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

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

Numeracy Stage 6 CEC
Teaching Guide

Module 1

 Effective from     Year 11, 2022

 Publication date   July 2021

 Updated            NA
Numeracy Stage 6 CEC Teaching Guide - Module 1 NSW Education Standards Authority - NESA
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Numeracy Stage 6 CEC Teaching Guide - Module 1 NSW Education Standards Authority - NESA
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.

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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?

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▪   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.

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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.

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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.

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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

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