cloud ........................................................................................................................ 26 Task 2: Line them up .................................................................................................................... 27 Suggested activities ........................................................................................................................... 27 Activity 1: What was the question? .............................................................................................. 27 Activity 2: Shopping ...................................................................................................................... 27 Activity 3: Mathematics problems that go viral ............................................................................ 28 Activity 4: Too much sugar! .......................................................................................................... 28 Activity 5: How many nuggets? .................................................................................................... 28

Activity 6: Working with numbers ................................................................................................. 28 Activity 7: All the people in the world ........................................................................................... 29 Activity 8: Moving inventory .......................................................................................................... 30 Activity 9: Mathematics as storytelling ......................................................................................... 30 1.3 Distance, area and volume .......................................................................................... 31 Contexts for distance, area and volume ........................................................................................... 31 Stimulus questions ............................................................................................................................. 31 Language and literacy ....................................................................................................................... 31 Common misconception .................................................................................................................... 31 Introductory task................................................................................................................................. 32 Task: Connecting distance and area ........................................................................................... 32 Suggested activities ........................................................................................................................... 32 Activity 1: How far is that? ............................................................................................................ 32 Activity 2: Estimating room sizes ................................................................................................. 33 Activity 3: How much space does a parked car need? ............................................................... 33 Activity 4: Understanding area ..................................................................................................... 34 Activity 5: Designing rectangles ................................................................................................... 35 Activity 6: The wrapping problem ................................................................................................. 36 Activity 7: Matchstick construction ............................................................................................... 36 Activity 8: NRMT – The doghouse ............................................................................................... 37 1.4 Time ............................................................................................................................... 38 Contexts for time ................................................................................................................................ 38 Stimulus questions ............................................................................................................................. 38 Language and literacy ....................................................................................................................... 39 Common misconceptions .................................................................................................................. 39 Introductory task................................................................................................................................. 40 Task: A lifetime ............................................................................................................................. 40 Suggested activities ........................................................................................................................... 41 Activity 1: TV schedule ................................................................................................................. 41 Activity 2: Trip planning ................................................................................................................ 41 Activity 3: Bus timetable ............................................................................................................... 41 Activity 4: Planning simple journeys ............................................................................................ 42

Activity 5: Calendars ..................................................................................................................... 42 Activity 6: Time families card games ........................................................................................... 42 Activity 7: Time hexasaws ............................................................................................................ 42 1.5 Data, graphs and tables ............................................................................................... 44 Contexts for data, graphs and tables ................................................................................................ 44 Stimulus questions ............................................................................................................................. 44 Language and literacy ....................................................................................................................... 45 Common misconceptions .................................................................................................................. 45 Introductory task................................................................................................................................. 46 Task: What does the data say? ................................................................................................... 46 Suggested activities ........................................................................................................................... 47 Activity 1: Interpreting points ........................................................................................................ 47 Activity 2: Heads and feet............................................................................................................. 47 Activity 3: Creating two-way tables .............................................................................................. 48 Activity 4: Heart rate ..................................................................................................................... 49 Activity 5: Presenting data ............................................................................................................ 49 NRMT activities ................................................................................................................... 49 Activity 1: A roasting outback town.................................................................................................... 49 Global warming – real or fake news? .......................................................................................... 49 Other related resources ................................................................................................................ 51 Activity 2: Who wakes up first? .......................................................................................................... 51 Activity 3: Raising world awareness .................................................................................................. 51 Activity 4: Time in Australia................................................................................................................ 51 Activity 5: Valued brands ................................................................................................................... 52 Activity 6: Sissa’s reward ................................................................................................................... 52 Embedded objects ............................................................................................................. 53 Web links ............................................................................................................................. 53

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 1, published July 2021 Page 7 of 54

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 1, published July 2021 Page 8 of 54

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 1, published July 2021 Page 9 of 54

Module 1 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 N6-3.1 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 1. From time to time it may be necessary to include aspects from the content listed in Modules 2 to 4 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’. In Module 1, students are introduced to the Numerical Reasoning and Mathematical Thinking (NRMT) process. Initially it is vital that the teacher guides the exploration of an activity through the use of key questions and student discussion in order to model the process. By naming the steps in the process explicitly, students are provided with a common language for numerical reasoning and mathematical thinking: interpreting, choosing, applying, reflecting, communicating; which will help them begin to apply the process autonomously. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 10 of 54

Introduction to the course A key element of the Numeracy CEC is for teachers to ascertain the entry knowledge, skills and understanding of their students, and to gain a deeper understanding of their interests and aspirations. The first week of the course has been set aside for these purposes. The Numeracy CEC classroom is a safe learning environment where collaboration is encouraged and the student experience is a positive one. Getting to know your students It may be useful to use a simple class discussion or survey where students rate statements such as: ▪ I like to work in a group. ▪ I am willing to try something new. ▪ I use technology confidently. Alternatively, students could respond to open-ended questions such as: ▪ I learn best when ▪ I find it difficult to learn when ▪ What I’d like to change about the way I learn is ▪ I’m really interested in ▪ My dream job is ▪ I’d also like to say Towards the end of the year, students could respond to similar prompts to help the teacher gauge changes in student perceptions towards their learning. The numeracy learning landscape Numeracy development influences student success in many areas of learning. The National Numeracy Learning Progression can assist in strengthening teacher knowledge and facilitating a shared professional understanding of numeracy development. The progression can be used to identify the numeracy development of students and the development that should follow. This assists teachers to differentiate teaching and learning experiences and to provide feedback to students about next steps in learning. The progression is used in conjunction with the syllabus, which remains the focus for planning, programming, teaching, learning and assessment. The progression is organised into elements and sub-elements that describe common developmental pathways as students become increasingly adept in particular aspects of numeracy. Each sub-element contains descriptions of observable student behaviours known as indicators. The indicators within each sub-element are grouped together to form developmental levels. The listing of indicators within a level is non-hierarchical as the levels are collections of indicators. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 11 of 54

The National Numeracy Learning Progression has three elements: Number Sense and Algebra; Measurement and Geometry; and Statistics and Probability. There are nine sub- elements in Number Sense and Algebra, four in Measurement and Geometry, and two in Statistics and Probability. Number Sense and Algebra is the broadest of the three elements, addressing the following sub-elements: Quantifying numbers; Additive strategies; Multiplicative strategies; Operating with decimals; Operating with percentages; Understanding money; Number patterns and algebraic thinking; Interpreting fractions and Comparing units. The progression can be used to develop a detailed map of student progress within and across the 15 sub-elements. Prior to using the progression, it is helpful to gain an overall sense of student progress on the numeracy learning landscape. One of the most significant landmarks on the numeracy learning landscape is multiplicative reasoning, and the progression can be used to help understand which students may not have yet successfully transitioned from additive thinking to multiplicative thinking. For example, when asked to describe the relationship between 2 and 10, a student using additive thinking is likely to state that it has increased by 8. A student who is using multiplicative reasoning will see 10 as 5 times 2, as well as 8 more than 2. That is, a multiplicative thinker can see situations of comparison in a multiplicative sense, not just an additive sense. Multiplicative thinking is essential for progressing to the higher levels of quantifying numbers on the progression, as well as the Multiplicative strategies and Comparing units sub-elements. It also applies to area and volume in Measurement as well as fractions and proportions in Statistics. Some researchers have suggested that multiplicative reasoning does not develop fully until about the age of 14 (Siemon et al., 2011)1. Further, as there are many different applications of multiplicative reasoning, it is seldom straightforward determining which students have made the transition from relying almost exclusively on additive reasoning to the appropriate use of multiplicative reasoning. Consequently, any classification of a student’s thinking as being multiplicative is at best tentative. Introductory tasks The Introductory tasks in the Teaching Guide have been designed to help teachers gain an initial overview of their class by highlighting the type of thinking and numeracy skills students are using in situations that are best thought of as requiring multiplicative reasoning. Assessing multiplicative reasoning No one problem can fully assess multiplicative reasoning. In fact, a developed sense of multiplicative reasoning is indicated by being successful at a range of problems from diverse contexts. However, teachers can glean information from a careful analysis of a single problem and productively use it to inform instructional decisions. 1 Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R. and Warren, E. (2011). Teaching mathematics: Foundations to middle years. South Melbourne: Oxford University Press. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 12 of 54

Additive and multiplicative strategies are described in section: 1.2 Operations with whole numbers. These can be employed to address issues identified by Tasks 1 to 5. Using the Introductory tasks Introductory tasks provide opportunities for informal identification of common misconceptions or numeracy ‘gaps’. The introductory tasks 1 to 4 are intended to provide a broad overview of how your students have progressed from additive to multiplicative reasoning. It is likely that students taking the Numeracy CEC will have a range of different needs. For example, students demonstrating multiplicative reasoning with whole numbers might not apply it to fractions or decimals. Introductory Tasks 1 and 4 could be completed as a class discussion, with Tasks 2 and 3 requiring a written response. Task 1 can be used to help explain to your students the difference between additive reasoning and multiplicative reasoning. A simple tally of the number of students who indicated the dogs grew by the same amount, before and after discussion, will provide the basis of feedback. Task 4 reinforces the need to understand the context. Being able to reason multiplicatively means that you can determine when it is appropriate to multiply. Tasks 2 and 3 provide specific information on students identifying relationships involving multiplying and dividing by 10 and 100. If you wish to mark these responses and return the questions to the students, simply indicate how many mistakes they made on each task and ask them to work out which answers were wrong and why. For students who answered all of Task 2 correctly, add in an additional question: ‘How many times bigger is 2 than 0.02?’ You might also provide feedback highlighting the problems associated with shortcuts like adding a zero when multiplying by 10 (eg 0.02 × 10 ≠ 0.020). Task 1: Growth This task can help the teacher assess whether the students have progressed from additive to multiplicative thinking. This task relates to the National Numeracy Learning Progression: Additive Strategies Level 4 (AdS4) and Multiplicative Strategies Level 5 (MuS5). Students are shown images of a pair of puppies from the same litter and are asked to decide which had grown more in the six months between measurements. The task is best completed individually by students before engaging the whole class in discussing the reason for the suggested answer. Growth Understanding student responses Both dogs have gained 3 kg in weight (an additive difference of 3 kg). However, Dog 1 has doubled in weight (a multiplicative relationship) whereas Dog 2 hasn’t. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 13 of 54

▪ Keep a tally of the answers provided before the class discussion. ▪ Does everyone in the class agree on an answer? [Either both dogs grew the same amount or Dog 1 grew more.] ▪ Does anyone believe Dog 2 grew more? ▪ After the class discussion, where students need to provide reasons for their answers, check if any students would like to change their answers. Task 2: How many times bigger? This task can help the teacher understand how a student interprets the relationship between adjacent positions in place value (QuN11). The questions in the embedded slide are intended to be answered without using a calculator. Using a calculator will not guarantee a correct answer. These are best completed by students individually. How many times bigger Understanding student responses It is not uncommon to have a mixture of correct and incorrect responses to this series of four questions. Some students will provide correct answers to the first and last question but not the middle two questions. Others may provide purely additive responses, such as 18, 180, 1800 or 198, to some or all of the questions. Task 3: A number table This task can help the teacher understand how a student interprets the relationship between adjacent positions in place value for decimals (QuN11). Students are presented with a number table and asked to work out the values of missing numbers based on patterns that they recognise. A number table Understanding student responses Students are more likely to get the first question correct than the second or third. Although all three questions relate to multiplicative reasoning, the final two answers also relate to understanding decimal place value. Students frequently produce negative answers for Questions 2 and 3, corresponding to additive reasoning. Compare students’ responses to Operating with Decimals Level 3: understands that multiplying and dividing decimals by 10, 100, 1000 changes the positional value of the numerals, on the National Numeracy Learning Progression. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 14 of 54

Task 4: Big brother This task can help the teacher understand how a student selects appropriate strategies with two-digit numbers (AdS7). Students are presented with the situation and question: ‘Tim is 6 and his sister Anthea is half his age. When Tim is 48, how old will his sister be?’ Big brother Understanding student responses Although most research emphasises students’ use of additive strategies in response to questions involving multiplicative relationships, students sometimes overgeneralise a multiplicative relationship. In this task, a response of 24 suggests an inappropriate use of multiplicative reasoning. The use of additive strategies to problems that are not additive can be seen as an example of the law of the hammer. The law of the hammer, sometimes known as Maslow’s hammer, is a cognitive bias that involves an over-reliance on a familiar tool. ‘If the only tool you have is a hammer, it is tempting to treat everything as if it were a nail.’ The law of the hammer is relevant because additive strategies develop before multiplicative reasoning. The incorrect use of the initial multiplicative relationship between the ages in this task is a problem of interpreting the context of the question. Should the answer 24 be popular in the class, follow up with the questions on the second slide: ▪ Can you think of a time when you were or will be half your mother’s age? ▪ Can you think of another time when you were or will be half your mother’s age? ▪ Why or why not? Encourage students to take some time to think about the questions. The following explanation may be helpful: The only time you can be half your mother’s age is when the same number of years have passed as the age your mother was when you were born. For example, if your mother was 25 when you were born, in 25 years when your mother is 50, you will be half your mother’s age. When your mother is 60, you will be 25 years younger (35). You will never be half your mother’s age again as growing older is an additive process. Task 5: Mr Tall and Mr Short This task can help the teacher understand how a student applies proportion (CoU3). Proportional reasoning relies on multiplicative reasoning but it is more complex. If you are short of time, leave Task 5 for later. This task presents a question long associated with investigating proportional reasoning Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 15 of 54

(Karplus, Karplus, & Wollman, 1974).2 Within the National Numeracy Learning Progression, the sub-element Comparing units (ratios, rates and proportion) relates strongly to student responses to this task. Mr Tall and Mr Short Understanding student responses Additive reasoning 1 ‘Mr Tall is eight paper clips high. Mr Short is four matchsticks high and six paper clips high. So, the matchsticks are two less than the paper clips. Since Mr Tall is six matchsticks high, he must be eight paper clips high. Additive reasoning 2 ‘Mr Tall is two more matchsticks taller than Mr Short so he will also be two more paper clips taller than Mr Short resulting in eight paper clips. Just estimated ‘Nine, I figured he would be a bit taller.’ It is important to understand a student’s reasoning, rather than simply relying on whether the answer is correct or not. The additive solution is by far the most common. It is a consistent, logical strategy, albeit incorrect, and for the values involved in the problem it produces a believable answer. This problem is often found to be more difficult than expected because it uses a scaling context with a non-integer ratio (112). Karplus, Karplus, and Wollman (1974) report that 37 per cent of 610 students (Grades 4–9) used correct reasoning and 32 per cent used additive reasoning. Understanding the proportion requires appreciating that the matchstick-to-matchstick relationship is the same as the paperclip-to-paperclip relationship. The second multiplicative relationship involved in the proportion is the matchstick-to-paperclip relationship for Mr Short, which is the same as the matchstick-to-paperclip relationship for Mr Tall. Typically, students succeed in solving the problem when they have at least a basic understanding of one of these multiplicative relationships. Where to next? Ongoing monitoring of student progressing from additive to multiplicative reasoning will need to draw on manageable classroom assessment techniques. Classroom assessment techniques can be used as formative practice. Black and Wiliam (2009, p. 9)3 described formative practice as using evidence about student achievement ‘…to make decisions about 2 Karplus, E. F., Karplus, R., & Wollman, W. (1974). ‘Intellectual Development beyond Elementary School IV: Ratio, the Influence of Cognitive Style.’ School Science and Mathematics 74 (6): 476–82. doi:http:// dx.doi.org/10.1111/j.1949-8594 .1974.tb08937.x 3 Black, P. J., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education: Principles Policy and Practice, 5(1), 7–73. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 16 of 54

the next steps in instruction that are likely to be better, or better founded, than the decisions they would have taken in the absence of the evidence that was elicited’. Manageable classroom assessment techniques provide the teacher with a quick overview of students’ knowledge and help identify common stumbling blocks in learning. That is, classroom assessment techniques (sometimes described as CATs) are in-class activities designed to give useful feedback on the teaching–learning process as it is happening. Manageable CATs 1.1 Whole numbers Contexts for whole numbers The following situations provide meaningful contexts for learning activities that involve whole numbers: ▪ interpreting numbers in news articles or catalogue texts ▪ comparing the population of different countries and tracking population change over time ▪ describing and comparing the number of users of social media platforms ▪ estimating crowd numbers ▪ evaluating the accuracy of media reports ▪ researching grandstand capacity, seating capacity of entertainment venues and the capacity of sporting events and developing a meaningful sense of audience sizes ▪ describing the size of payouts in Lotto wins ▪ estimating the number of cars in a carpark ▪ comparing the total capacity of buses, trains, ferries and boats ▪ reading temperature scales ▪ ordering locations by average temperature, or minimum and maximum temperatures ▪ researching the wealth or debt of individuals or countries ▪ naming levels in a carpark to represent below ground, eg B1 is above B2. Stimulus questions ▪ How big is a million/billion? ▪ What is a gazillion? ▪ How did Google get its name? ▪ How many minutes from one birthday to the next? More than a million? How would you develop an estimate? ▪ Exactly how old are you in months, weeks, days, hours, minutes or seconds? Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 17 of 54

▪ What are the five countries in the world with the highest population? 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 meaning of commonly used counting numbers, eg thousands, millions, billions ▪ words and phrases that indicate relative size, eg more, greater, less, lower, maximum, minimum ▪ terms that are used to imply negatives, eg above and below sea level or freezing ▪ a ‘gazillion’ is not a real number. A ‘billion’ at one stage had different meanings in the UK and USA. Due to global finances, the USA definition of a ‘billion’ as a thousand million is now recognised as the standard definition. Common misconceptions 1. A common misunderstanding of the magnitude of a number is revealed when reading the numerals used to represent a number. For example, 1002 is confused with 102 because the first three digits of 1002 can be seen to represent 100 and the last as 2. Students who make this error will need to develop the capability to rename the groups of three digits within a number to form hundreds, thousands, hundreds of thousands etc, to enable them to read the number. 2. Australia uses the International Standard for writing numbers that consist of long sequences of digits: separate numerals into groups of three using a space to indicate thousands and millions. For example, twelve million is written as 12 000 000. This standard is often ignored, particularly when large numbers appear in newspaper articles or on the internet. Consequently students should also be familiar with the use of the comma to separate groups of three digits. 3. This may cause confusion because a comma is also used as the decimal sign in other countries. When writing large numbers, students should be encouraged to adhere to the Australian standard and use the space as the numeral grouping indicator. 4. The negative symbol can be used to indicate the operation of subtraction or to indicate a negative number. This can result in confusion in the interpretation of negative numbers when used in context. 5. Students may misunderstand the relative size of negative numbers. For example, −4 is sometimes thought to be larger than −2. Suggested activities Activity 1: What is the largest number you know you can say? In this activity, students apply their knowledge of place value and the prefixes used to say a number. Teachers present students with a list of numbers of increasing magnitude to determine which number is the largest they can read out. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 18 of 54

Activity 2: Which number can you read? In this activity, students will apply their understanding of place value. Students access a website such as Australian Debt Clock to view large numbers used in a real context. While the clock is running, the teacher asks students what they observe. Student responses may include values increasing, decreasing or remaining static. The teacher invites students to share ideas for why they are seeing the trends they have noted. After observing the clock for a short period of time, the teacher takes a screenshot of the page so that students can focus their attention on specific numbers on the page. Students are asked to record the numbers on the debt clock that they can read or write in words and then record those that they cannot yet read. The numbers students cannot read are then noted by the teacher and are used to inform explicit teaching or to prompt a class discussion covering the naming and reading of very large numbers. Activity 3: Creating large numbers In this activity, students use cards to create, read, compare and order large whole numbers. They work cooperatively in groups of two to four to achieve the randomised goals of the game. Students either self-correct their work or use technology to confirm results. The teacher may need to ask clarifying questions of students while they play the game but should avoid correcting student responses. The teacher may wish to play the game with their students, in which case student and teacher roles during the playing of the game are different. Instructions for playing and scoring the game and a template of the playing cards for printing are provided in the embedded objects below. Cards could be printed onto cardboard and laminated. Alternatively, write the digits on blank playing cards to create sets of 72 cards. Note that it is easier to manage the collection of cards after the game if each set is printed on a different colour and each set of cards is stored in a zip-lock bag for storage. Make sure the colours used are accessible, with an appropriate background and text contrast ratio. This set of cards will be used for other games in this course and consequently time spent making permanent sets will prove valuable in the future. Creating large Creating large numbers card sets numbers Self-correcting with technology Students could type in the number they have formed and it can be read out loud using a text to speech reader. Students could check how to say large numbers using websites such as Really Big Numbers or Numbers to Words Calculator. Note that these websites use American pronunciation – something that could be discussed with the class. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 19 of 54

Explicit teaching This activity may indicate the need to differentiate learning experiences for students. Some students may require explicit teaching of concepts while others engage in activities to consolidate or extend their skills. Consolidation activity Kim thinks that the number 400 027 is read and written in words as ‘four thousand and twenty- seven’. Compose a short letter to Kim explaining how the number should be written in words. Possible variations to the game 1. Simplify the game: If the size of the numbers causes students to experience difficulties in reading or operating with the numbers, reduce the number of cards used to achieve the goals. For example, begin with three-digit numbers. 2. Let’s create decimal numbers: Create the cards labelled ‘decimal expansion pack playing cards’ that are included in the Resources section of this document. To play, each student is issued with two additional zero cards and a decimal point card which they cannot exchange or return to the pile. Design appropriate ‘decimal expansion pack playing cards’ to play the game. 3. Operations with numbers (or decimals): Use a combination of numbered playing cards with or without the decimal point cards, the ‘operations expansion pack cards’ that are included in the Resources section of this document, and appropriate goal cards. Students could play a game where they select a combination of numbered cards and operation cards, the aim being to form number sentences to achieve the given goal. Activity 4: People count This activity provides a real context for students to engage with numbers of different magnitudes. Students are presented with information about census figures from China during the Han Dynasty in the year AD 2. They use the internet to explore the recent population of Chengdu, China. The linked site indicates a population of 2 639 407 in 1985 and a population of 2 954 909 in 1990. Students respond to questions involving these numbers and also use the internet to find information about giant panda populations. People count Activity 5: Explaining estimations This activity addresses sub-elements of both Quantifying numbers and Multiplicative strategies of the National Numeracy Learning Progression. As the initial estimations in this activity assume that students are familiar with ‘Hundreds and Thousands’, it could be introduced with a fairy bread snack. Note that students with specific dietary needs may need gluten-free bread and non-dairy spread. Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 20 of 54

Role-play Students can develop a deeper understanding of a concept when they have to explain it to someone else. For this activity, students imagine that they are the teacher of a primary class and need to explain estimation techniques to the children. The focus of this activity is not the answer, but the techniques described to develop an estimate of a quantity. Discussion prompts ▪ If you wanted to estimate how many peas were in a picture, how can you work out the answer? What processes would you undertake? ▪ What do you need to know to do an estimation? Other resources Using examples in children’s books, students could develop some strategies for helping young children to estimate a quantity. For example, How many jelly beans? by Andrea Menotti which has a foldout display of one million jelly beans, Counting on Frank by Rod Clement, or How many seeds in a pumpkin? by Margaret McNamara. Extension The video The Time You Have (in Jelly Beans) provides an opportunity for students to visualise large numbers with the aid of jelly beans. Teachers could use this video as a stimulus in later content areas including Time; Data, graphs and tables; or Rates and ratios, which are included in Module 4. Key questions associated with the video are included in the embedded slide show. Explaining estimations Activity 6: It’s hot (or cold) outside This activity provides a real context for students to engage with directed numbers. Show videos of a cup of freezing boiling water instantly and frying an egg on the bonnet of a car as an introduction to directed number. Students identify up to five towns or cities in both hemispheres that they have visited, aspire to travel to, or are interested in knowing more about. The teacher may include some locations such as Kangerlussuaq (Greenland), Etosha (Namibia) to ensure that temperatures include both positive and negative recordings. Students then use the internet or a weather app to find the current temperature for their locations. Students work collaboratively to: ▪ order all the locations from the class from hottest to coldest ▪ calculate the temperature difference between pairs of given locations ▪ find the greatest difference in temperatures for the listed locations ▪ find the two locations that are closest in temperature. Note that at this stage it may be useful to discuss the location of the places in relation to their Numeracy Stage 6 CEC: Teaching Guide Module 1, published July 2021 Page 21 of 54

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