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MATHEMATICS YEAR 1

TEACHERS’ MANUAL 2013

ISBN

Curriculum Materials and Assessment Division Ministry of Education, Sports and Culture

This Teachers’ Manual was prepared and written by Professor Karoline Afamasaga-Fuata’i after

consultations with the Mathematics Subject Committee and others. Tafaomālō Sione was the CMAD Curriculum Officer.

Utumoa S.F. Oloapu

Tuaia Isaia Naomi Tavila

Vine S. Maulolo Viesea Isaako Tuitamai Lui

Luapi Reti Perenise Sufia

Funding provided by AusAid, NZAID, ADB under the Education Sector Programme II.

January 2013

TABLE OF CONTENTS SECTION I INTRODUCTION ........................................................................................................................... 1

PURPOSE ............................................................................................................................................................ 1 STRUCTURE ........................................................................................................................................................ 1 PRINCIPLES ......................................................................................................................................................... 1

SECTION II GENERAL INFORMATION ............................................................................................................ 2 BACKGROUND .................................................................................................................................................... 2 APPROACHES TO TEACHING AND LEARNING MATHEMATICS ............................................................................ 2

Key Principles ................................................................................................................................................ 2 Essential Skills................................................................................................................................................ 2

HOW TO USE THE TEACHERS’ MANUAL ............................................................................................................. 3 Year Plan ....................................................................................................................................................... 3

SECTION III YEAR LEVEL ............................................................................................................................... 4 LEARNING OUTCOMES AND CONTENT OVERVIEW ............................................................................................ 4

Outcome Codes ............................................................................................................................................. 4 YEAR FOUR LEARNING OUTCOMES AND KEY IDEAS ........................................................................................... 5 UNIT PLAN .......................................................................................................................................................... 6 LESSON PLAN TEMPLATE .................................................................................................................................... 7 LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES ........................................................................... 8 ASSESSMENT FOR LEARNING INDICATORS ...................................................................................................... 10 ANNUAL PLAN .................................................................................................................................................. 10 WORKING MATHEMATICALLY .......................................................................................................................... 11

WM4.1 Interpreting &/or Posing Questions ............................................................................................... 12 WM4.2 Strategically Thinking & Representing ........................................................................................... 12 WM4.3 Reasoning & Justifying ................................................................................................................... 12 WM4.4 Reflecting & Evaluating .................................................................................................................. 13 WM4.5 Communicating Mathematically .................................................................................................... 13

NUMBER & OPERATIONS ................................................................................................................................. 14 NR4.1 Whole Numbers ................................................................................................................................ 15 NR4.2 Addition & Subtraction ..................................................................................................................... 19 NR4.3 Multiplication & Division .................................................................................................................. 25 NR4.4 Fractions & Decimals ........................................................................................................................ 31 NR4.5 Chance .............................................................................................................................................. 36

PATTERNS AND ALGEBRA ................................................................................................................................. 40 PA4.1a Number Patterns ............................................................................................................................. 41 PA4.1b Number Relationships ..................................................................................................................... 44

DATA ANALYSIS ................................................................................................................................................ 48 DA4.1 Data .................................................................................................................................................. 48

MEASUREMENT ............................................................................................................................................... 52 MS4.1 Length .............................................................................................................................................. 53 MS4.2 Area .................................................................................................................................................. 57 MS4.3 Volume & Capacity ........................................................................................................................... 61 MS4.4 Mass ................................................................................................................................................. 65 MS4.5 Time ................................................................................................................................................. 67

SPACE AND GEOMETRY .................................................................................................................................... 70 SG4.1 Three Dimensional Space .................................................................................................................. 70 SG4.2a Two Dimensional Space .................................................................................................................. 76 SG4.2b Two Dimensional Space cont’d ...................................................................................................... 87 SG4.3 Position ............................................................................................................................................. 89

GLOSSARY ...................................................................................................................................................... 92 APPENDICES................................................................................................................................................. 101

APPENDIX I ANNUAL PLAN .......................................................................................................................... 101

Mathematics Year 4 Teachers’ Manual 2013

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SECTION I INTRODUCTION PURPOSE The Teachers’ Manual is intended for teachers of primary schools and supports the implementation of the new primary curriculum. It is designed to extend teachers’ understanding of how they can use the curriculum to create, deliver and assess effective teaching and learning programmes for primary school students in all subjects. It includes integrated units where cross-curriculum learning is deemed logical and appropriate. It gives suggestions of sequenced instructional activities related to learning outcomes. Along with instructional activities, sample assessment tasks and student work samples will assist in benchmarking student performance and help teachers to assess student progress and growth.

Essentially the Teachers’ Manual presents ways in which teachers can focus programmes on the learning of students so that they can achieve learning outcomes.

The Teachers’ Manual is aimed at

increasing the content knowledge of teachers

increasing teachers’ ability to plan, and deliver learning on the basis of the curriculum statements

developing teachers’ assessment knowledge and practice through gathering and interpreting evidence of student achievement in order to decide on next instructional steps

having teachers use active and interactive methodologies to engage students in learning

enabling teachers’ to attend to the language and literacy demands of different subjects

challenging teachers to reflect and evaluate their practice and to make the necessary adjustments

The Teachers’ Manual is consistent with the Ministry of Education Sports and Culture’s National Curriculum Policy Framework and the Subject Curriculum Statements.

STRUCTURE The Teachers’ Manual consists of three sections. This section is Section 1 - Introduction. Section 2 provides general information on the Teachers’ Manual, approaches to teaching and learning in Mathematics and how to use the guide. Section 3 includes specific examples of units of work that focus on learning activities to achieve the learning outcomes, and assessment.

PRINCIPLES The units of work are underpinned by specific principles based on the idea that learners make meaning from what they experience in supportive environments. The units of work therefore have these features:

1. Problem-focused. Units are problem-focused, requiring students to solve open-ended contextualised problems.

2. Generative knowledge. Units enable students to have access to research and other knowledge in solving problems (generative knowledge).

3. Learning strategies. Students have opportunities for learning how to learn through the use of matrices, and web diagrams, self reflection and goal setting, formulating questions, relating and applying learning to own context.

4. Scaffolding. Students have the necessary scaffolding or structure throughout units. 5. Learning as a social process. Because learning is a social process, units of work ensure that

students spend at least part of their time in group formats, such as cooperative learning. 6. Demonstrate learning. Units require students to demonstrate learning in some authentic

manner.


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SECTION II GENERAL INFORMATION

BACKGROUND The Year Four Teachers’ Manual (TM) was developed specifically to further elaborate the Learning Outcomes (LO) and Working Mathematically (WM) Outcomes that are given in the Mathematics Curriculum Book: Mathematics Years 1-8 Primary School Curriculum (pp. 27-32) and as mapped out in the Mathematics Chart: Mathematics Progression by Strand and Year Level. The elaborations presented in this TM are in terms of knowledge and skills LO including the mathematical processes and problem strategies that should underpin all learning experiences. Also included are sample learning experiences and assessment opportunities as well as links to the available PEMP materials. Teachers are encouraged to use these to further develop, enrich and broaden your students’ mathematical understanding. The importance of linking to other units and/or learning areas is recommended and encouraged with the provision of a ‘Links’ section. Your students’ ability to communicate effectively in mathematics using the appropriate language and terminology should be explicitly developed. The required keywords are also provided for each unit.

APPROACHES TO TEACHING AND LEARNING MATHEMATICS Key Principles The key principles that should underpin the teaching of Mathematics are as follows:

All students can be successful learners when they are provided with sufficient time and support.

Students need to be engaged with learning experiences that are related to their interests, daily needs and learning styles in order to motivate and challenge their mathematical thinking and reasoning.

Programs must be carefully planned to achieve the listed learning and working mathematically outcomes for each unit, consolidate links between units and across different learning areas and use a range of teaching approaches to cater for the various learning styles of students.

Programs must be broad and balanced and provide opportunities for the intellectual, social and cultural disposition of each student to develop wholistically so when students complete their schooling they are well prepared for work and further study.

Monitoring, assessment and reporting practices help teachers evaluate the effectiveness of their teaching practices as well as provide an indication of student achievement against established standards.

Teachers make a difference in student achievement and effective teaching ensures quality outcomes for students.

Community involvement assists student learning so is the use of fa’asamoa practices to illustrate and elaborate mathematical ideas and concepts.

Students must be engaged with authentic (or real-life) learning experiences that raise their awareness about, and enhance their knowledge and skills to develop environmentally, socially, culturally and economically, sustainable and acceptable practices.

Essential Skills Essential skills are the broader skills that are developed throughout the years of schooling as a result of the quality of the experiences provided in all classroom and school activities. These skills are used by students in all school activities as well as in their social and cultural world outside the school. For example, the broader essential skills that are encouraged include: communicating effectively, solving problems, utilising aesthetic judgment, developing social and cultural skills and attributes, managing oneself and developing work and study skills, and effectively using technology. In teaching Mathematics, the specific knowledge, skills and understanding to be developed include:


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through inquiry, application of problem-solving strategies including the selection and use of appropriate technology, communication, reasoning, justifying and reflection;

in mental and written computation and numerical reasoning;

in patterning, generalisation and algebraic reasoning;

in collecting, representing, analysing and evaluating information;

in identifying and quantifying the attributes of shapes and objects and applying measurement strategies;

in spatial visualisation and geometric reasoning.

HOW TO USE THE TEACHERS’ MANUAL Year Plan The design of a year plan to teach all of the Year Four 16 substrands and 5 WM subprocesses has been made easier for you with the inclusion of a section: ‘Year Level’ in the Mathematics Curriculum Book (pp. 27-32) that is sub-divided into Years One to Eight. Within each year level sub-section are tables for each of the content substrands and working mathematically subprocesses. Displayed in each substrand table are the relevant Achievement Objectives (AO), Learning Outcomes (LO) and Key Ideas (KI) with a separate table showing the relevant WM AO and LO. The links between these tables and the TM are through the substrands and relevant LO and KI and WM LO. Table 1 shows a summary of Year Four substrands and subprocesses to guide the development of your annual plan.

The next section provides the structure of the Year Four Teachers’ Manual as well as sample activities to support the development of students’ understanding, knowledge and skills as described by the relevant learning and working mathematically outcomes.

Table 1 Year Four Content Substrands and WM Subprocesses

Number and Operations

Measurement cont’d

1 NR4.1 Whole Numbers 11 MS4.3 Volume & Capacity

2 NR4.2 Addition and Subtraction 12 MS4.4 Mass

3 NR4.3 Multiplication and Division

13 MS4.5 Time

4 NR4.4 Fractions and Decimals Space and Geometry

5 NR4.5 Chance 14 SG4.1 Three Dimensional Space

Patterns and Algebra 15 SG4.2 Two Dimensional Space

6 PA4.1a Patterns 16 SG4.3 Position

7 PA4.1b Number Relationships Working Mathematically

Data Analysis 1 WM4.1 Interpreting &/Or Posing Questions

8 DA4.1 Data Representation 2 WM4.2 Strategically Thinking & Representing

Measurement 3 WM4.3 Reasoning & Justifying

9 MS4.1 Length 4 WM4.4 Reflecting & Evaluating

10 MS4.2 Area 5 WM4.5 Communicating Mathematically


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SECTION III YEAR LEVEL

LEARNING OUTCOMES AND CONTENT OVERVIEW The Learning Outcomes and Content Overview for Year Four Mathematics are found in the Mathematics Curriculum Book (page 46-52) and also available from the chart: Mathematics Progression by Strand and Year Level. Figure 1 below presents an overview map of the Year Four 16 content substrands (i.e. units) and 5 WM subprocesses. Years One to Six has the same overview content and WM map. Provided in Table 1 are the Year Four content substrands and subprocesses including their respective codes for ease of cross-referencing.

Outcome Codes The following codes are used when referring to substrands, units or outcomes.

Strands WM – Working Mathematically NR – Number and Operations PA – Patterns and Algebra DA – Data Analysis MS – Measurement SG – Space and Geometry

Outcomes K&S - Knowledge and Skills WMO – Working Mathematically Outcomes

Examples of Codes: NR4.2 is read as NR for Content Strand: Number and Operations, 4 is for Year Level Four and 2

denotes Unit 2. Therefore NR4.2 refers to the second unit in Year Four for the Number and Operations strand.

PA7.3 is read as PA for Content Strand: Patterns and Algebra, 7 is for Year Level Seven and 3 denotes Unit 3. Therefore PA7.3 refers to the third unit in Year Seven for the Patterns and Algebra strand.

The following codes are used when referring to K&S LO, WMO and activities. K&S 3 refers to Knowledge & Skills Learning Outcome number 3 in the substrand table. WMO NR4.3.d refers to WMO number d in the substrand NR4.3 Activity NR4.3.4 refers to Activity 4 in the substrand NR4.3

These codes are used to cross-reference the appropriate items in the following sections.


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Figure 1: Years One to Six Content Strands and Working Mathematically Process Strand

YEAR FOUR LEARNING OUTCOMES AND KEY IDEAS This Teachers’ Manual takes the listed ‘Year Four Learning Outcomes and Key Ideas’ (see the Mathematics Curriculum Book, pp. 27-32) and elaborates them further. The following sections outline only the Year Four content so that teachers can meet the learning needs of their students.

Within each Substrand (or Unit), the outcomes, key ideas, content, background information, and advice about language are presented in tables as follows. The content is comprised of the statements of knowledge and skills (i.e., Knowledge & Skills) in the left hand column and the statements about Working Mathematically in the right hand column. Also included in the table are suggested resources for the sample activities provided plus links to other learning areas and mathematics substrands or units. The generic structure of the substrand tables and relevant descriptions are given in Table 2 below.


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Table 2 Year Four Content Substrands and WM Subprocesses

Outcome Code Substrand Title: A Statement of the Substrand/Unit Learning Outcome

Key Ideas An overview of key ideas to be covered as suggested below in Knowledge & Skills and Working Mathematically Learning Outcomes

Attitudes & Values A list of attitudes and values to be developed through the learning of knowledge and skills and implementation of working mathematically processes

Knowledge and Skills Students learn about A set of statements outlining the knowledge and skills students need to understand and apply in order to achieve the outcome. These are generally presented as a hierarchy of concept development; however, separate statements would typically be grouped and addressed together when planning teaching and learning experiences. The content is written for a whole school year.

Working Mathematically Students learn to

A sample set of statements that incorporate Working Mathematically processes into the knowledge and skills listed in the left hand column.

Teachers are encouraged to extend this list of statements by creating their own Working Mathematically experiences for students to engage with each of the five processes (Interpreting &/or Posing Questions, Strategically Thinking & Representing, Reasoning & Justifying, Reflecting & Evaluating, and Communicating Mathematically).

Understanding is encompassed in the development of concepts and processes in both of these columns.

Background Information Provides background knowledge for teachers to assist with planning programs of study for students.

Language Advice about language and literacy that may assist student engagement and understanding of the content in the unit.

Keywords to be used: Language expected to be used by students in the unit.

Resources Resources that could be used in the unit.

Links Links to other learning areas and Mathematics Years 1-8 substrands.

UNIT PLAN For each substrand learning outcome, there is a related set of content (i.e. K&S LO) and WMO which may be further sub-divided into groups of twos or threes K&S LO to form individual lessons. Thus a unit plan may consist of at least one and up to 6 lessons depending on the number of K&S LO and the ability of the students in your class. It is important that teachers use the content (i.e. K&S LO) section for programming since this includes all of the key ideas as well as a comprehensive list of the knowledge and skills, and suggestions for the integration of WM processes.

As an example, consider the substrand NR4.1: Whole Numbers with 10 K&S LO and 6 WMO. A unit plan on Whole Numbers can consist of up to 6 lessons with the grouping of K&S LO as suggested in Table 3.

Table 3 Unit Plan Example – NR4.1: Whole Numbers Strand: Number and Operations Substrand/Unit: NR4.1 Whole Numbers


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Lesson K&S LO WMO Activity

1 & 2 K&S 1: counting forwards to, and backwards from, a given four-digit number, by ones, twos, fives, tens, hundreds, or thousands and on and off the decade e.g., 1220, 1230, 1240 (on the decade); 2423, 3323, 5223 (off the decade)

K&S 2: representing numbers up to four digits using numerals, words, objects and digital displays

K&S 3: identifying the number before and after a given four-digit number

K&S 4: using a number line to assist with counting and ordering

NR4.1.a NR4.1.b NR4.1.c NR4.1.d NR4.1.f

NR4.1.1 NR4.1.5 NR4.1.6 NR4.1.7

3 K&S 5: applying an understanding of place value and the role of zero to read, write and order numbers up to four digits

K&S 6: stating the place value of digits in up to four-digit numbers e.g., ‘in the number 3426, the 3 represents 3000 or 3 thousands’

K&S 4: using a number line to assist with counting and ordering

NR4.1.a NR4.1.b NR4.1.c NR4.1.d NR4.1.e NR4.1.f

NR4.1.2 NR4.1.5 NR4.1.7

4 K&S 7: ordering a set of four-digit numbers in ascending or descending order

K&S 8: using the symbols for ‘is less than’ (<) and ‘is greater than’ (>) to show the relationship between two numbers

Apply K&S 4: using a number line to assist with ordering (and to show relationship between numbers)


NR4.1.3 NR4.1.5 NR4.1.8

5 & 6 K&S 9: recording numbers up to four digits using expanded notation e.g., 5429 = 5000 + 400 + 20 + 9

K&S 10: rounding numbers to the nearest ten, hundred or thousand when estimating

Apply K&S 4: using a number line to assist with (rounding) and (estimating)


NR4.1.4 NR4.1.5 NR4.1.6 NR4.1.9 NR4.1.10

Also included are recommended activities and accompanying WMO. You are also encouraged to

use additional resources (including PEMP materials) to find/design your own activities that are appropriate to achieve the identified K&S LO to supplement the sample activities provided, or when none of the sample activities is suitable.

LESSON PLAN TEMPLATE A one-page, lesson plan template (see Figure 2) displays the recommended components that should form part of your lesson plan. Completing the required lesson components for each lesson is encouraged. Doing so will be helpful in checking before the lesson is actually taught, that your planned teaching approach, developmental strategies and identified activities can link students’ prior knowledge to the new knowledge as stated in the lesson’s LO and WMO.

Your ‘Development/Process’ component should include review or short questions that formatively assess students’ existing K&S to ensure that students have the required prior mathematical knowledge and skills described in the ‘Prior Knowledge’ component. These prior K&S should form the bases upon which new K&S will be developed as your students engage with the identified activities. Each activity should include guiding questions to encourage students’ engagement with all or most of the Working Mathematically subprocesses, namely, interpreting


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and/or posing questions, strategically thinking and representing, reasoning and justifying, reflecting and evaluating, and communicating mathematically.

The ‘Assessment’ component describes your proposed ‘assessment for learning’ and ‘assessment of learning’ strategies to determine whether or not your students have achieved their learning outcomes as you described in the lesson’s K&S LO and WMO.

The ‘Extension’ and ‘Links’ components indicate your planned intentions of extending/linking and connecting the current lesson to other topics/curricular areas.

There is space at the bottom of the 1- page template for your own reflections and self-evaluation of the lesson including an indication of what you plan to teach next.

Completing a lesson template for each lesson of a unit plan facilitates the evaluation of whether or not your series of lessons (i.e. lesson series) aligns with your proposed unit plan.

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES The sample activities for Learning Experiences and Assessment Opportunities (LEAO) provided have been developed to demonstrate ways in which teachers can design a range of activities to ensure coverage of the Knowledge & Skills content (left-column) and Working Mathematically processes (right-column) for each topic/unit learning outcome.

The Working Mathematically (WM) outcomes are listed so that teachers consider the development and assessment of these outcomes as well as the Knowledge & Skills content outcomes. Teachers should also incorporate into students’ learning experiences a selection of problem solving strategies as recommended in the many of the WMO. Figure 3 provides an overview of a number of problem solving strategies that teachers should explicitly teach and develop as a routine part of students’ learning experiences.

The sample LEAO provided should enable students to develop and demonstrate one or more of the five Working Mathematically subprocesses (Interpreting &/or Posing Questions, Strategically Thinking & Representing, Reasoning & Justifying, Reflecting & Evaluating, and Communicating Mathematically) have been labeled with WM. The sample activities contain examples of the types of activities teachers might employ to cover the content in the Mathematics Year Four Syllabus.


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Figure 2: Lesson Plan Template


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Figure 3: An Overview of Problem Solving Strategies The sample activities are not mandatory. It is expected that teachers and schools will adapt the

activities according to the needs of their students, the availability of or preference for particular resources and the nature of school policies and priorities. This might mean that teachers and schools:

implement all of the units as outlined plus additional school-designed units to cover all syllabus requirements for Year Four

implement some of the units and develop school-designed units to complement them

use the Outcomes and Content sections of the Mathematics Year Four Syllabus as the basis for planning, making use of their own units.

ASSESSMENT FOR LEARNING INDICATORS Each outcome is accompanied by a sample set of indicators. An indicator is a statement of the behaviour that students might display as they work towards the achievement of a syllabus outcome. Indicators are included in this syllabus (after the list of sample LEAO) to help exemplify the range of observable behaviours that contribute to the achievement of outcomes linked to the content. They can be used by teachers to monitor student progress within Year Three and to make on-balance judgements about the achievement of outcomes at the end of the year. Teachers may wish to develop their own indicators or modify the syllabus indicators, as there are numerous ways that students may demonstrate what they know and can do. Indicators are not content. The relevant content sections are described by the Sub-strand and Topic Learning Outcomes, Key Ideas, ‘Knowledge & Skills’ Learning Outcomes and ‘Working Mathematically’ Outcomes. It is important that teachers use the content section for programming since this includes all of the key ideas as well as a comprehensive list of the knowledge and skills, and suggestions for the integration of Working Mathematically processes.

ANNUAL PLAN Appendix 1 includes an annual plan for each year level. Years 1 to 6 follow the same coverage of topics while Years 7 and 8 show different emphasis within the strands. The plan includes both a three term year and the four term year. This is to show teachers how learning in Mathematics that was previously organised in three terms can be organised in four terms, each term having ten weeks. The annual plan is divided according to strands with each strand having a number of units. Completion and review times are also allowed on the plan.


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WORKING MATHEMATICALLY Working Mathematically (WM) encompasses five interrelated processes. These processes underpin the teaching and development of new skills and concepts and also when applying existing knowledge to solve routine and application problems both within and beyond mathematics. Whilst the focus may be on a particular process or group of processes some of the time, oftentimes the five processes overlap. With its own set of separate outcomes, the WM processes are integrated into the content listed for each of the five content strands in the syllabus. Working Mathematically provides opportunities for students to engage in genuine mathematical activity such as problem solving and investigation and to develop the skills to become fluent and confident users of mathematics. The five processes for Working Mathematically are:

Interpreting &/ Posing Questions Students interpret and/ pose questions in relation to mathematical situations and their mathematical experiences. Encouraging students to interpret given questions or pose their own extends and motivates their curiosity and interest in mathematics. ‘I wonder if’ and ‘what if’ types of questions encourage students to make conjectures and/or predictions.

Strategically Thinking & Representing Based on their interpretations of mathematical situations, questions or problems, students strategically select multiple strategies, model and represent their interpretations, including the selection and use of appropriate technology, to investigate and to generate plausible solutions to mathematics problems.

Reasoning & Justifying Students develop and use processes for exploring relationships, checking solutions and giving reasons to support their conclusions. Students also need to develop and use logical reasoning, proof and justification.

Reflecting & Evaluating Students reflect on their experiences and critical understanding to make connections with, and generalisations about, existing knowledge and understanding and evaluate how this knowledge and understanding can be applied to solve mathematics problems. Students make connections with the use of mathematics in the real world by identifying where, and how, particular mathematical ideas, processes and concepts are being used and applied in the authentic context.

Communicating Mathematically Students develop and use appropriate language and representations to formulate and express mathematical ideas in written, oral and diagrammatic form.

Examples of learning experiences for each of the processes for Working Mathematically are embedded in the right-hand column of the content for each outcome in the Number and Operations, Patterns and Algebra, Data Analysis, Measurement, and Space and Geometry strands. Working Mathematically is embedded in all content.


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ASSESSMENT FOR LEARNING INDICATORS WM4.1 Interpreting &/or Posing Questions Students interpret and/or pose questions, when investigating mathematical situations and their mathematical experiences using Year 4 content. WM4.1 Interprets and/or poses questions, when investigating mathematical situations and their mathematical experiences using Year 4 content. The student, for example:

poses questions that clarify a mathematical situation and enable progression towards a solution

generates questions when considering a number pattern e.g., ‘What is the next term in the pattern?’

poses questions about a collection of items e.g., ‘Is it possible to find one-fifth of this collection of objects?’

poses problems based on number relationships e.g., ‘What mental strategies can you use with multiples of 3 to generate multiples of 6?’

poses questions that can be answered using the information from a table or graph e.g., ‘What is the highest value shown?’

poses a suitable focus question to guide a survey

questions why two students may obtain different measurements for the same length, perimeter, area, volume, capacity or mass

ASSESSMENT FOR LEARNING INDICATORS WM4.2 Strategically Thinking & Representing Students strategically select and use appropriate mental or written strategies, or technology, to examine and analyse, to represent their interpretations of, and to solve, mathematical problems. WM4.2 Selects and uses appropriate mental or written strategies, or technology, to examine and analyse, to represent their

interpretations of, and to solve, mathematical problems. The student, for example:

solves a variety of problems using strategies such as trial and error, drawing a diagram, and looking for patterns

uses problem-solving strategies including those based on selecting key information and acting it out, working backwards, and using a table

uses a calculator to create patterns involving decimals

uses efficient strategies to estimate and measure quantities to the nearest 100 mL and/or nearest 10 mL

designs a table to enter data and then create a graph of the data

selects and uses the best measuring tool for a given task e.g., finding the area of a variety of shapes

ASSESSMENT FOR LEARNING INDICATORS WM4.3 Reasoning & Justifying Students integrate mathematical ideas and make connections with, and generalisations about, existing knowledge and understanding in terms of Year 4 content to justify their conjectures, answers and solutions. WM4.3 Integrates mathematical ideas and makes connections with, and generalisations about, existing knowledge and understanding in terms of Year 4 content to justify their conjectures, answers and solutions. The student, for example:

checks whether the answer is correct by using an alternative strategy

checks the answer to a subtraction problem using addition

checks the reasonableness of a solution to a problem by relating it to an original estimation

compares tables and graphs constructed from the same data to determine which is the most appropriate method of display

checks predictions when extending patterns by using various strategies


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compares tables and graphs constructed from the same data to determine which is the most appropriate method of display

explains why two students may obtain different results for the same measurement

explains why a given length is equal to, greater than or less than 1 metre or 1 yard

uses two different methods to confirm an answer

ASSESSMENT FOR LEARNING INDICATORS WM4.4 Reflecting & Evaluating Students reflect critically upon the results of their conjectures, explain their results and evaluate conclusions. WM4.4 Reflects critically upon the results of their conjectures and explains their answers and evaluates conclusions. The student, for example:

identifies and describes the use of mathematics in everyday contexts

discusses strategies used to estimate area in square yards or square meters

interprets information about capacity and volume on commercial packaging

discusses the advantages and disadvantages of different representations of the same data

relates the millilitre to familiar everyday containers

compares own drawings of 3D objects with other drawings and photographs of 3D objects

interprets the everyday use of fractions, decimals and percentages

recognises that a particular shape can be represented in different sizes and orientations

ASSESSMENT FOR LEARNING INDICATORS

WM4.5 Communicating Mathematically Students use some appropriate terminology to describe their mathematical ideas and to make connections between their learning and other experiences. WM4.5 Uses some appropriate terminology to describe mathematical ideas and to make connections between their learning and other experiences The student, for example:

interprets graph found on the internet, newspapers and factual text

writes a procedure to outline the method used to solve a problem

uses and follows positional and directional language

selects a shape from a description of its features

explains the difference between expected results and actual results in simple chance experiments

discusses and compares areas using some mathematical terms

discusses different methods for solving a given problem

explains the mental strategy used to solve a problem

generates number patterns, by choosing a variety of starting numbers, and using the process of repeatedly adding or subtracting the same number either mentally or on a calculator

explains how an answer was obtained and compared own method/s of solution to a problem with those of others


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NUMBER & OPERATIONS The skills developed in the Number & Operation strand formed the basis to all other strands of this Mathematics syllabus and are developed across the years from Years 1 to 8. Numbers, in their various forms, are used to quantify and describe the world. From Year 1 there is an emphasis on the development of number sense and confidence and competence in using mental, written and calculator techniques for solving appropriate problems. Formal algorithms are only introduced after students have gained a firm understanding of basic concepts including place value, and have developed mental strategies for computing with two- and three-digit numbers. Approximation is important and the systematic use of estimation is to be encouraged at all times. Students should always check that their answers ‘make sense’ in the context of the problems they are solving. The use of mental computation strategies should be developed at all levels. Calculators can be used to investigate number patterns and relationships and facilitate the solution of real problems with measurements and quantities not easy to handle with mental or written techniques. The Number & Operations strand for Years 1 to 8 is organised into five substrands:

Whole Numbers

Addition and Subtraction

Multiplication and Division

Fractions and Decimals

Chance. Whole Numbers includes counting strategies, number relationships and the concept of place value. The

operations are paired in the substrands Addition and Subtraction, and Multiplication and Division, to emphasise the importance of developing awareness of the inverse relationships between these operations.

In Fractions and Decimals, students are introduced to the concept of a fraction through everyday experiences. Development of the idea of division of a whole and collections or sets of objects into equal parts, leads to equivalence relationships and simple operations including addition and subtraction of fractions with denominators that are multiples of each other and multiplication of fractions by whole numbers. Students also develop an understanding of decimals and perform calculations with decimals up to three-decimal places. Percentages are introduced to enable interpretation of their use in everyday contexts.

The substrand Chance has been included from Year 1 to enable the development of understanding of chance concepts from an early age. Early emphasis in the Chance substrand is on understanding the idea of chance and the use of informal language associated with chance. The understanding of chance situations is further developed through the use of simple experiments which produce data so that students can make comparisons of the likelihood of events occurring and begin to order chance expressions on a scale from zero to one.

Development of an understanding of the monetary system and computation with money is integrated into the substrands of Whole Numbers, Addition and Subtraction, Multiplication and Division, and Fractions and Decimals.

This section presents the Year 4 outcomes, key ideas, knowledge and skills, and Working Mathematically statements in each substrand.


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NR4.1 Whole Numbers NR4.1 Substrand: Whole Numbers Recognises, counts, orders, reads and records numbers up to 9999

Key Ideas Attitudes & Values

Count forwards to, and backwards from, 9999 by hundreds and thousands, on and off the decade Recognise, read, partition, regroup, represent and order numbers up to 9999 using place value

Logical and critical thinking Communication Consistency Work things out Investigation Accuracy Verification Proof and reflection

Knowledge and Skills Students learn about 1. counting forwards to, and backwards from, a

given four-digit number, by ones, twos, fives, tens, hundreds, or thousands and on and off the decade e.g., 1220, 1230, 1240 (on the decade); 2423, 3323, 5223 (off the decade)

2. representing numbers up to four digits using numerals, words, objects and digital displays

3. identifying the number before and after a given four-digit number

4. using a number line to assist with counting and ordering

5. applying an understanding of place value and the role of zero to read, write and order numbers up to four digits

6. stating the place value of digits in up to four-digit numbers e.g., ‘in the number 3426, the 3 represents 3000 or 3 thousands’

7. ordering a set of four-digit numbers in ascending or descending order

8. using the symbols for ‘is less than’ (<) and ‘is greater than’ (>) to show the relationship between two numbers

9. recording numbers up to four digits using expanded notation e.g., 5429 = 5000 + 400 + 20 + 9

10. rounding numbers to the nearest ten, hundred or thousand when estimating

Working Mathematically Students learn to a. pose problems involving four-digit

numbers ‘How many more is 4567 from 4089?’ (Interpreting &/or Posing Questions)

b. identify some of the ways numbers are used in our lives (Reflecting & Evaluating)

c. interpret four-digit numbers used in everyday contexts (Communicating Mathematically)

d. compare and explain the relative size of four-digit numbers (Strategically Thinking & Representing, Communicating Mathematically)

e. make the largest and smallest number given any four digits (Strategically Thinking & Representing)

f. solve a variety of problems using problem-solving strategies, including:

- trial and error - drawing a diagram - working backwards - looking for patterns

- using a table (Strategically Thinking & Representing, Communicating Mathematically)

Background Information The convention for writing numbers of more than four digits requires that they have a comma to the left of each group of three digits, when counting from the Units column. Students need to develop an understanding of place value relationships such as 1 thousand = 10 hundreds = 100 tens = 1000 ones.

The abbreviation K comes from the Greek word khilioi meaning thousand. It is used in many job advertisements (e.g., a salary of $70K) and as an abbreviation for the size of computer files eg 26K (kilobytes).


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Language ‘3629 is the same as 36 hundreds, 2 tens and 9 units.’ ‘three thousand, six hundred and twenty-nine.’

Keywords zero, digit, number, units, before, after, ones, tens, hundreds, thousands, place value, less than, forwards, backwards, greater than, largest, smallest, highest, lowest, trading, on/or the decade, rounding, estimating, less than, greater than, represent, ascending, descending

Resources dice, number cards, popsticks, Base 10 material, numeral expanders, calculators, place value chart, newspaper, internet

Links Addition and Subtraction Multiplication and Division Data Social Studies

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES NR4.1 Recognises, counts, orders, reads and records numbers up to 9999

ACTIVITY NR4.1.1: COUNTING BY HUNDREDS AND THOUSANDS COUNTING RACES Students are divided into two groups. The teacher nominates a starting number e.g., 124. One group counts by hundreds, while the other counts by thousands from the starting number. Both groups start counting and are asked to stop at the same time.

Before commencing the activity, students discuss:

will both groups start/finish on the same number? Why?

which group will stop on the highest number? Why?

will both groups count number 1392? Why?/Why not?

what are some of the numbers both groups will count? Why?/Why not?

what is a number only your group will count?

Variation: Students play ‘Buzz’1 counting by hundreds on and off the decade. They ‘buzz’ on the thousands.

1 Have the players sit in a circle and begin to count in turn, but

when the count reaches thousands, they say "Buzz," instead of whatever the number may be.

ACTIVITY NR4.1.2: FOUR-DIGIT NUMBERS In small groups, students use a pack of playing cards with the tens and picture cards removed. The Aces are retained and count as 1 and the Jokers are retained and count as 0. Student A turns over the first 4 cards and each player makes a different four-digit number. Student A records the numbers and puts the cards at the bottom of the pile. They each take a turn turning over four cards and recording the group’s four -digit numbers. When each student has had a turn they sort and order their numbers. Possible questions include:

Can you read each number aloud?

Can you order the numbers in ascending and descending order?

Can you state the place value of each numeral?

What is the largest/smallest number you can make using four cards?

What is the next largest/smallest number you can make using four cards?

Can you identify the number before/after one of your four-digit numbers?

Can you find a pattern? How can you describe your pattern? How can you continue the pattern?

How many different ways can you represent each number? (expanded notation, in words)

Can you count forwards/backwards by tens/hundreds/thousands from one of your four-digit numbers?


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Can you round one of your four-digit numbers to the nearest hundred? nearest thousand?

Variation: Students could represent numbers using Base 10 material, or expanded notation, to show place value.

ACTIVITY NR4.1.3: LESS THAN AND GREATER THAN, ORDERING

Part A In pairs, students are given four different-coloured dice, representing thousands, hundreds, tens and ones. Students take turns to throw the dice, record their four-digit number and state the number before and after.

Part B In pairs, students are given four different-coloured dice, representing thousands, hundreds, tens and ones. Students take turns to throw the dice and record their four-digit number. Students nominate whether they are ‘greater than’ or ‘less than.’ They compare their numbers by showing the relationship between the two four-digit numbers they have made by using a < or > sign e.g., Student A rolls 2531 and is ‘greater than’ and Student B rolls 1126 and is ‘less than’. Student B wins the point. The winner is the first to 20. Extension. Students keep a record of all the four-digit numbers they rolled. Their next task is to place and order their four-digit numbers on an empty number line.

ACTIVITY NR4.1.4: WIPE-OUT Students are asked to enter a four-digit number into a calculator e.g., 2541. The teacher then asks the students to ‘wipe out’ one digit i.e. change it to a zero. In the example above, ‘wiping out the 4’ would require a student to change the number to 2501 by subtracting 40. Students could demonstrate this using Base 10 material.

Variation. Students repeat the activity to make the other digits zero (i.e., change 5 to 0, then change 2 to 0) using two different strategies. They can continue with the activity using different four-digit numbers. Students

record their strategies using numerals and empty number lines.

ACTIVITY NR4.1.5: PROBLEM SOLVING AND PROBLEM POSING Students solve a variety of problems using a large number of strategies. The teacher should encourage students to pose their own problems involving numbers of up to four digits.

ACTIVITY NR4.1.6: HOW MANY WAYS? The teacher selects a four-digit number and records it on the board. Students express and/or present the number in as many ways as they can (a time limit may be imposed) with Base 10 material or expanded notation e.g., 1263 one thousand two hundred and sixty three 1000 + 200 + 60 + 3 1200 + 60 + 3 1200 + 63 1000 + 263

ACTIVITY NR4.1.7: CALCULATOR BROKEN KEY The teacher tells students the 6 key on their calculator is broken.

Students are asked to make the calculator display show 6666 without pressing the 6 key. Students share their solutions.

Students are asked which solutions they like best and why.

Variation: Students could repeat the activity using different target numbers and different ‘broken’ keys.

ACTIVITY NR4.1.8: HIGHER OR LOWER Students play in groups of four (2 players and 1 adjudicator) ‘Higher or Lower’. The adjudicator records a ‘secret’ four-digit number on a card and states the boundaries for the number e.g., the number is between 2000 and 6000.’ Students draw their own number line, marking the boundaries for the number.

The first player chooses a number in the range and the adjudicator responds by stating


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whether the number is higher or lower than the one chosen. The players record the response on their number line. The second player then states a number and the adjudicator responds with ‘higher’ or ‘lower’. The game continues until a player gives the correct number.

Students discuss the strategies they used to determine the secret number.

ACTIVITY NR4.1.9: ROUNDING Students use number cards 0 to 9 to create four-digit numbers. They randomly allocate a card to each place value column. They round the numbers to the nearest

ten

hundred

thousand. In pairs, students take turns in asking their partner to round a number. The partner explains strategies used.

ACTIVITY NR4.1.10: ESTIMATING HOW MANY Students are asked to estimate:

how many students could sit comfortably in a 10 metre by 10 metre marked area?

how many mats could fit on the floor of their classroom?

how many pencils will fill a large container visible in the room?

Students are asked to estimate a range by stating ‘I think that there will be at least______ but not more than ______’. They discuss how they can refine their estimate

and make it more accurate without actually completing the task. Students revise their estimate. Students are encouraged to pose their own problems.

ASSESSMENT FOR LEARNING INDICATORS NR4.1 Recognises, counts, orders, reads and records numbers up to 9999 The student for example:

counts forwards to, or backwards from, a given four-digit number by ones, twos, fives, tens and hundreds and thousands and on and off the decade

identifies the number before and after a given four-digit number

represents four-digit numbers using numerals, words and objects

applies an understanding of place value and the role of zero to read, write and order four-digit numbers

orders a set of four-digit numbers in ascending or descending order

states the place value of digits in four-digit numbers

uses the terms ‘is more than’ and ‘is less than’ to compare numbers

counts and represents large sets of objects by systematically grouping in tens, hundreds, and thousands

uses a number line to assist with counting and ordering

rounds numbers to the nearest ten, hundred or thousand when estimating


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NR4.2 Addition & Subtraction NR4.2 Substrand: Addition & Subtraction Models and represents addition and subtraction involving two-, three- and four-digit numbers up to 9999 by applying mental, written and formal strategies


Model addition and subtraction involving up to four-digit numbers by applying a range of mental and written strategies Describe, justify and record methods for adding and subtracting Represent and record addition and subtraction by applying a formal written algorithm


Knowledge and Skills Students learn about 1. using mental strategies for addition and

subtraction involving up to four-digit numbers, including

-- the jump strategy -- the split strategy -- the compensation strategy -- bridging the decades -- changing the order of addends to

form multiples of 10 -- using patterns to extend number

facts 2. recording mental strategies descriptively

in words or diagrammatically on an empty number line

3. adding and subtracting two or more numbers, with and without trading, using sets and/or abacus and recording their method

4. using a formal written algorithm and applying place value to solve addition and subtraction problems, involving up to four-digit numbers

Working Mathematically Students learn to a. pose problems that can be solved using

addition and subtraction, including those involving money (Interpreting &/or Posing Questions)

b. ask ‘What is the best method to find a solution to this problem?’ (Interpreting &/or Posing Questions)

c. select and use mental, written or calculator methods to solve addition and subtraction problems (Strategically Thinking & Representing)

d. solve a variety of problems using problem-solving strategies, including:

-- trial and error -- drawing a diagram -- working backwards -- looking for patterns -- using a table

(Strategically Thinking & Representing, Communicating Mathematically)

e. use estimation to check solutions to addition and subtraction problems, including those involving money (Reflecting & Evaluating, Strategically Thinking & Representing)

f. check the reasonableness of a solution to a problem by relating it to an original estimation (Reasoning & Justifying)

g. check solutions using the inverse operation or a different method (Strategically Thinking & Representing, Reasoning & Justifying)

h. explain how an answer was obtained for an addition or subtraction problem (Communicating Mathematically, Reasoning & Justifying)

i. reflect on own method of solution for a


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problem, considering whether it can be improved (Reflecting & Evaluating)

j. use a calculator to generate number patterns, using addition and subtraction (Strategically Thinking & Representing)

Background Information Students should be encouraged to estimate answers before attempting to solve problems in concrete or symbolic form. There is still a need to emphasise mental computation even though students can now use a formal written method. The following formal methods may be used. Decomposition The following example shows a suitable layout for the decomposition method.

Equal Addends For students who have a good understanding of subtraction, the equal addends algorithm may be introduced as an alternative, particularly where very large numbers are involved. There are several possible layouts of the method, of which the following is only one and not necessarily the best. The expression ‘borrow and pay back’ should not be used. ‘Add ten ones’ and ‘add ten’ is preferable.

When developing a formal written algorithm, it will be necessary to sequence the examples to cover the range of possibilities that include with and without trading in one or more places, and one or more zeros in the first number.

Language ‘One thousand, four hundred and twenty-seven people are going to the show. ‘One thousand, two hundred and five have collected their tickets. Twenty two more makes one thousand two hundred and twenty seven then another two hundred more makes two hundred and twenty two. So two hundred and twenty two still have to collect their tickets.’ ‘I left a space to show the thousands space.’ ‘I can add four thousand and eight thousand in my head.’

Keywords to be used: place value, formal algorithm, addition, subtraction, solution, answer, digit, trade, jump strategy, split strategy, compensation strategy, bridging to decades, number line, difference, multiples, exchange, swap, greater, altogether, total

Resources number cards 1 to 9, calculator, paper, Base 10 material, place value chart, dice, playing cards

Links Multiplication and Division Whole Numbers

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES NR 4.2 Models and represents addition and subtraction involving two-, three- and four-digit numbers up to 9999 by applying mental, written and formal strategies

ACTIVITY NR4.2.1: WM MENTAL STRATEGIES

Students are asked to calculate 125 + 47 in their heads. They are then asked to describe their strategies in writing and using empty number lines. This process is repeated for other problems, such as: 173 – 25; 1462 – 269; 263 + 129; 2188 – 489 Students discuss which methods are the most efficient.

Extension: Students are given increasingly more difficult problems to solve mentally. Students explain and discuss the strategies


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they use e.g., for ‘3188 – 265 = ?’ A student

may say, ‘I split 3188 into 3100 and 80 and 8 and 265 into 200, 60 and 5 so I subtracted

200 from 3100 leaves 2900 and 60 from 80 is 20 and 5 from 8 is 3 so the answer is 2900 plus 20 is 2920 and 3 more is 2923, so the answer is 2923.’ Students record the mental strategies they use. Possible questions include:

Is there a better strategy?

What is the best method to find a solution to this problem?

Students are asked to create a story problem (or word problem) whose solution and answer is provided by the computation.

ACTIVITY NR4.2.2: RECORDING ON EMPTY NUMBER LINES Students are shown the number sentence 1356 + 227 and an empty number line. The teacher marks the number 1356 on the number line. Possible questions include:

What is the next multiple of ten after 1356?

How many do you add on to get that number?

Students record their answers on the number line.

Possible questions include:

Can you work it out with fewer steps?

Can you visualise the number line in your head and do it?

Can you write the numbers on paper to help you keep track?

Students are asked to create a story problem (or word problem) whose solution and answer is provided by the computation.

ACTIVITY NR4.2.3: DIFFERENCES ON NUMBER LINES In pairs, students draw an empty number line. Student A chooses two four-digit numbers and places them on the number line.

Student B uses the number line to work out and record the difference between the two numbers. Students explain the mental strategies they used to find the answer. They

reflect on their method, considering whether it can be improved.

ACTIVITY NR4.2.4: APPROPRIATE CALCULATIONS Students are given a calculation such as 1350 – 524 = 826 and are asked to create a number of problems where this calculation would be needed. Students share and discuss responses.

ACTIVITY NR4.2.5: BASE 10 MATERIAL Students use 4 dice to generate a four-digit number and then represent this number using Base 10 material. Students then generate a second, smaller number by rolling 3 dice. Students represent this number using Base 10 material, then add the two numbers and show the result using Base 10 material.

Students repeat this process, subtracting the second number from the first. Students record their solutions.

ACTIVITY NR4.2.6: LINKING 3 Students record sixteen different numbers between 1 and 120 in a 4 × 4 grid e.g.,

101 108 107 96

24 118 91 59

112 18 106 109

105 86 46 78

Students link and add three numbers vertically or horizontally. Possible questions include:

Can you find links that have a total of more than 220?

Can you find links that have a total of less than 220?

How many links can you find that have a total that is a multiple of 10?

What is the smallest/largest total you can find?

Can you find ten even/odd totals?

ACTIVITY NR4.2.7: ESTIMATING DIFFERENCES The teacher shows a card with the subtraction of a pair of three-digit numbers e.g., 268 –


22

192. Students estimate whether the difference between the numbers is closer to 10, 20, 30, 40, 50, 60, 70, 80 or 90 and give reasons why. The teacher shows other cards e.g., 531 – 489, 523 – 477, 436 – 385, 351 – 299. Students estimate the differences and discuss their strategies. They are asked to think about rounding numbers on purpose.

For example for 268 – 192, students may round 268 up to 270 and 192 down to 190.

ACTIVITY NR4.2.8: PLAYING CARDS & MENTAL STRATEGIES Students work in pairs with a deck of cards to mentally add or subtract three-digit numbers and to record and evaluate their mental strategies.

Students are given a pack of playing cards with the tens and the picture cards removed. The Aces are retained and represent 1 and the Jokers are retained and represent 0.

Each student flips three cards and assigns place values to the numbers turned over. Students estimate their answer mentally and then use formal written algorithms. Students could use a calculator to check their answer.

Each pair of student should answer the questions:

What is the sum of the two numbers? What mental strategy did you use?

What is the difference between the two numbers? What mental strategy did you use?

Students are encouraged to pose problems, including money problems, using their numbers.

Students record their (a) mental strategies using words and empty number lines, (b) formal algorithms, and (c) story problems for each pair of numbers.

The activity may be repeated to generate more pairs of numbers.

ACTIVITY NR4.2.9: ESTIMATING ADDITION OF FOUR-DIGIT NUMBERS The teacher briefly displays the numbers 2345, 2341, 2340, 2346, 2342 on cards, then turns the cards over so that the numbers cannot be seen. Students are asked to estimate the total and give their reasons. The

teacher reveals the numbers one at a time so that the students can find the total. The task could be repeated with other four-digit numbers.

ACTIVITY NR4.2.10: TAKE-AWAY REVERSALS In pairs, students choose a four-digit number without repeating any digit and without using zero e.g., 7281. The student reverses the order of the digits to create a second number i.e. 1827. The student subtracts the smaller number from the larger and records this as a number sentence. The answer is used to start another reversal subtraction. Play continues until zero is reached.

Students discuss their work and any patterns they have observed.

The process could be repeated for other four-digit numbers.

Students record their working, number of subtraction steps to reach zero and patterns identified for each number.

ACTIVITY NR4.2.11: WHAT WENT WRONG? Students are shown a number of completed subtraction problems with a consistent error e.g., subtracting the smaller number in a column from the larger number. Students correct the calculations and describe the error that was made. e.g., Students plan how to teach a person who made this mistake a correct method for obtaining solutions.

ACTIVITY NR4.2.12: THE ANSWER IS … Students construct subtraction number sentences with the answer 1057. Students are challenged to include number sentences involving four-digit numbers.

5765 –

2394

3431

6327 –

2168

4241

2738 –

1651

1127


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Player A Start number 1235 1235 + 200 = 1435 1435 + 265 = 1700 1700 + 300 = 2000 2000 + 500 = 2500 2500 + 10 = 2510 2510 + 5 = 2515 2515 + 5 = 2520 2520 + 5 = 2525 Player A wins

ACTIVITY NR4.2.13: HOW MANY DAYS HAVE YOU BEEN IN SCHOOL? The teacher poses the question ‘If you started school when you were 5 years old, how many days have you been in school?’ Students solve this problem using a calculator.

Students record their solutions and compare the methods they used to solve the problem.

Students are encouraged to pose and solve similar problems and evaluate their strategies.

ACTIVITY NR4.2.14: WM WHICH WAY IS BEST? Students are asked to solve problems in three different ways: using a mental strategy, a formal written algorithm, and a calculator e.g., ‘Our class has 4329 points and another class has 6258 points. How many points do we need to catch up?’

Students compare the strategies used and discuss the advantages and disadvantages of each method. If students come up with different answers, they are asked to show which answer is correct.

Variation: Students write their own problems and swap with others.

ACTIVITY NR4.2.15: NUMBER CARDS Students make number cards from 1 to 9 as shown.

Students use these cards to make two four-digit numbers that add to give the largest total possible and the smallest total possible e.g., Given 3, 6, 8, 9 and 2, 1, 5, 4:

Largest total possible is 9863 + 4521 = 14384

Smallest total possible is 3689 + 1254 = 4934

Students arrange the cards to make two four-digit numbers that add up to 9999. Students are challenged to find as many solutions as they can.

ACTIVITY NR4.2.16: ESTIMATING TO THE NEAREST 1000 The teacher displays three cards with the following amounts written on them: $8245, $6475, $3750. Students estimate the total to the nearest $1000 and explain their strategies.

ACTIVITY NR4.2.17: CROSS-OVER In pairs, students each choose a number between 1 and 9999.

The student with the larger number always subtracts a number from their chosen number. The student with the smaller number always adds a number to their chosen number. The student who is adding must always have a number less than their partner’s answer. The student who is subtracting must always have a number more than their partner’s answer. Play continues until one student is forced to ‘cross over’ their partner’s number.

The student who crosses over their partner’s number loses the game.


What strategy did you use in solving the addition or subtraction problems?

Can you find a quicker way to add/subtract?

Can you explain to a friend what you did?

How can you show that your answer is correct?

Does the rule always work?

1

Player B Start number 7899 7899 - 899 = 7000 7000 - 3000 = 4000 4000 - 1000 = 3000 3000 – 100 = 2900 2900 – 100 = 2800 2800 – 100 = 2700 2700 – 100 = 2600 2600 – 100 = 2500 Player B wins

2 3 4 5

6 7 8 9


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Can you use a different method?

ACTIVITY 18: HOW MUCH? Students are told that a TV set and a table cost $1116. If the sofa costs $800 more than the table, how much does the TV set cost? Students discuss. Students could pose other similar problems to solve such as ‘What does each item cost if together they cost $1458 and one was $937 more than the other?’


What strategy did you choose to use and why?

What was the key word/s in understanding the problem?

How could you check that you have the correct solution?

Could there be more than one solution?

ACTIVITY NR4.2.19: WM MISSING DIGITS Students are shown a calculation to find the sum of two four digit numbers, with some of the digits missing, as shown below.

Students investigate possible solutions for this problem. Students are encouraged to design their own ‘missing digits’ problems. This activity should be repeated using subtraction.

ACTIVITY NR4.2.20: ELECTRICITY BILLS Five local companies received their electricity bills all on the same day. The following table shows the amounts they had to pay to EPC in 30 days or face power disconnection.

Company Billed Amount

Final Amount After discount

Asa Ltd $1205.00

IliAfi Company $3529.00

Mosooi Cakes $1324.00

Lila Uila Ltd $3124.00

Vai Malu Waters

$2967.00

Students are asked to work out the final

amounts for each company’s bill if EPC is offering a $250.00 discount provided payments are made within two weeks after receiving the notice.

Students show their strategies in working out the final amounts with the table entries completed.

ASSESSMENT FOR LEARNING INDICATORS NR 4.2 Models and represents addition and subtraction involving two-, three- and four-digit numbers up to 9999 by applying mental, written and formal strategies The student for example:

uses patterns to extend number facts

explains and records methods for adding and subtracting

uses a split strategy for addition or subtraction

uses an empty number line and jump strategies to represent solutions to addition and subtraction problems involving up to four-digit numbers

adds or subtracts two numbers, with and without trading, using concrete materials

uses the formal written algorithm to solve addition or subtraction problems

uses a calculator to solve addition and subtraction problems that include larger numbers contained in a problem context

5

4 7

6 1

1


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NR4.3 Multiplication & Division NR4.3 Substrand: Multiplication & Division Models and represents multiplication and division by applying various mental and formal written strategies


Develop mental fluency for number facts up to

12 12 Find factors and cubes of numbers Interpret remainders in division problems and as fractions and decimals Use efficient mental and informal written strategies for multiplying or dividing a two-digit number by a one-digit operator using multiplication facts up to 12 x12


Knowledge and Skills Students learn about

1. using mental strategies to multiply a one-

digit number by a multiple of 10 (e.g., 3 20) by: repeated addition (20 + 20 + 20 = 60) using place value concepts

(3 2 tens = 6 tens = 60)

factoring (3 2 10 = 6 10 = 60) 2. using mental strategies to multiply a two-

digit number by a one-digit number, including using known facts

e.g., 11 9 = 99 so 14 9 = 99 + 9 + 9 + 9 multiplying the tens and then the units

e.g., 8 17 is (8 10) + (8 7) = 80 + 56 = 136 the relationship between multiplication facts

e.g., 23 4 is double 23 and double again

factorising e.g., 18 5 = 9 2 5 = 9 10 = 90

3. using mental strategies to divide by a one-

digit number multiplication facts up to 12 12, in problems for which answers include a remainder

e.g., 136 ÷ 3; if 45 3 = 135 and 46 3 = 138 the answer is 45 remainder 1

4. recording remainders to division problems e.g., 19 ÷ 4 = 4 remainder 3

5. recording answers, which include a remainder, to division problems to show the connection with multiplication e.g.,147 = 14

10 + 7 6. interpreting the remainder in the context of

Working Mathematically Students learn to a. pose and solve multiplication and division

problems (Interpreting &/or Posing Questions, Strategically Thinking & Representing)

b. select and use mental, written and calculator strategies to solve multiplication or division problems e.g., ‘to multiply by 12, multiply by 6 and then double’ (Strategically Thinking & Representing)

c. solve a variety of problems using problem-solving strategies, including:

- trial and error - drawing a diagram - working backwards - looking for patterns - using a table


d. identify the operation/s required to solve a problem (Strategically Thinking & Representing)

e. check the reasonableness of a solution to a problem by relating it to an original estimation (Reasoning & Justifying)

f. explain how an answer was obtained and compare own method/s of solution to a problem with those of others (Communicating Mathematically, Reflecting & Evaluating)

g. use multiplication and division facts in board, card and computer games (Strategically Thinking & Representing)


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the word problem 7. describing multiplication as the product of

two or more numbers 8. describing and recording methods used in

solving multiplication and division problems 9. determining factors for a given number

e.g., factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

10. finding cubes of numbers using squares of numbers and one more multiplication and/or multiplying number by itself three times

h. apply the inverse relationship of multiplication and division to check answers

e.g., 56 ÷ 8 is 7 because 8 7 = 56 (Strategically Thinking & Representing, Reflecting & Evaluating)

i. explain why a remainder is obtained in answers to some division problems e.g., 67 ÷

8 is 8 remainder 3 because 8 8 = 64 plus 3 is 67 (Communicating Mathematically, Reasoning & Justifying)

j. apply factorisation of a number to aid mental

computation e.g., 16 25 = 4 4 25 = 4 100 = 400 (Strategically Thinking & Representing)

Background Information At Years 3 and 4, the emphasis in multiplication and division is on students developing mental strategies and using their own (informal) methods for recording their strategies. Comparing their method of solution with those of others, will lead to the identification of efficient mental and written strategies. One problem may have several acceptable methods of solution.

Linking multiplication and division is an important understanding for students at Years 3 and 4. Students should come to realise that division ‘undoes’ multiplication and multiplication ‘undoes’ division. Students should be encouraged to check the answer to a division question by multiplying their answer by the divisor. To divide, students may recall division facts or transform the division into a multiplication and use multiplication facts e.g., 40 ÷ 8 is the same as

8

Language When beginning to build and read multiplication tables aloud, it is best to use a language pattern of words that relates back to concrete materials such as arrays. As students become more confident with recalling multiplication number facts, they may use less language. For example, ‘seven rows (or groups) of three’ becomes ‘seven threes’ with the ‘rows of’ or ‘groups of’ implied. This then leads to:

- one three is three - two threes are six - three threes are nine, and so on.

Examples of explanations: ‘The pattern for sixes is twice as big as the pattern for threes. If you double the threes pattern you get the sixes pattern.’ ‘I found out that sixty-four is a multiple of eight. It is also a multiple of four and two.’ ‘Three times as many means three times or double plus the number.’ ‘I think there’ll be four groups of five in twenty.’

Keywords to be used: multiplication, division, inverse relationship, arrays, groups of, skip counting, factors, number facts, multiples, estimate, product, remainder, number pattern, multiplied by, trade, twice as many, three times as many


27

‘Thirty-six shared between five is equal to seven and one remaining.’ ‘Forty-nine divided by six is equal to eight with one remaining.’ ‘I remember the multiplication and reverse it.’

Resources calculators, multiplication and division grid, interlocking cubes, flash cards, string, envelope, hundreds chart, Base 10, material, place value chart, dice, counters

Links Whole Numbers Addition and Subtraction Area Volume and Capacity

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES NR 4.3 Models and represents multiplication and division by applying various mental and formal written strategies

ACTIVITY NR4.3.1: WM MENTAL STRATEGIES MODELS OF THE MULTIPLICATION FACTS OF AT LEAST 100

Part A Students construct models of the multiplication facts by building a staircase e.g., with 10 blocks in the first step, 20 in the second etc, to represent the multiplication facts for 10. Alternatively, students use drawings on 1 cm grid paper. Students use a 12 × 12 grid to record their answers.

Part B Students model the multiplication facts between 100 and 144 using rectangular arrays and record the associated inverse relationships

11 groups of 10 is 110

110 shared among

11 is 10

11 10 = 110

110 ÷ 11 = 10

Variation: Students are given a number (e.g., 124) and asked to represent all its factors using arrays.

Extension: Students are given numbers beyond 144 to represent its factors using arrays.

ACTIVITY NR4.3.2: MULTIPLICATION FACTS Students write the multiplication facts on

flash cards from 0 1 up to 12 12. In pairs, students test each other to find which facts they can immediately recall and put these into the ‘known’ pile. The others are put into the ‘unknown’ pile. Each day the students concentrate on learning from their ‘unknown’ facts.

Students could repeat this activity with division facts.

Variation: Students play ‘Bingo’ using multiplication and division facts.

ACTIVITY NR4.3.3: TABLES RACES Students make up cards for particular multiplication facts for particular numbers

including up to 12 12, shuffle them and put them into an envelope e.g.,

In groups, students are given an envelope of cards. Students race each other to put the cards into order, skip counting aloud. Students state which number has the multiplication facts their cards represent.

Variation: Students write numbers in descending order.

96 12 72

108

60

48 120 36 132

24


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ACTIVITY NR4.3.4: 12 12 MULTIPLICATION GRID Students keep a multiplication grid, as shown below. When students are sure they have learnt particular multiplication facts, they fill in that section of the grid.

1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

5

6

7

8

9

10

11

12 Students are encouraged to recognise that

if they know 11 12 = 132 they also know 12

11 = 132, and so they can fill in two squares on the grid.

Extension: Using the completed multiplication grid, find all factors (numbers used in multiplying) to give a particular product such as 6, 8, 10 and 12.

Possible questions to ask:

Which numbers have no other factors except themselves and 1? Colour these numbers.

What are such numbers called? What is the next such number (i.e., number with no other factors except itself and 1) after 144?

ACTIVITY NR4.3.5: PATTERNS Students investigate patterns in the multiplication grid. Students discuss these patterns and record their observations. For example, students compare the multiplication facts for 3 and 6 and the multiplication facts for 12. They then investigate the multiplication facts for 9 and 12.

Students colour multiples on a hundreds chart and are encouraged to describe the patterns created.

Extension: Students are asked to pick a number on a hundreds chart and count (a) 10 more, (b) 20 more, and (c) 50 more and describe any observed patterns.


How can you use the chart to mentally add 50 more?

Given the following portion of a hundreds chart, what are the values of A, B, C and D? How do you know? Explain how you got each value.

D

46

C

B

A

Is there another way of finding C? Why or why not?

Variation: Students create their own portions of the hundreds chart with missing values and exchange these with other students to solve.

ACTIVITY NR4.3.6: MASI POPO BOXES The teacher poses the problem: ‘Imagine you had the job of designing a masi popo (coconut biscuits) box. There are to be 120 masi popo in the box. The box can be more than 2 layers high.

Students answer the question:

How many ways could you arrange the masi popo in the box?’

Students draw or make models of their solutions and discuss these in terms of multiplication and division facts.

ACTIVITY NR4.3.7: CUBES Students work in small groups. A student chooses a small whole number (between 2 and 8) and the next student cubes it (i.e. multiply the number by itself three times). They take turns to keep cubing the number. A student checks the results with a calculator. In the next round they start with a different number.

The teacher introduces the term “cubes” as a label for products resulting from multiplying a whole number by itself three times.

Extension. Link this activity to the MS4.3 Learning Outcome on Volume and Capacity


29

and ask students to propose concrete models for these numbers (i.e. cubes of whole numbers).

ACTIVITY NR4.3.8: MULTIPLICATION AND DIVISION TRADING GAME Students play the trading game ‘Race to and from 10000’ with the following variation. Students throw two dice, one numbered 5 to 9 and the other numbered 8 to 12. They multiply the numbers thrown and record it. They continue to throw and keep a record of their running totals. The winner is first to 10,000.

ACTIVITY NR4.3.9: THIRDS Students work in small groups. One student chooses a number. The next student finds one-third of it. Students take turns as they keep one-thirding. The teacher asks how far they think they can go. A student checks the results with a calculator. Students try starting at a different number when playing the next round. Extension: Students are asked to design their own games involving multiplication and division number facts.

ACTIVITY NR4.3.10: CREATING SEVERAL ARRAYS - FACTORS Students use counters to make an array for a particular number. They create new arrays for this number. Students record their findings e.g., 72 can be 8 rows of 9 or 4 rows of 18.


How many different arrays can you make?

How many rows do I have if there are 6 counters in each row?

Students record their different array arrangements for each number or product. Students are then introduced to the term “factors” the name for the numbers used in the multiplication to produce the “product”.

Extension: Students repeat the activities for other two- and three-digit products such as 96, 81, 108, 132, 144.

ACTIVITY NR4.3.11: REDUCING MULTIPLES EXTENDED

Students use part of their completed 12 x 12 multiplication grid (from Activity 4) to investigate for more patterns. The idea, first introduced in Year 3 (Activity NR3.3.15: Reducing Multiples), is based on an ancient Chinese unique “one digit” table. A portion

(i.e. 9 9) of the completed 12 x 12 multiplication grid is further investigated for more patterns.

Students are asked to reconstruct their 9 9 grid so that only one-digit entries appear. For example, each two-digit product is reduced to one digit as follows: e.g., 96336 so where 36 was, it is now 9. The one-digit products are left alone.

Examining their resulting 9 9 grid of “one-digit” entries, students discuss the following questions:

How many patterns can you see? Describe as many as you can.

What happens if you connect all the 8’s? What shape does it form?

What is the pattern if you connect all the 7’s? other digits? Is the pattern the same for all digits?

Can you spot any other “squares” enclosing numbers that are reflecting off a diagonal of numbers such as the one below?

List as many squares you can find.

9 3 6 9

3 7 2 6

6 2 7 3

9 6 3 9


30

ACTIVITY NR4.3.12: COUNTING FORWARDS AND BACKWARDS Students count forwards and backwards by fives, tens, and twenties in the context of money.

Students are asked:

How many 5 sene pieces in 80 sene? $2.00? $5.00? How do you know?

How many 20 sene pieces in $1.00? $2.00? $5.00? $10.00? Explain your answers.

How many 50 sene pieces in $1.00? $2.00? $5.00? $10.00? Justify your answers.

Students record their strategies and answers using drawings, numerals and words.

ACTIVITY NR4.3.13: FACTOR GAME The teacher prepares two dice, one with faces numbered 1 to 6 and the other with faces numbered 5 to 10. Each student is given a blank 6 x 6 grid on which to record factors from 1 to 60. Students work in groups and take turns to roll the two dice and multiply the numbers obtained. For example, if a student rolls 5 and 8, they multiply the numbers together to obtain 40 and each student in the group places counters on all of the factors of 40 on their individual grid i.e. 1 and 40, 2 and 20, 4 and 10, 5 and 8. The winner is the first student to put three counters in a straight line, horizontally or vertically.

ACTIVITY NR4.3.14: TAG Students find a space to stand in the classroom. The teacher asks students in turn to answer questions e.g., ‘What are the factors of 16?’ If the student is incorrect they sit down. The teacher continues to ask the same question until a correct answer is given.

When a student gives a correct answer, they take a step closer to another student and may tip them if within reach. The ‘tipped’ student sits down. The question is then changed.

Play continues until one student remains, who then becomes the questioner. This game is designed for quick responses and repeated games.

9ASSESSMENT FOR LEARNING INDICATORS NR 4.3 Models and represents multiplication and division by applying various mental and formal written strategies The student, for example;

uses mental strategies to recall

multiplication facts to 12 12

uses multiplication facts to work out division facts

explains the relationship between multiplication facts e.g., explains how the 3, 6, 9, and 12 times tables are related

uses mental strategies to divide a two-digit number by a one-digit number

describes and records methods used to solve a multiplication or division problem

identifies cubes for a given number

uses mental strategies to multiply a two-digit number by a multiple of 10

uses mental strategies to multiply a two-digit number by a one-digit number

explains and records remainders to division problems e.g., 98 ÷ 12 = 8 remainder 2

determines factors for a given number


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NR4.4 Fractions & Decimals NR4.4 Substrand: Fractions and Decimals Models, compares and represents simple fractions (including fifths, tenths, hundredths and sixths) and decimals, multiplies and divides decimals with two decimal places, and interprets everyday percentages


Model, compare and represent fractions including sixths Find equivalence between halves, tenths and hundredths; fifths, tenths and hundredths and thirds and sixths Model, compare and represent decimals to 2 decimal places Multiply and divide decimals with the same number of decimal places (to 2 decimal places) Recognise percentages in everyday situations. Relate a common percentage (benchmark) to a fraction or decimal Solve problems involving calculations with money


Knowledge and Skills Students learn about 1. modelling, comparing and representing

simple fractions including those with denominators 3 and 6 finding equivalence between thirds and

sixths using concrete materials and diagrams, by re-dividing the unit

e.g.,

=

31

62

placing halves, quarters and eighths on a number line between 0 and 1 to further develop equivalence

e.g.,

0 1

2 1

0 2

4 1 3

4 1

4

0 4

8 1 6

8

2

8

7

8

5

8

3

8

1

8 counting by thirds, quarters, sixths and

eighths e.g., 0,31 , ,3

2 1,

,1 31 ,1 3

2 2, …

modelling mixed numerals e.g.,

21

2

placing halves and quarters on a number line beyond 1

e.g. 0 1 2 2

4

2

4

3

4

1

4

3

4

1

4

1 1 1

2. modelling, comparing and representing

Working Mathematically Students learn to a. pose questions about a collection of items

e.g., ‘Is it possible to show one-sixth of this collection of objects?’ (Interpreting &/or Posing Questions)

b. check whether an answer is correct by using an alternative method e.g., use a number

line or calculator to show that 31 is the same

as 0.33 and 62

(Reasoning & Justifying) c. interpret the everyday use of fractions,

decimals and percentages, such as in advertisements (Reflecting & Evaluating)

d. interpret a calculator display in the context of the problem e.g., 2.6 means $2.60 when using money (Strategically Thinking & Representing, Communicating Mathematically)

e. apply decimal knowledge to record measurements e.g., 2.5 yds = 2 yds 1.5 ft (Reflecting & Evaluating)

f. explain the relationship between fractions

and decimals e.g., 51 is the same as 0.2

(Reasoning & Justifying, Communicating Mathematically)

g. round an answer obtained by using a calculator, to one or two decimal places (Strategically Thinking & Representing)


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fractions with denominators 3, 6, and then 5, 10 and 100 by extending the knowledge and skills

covered above from thirds to sixths, to fifths and then to tenths and hundredths

modelling fifths, sixths, tenths, and hundredths of a whole object or collection of objects

3. ordering decimals with the same number of decimal places (to 3 decimal places) on a number line

4. rounding a number with up to 3 decimal places to the nearest whole number

5. recognising the number pattern formed when decimal numbers are multiplied or divided by 10 or 100

6. multiplying and dividing decimals with the same number of decimal places (to 2 decimal places)

7. recognising that the symbol % means ‘percent’

8. relating a common percentage (benchmark) to a fraction or decimal e.g., ‘25% means 25 out of 100 or 0.25’ equating 10% to

1

10, 25%

to

1

4 and 50% to

1

2

h. use a calculator to create patterns involving decimal numbers e.g., 1 ÷ 10, 2 ÷ 10, 3 ÷ 10 (Strategically Thinking & Representing)

i. perform calculations with money (Strategically Thinking & Representing)

Background Information At Years 3 and 4, fractions are used in two different ways:

- to describe equal parts of a whole, and - to describe equal parts of a collection of objects.

Fractions refer to the relationship of the equal parts to the whole unit. When using collections to model fractions it is important that students appreciate the collection as being a ‘whole’ and the resulting groups as ‘parts of that whole’. It should be noted that the size of the resulting fraction will depend on the size of the original whole or collection of objects.

Language At Years 3 and 4, the term ‘three-quarters’ may be used informally to name the remaining parts after one-quarter has been identified. Examples of students’ explanations: ‘When a whole number is divided by 10, the answer has one decimal place. When a whole number is divided by 100, the answer has 2 decimal places.’ ‘one-fifth is the same as 0.2 because one-fifth is equivalent to two-tenths and two divided by 10 is 0.2.’ ‘There are twelve objects in this collection, if the whole collection is divided into six equal groups, each group will have 2 objects.’ ‘2.5 yards means 2 yards and half-a-yard, half-a-yard in feet is 1.5 feet therefore 2.5 yards is the same as 2 yards and 1.5 feet.’

Keywords to be used: fraction, decimal, percentage, thousandth, hundredth, tenth, decimal places, whole, part of, half, quarter, third, sixth, eighth, mixed numeral, proper fraction, improper fraction, denominator, numerator, equivalence, percent, benchmark percentage


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Resources fraction kits, pattern blocks, fraction cards, paper, calculators

Links Addition and Subtraction Multiplication and Division Patterns and Algebra Chance Data

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES NR 4.4 Models, compares and represents simple fractions (including fifths, tenths, hundredths and sixths) and decimals, multiplies and divides decimals with two decimal places, and interprets everyday percentages

ACTIVITY NR4.4.1: SHARING SIXTHS Students form groups of 6 and share a slice of bread so that each person gets the same amount and there is none left over.

Each group discusses how they shared the bread and names the pieces ‘sixths’. Students regroup into groups of 3, and then into groups of 2, and repeat the activity, naming the pieces ‘thirds’ or ‘halves’. Students compare the relative sizes of the fractions and then order them according to their size. Students record their findings using drawings and the new terms.

ACTIVITY NR4.4.2: COMPARING AND ORDERING SIXTHS Students are provided with four sets of cards representing the same fractions (include sixths, fifths, and tenths). The first set has the fractions represented in fraction notation, the second set has the fractions represented in words, the third set has the fractions represented as shaded regions and the fourth set has the fractions represented as the shaded part of a collection. The cards are randomly distributed to students who must find other students with the same fraction represented. Students then place the sets of fraction cards in order. Students justify their order to others.

ACTIVITY NR4.4.3: MATCH UP TENTHS The teacher provides two sets of cards, the first with tenths expressed in fraction notation, and the second with tenths expressed in decimal notation. The teacher distributes the cards randomly to the students who then find the student/students with the same fraction represented.


How many of the same fractions/decimals did you find?

How can you check if there are any more?

Is there another way to write that fraction/decimal?

Variations: This activity should be repeated using cards with hundredths and a mixture of tenths and hundredths.

ACTIVITY NR4.4.4: CLOTHES LINE - HUNDREDTHS

Part A The teacher provides cards each naming a different fraction with the same denominator (choose from halves, thirds, quarters, fifths, sixths, eighths, tenths or hundredths). Students choose a card and peg it on a string number line in the appropriate place.

Part B The teacher provides cards, each naming a different decimal to 3 decimal places. Students choose a card and peg it on a string number line in the appropriate place.

Variation: Students make their own cards and arrange them on their desk or a sheet of paper.


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ACTIVITY NR4.4.5: IS IT POSSIBLE? Students are given 18 counters and need to

determine whether it is possible to find 31

21 , ,

or .61

e.g., I can find

21 of 18 (9)

31 of 18 (6)

61

of 18 (3).

Students record their findings. The activity should be repeated using different numbers of counters and extended to include fractions with denominator of 8, 5, 10 and 100.

ACTIVITY 6: DIFFERENT GROUPINGS – INCLUDING SIXTHS Students move about in an open space in a group of 24. The teacher asks the group to divide into halves, thirds, quarters, sixths or eighths. Any remaining students check the groupings. The activity should be repeated using groups of different sizes. Extension: The activity could be repeated using different numbers of students and extended to enable division of group into fifths, quarters and halves. Possible questions include:

What is a reasonable group size to enable division into fifths, quarters and halves without incurring a remainder? How do you know?

What are other reasonable group sizes to enable group division into fifths, quarters and halves without incurring a remainder?

Is it possible to find tenths of these group sizes? Explain your answer.

What group size will enable division into halves, quarters, fifths and tenths as well as eighths without a remainder? Justify your answer.

ACTIVITY NR4.4.7: DESIGN A MENU Students design a menu for a local take-away food shop. Students investigate different selections from the menu that total different amounts e.g., $10.50, $20.50, $10.00, $25.50. Possible questions include:

How much would it cost to feed yourself; yourself and a friend; or yourself and your family?

What is the change from $20/$60/$120 after the purchases?

Students pose their own questions based on their own menu and then their own orders.

ACTIVITY NR4.4.8: TWO DECIMAL PLACES GAME – 1 TO 2

The teacher makes a die writing a decimal (between 1 and 2) to two places on each face. Students use a 10 × 10 grid as a score sheet. Students take turns to throw the die and colour the appropriate section on their grid. The winner is the first player to colour their 10 × 10 grid completely.

Variation: Students can make their own dice labeling them using common fractions, decimals or a combination of fractions and decimals.

Extension: Students record the decimal thrown and add decimals together after each throw. Students colour each throw differently.

ACTIVITY NR4.4.9: ADDING AND SUBTRACTING DECIMALS (3 DP2) In pairs, students are provided with a pack of playing cards with the tens and picture cards removed. The Aces are retained and represent 1 and the Jokers are retained and represent 0.

Student A flips three cards and places them together to form a decimal to three decimal places. Student B flips two cards and places them together to form a decimal to three decimal places.

Student A copies down the decimals and uses a written algorithm to find their sum. Student B checks Student A’s answer. Students swap roles and the activity is repeated.

Variation: The activity is repeated to involve subtraction of decimals to three decimal places.

2 dp is decimal places


35

ACTIVITY NR4.4.10: PAPER FOLDING – FIFTHS AND TENTHS

Students are given four strips of different-coloured paper of the same length. The first strip represents one whole. The second strip is folded into halves and labeled. The third strip is folded into fifths and labeled. The fourth strip is folded into tenths and labeled. Students line up the four strips and discuss.


What can you tell about the size of each fraction and the denominator?

What strategies did you use to create your fractions?

What strategies did you use to fold your strip into equal parts?

Variation: Students cut the folded strips into halves, fifths and tenths and order the strips from smallest to largest parts.

They discuss their findings. Extension: Students are given another set of

coloured strips to represent and compare thirds and sixths.

ACTIVITY NR4.4.11: FRACTION POSTERS Students choose a fraction and create a poster, writing everything they know about that fraction. Students report back to the group their findings about their fraction.

Variation: Students repeat the activity but using a different fraction this time.

ACTIVITY NR4.4.12: BIGGEST OR SMALLEST – 3 DP The teacher places cards with the digits 0 to 9 into a bag. In pairs, students randomly select three cards from the bag.

Students use the digits to make a decimal number less than 1 e.g., if 6, 1 and 4 are selected the students record 0.164. Students use the three digits to make two new decimals i.e. 0.416 and O.641


Which decimal is the larger? smaller?

How do you know?

How can you show this? The number cards are replaced and the

activity repeated.

Students record the decimal numbers on a number line.

ASSESSMENT FOR LEARNING INDICATORS NR 4.4 Models, compares and represents simple fractions (including fifths, tenths, hundredths and sixths) and decimals, multiplies and divides decimals with two decimal places, and interprets everyday percentages The student, for example:

models and compares fractions with denominators 3, 6, and then 5, 10 and 100

models and represents fifths, sixths, tenths and hundredths of an object or collection of objects

interprets decimal notation for tenths or hundredths

finds equivalence between halves, quarters and eighths; thirds and sixths; halves, tenths and hundredths; and fifths, tenths and hundredths of an object or collection of objects

orders decimals with the same number of decimal places (to 3 decimal places) on a number line

rounds a number with up to 3 decimal places to the nearest whole number

recognises the number pattern formed when decimal numbers are multiplied or divided by 10 or 100

multiplies and divides decimals with the same number of decimal places (to 2 decimal places)

recognises that the symbol % means ‘percent’

relates a common percentage (benchmark) to a fraction or decimal e.g.,


36

NR4.5 Chance NR4.5 Substrand: Chance Describes, orders and compares likelihood of events with chance experiments and recognise that there will be variation in results and expected outcomes


Predict the outcomes of chance experiments involving equally likely events Collect and organise data to compare likelihood of events under various conditions Determine the likelihood of outcomes of experiments with small numbers of trials


Knowledge and Skills Students learn about 1. listing all the possible outcomes in a simple chance

situation 2. predicting and recording all possible outcomes in a

simple chance experiment 3. using the language of chance in everyday contexts 4. predicting and recording all possible combinations

e.g., the number of possible outfits arising from three different t-shirts and two different pairs of shorts

5. conducting simple experiments with random generators such as coins, dice or spinners to inform discussion about the likelihood of outcomes e.g., roll a die fifty times, keep a tally and graph the results

Working Mathematically Students learn to a. discuss the ‘fairness’ of simple games involving

chance (Communicating Mathematically)

b. compare the likelihood of outcomes in a simple chance experiment (Reasoning & Justifying)

c. apply an understanding of equally likely outcomes in situations involving random generators such as dice, coins and spinners (Reflecting & Evaluating)

d. make statements that acknowledge ‘randomness’ in a situation e.g., ‘the spinner could stop on any colour’ (Communicating Mathematically, Reflecting & Evaluating)

e. explain the differences between expected results and actual results in a simple chance experiment (Communicating Mathematically, Reflecting & Evaluating)

Background Information When a fair coin is tossed, theoretically there is an equal chance of a head or tail. If the coin is tossed and there are five heads in a row there is still an equal chance of a head or tail on the next toss, since each toss is an independent event.

Language Examples of explanations: ‘It’s most likely it will happen.’ ‘It’s more likely that you will pick a red counter because there are more of them in the bag.’ ‘The most likely number to come up is 6 as there are more 6s on the die than any other number.’

Keywords to be used: likelihood, always, probably, certain, uncertain, possible, predict, simple chance experiment, events, equally likely, more likely, less likely, least likely, most likely, random, fairness, dice, spinner, random generators, tally, combination, expected, outcomes, actual

Resources coins, coloured counters, coloured blocks, bottle lids, spinner, dice, simple game boards, bucket of pegs, Snakes and Ladders etc, textas

Links Whole Numbers Addition and Subtraction Data Analysis Literacy


37

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES NR 4.5 Describes orders and compares likelihood of events with chance experiments and recognise that there will be variation in results and expected outcomes

ACTIVITY NR4.5.1: SAME BIRTHDAY Students are asked to find out the birthdays of as many members of their family as possible. Do any of them have birthdays on the same day of the year? Now try the experiment with all members of your class. We will see how likely it is that two members of your class have the same birthday. Consider each member of your class. Possible questions include:

What is the likelihood that the second person has the same birthday as the first one?

What is the likelihood that the third person will have a different birthday from the first two?

What is the likelihood that two out of three people have the same birthday?

ACTIVITY NR4.5.2: EVENS AND ODDS This is a simple game where you throw a dice which controls the position of your counter on a 3 x 3 board

FINISH

START

Students put their counter at the START

square and then throw a dice. If the dice shows an EVEN number, then the

counter is moved upwards. If dice shows an ODD number, then the

counter is moved left If the counter moves off any side of the

board, the student loses. If the counter reaches the FINISH square,

the student wins. Play the game a few times and see if you

win. Possible questions include:

How many “odds” and how many “evens” do you need to win?

What is the likelihood that you will win? Extension: Students repeat the game but

use a 4 x 4, 5 x 5 board and answer the same questions. Describe any patterns you notice.

ACTIVITY NR4.5.3: MOST LIKELY AND LEAST LIKELY TOTALS Students, in pairs, are asked to toss two dice, add the two numbers and record the number of times each total occurs.

Before tossing the dice, the teacher initiates a whole class discussion around the following questions:

What total(s) is (are) likely to occur the most? How do you know?

Which total(s) is (are) likely to occur the least? Why?

What is the range of all possible totals from two dice? Justify your answer.

In pairs, students collect their data from 20 tosses of the dice to answer the following questions:

How do you organize your results so that it is easy to present to another pair?

What is (are) the most occurring total(s)? Justify your answer.

What is (are) the least occurring total(s)? How do you know?

What is an appropriate graph to display your results?

Are your pair’s actual results the same as the class’ predicted results? Why or why not?

What can you do to improve your actual results compared to the predicted results?

Extension. Pair results are pooled to form a class data set to be used further as an activity for DA4.1 Data Analysis Learning Outcomes.

ACTIVITY NR4.5.4: COMBINATION DRESSING Students are told that they will be given three t-shirts and two pairs of trousers and are asked to predict how many different combinations of clothes they could make from them. They work out a strategy and


38

follow it to calculate the number of combinations and compare the results to their predictions.

Extension. Students repeat the activity with 3 t-shirts and 4 pairs of trousers. Students record their strategies for working out the different combinations and the number of each combination.

ACTIVITY 5: A DIE AND A COIN. Students are given a die to roll and a coin to toss 10 times. Working in pairs, one student tosses the coin and rolls the die at the same time 5 times while the other records after which they switch roles.

Before beginning the experiment, each pair discusses answers to these questions:

How many different possible outcomes can this experiment have?

How can I confirm all the possible outcomes?

Which outcome do we predict is most likely to occur? Which is the least likely? Explain your answers.

After the collecting the data, each pair answers the questions:

How can you organize your results so that it is easy to present to another pair?

What is (are) the most occurring outcome(s)? Justify your answer.

What is (are) the least occurring outcome(s)? How do you know?

What is an appropriate graph to display your results?

Are your actual results the same as your predicted results? Why or why not?

Comparing your actual results to another pair’s results, are yours the same? similar? very different from their’s? Explain in your own words

ACTIVITY NR4.5.6: GET ALL 6! Ask students to list the number 1 to 6 at the bottom of a frequency table as shown below, to record the results of rolling a die.

What do you predict the likelihood is for each outcome to occur?

1 2 3 4 5 6

Students are asked to mark an x in each

square over the number until they have rolled each number at least once. Repeat five or six times.

Discuss how the frequency charts from all the pairs compare.

Are they all the same, different or similar? [Students should observe a lot of variations in the results across the pairs. Some will take 20 rolls, 25 rolls or more before getting all numbers, while in other cases, they got all the numbers in only 10 rolls.]


What happens to this variation in results if all pairs’ results are pooled together? Explain in your own words.

What are the most likely and least likely outcomes? Or are the results more or less evened out? Explain why or why not?

Are the predicted and actual results similar when pooled together? Explain why or why not.

ACTIVITY NR4.5.7: SPINNERS Students design their own template for spinners, predict the likelihood of outcomes and test their predictions.

Show students a spinner with all numbers having an equal chance of coming up. They are asked:

Why is this spinner a fair one? Students discuss their ideas with each other

and the whole class. Challenge the students to design two

different spinner templates on which the numbers (or colours) do not have an equal chance of coming up. They should use three to six numbers (or colours) for the spinner


39

template. They can work in small groups of 4 to 5. Once a group designs a new spinner, a question to ask includes:

How often do you predict the spinner to land on each number (colour) after 100 spins? Why?

Each group should build identical templates of their spinner for each person in the group. Then the group should make 100 spins and collect data to verify their predictions.

ASSESSMENT FOR LEARNING INDICATORS NR 4.5 Describes, orders and compares likelihood of events with chance experiments and recognise that there will be variation in results and expected outcomes

The student, for example:

lists all the possible outcomes in a simple chance situation

predicts and records all possible outcomes in a simple chance experiment

predicts and records all possible combinations

conducts simple experiments using coins, dice or spinners and records the results

explains the differences between expected results and actual results in a simple chance experiment

using the language of chance in everyday contexts


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PATTERNS AND ALGEBRA The inclusion of Patterns and Algebra strand in the primary curriculum is to demonstrate the importance of early number learning in the development of algebraic thinking. This strand emphasises number patterns and number relationships leading to an investigation of the way that one quantity changes relative to another. The Patterns and Algebra strand extends from Year 1 to Year 8. In the early Years students explore number and pre-algebra concepts by pattern making, and discussing, generalising and recording their observations. Separating these concepts into a distinct strand is intended to demonstrate the connections between these early understandings and the algebra concepts that follow. The Patterns and Algebra strand links with the Number strand and it is recommended that it be taught in conjunction with the development of number concepts.

One important aspect of algebraic thinking is the development of students’ abilities to replicate, complete, continue, describe, generalise and create repeating patterns and number patterns that increase or decrease (i.e. growing patterns). These number patterns can be formed using rhythmic or skip counting. Repeating patterns can be created using sounds, actions, shapes, objects, stamps, pictures and other materials. Children could be encouraged to create a wide variety of such patterns and then to describe and label them using numbers. Repeating patterns can be described using numbers that indicate the number of elements that repeat. For example, A, B, C, A, B, C, … has three elements that repeat and is

referred to as a ‘three’ pattern; , , , , , , … is also a three pattern because there is a sequence of three repeating elements. Another important aspect of algebraic thinking is the ability to recognise and use number relationships and to be able to make generalisations about number relationships. From Year 1, children should be encouraged to describe number relationships and to make generalisations when appropriate. In addition, finding unknowns or missing elements in number sentences needs to be addressed from an early Year. This is associated with the concept of equality and the need to develop an understanding that the equals sign also means ‘is the same as’. This section presents the Year 4 outcomes, key ideas, knowledge and skills, and Working Mathematically statements in one substrand.


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PA4.1a Number Patterns PA4.1a Substrand: Number Patterns Generates, describes and records number patterns using a variety of strategies and using change, finds missing terms and makes predictions


Create, describe and extend number patterns and complete simple number sentences using various strategies Analyse and describe change in growing patterns and use tables and graphs to base conclusions


Knowledge and Skills Students learn about 1. identifying and describing patterns when

counting forwards or backwards by up to eights, nines or tens

2. creating, with materials or a calculator, a variety of patterns using whole numbers or decimals e.g., 2.2, 2.0, 1.8, 1.6, …

3. finding a higher term in a decreasing number pattern given the first four terms e.g., determine the 8th term given a number pattern beginning with 50, 47, 44, 41, … or the 9th term given 72, 66, 60, 54, 48, ...

4. describing a simple decreasing number pattern in words

5. analyse, identify and describe change between terms in growing patterns (both increasing and decreasing) and use tables of values to base conclusions


a. pose problems based on number patterns (Interpreting &/or Posing Questions)

b. solve a variety of problems using problem-solving strategies, including: - trial and error - drawing a diagram - working backwards - looking for patterns - using a table


c. ask questions about how number patterns have been created and how they can be continued (Interpreting &/or Posing Questions)

d. generate a variety of number patterns that increase or decrease and record them in more than one way (Strategically Thinking & Representing, Communicating Mathematically)

e. generate number patterns, by choosing a variety of starting numbers, and using the process of repeatedly adding or subtracting the same number either mentally or on a calculator (Communicating Mathematically)

f. model and then record number patterns using diagrams, words, change and/or symbols (Communicating Mathematically)

g. use change to describe, generate, generalise, and/or extend patterns, find missing terms and make predictions (Strategically Thinking & Representing, Communicating Mathematically)


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Background Information At Year 4, students continue to explore number patterns that increase and also further develop their understanding of decreasing patterns with particular attention to identifying amount of changes between consecutive terms of a number pattern that are both whole numbers and decimals. Patterns could include any patterns observed on a hundreds chart and those patterns created by counting forwards and backwards in ones, twos and up to tens This links closely with the development of Whole Numbers and Multiplication and Division.

Language At Year 4, students continue to describe number patterns in words in terms of the amount of change between consecutive terms (initially informally introduced as ‘count forwards by …’ and ‘count backwards by ….”), the starting number and at times, the number of terms Such an approach will assist in developing students’ understanding of functions later on.

Keywords to be used decreasing, increasing, change, jump, change, count by, count forwards, counting backwards, starting number, terms, sequence, empty number line

Resources calculators,

Links Whole Number Addition and Subtraction Multiplication and Division Data Analysis Social Studies Science

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES PA4.1a Generates, describes and records number patterns using a variety of strategies and using change, finds missing terms and makes predictions

ACTIVITY PA4.1A.1: GENERATE A NUMBER PATTERN Students are asked to generate a number pattern that increases by eight from a given even two-digit starting number for 5 terms.

They are asked to:

describe their number pattern in words and record these words

determine the amount of change between terms

continue their number pattern for 5 more terms

explain why a particular number (choose a number) is or is not used in their number pattern, explain why?

predict the fifteenth term of the number pattern and justify answer using the amount of change and an empty number line

describe any pattern formed by the last digits of the terms of the number pattern

create another number pattern that has a particular number in it e.g., ‘create a similar number pattern with the number 37 in it’. How do you do that? Show on an empty number line

Variation. Students are asked to repeat the activity but using an odd number (e.g. 17 or 23) as the new starting number and count by eights. They record their strategies using drawings, words and empty number lines.

Extension. Students generate a pattern by counting by nines and using their own choice of a two-digit starting number and then a three-digit starting number. In each case, predict the 20th term. Students describe their pattern and strategies using change and record them in words and using empty number lines.


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ACTIVITY PA4.1A.2: MAKE THE CALCULATOR COUNT In pairs, students are given a calculator and are asked to generate a number pattern by repeatedly adding 10 to a non-zero two-digit number (e.g., 22)

Possible questions to ask include:

What pattern do you see with the numbers?

What is the pattern, if any, of the last digits of the terms? What about the tens digit and the hundreds digit? Describe in your own words.

Students record their number pattern on an empty number line and describe the pattern in terms of change and in words.

Variation. Students are asked to use the calculator generate to generate a number pattern by repeatedly subtracting 9 from any two-digit number. They should use an empty number line to show and record the number pattern.

ACTIVITY PA4.1A.3: DECIMAL NUMBER PATTERNS Students are asked to generate a number pattern starting at any two-digit number and repeatedly subtracting 0.8.

Possible questions include

What is the 10th term of the pattern? How do you know?

Without using the calculator, what do you predict the 15th term to be? Check your answer using the calculator. Are they the same or different? Explain why or not.

What strategy did you use to make your prediction? Does it work if you predict for the 20th term of the pattern? Justify your answer.

Student record their pattern on an empty number line and describe strategies used in making predictions using drawings and words.

Variation. Students are asked to generate a decreasing number pattern using the same two-digit starting number but repeatedly subtracting 0.2 to obtain 10 terms. They record the pattern on an empty number line before answering the question:

Are there any other common terms between the two sequences within

their first ten terms? Explain your answer and strategies using drawings and empty number lines.

ACTIVITY PA4.1A.4: CREATE YOUR OWN PATTERNS Students are asked to create their own decreasing patterns using whole number and decimal jumps, with and without the calculator, and using empty number lines to illustrate their strategies and number patterns.

ACTIVITY PA4.1A 5: SOIL EROSION Sione’s grandfather Lata built his house by the sea many years ago and he says that the sea has been eroding his land since then.

When Lata first built his house in 1970, the coastline was about 1.5 metres from his front step. Forty years later, the coastline is about 0.75 metres from his house. The following table shows the distance of the coastline from Lata’s house as measured by Sione and his father after every ten years.

What do you predict will be the distance of Lata’s front step from the coastline after the fifth ten years assuming that the rate of erosion remains the same? Explain your reasoning.

Number of Ten Years

First Second

Third

Fourth

Distance from House (m)

1.50 1.25 1.00 0.75

If Lata does not build any seawall soon, when do you predict the sea erosion will cause Lata’s house to topple down? Justify your answer. Use an empty number line to illustrate your strategy


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ASSESSMENT FOR LEARNING INDICATORS PA4.1a Generates, describes and records number patterns using a variety of strategies and using change, finds missing terms and makes predictions The student, for example:

identifies and describes patterns when counting forwards or backwards by up to eights, nines or tens

creates, with materials or a calculator, a variety of patterns using whole numbers or decimals

finds a higher term in a decreasing number pattern given the first four terms

describes a simple decreasing number pattern in words

analyses, identifies and describes change between terms in growing patterns (both increasing and decreasing) and use tables of values to base conclusions

PA4.1b Number Relationships PA4.1b Substrand: Number Relationships

Models quantitative relationships involving multiplication and division number facts to at least 12 12, using objects, pictures and/or numbers and completes complex number sentences by calculating missing values


Model and extend quantitative relationships involving multiplication and division facts to at

least 12 12 Determine the value of a missing number in complex number sentences involving two operations



1. using the equals sign to record equivalent number relationships and to mean ‘is the same as’ rather than as an indication to perform an operation

e.g., 13 15 = 39 5 2. building the multiplication facts to at least

12 x 12 by recognising and describing patterns and applying the commutative

property e.g., 7 9 = 9 7 3. forming arrays using materials to

demonstrate multiplication patterns and relationships

4. relating multiplication and division facts 5. applying the associative property of

addition and multiplication to aid mental computation

e.g., 2 + 3 + 8 = 2 + 8 + 3; 2 3 5 = 2 5 3

6. completing number sentences involving two operations by calculating missing values


a. check solutions to missing elements in patterns by repeating the process (Reasoning & Justifying)

b. play ‘guess my rule’ games e.g., 1, 4, 7: what is the rule? (Strategically Thinking & Representing)

c. describe what has been learnt from creating patterns, making connections with addition facts and multiplication facts (Communicating Mathematically, Reflecting & Evaluating)

d. explain the relationship between multiplication facts e.g., explain how the 3 and 6 times tables are related (Reflecting & Evaluating)

e. make generalisations about numbers and number relationships e.g., ‘It doesn’t matter what order you multiply two numbers because the answer is always the same.’ (Reflecting & Evaluating)


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- 2 = 18,

7 + 3 7. transforming a division calculation into a

multiplication problem.

f. check number sentences to determine if they are true or false, and, if false, explain why (Strategically Thinking & Representing, Reasoning & Justifying)

g. justify a solution to a number sentence (Reasoning & Justifying)

h. use inverse operations to complete number sentences (Strategically Thinking & Representing)

i. describe strategies for completing complex number sentences (Communicating Mathematically)

Background Information At Year 4, describing number relationships and making generalisations is encouraged as appropriate The concept of equality and the understanding that the ‘equal’ sign also means ‘is the same as’ is important and should be consolidated. Linking multiplication and division is an important understanding for students at Years 3 and 4. Students should come to realise that division ‘undoes’ multiplication and multiplication ‘undoes’ division. Students should be encouraged to check the answer to a division question by multiplying their answer by the divisor

Language Students continue to use array arrangements of a group/set of objects to model multiplication. The ‘rows and columns’ arrangements provide a useful model or bridge to the concept of area later on.

Keywords to be used: ‘is the same as’, equal, equivalent, commutative, rectangular arrays, rows, columns, arrangement, multiplication, division, addition, reverse operations, missing terms, number sentences, grid, decade, ordinal numbers

Resources bottle lids, counters, 10 x 10 grids,

Links Whole Number Multiplication and Division Addition and Subtraction

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES PA4.1b Models quantitative relationships involving multiplication and division number

facts to at least 12 12, using objects, pictures and/or numbers and completes complex number sentences by calculating missing values

ACTIVITY PA4.1B.1: USE THOSE DIGITS! Students, working in pairs, are asked to use any THREE of the digits 1, 2, 3, 4, 5, 6, 7, 8 or 9 to create number sentences to form the numbers 11 to 20 as the answer. Students

may use any of the four operations: +, x,

and . They cannot use the same operation or the same digit twice.

An example number sentence is provided to form 11: 11 = 5 x 2 + 1 but you must make your own number sentence.

11 =

12 =

13 =

14 =

15 =

16 =

17 =

18 =

19 =

20 =


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Two pairs exchange their completed work, discuss strategies and check each other’s work.

The teacher facilitates a whole class discussion in which all pairs contribute their own different ways of generating each number.

Extension. Students, working in pairs, are asked to use any FOUR of the digits 1, 2, 3, 4, 5, 6, 7, 8 or 9 to create number sentences to form the numbers 21 to 30 as the answer. Students may use any of the four operations:

+, x, and . They cannot use the same operation or same digit twice. An example number sentence is provided to form 21: 21 = 7 x 2 + 8 – 1.

ACTIVITY PA4.1A.2: 12 X 12 MULTIPLICATION GRID This activity extends students’ 10 x 10 multiplication grid completed in Year 3. Students are asked to use any patterns and/or strategies they already know to complete the rest of the 12 x 12 multiplication facts.

As an example, students already know or should know that, starting at a particular number and skip-counting by the same amount would generate multiples of the start number (e.g., start at 6 and skip count by sixes generates the multiples of 6; similarly starting at 7 and skip counting by sevens generates the multiples of 7, etc). Can you apply the same strategy to generate multiples of 11 and multiples of 12?

1 2 3 4 5 6 7 8 9 10 11 12

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

11

12 Possible questions include:

If you know from skip counting the product 11 x 12 or the multiple of 11 that corresponds to 11 x 12 (at the 12th decade [row]), how do you use this fact to generate the product for 12 x 11? What do you call this rule?

What mental strategy can you use with the multiples of 6 to generate the multiples of 12?

What about the multiples of 4? What mental strategy can you use to generate the multiples of 12? (e.g., double plus one more.)

What mental strategies can you use with the multiples of 3 and 2 to generate the multiples of 12? Explain your strategies.

If you compare any two consecutive columns of multiples, what pattern do you notice with the differences of numbers on the same row as you go down the decades (rows)? For example, for multiples of 6 and 7 down the decades: 6 and 7, 12 and 14, 18 and 21, 24 and 28 all the way to 54 and 63, and 60 and 70).

What is the pattern of differences between the multiples as you move down the rows? List the differences as a number pattern.

How can you use this number pattern of differences and multiples of 10 to generate the multiples of 11? Describe in your own words.

What do you predict the differences (i.e., terms of the number pattern above) would be to generate the multiples of 11 and 12 corresponding to the 11th and 12th decades (rows)?

Extension. Students are asked to use objects and drawings to support, and use number sentences to record, their multiplication facts for x 11 and x 12.

ACTIVITY PA4.1A.3: REGROUPING MENTAL STRATEGIES Students are asked to solve the following problems using the mental strategies of splitting (if applicable) and regrouping.


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For example: Mina has 24 boxes of chocolate and each box has 25 bars. How many bars does Mina have? Solution: 24 x 25 = 4 x 6 x 25 = 6 x 4 x 25 = 6 x 100 = 600 bars a. Toma and Pati each planted 3 taro tops a

day for 11 days. How many taro tops did Toma and Pati plant altogether? (e.g., 3 x 11 x 2 = 3 x 2 x 11 = 6 x 11 = 66)

b. Malia’s large family planned to have a reunion over the Christmas period for 5 whole days. If it is estimated that they would need 4 dozen eggs for breakfast and another 4 dozen eggs for dinner each day, how many dozens of eggs should they order for the reunion? (e.g., 5 x 4 x 2 = 5 x 2 x 4 = 10 x 4 = 40)

c. Four parents were watching a rugby practice after school before three more parents arrived and the last to arrive were six other parents. How many parents altogether watched the rugby practice?

d. 32 x 13 = ? e. 26 x 17 = f. 127 + 53 = g. 43 + 16 =

Students are asked to create story problems whose answers are provided by questions d, e, f and g above.

Students are asked to pose their own problems similar to the ones provided and to apply any mental strategies but that one of the strategies is regrouping.

ACTIVITY PA4.1A.4: SELLING A PIG Pita buys a pig for $60, sells it for $70, buys it back for $80, and sells it for $90. How much money does Pita make or lose in the pig trading business? Students are asked to solve this problem 3 different ways using objects, drawings, words, empty number lines and/or number sentences.

ASSESSMENT FOR LEARNING INDICATORS PA4.1b Models quantitative relationships involving multiplication and division number

facts to at least 12 12, using objects, pictures and/or numbers and completes

complex number sentences by calculating missing values The student, for example,

uses the equals sign to record equivalent number relationships and to mean ‘is the same as’ rather than as an indication to perform an operation e.g.,

13 15 = 39 5

builds the multiplication facts to at least 12 x 12 by recognising and describing patterns and applying the commutative

property e.g., 7 9 = 9 7

forms arrays using materials to demonstrate multiplication patterns and relationships

relates multiplication and division facts

applies the associative property of addition and multiplication to aid mental computation

e.g., 2 + 3 + 8 = 2 + 8 + 3; 2 3 5 = 2

5 3

completes number sentences involving two operations by calculating missing values e.g., find - 2 = 18,

7 + 3

transforms a division calculation into a multiplication problem.


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DATA ANALYSIS Everyday we are exposed to a constant stream of information delivered through television, newspapers, radio, business transactions, sales, family and community affairs, and the internet to name a few thus there is a constant need for all people to understand, interpret and analyse information displayed in tabular or graphical forms. Students need to recognise how information may be displayed in a misleading manner resulting in false conclusions.

The Data strand extends from Year 1 to Year 8 and includes the collection, organisation, display and analysis of data. Early experiences are based on real-life contexts using concrete materials. This leads to data collection methods and the display of data in a variety of ways. Students are encouraged to ask questions relevant to their experiences and interests and to design ways of investigating their questions. Students should be aware of the extensive use of statistics in society. Print and Internet materials are useful sources of data that can be analysed and evaluated. Tools such as spreadsheets and other software packages, if available, may be used where appropriate to organise, display and analyse data. If computers are not available, manually organise, display and analyse data. This section presents the Year 4 outcomes, key ideas, knowledge and skills, and Working Mathematically statements in one substrand.

DA4.1 Data DA4.1 Substrand: Data Conducts an investigation, collects and organises data into tables, uses picture graphs, column graphs and grid paper to display data and interprets the results


Plan and undertake investigations to answer questions about familiar situations, classify and organise data using tables Represent data using column graphs (i.e., vertical and horizontal) and picture graphs on grid paper with labeled axes and scale of one-to-one between data and symbols Read and interpret data presented in tables, column graphs and picture graphs


Knowledge and Skills Students learn about 1. conducting surveys to collect data and

organizing them in tables 2. constructing vertical and horizontal column

graphs and picture graphs on grid paper using one-to-one correspondence

3. marking equal spaces on axes, labeling axes and naming the display

4. interpreting information presented in column graphs and picture graphs

5. representing the same data in more than one way e.g., tables, column graphs, picture graphs

6. interpreting information presented in different types of graphs

Working Mathematically Students learn to a. pose a suitable question to be answered using

a survey e.g., ‘What is the most popular Samoan idol among students in our class?’ (Interpreting &/or Posing Questions)

b. pose questions that can be answered using the information from a table or graph (Interpreting &/or Posing Questions)

c. design a table to enter data and then create a graph of the data (Strategically Thinking & Representing)

d. interpret graphs found on the Internet, in newspaper and in factual texts (Strategically Thinking & Representing, Communicating Mathematically)

e. discuss the advantages and disadvantages of different representations of the same data (Communicating Mathematically, Reflecting & Evaluating)

f. compare tables and graphs constructed from the same data to determine which is the most


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appropriate method of display (Reasoning & Justifying)

Background Information This topic provides many opportunities for students to collect information about a variety of areas of interest and can be readily linked with other key learning areas such as Social Studies and Science.

Language Column graphs that display categorical data (e.g. colours, favourite drinks) with no numeric ordering, use vertical or horizontal bars that are separated because these categories have no numeric ordering. However when the data that is collected is grouped along a continuous scale (e.g., ages, marks, heights, temperatures), then they should be ordered along a number line. Therefore, the column bars used touch each other (no gaps) and the column graph is called a histogram.

Keywords to be used: surveys, rows, columns, column graph, picture graphs, histograms, one-to-one correspondence, equal spaces, axes, grid paper, representation, symbol, horizontal, vertical

Resources grid paper, coloured blocks/counters

Links Whole Numbers Addition and Subtraction Social Studies Science

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES DA4.1 Conducts an investigation, collects and organises data into tables, uses picture graphs, column graphs and grid paper to display data and interprets the results

ACTIVITY DA4.1.1: ROLLING AND RECORDING Students choose a partner, and each student makes a chart like the one shown below. 2 3 4 5 6 7 8 9 10 11 12

Each student takes a turn to roll two dice, finds the sum of the two numbers rolled and places a tally mark in that column in his/her chart.

Students continue taking turns until one of them has 10 tally marks in one column. Questions to be answered by the pair include:

Why doesn’t the chart need a ones column? A thirteens column?

In which column did you or your partner reach 10?

Comparing your results, what are any similarities? Any differences? Explain.

Why would you expect more sums of 7 than 2? Illustrate your answer with examples.

Which sum would you expect to have the same number of sums as 4? Why?

Would you expect to get about the same number of even sums as odd sums? Explain why or why not.

Each pair is asked to display their individual results using a vertical column graph and for each student to find the most common sum and the least common sum.

Next the pair pooled their results in an appropriate table format before graphing the pooled results using a column graph of their choice. The students are again asked:

What is the most common sum? Least common sum?

Are they the same as individual findings? Why or why not?


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Extension. Students are asked to multiply the two numbers rolled instead of adding them. Questions to ask include:

How would the values along the top of the chart change?

How many values (that is, different products) would be needed?

Which values would be least likely?

Would you expect to get about the same number of even values as odd values? Why or why not.

Students draw a graphs to display their individual and pooled results. They should discuss any similarities and differences shown in relation to expected results for the different products.

ACTIVITY DA4.1.2: CONDUCT A SURVEY Students, in small groups of 3, are asked to design a question to be answered by a class survey. This could be anything to do with themselves (e.g., the type of hair accessories they are wearing, their birthdays, their ages, number of letters in their names or number of siblings they have, etc) or something from their village or home environment (e.g., type of buildings in the village, number of houses per family, number of family members etc).


What is an appropriate focus question for the survey?

What sort of data should we collect? How do we collect the data?

How do we record and organise the data collected using tables?

What is another different way of recording and organising the same data?

Which is the more appropriate table format? Why?

Can we organise our data as a list? Explain why or why not.

How are we going to display our data? Which graph is most suitable? Why?

What is another type of graph to display the same data?

What are the advantages and disadvantages of each of the two types of data display?

In your own words, what is the answer to the focus question of the survey based on the data collected?

Students are asked to use an A3 paper as a poster to record the results of their survey using the questions above as sections of the poster with their answers (e.g., findings, tables and displays) inserted after each question.

ACTIVITY DA4.1.3: AGES IN A CLASS Lisa’s class collected data from her class so that they could draw a graph. She suggested that they should find out the ages of all the children in their class. After a class discussion, they decided to collect years and months but not days. The results from the class survey are shown in the table below. Age of students in the class

Age Number

8 years 7 months 3

8 years 8 months 4

8 years 9 months 2

8 years 10 months 3

8 years 11 months 3

9 years 0 months 2

9 years 1 months 1

9 years 2 months 4

9 years 3 months 0

9 years 4 months 3

9 years 5 months 2

9 years 6 months 1

9 years 7 months 2

Students are asked to draw a histogram to

display Lisa’s class data. Possible questions include:

What do the two axes of the histogram represent? What are their labels?

What scale are you going to use for the axis representing the ages? What is the lowest value and what is the highest value?


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How many divisions on the age axis in order to accommodate the different age categories in the table?

What does one division of the age axis represent?

What scale is needed for the number for each age axis? What is the highest value and what is the lowest value? Explain why.

How many divisions for the number for each age axis in order to accommodate all the values in the table? What does one division represent?

What is an appropriate title for the histogram?

What age appeared more common in the class? Least common?

ASSESSMENT FOR LEARNING INDICATORS DA4.1 Conducts an investigation, collects and organises data into tables, uses picture graphs, column graphs and grid paper to display data and interprets the results The student, for example:

conducts surveys to collect data and organizes them in tables

constructs vertical and horizontal column graphs and picture graphs on grid paper using one-to-one correspondence

marks equal spaces on axes, labeling axes and naming the display

interprets information presented in column graphs and picture graphs

represents the same data in more than one way e.g., tables, column graphs, picture graphs

interprets information presented in different types of graphs


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MEASUREMENT Measurement enables the identification and quantification of attributes of objects so that they can be compared and ordered. All measurements are approximations; therefore opportunities to develop an understanding of approximation are important. Estimation skills are essential, particularly in situations where it is not convenient or necessary to use measuring devices. Accuracy in estimated measurement is obtained through extensive practice using a variety of units of measure and in a variety of contexts.

The Measurement strand for Year 1 to Year 6 is organised into five substrands that each focus on a particular attribute: Length, Area, Volume and Capacity, Mass and Time.

The development of each of these attributes progresses through several processes including identifying the attribute and making comparisons, using informal units, using formal units, and applying and generalising methods.

Identifying the attribute and comparison The first stage is recognising that objects have attributes that can be measured. Students begin by looking at, touching or directly comparing two or more objects in relation to a particular attribute. Through conversation and interpreting and/or posing questions students develop some of the language used to describe these attributes.

Informal units Students then continue to develop the key understandings of the measurement process using repeated informal units. Understandings include

the need for repeated units that do not change

the appropriateness of a selected unit

the need for the same unit to be used to compare two or more objects

the relationship between the size of the unit and the number required to measure, and

the structure of the repeated units (for length, area and volume).

Formal units Discussions and comparisons of measurement with informal units will lead to the realisation that there is need for a standard unit. Given the prevailing practice in Samoa, both metric and imperial units are included in the syllabus. Experiences with formal units should allow students to:

become familiar with the relative size of the unit

determine the degree of accuracy required

select and use the appropriate attribute and unit of measurement

select and use the appropriate measuring device

record and recognise the abbreviations, and

convert between units.

Applications and generalisations Finally students apply this knowledge in a variety of contexts and begin to generalise their methods to calculate perimeters, areas and volumes.



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MS4.1 Length MS4.1 Substrand: Length Uses formal units to estimate, measure, order, compare and record lengths, distances and perimeters and converts between units within each measurement system (between metres, centimetres and millimetres & between yards, feet and inches)

Key Ideas Use formal units: metres, centimetres and millimetres & yards, feet and inches to estimate, measure, compare, order and record lengths and distances Carry out simple unit conversion within each measurement system (e.g., between metres, centimetres and millimetres & between yards, feet and inches) Develop ‘real-life’ benchmarks for a length of one millimetre and one inch Estimate using benchmarks and measure using formal units the perimeter of two-dimensional shapes Use decimal notation to two places to record lengths and distances

Attitudes & Values Logical and critical thinking Communication Consistency Work things out Investigation Accuracy Verification Proof and reflection

Knowledge and Skills Students learn about 1. recognising that ten millimetres equal one

centimetre and describing one millimetre as one tenth of a centimetre

2. recognising that 12 inches equal one foot and describing one inch as one twelfth of a foot

3. using the abbreviations for millimetre (mm) and inch (in)

4. recording lengths or distances using centimetres and millimetres or feet and inches e.g., 5 cm 3 mm or 2 ft 6 in

5. converting between metres and centimetres, and centimetres and millimetres

6. converting between yard and feet, and feet and inches

7. recording lengths or distances using decimal notation to two decimal places e.g., 1.25 m, 2.50 yd

8. recognising the features of an object associated with length, that can be measured e.g., length, breadth, height, perimeter

9. using the term ‘perimeter’ to describe the total distance around a shape

10. estimating and measuring the perimeter of 2D shapes

11. using a tape measure, ruler or trundle wheel to measure lengths or distances

Working Mathematically Students learn to a. question and explain why two students may

obtain different measures for the same length, distance or perimeter (Interpreting &/or Posing Questions, Communicating Mathematically, Reasoning & Justifying)

b. explain the relationship between the size of a unit and the number of units needed e.g., more centimetres than metres will be needed to measure the same length (Communicating Mathematically, Reflecting & Evaluating)


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Background Information At Years 3 and 4, measurement experiences enable students to: - develop an understanding of the size of the

metric units (metre, centimetre and millimetre) and imperial units (yard, feet and inches) for length and distance

- estimate and measure using these units, and - select the appropriate unit and measuring

device.

A trundle wheel is a simple device for measuring distances that are too long for a tape measure, or that aren't relatively straight. The trundle wheel is composed of a wheel, a handle which is attached to the axle allowing the trundle wheel to be held easily, and a clicking device which is triggered once per revolution of the wheel. Trundle wheels are not as accurate as other methods of measuring distance but are a good way to get a rough estimation of a fairly long distance over a good surface. It works by having a wheel which has a circumference of exactly 1 metre, hence one revolution of the wheel equates to 1 metre of distance traveled on the ground if there is no slip.

Language Perimeter’ comes from the Greek words that mean to measure around the outside

Keywords to be used: centimeter one hundredth, metre, yard, one-third, foot, feet, millimetre, inches, metric units imperial units, perimeter, total distance, centimetre, circumference, estimating, measuring, two-dimensional, decimal point, around, outside, ruler, equal lengths, longer, shorter, standard, formal unit

Resources strings, sinnet (afa), butcher paper, paper strips, metre ruler, foot ruler with inch markings, yard ruler, metre ruler, yard ruler, tape, toothpicks, environmental materials, cards, grid paper, string, plasticine, paper, streamers, book cover, art paper, leaves, grid paper, dot paper

Links Whole Numbers Fractions and Decimals Addition and Subtraction Two-dimensional Space Area

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES MS4.1 Uses formal units to estimate, measure, order, compare and record lengths, distances and perimeters and converts between units within each measurement system (between metres, centimetres and millimetres & between yards, feet and inches)

ACTIVITY MS4.1: WHEN IS A FOOT A FOOT? Students are asked to find lengths on their body to serve as approximations of 1 inch and 1 foot. Students are given 1 foot rulers with inch markings.

Students are told that the second knuckle of their index finger is a good approximation for an inch.

The activity requires that students use their two body measurements to find the width of their desk (inch unit) and the length of the blackboard (foot unit).

Students record their data in a table and then using their 1 foot ruler, they find a more accurate measure of the width of their desk and length of blackboard.

ACTIVITY MS4.2: INVESTIGATING PERIMETERS Students use geoboards to investigate perimeters of shapes.


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They use shapes that have square corners. Students construct shapes that have perimeters of 4 units, 6 units, 8 units, etc. They record the shapes on dot or square paper. Students try to make different shapes that have the same perimeters.


Is it possible to make shapes which have a perimeter of 3 units? 5 units? 7 units? etc. Explain how or why not.

Students use the geoboard to make a shape which has:

the smallest perimeter

the largest perimeter.

ACTIVITY MS4.3: PERIMETER MATCH In pairs, students are given a length (e.g., 16 cm or 2 feet) and are required to construct a two-dimensional shape on a card with this perimeter. The teacher collects, shuffles and re-allocates cards to each pair. Students estimate and then measure the perimeter of their allocated shape. They then find their partner and compare and contrast their shapes.

ACTIVITY MS4.4: PERIMETERS Students estimate and then measure, to the nearest centimetre and nearest foot, the perimeters of small items such as book covers, art paper, leaves. Students record the results using words and drawings and discuss the differences between their estimates and actual measurements.

Extension. Ask students to investigate different ways to find the perimeter of their classroom door.

ACTIVITY MS4.5: CHANGING UNITS. Students are asked to measure a length with a specified unit. Then they are provided with a new unit that is twice as long or half as long as the original unit.

Their task is to predict the measure of the same length using the new unit.

Students should write down their estimations and explanations of how they

were made before they actually measure the length.


What is the length using the first unit?

What is the length using the new unit?

Are the two measurements the same? Why or why not?

Variation. Students repeat the activity but they start off measuring the same length using a 1 yard string (or 1 metre string) before they are given a 1 foot ruler (or 10 cm ruler).

ACTIVITY MS4.6: UNIT HUNT Students, in pairs, are asked to take a 1 yard ruler, stick or string and find and make a list of 5 things in the room that are shorter than, longer than, or about the same length as their target unit.

Students repeat the activity using another standard unit such as a metre stick, ruler or string.

ACTIVITY MS4.7: CROOKED PATHS Make some crooked or curvy paths on the floor (or outside) with chalk. Students are asked to determine which path is longest, next longest, and so on.


How do we measure these curvy paths so that we can compare them easily?

What device can we use?

What units do we use (metric or imperial units)?

Which is the longest path? Shortest path?

What is the ascending order of path lengths?

Students discuss their answers and provide justifications for them.

If the teacher wishes to offer a hint, provide pairs of students with a long piece of rope. The task is easier if the rope is longer than the crooked path.

Students are asked to explain in writing how they solved the problem with path lengths in ascending order and using the appropriate units of length.

Variation. Students repeat the measurement of the paths using a different system of units from the one used before.


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ASSESSMENT FOR LEARNING INDICATORS MS4.1 Uses formal units to estimate, measure, order, compare and record lengths, distances and perimeters and converts between units within each measurement system (between metres, centimetres and millimetres & between yards, feet and inches) The student, for example:

recognises that ten millimetres equal one centimetre and describes one millimetre as one tenth of a centimetre, and that 12 inches equal one foot and describing one inch as one twelfth of a foot

uses the abbreviations for millimetre (mm) and inch (in)

records lengths or distances using centimetres and millimetres or feet and inches

converts between metres and centimetres, and centimetres and millimetres

converts between yard and feet, and feet and inches

records lengths or distances using decimal notation to two decimal places

recognises the features of an object associated with length, that can be measured

uses the term ‘perimeter’ to describe the total distance around a shape

estimates and measures the perimeter of 2D shapes

uses a tape measure, ruler or trundle wheel to measure lengths or distances


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MS4.2 Area MS4.2 Substrand: Area Understands the need for, and uses, larger formal units (square metres and square feet) to estimate, measure, compare and record the areas of surfaces and converts between units within each system


Understand the need for, and use, larger formal units: square metres and square feet to measure area Convert between area units within each measurement system Use square metres and square feet to estimate, measure, compare and record areas Explore what happens to perimeters and areas of rectangles when the shape is changed in some ways



1. recognising the need for a unit larger than a square centimetre and square inch

2. constructing a square metre and a square yard

3. estimating, measuring and comparing areas in square metres and square yards

4. recording area in square metres and square yards e.g., 5 square metres, 3 square yards

5. using the abbreviations for square metre (m2) and square yard (yd2)

Working Mathematically Students learn to a. question why two students may obtain

different measurements for the same area (Interpreting &/or Posing Questions)

b. discuss and compare areas using some mathematical terms (Communicating Mathematically)

c. discuss strategies used to estimate area in square centimetres or square metres and square inch or square foot e.g., visualising repeated units (Communicating Mathematically, Reflecting & Evaluating)

d. apply strategies for measuring the areas of a variety of shapes (Strategically Thinking & Representing)

e. explain where square metres or square feet are used for measuring in everyday situations e.g., floor coverings (Communicating Mathematically, Reflecting & Evaluating)

f. recognise areas that are ‘smaller than’, ‘about the same as’ and ‘bigger than’ a square metre or square foot (Strategically Thinking & Representing)

Background Information At Years 3 and 4, students should appreciate that a formal unit allows for easier and more accurate communication of area measures. Measurement experiences should enable students to develop an understanding of the size of units, select the appropriate unit, and estimate and measure using the unit.

An important understanding at this Year level is that an area of one square metre (or square feet) need not be a square. It could, for example, be a rectangle, two metres (feet) long and half a metre (foot) wide.


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Language The abbreviation m2 is read ‘square metre(s)’ and not ‘metre squared’ or ‘metre square’. The abbreviation ft2 is read ‘square feet’ and not ‘feet squared’ or ‘feet square’.

Keywords to be used: square metre, square centimetre, square inch, square feet, grid overlay, overlap, same as, more than, less than, smaller than, about the same, bigger than, grid, less than, more than, square tile

Resources newspaper, scissors, square centimetre tiles, geoboards, one metre ruler, tiles, elastic bands, 1

cm grid paper, grid overlays, 10 cm 10 cm grids,

1 cm 1 ft grids, tennis balls or stones or small boxes, paint

Links Multiplication and Division Fractions and Decimals Length


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LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES MS4.2 Understands the need for, and uses, larger formal units (square metres and square feet) to estimate, measure, compare and record the areas of surfaces and converts between units within each system Activity MS4.2.1: Constructing a Square Metre In groups, students make a one square metre model out of newspaper sheets taped together. Students then discuss different shapes that could be created by cutting and rearranging the pieces.

Students display the different shapes formed and label their areas ‘One square metre’. Students examine the shapes.


How can you fit the most people into a square metre?

Does an area of one square metre need to be shaped like a square? Why?

What did you notice about the area of the newspaper when it was changed to a rectangular shape?

Can you name some other dimensions for a square metre?

When you measured the area of your square, did you get the same answer as the person next to you? Why? Why not?

Variation. Students are asked to construct a one square foot model and to repeat the

activity. They answer the same questions

but replace square metre with square feet.

ACTIVITY MS4.2.2: CM2 AND M2

The teacher provides students with a collection of materials of various sizes. In pairs, students select the appropriate unit (cm2 and m2) and estimate the area of each item. Students check their estimates by measuring areas using square centimetre tiles/grids or square metre templates. Students then record their results in a table.

Item cm2 or m2

Estimate Measurement


How did you decide when to use cm2?

What strategy did you use to estimate the areas?

Were your estimates close to the actual measurements?

What device did you select to measure? Why?

Could you estimate, measure and record the area of six different surfaces or shapes?

Can you compare the measurements of each shape or surface?

ACTIVITY MS4.2.3: IN2 AND FT2 The teacher provides students with a collection of materials of various sizes. In pairs, students select the appropriate unit (in2 and ft2) and estimate the area of each item. Students check their estimates by measuring areas using square inch tiles/grids or square feet templates. Students then record their results in a table.

Item in2 or ft2

Estimate Measurement


How did you decide when to use in2?

What strategy did you use to estimate the areas?

Were your estimates close to the actual measurements?

What device did you select to measure? Why?

Could you estimate, measure and record the area of six different surfaces or shapes?

Can you compare the measurements of each shape or surface?


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ACTIVITY MS4.2.4: ESTIMATION Students estimate, and then use square metre (or square foot) templates, to measure a variety of floor areas.


How can you measure an area that is not quite a square metre (or square foot)?

Did you have any problems with overlapping?

What did you do about it?

What do you recommend should be done to get a more accurate measure? Explain.

Students record their strategies in writing and results in a table showing estimates and actual measurements using the templates.

ACTIVITY MS4.2.5: DIRECT COMPARISON Students find things that are:

smaller than a square metre

equal to (or almost equal to) a square metre

larger than a square metre. Students record their findings in a table using the abbreviation m2 e.g.,

Less than 1 m2

About 1 m2 More than 1 m2

Variation. Students repeat the activity but using a square foot template. They can sort the same objects above.

ACTIVITY MS4.2.6: PLAY AREA Students work in small groups to estimate the area of the school’s play area in square metres. Students use a square metre template to measure the area.


What strategy did you use to estimate the area?

What strategy did you use to measure the area?

Would it be easier to measure the area in square centimetres? Why?

How many classrooms would we need to put together to make a similar area?

Variation: Students find the area of their classroom or other large areas in the school.

ACTIVITY MS4.2.7: COVERING A SQUARE METRE The teacher poses the problem ‘How many students do you think will fit onto a square metre?’ Students record results for standing, sitting, lying down, etc. Students repeat the activity using a square metre in different shapes.


Were the results the same for the different shapes?

Why might there be a variety of results? Students repeat the investigation with

students from different classes and compare results.

Variation: Students repeat the activity but using the imperial unit square yard and answer the same questions.


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ASSESSMENT FOR LEARNING INDICATORS MS4.2 Understands the need for, and uses, larger formal units (square metres and square feet) to estimate, measure, compare and record the areas of surfaces and converts between units within each system The student, for example:

recognises the need for a unit larger than a square centimetre and square inch

constructs a square metre and a square foot

estimates, measures and compares areas in square metres and square feet

records area in square metres and square feet e.g., 5 square metres, 3 square feet

uses the abbreviations for square metre (m2) and square feet (feet2)

MS4.3 Volume & Capacity MS4.3 Substrand: Volume & Capacity Use smaller formal units (milliliters and pints and fluid ounces) to estimate, measure, compare and record volumes and capacities and convert between units within each system.


Understand the need for, and use, smaller formal units: millilitres and pints to measure capacity and volume Use formal units to estimate, measure and compare capacity and volume Compare, estimate and measure volume of objects using rise in water level or overflows Convert measurements from one unit to another within each measurement system Convert between pints, fluid ounces, millilitres and cubic inches Use formal units to record measurements using decimal notation to one decimal place



1. recognising the need for a unit smaller than the litre or quart

2. estimating, measuring and comparing volumes and capacities using millilitres and pints and fluid ounces

3. making a measuring device calibrated in multiples of 100 millilitres or 1 pint (or multiples of 1 fluid ounce [oz])

4. using a measuring device calibrated in millilitres e.g., medicine glass, measuring cylinder

5. using a measuring device calibrated in pints (or fluid ounces) e.g., baby milk bottles

6. using the abbreviations for millilitre (mL), pint (pt) and fluid ounce (oz)

7. recognising that 1000 millilitres equal one litre and 2 pints equal 1 quart and 1 quart

Working Mathematically Students learn to a. explain the need for a standard unit to

measure the volume of liquids and the capacity of containers (Communicating Mathematically)

b. estimate and measure quantities to the nearest 100 mL and/or to the nearest 10 mL or nearest 20 ounces (Strategically Thinking & Representing)

c. interpret information about capacity and volume on commercial packaging (Communicating Mathematically, Reflecting & Evaluating)

d. estimate the volume of a substance in a partially filled container from the information on the label detailing the contents of the container (Strategically Thinking & Representing)

e. relate the millilitre to familiar everyday


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equals 20 fluid ounces. 8. converting between millilitres and litres

e.g., 1250 mL = 1 litre 250 millilitres 9. converting between pints and quarts and

ounces e.g., 7 pints = 3 quarts 1 pint = 140 fluid ounces

10. comparing the volumes of two or more objects by marking the change in water level when each is submerged in a container

11. measuring the overflow in millilitres (or ounces) when different objects are submerged in a container filled to the brim with water

containers and familiar informal units e.g., 1 teaspoon is approximately 5 mL, 250 mL fruit juice containers (Reflecting & Evaluating)

f. estimate the change in water level expected when an object is submerged (Strategically Thinking & Representing)

Background Information At Year 4, students should appreciate that a smaller formal unit than a litre or a quart allows for easier and more accurate communication of measures and are introduced to the millilitre and pint and ounce. Measurement experiences should enable students to develop an understanding of the size of the unit, estimate and measure using the unit, and select the appropriate unit and measuring device. The displacement strategy for finding the volume of an object relies on the fact that an object displaces its own volume when it is totally submerged in a liquid. The strategy may be applied in two ways: - using a partially filled, calibrated, clear

container and noting the change in the level of the liquid when the object is submerged, or

- submerging an object into a container filled to the brim with liquid and measuring the overflow.

The following conversions are provided to assist with converting between measurement systems and within system where appropriate. 2 pints = 1 quart; 4 quarts = 1 gallon; 8 pints = 1 gallon; 1 pint = 20 fluid ounce = 568.26 millilitres. 1 mL = 1 cm3 = 0.061 in3 1 oz = 28.4 mL = 1.73 in3

Language The abbreviation cm3 is read ‘cubic centimetre(s)’ and not ‘centimetre cubed’. Similarly, in3 is read ‘cubic inch(es)’ and not ‘inch cubed’ Examples of students’ explanations: ‘The overflow is about 10 millilitres therefore the volume of the object is about 10 cubic centimetres.’ ‘The first object has a larger volume than the second one because the water level rose much higher for the first object than for the second object.’

Keywords to be used: pints, millilitres, fluid ounces, overflows, water level, displacement, decimal place, decimal, calibrated, medicine cylinder, milk bottle, submerged, partially filled


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Resources measuring cups, medicine glass, baby bottle, empty spring water bottles of different capacities,

Links Multiplication and Division Addition and Subtraction Data

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES MS4.3 Use smaller formal units (milliliters and pints and fluid ounces) to estimate, measure, compare and record volumes and capacities and convert between units within each system.

ACTIVITY MS4.3.1: DISPLACEMENT Students are provided with a variety of materials to place in water, small identical cups to collect the overflow, and a measuring cylinder (or a calibrated tall thin glass container).

Part A Students stand a large container in a tray and fill it to the brim. Students predict what will happen when an object is placed in the container. They collect the overflow and pour it into a cup. They repeat the activity using different materials, each time collecting the overflow in separate cups. Students compare the cups and form initial conclusions. Students then use the measuring cylinder (or calibrated clear container) to obtain more accurate measurements (in milliliters or pints or fluid ounces) for the overflows.


What is the volume of the overflow? How do you know?

What units of volume are you using?

What is the volume using a different unit?

Students record their measurements using appropriate units and form conclusions about the order of volumes of objects.

Part B Students place 10 large interlocking cubes or blocks individually into a clear container filled to the brim with water, and collect the overflow. Students use the measuring cylinder to measure the capacity of each overflow.

Students then make a model using the 10

cubes or blocks and repeat the activity. Possible questions include:

Do you get the same result when you put the cubes in individually? Justify your answer using actual measurements.

How much water was displaced each time? Justify using actual measurements.

Students record their findings using drawings, words and actual measurements.

ACTIVITY MS4.3.2: MAKE YOUR OWN MEASURE Students make their own measuring cylinder using a tall thin clear container or bottle. Materials needed include a large glass, spoon, small container or small 5-mL medicine cup, masking tape, water, and felt-tip pen.

Students put a piece of tape on the side of the clear container. They fill the small container/medicine cup with spoonfuls. Count how many it takes. Empty the small container into the clear container. Mark the level of water and the number of spoonfuls on one side of the mark and the number of mLs on the other (similar to the photo below of milk bottle using two types of units).

Students continue filling the small container and emptying into the clear container; each time the number of spoonfuls and mLs are noted alongside each mark. This continues until the water level reaches the top.


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Students use this calibrated clear container to measure how much other containers hold.

ACTIVITY MS4.3.3: DISPLACEMENT Students are provided with a variety of materials to place in water, and small identical cups to collect the overflow.

Students stand a large container in a tray and fill it to the brim. Students collect the overflow and determine how they would measure the overflow amount in terms of the appropriate:

device to use and

units. They repeat the activity using different

materials, each time collecting and measuring the amount of overflow using their selected device and units.

Students record the volumes of all objects using their selected units before ordering the objects from largest to lowest volume.

Variation. Students brainstorm strategies of measuring volume of much bigger irregular objects before assigning each pair of students to measure volume of two irregular solid objects.

Students record their strategies and measured volume of the two objects. By presenting the results to the whole class, students discuss and evaluate each others’ strategy and compare the accuracy of their measurements.

Extension. Students express the volumes of the objects in terms of a different unit within the same measurement system.

ASSESSMENT FOR LEARNING INDICATORS MS4.3 Use smaller formal units (milliliters and pints and fluid ounces) to estimate, measure, compare and record volumes and capacities and convert between units within each system. The student, for example:

recognises the need for a unit smaller than the litre or quart

estimates, measures and compares volumes and capacities using millilitres and pints

makes a measuring device calibrated in multiples of 100 millilitres or 1 pint (or multiples of 1 fluid ounce [oz])

uses a measuring device calibrated in millilitres

uses a measuring device calibrated in pints (fluid ounces) e.g., baby milk bottles

uses the abbreviations for millilitre (mL), pint (pt) and fluid ounce (oz)

recognises that 1000 millilitres equal one litre and 2 pints equal 1 quart and 1 quart equals 20 fluid ounces.

converts between millilitres and litres

converts between pints and quarts and ounces

compares the volumes of two or more objects by marking the change in water level when each is submerged in a container measures the overflow in millilitres (or ounces) when different objects are submerged in a container filled to the brim with water


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MS4.4 Mass MS4.4 Substrand: Mass Estimates, measures, compares and records masses using grams and ounces


Understand the need for, and use, smaller formal units (grams and ounces) to measure mass Use formal units to estimate, measure and compare masses Use formal units to record mass using decimal notation to one decimal place



1. recognising the need for a unit smaller than the kilogram or pound

2. measuring and comparing the masses of objects in kilograms and grams or pounds and ounces using a set of scales

3. using the abbreviations for grams (g) and ounces (oz)

4. recognising that 1000 grams equal one kilogram and 16 ounces equal one pound

5. interpreting commonly used fractions of a kilogram including

1

2, 14, 34

and relating these to

the number of grams 6. interpreting commonly used fractions of a pound

including

1

2, 14, 34

and relating these to the number

of ounces


a. recognise that objects with a mass of one gram or a mass of 1 ounce can be a variety of shapes and sizes (Reflecting & Evaluating)

b. interpret statements, and discuss the use of grams or ounces on commercial packaging (Communicating Mathematically)

c. discuss strategies used to estimate mass e.g., by referring to a known mass (Communicating Mathematically)

d. question and explain why two students may obtain different measures for the same mass (Interpreting &/or Posing Questions, Communicating Mathematically)

e. solve problems including those involving commonly used fractions of a kilogram or pound (Strategically Thinking & Representing)

Background Information There are 16 ounces in one pound and 1000 grams in one kilogram.

Language Students should experience and develop facility and fluency in describing weights of an object using two types of units within one measurement system and across systems. For example, an object that weighs 2 kilograms in the metric system would weigh about 4.4 pounds.

Keywords to be used kilograms, grams, pounds, ounces, mass, equal arm balance, weight, hefting, balanced, level balance, greater than, less than, more than, differences


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Resources Standard weights, equal arm balance, a variety of objects

Links Whole Numbers Addition and Subtraction Multiplication and Division

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES MS4.4 Estimates, measures, compares and records masses using grams and ounces

ACTIVITY MS4.4.1: WEIGHT OF SAMOAN COINS Students are asked to work in groups of 4 to determine the weight of each of the coins; 5 sene, 10 sene, 20 sene, and one tala using an equal arm balance and standard weights.

Students record their findings in a table. Extension. Students use their coins and

respective known weights to determine weights of a number of objects. For example, an eraser weighs the same as two $1 tala coins plus one 20 sene; therefore the weight of my rubber is the same as the sum of the weight of two $1 tala coins and weight of one 20 sene.

Students record their findings in a table, using drawings and actual weights.

ACTIVITY MS4.4.2: WEIGHING WITH GRAMS OR OUNCES Students use an equal arm balance and standard weights to measure the weights of a variety of objects, for example, a banana, a mango, duster, chalk or a handful of cocoa beans.


What is the weight of a banana in grams? In ounces?

What is the weight of a duster in grams? In kilograms?

If you know the weight of one duster, what is the weight of 4 dusters?

Approximately how many bananas would have a total weight close to one pound? How do you know?

Approximately how many bananas would have a total weight close to one kilogram? How do you know?

Students record their findings in a table and their working out to justify their answers.

Variation. Students are asked to pose their

own questions and to provide their own solutions.

ASSESSMENT FOR LEARNING INDICATORS MS4.4 Estimates, measures, compares and records masses using grams and ounces The student, for example:

recognises the need for a unit smaller than the kilogram or pound

measures and comparing the masses of objects in kilograms and grams or pounds and ounces using a set of scales

uses the abbreviations for grams (g) and ounces (oz)

recognises that 1000 grams equal one kilogram and 16 ounces equal one pound

interprets commonly used fractions of a kilogram including

1

2, 14, 34

and relating

these to the number of grams

interprets commonly used fractions of a pound including

1

2, 14, 34

and relating

these to the number of ounces


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MS4.5 Time MS4.5 Substrand: Time Reads digital and analogue clocks to the minute, records time using the correct notation, understands equivalent representations of time, and makes comparisons between time units


Understand, recognise, read and record time in one-minute intervals Make comparisons between time units Use digital and analog notations to read and record time Convert between units of time Tell time to the minute on digital and analog clocks Read and interpret simple timetables, timelines and calendars of real-life situations



1. recognising the coordinated movements of the hands on an analog clock, including

(i) how many minutes it takes for the minute hand to move from one numeral to the next

(ii) how many minutes it takes for the minute hand to complete one revolution

(iii) how many minutes it takes for the hour hand to move from one numeral to the next

(iv) how many minutes it takes for the minute hand to move from the twelve to any other numeral

(v) how many seconds it takes for the second hand to complete one revolution

2. relating analog notation to digital notation e.g., ten to nine is the same as 8:50

3. converting between units of time e.g., 60 seconds = 1 minute 60 minutes = 1 hour 24 hours = 1 day

4. solving problems and interpreting simple timetables, timelines and calendars


a. recall time facts e.g., 24 hours in a day (Communicating Mathematically, Strategically Thinking & Representing)

b. discuss time using appropriate language (Communicating Mathematically)

c. solve a variety of problems using problem-solving strategies, including:

trial and error drawing a diagram working backwards looking for patterns using a table


d. record in words various times as shown on analog and digital clocks (Communicating Mathematically)

e. compare and discuss the relationship between time units eg an hour is a longer time than a minute (Communicating Mathematically, Reflecting & Evaluating)

Background Information At Year 4, ‘telling time’ focuses on reading time to the minute on both analog and digital clocks. Students need to be aware that there are three ways of expressing the time. Note: When writing digital time, two dots should separate hours and minutes e.g., 4:24.


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Language Students practice describing telling time and measuring duration of events using to the minute time including twenty to and twenty past, five to and five past, and ten to and ten past.

Keywords to be used analog, digital, seconds, minutes, hours, days, weeks, year, time, clock, timetable, timeline, calendar, relationship, five to, five past, half past, ten to, ten after, twenty to, twenty past, clockwise, revolution, minute hand, hour hand, revolution, second hand, intervals

Resources shipping guides (newspapers), time cards (e.g., three to nine, 8:57), timelines, calendars, analog clocks, digital clocks, sets of numeral cards, cardboard, split pins stopwatch or cell phones

Links Whole Numbers Addition and Subtraction Multiplication and Division

LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES MS4.5 Reads digital and analogue clocks to the minute, records time using the correct notation, understands equivalent representations of time, and makes comparisons between time units

ACTIVITY MS4.5.1: HOW MANY DAYS? The teacher poses the problem ‘How many days have you attended school this term/year?’ Students calculate a solution.

Students are asked ‘How many other ways can you express this information?’ e.g., in hours, in minutes. Students use a calculator to check their answers.

This activity could be extended by asking ‘How many hours have you spent at recess and lunch this week?’ Students could record information in days, hours or minutes on a spreadsheet or table and then draw a graph.

ACTIVITY MS4.5.2: MATCHING TIMES The teacher provides students with sets of matching time cards in both analogue and digital notation (e.g., forty past six, 6:40). In small groups, students jumble the cards and place them face down. In turns, students turn two cards over. If the cards match, the student keeps them. The winner is the student with the most cards. Following the game, students record times in other ways and make additional cards for the

game. Students then repeat the game with the additional cards. Possible questions include:

Can you read the time on each card?

Can you record the time on each card in another way?

Can you explain the relationship between the time units?

ACTIVITY MS4.5.3: BARRIER GAME Students form pairs. Student A is provided with a series of digital times recorded on cards. Student B is provided with an analog clock. Student A selects a card and explains to Student B where to position the hands on their clock to make a matching time. Student B records the time they have made both in analog notation and in digital notation e.g., twenty to eleven and 10:40. Student A checks the digital time with their card. Students swap roles and repeat the game.

ACTIVITY MS4.5.3: THE MINUTE AND HOUR HANDS Students observe and discuss the position of the hour hand at twenty after, twenty to and ten to and ten after. Students construct an analog clock with an hour hand only. In pairs, students position the hour hand anywhere on their clock and swap clocks with their partner. Students are then asked to identify the time represented on their partner’s clock and give reasons.

Students are asked to display and name as many different times as possible using the minute and hour hands.


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ACTIVITY MS4.5.4: TIME BINGO Students are given a page of blank analog

clocks. They record their own times (to the hour, half-hour and quarter-hour including ten after, ten to, twenty to and twenty after, five to and five after and any to the minute time) on the clocks. The teacher calls out various times (using the hour, half-hour and quarter-hour including ten after, ten to, twenty to and twenty after, five to and five after and any to the minute time). A counter is placed on a clock with the matching time.

When all clocks are covered the student calls out ‘Bingo.’

Extension: Students are given a page with both analogue and digital clocks. They record various times in both forms. The teacher calls out a time e.g., a twenty past 12. Students place a counter on the corresponding time, analogue or digital i.e. a ten past 12 or 12:10. When all the times are covered the student calls out ‘Bingo’.

ACTIVITY MS4.5.5: TELEVISION VIEWING Students collect a variety of television guides from different sources e.g., magazines, newspapers. Students identify and discuss common features. Students then plan an evening of television viewing and record their plan in a table as shown below. Students use a simple timetable.


Can you convert the digital times to analog times?

What information can you interpret from a timetable?

Program Channel Program begins

Program finishes

Students use the information in the table to draw a timeline.

ASSESSMENT FOR LEARNING INDICATORS MS4.5 Reads digital and analogue clocks to the minute, records time using the correct notation, understands equivalent representations of time, and makes comparisons between time units The student, for example:

recognises the coordinated movements of the hands on an analog clock, including o how many minutes it takes for the

minute hand to move from one numeral to the next

o how many minutes it takes for the minute hand to complete one revolution

o how many minutes it takes for the hour hand to move from one numeral to the next

o how many minutes it takes for the minute hand to move from the twelve to any other numeral

o how many seconds it takes for the second hand to complete one revolution

relates analog notation to digital notation e.g., ten after seven is the same as 7:10

converts between units of time, seconds, minutes, hours and days

solving problems and interpreting simple timetables, timelines and calendars


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SPACE AND GEOMETRY Space and Geometry is the study of spatial forms. It involves representation of shape, size, pattern, position and movement of objects in the three-dimensional (3D) world, or in the mind of the learner.

The Space and Geometry strand for Year 1 to Year 6 is organised into three substrands: Three-dimensional space, Two-dimensional space and Position.

The Space and Geometry strand enables the investigation of 3D objects and two-dimensional (2D) shapes as well as the concepts of position, location and movement. Important and critical skills for students to acquire are those of recognising, visualising and drawing shapes and describing the features and properties of 3D objects and 2D shapes in static and dynamic situations. Features are generally observable whereas properties require mathematical knowledge e.g., ‘a rectangle has four sides’ is a feature and ‘a rectangle has opposite sides of equal length’ is a property. Manipulation of

a variety of real objects and shapes is crucial to the development of appropriate levels of imagery, language and representation.

When classifying quadrilaterals, teachers need to be aware of the inclusivity of the classification system. That is, trapeziums are inclusive of the parallelograms, which are inclusive of the rectangles and rhombuses, which are inclusive of the squares. These relationships are presented in the following Venn diagram, which is included here as background information.

For example, a rectangle is a special type of parallelogram. It is a parallelogram that contains a right angle. A rectangle may also be considered to be a trapezium that has both pairs of opposite sides parallel and equal.


SG4.1 Three Dimensional Space SG4.1 Substrand: Three Dimensional Space Compares, explains and models 3D objects including cylinders, cones and spheres; constructs nets from everyday packages; and describes cross-sections of 3D objects


Name, explain, classify, model and draw prisms, cylinders, cones and spheres showing depth Examine and construct nets from everyday packages Explore, identify and describe cross-sections of 3D objects


Quadrilaterals

Trapeziums

Parallelograms

Rectangles

Rhombuses

Squares


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1. comparing and describing features of prisms, pyramids, cylinders, cones and spheres

2. identifying and naming 3D objects as prisms, pyramids, cylinders, cones and spheres

3. recognising similarities and differences between prisms, pyramids, cylinders, cones and spheres

4. identifying 3D objects in the environment and from drawings, photographs or descriptions

5. making models of cylinders, cones and spheres given a 3D object, picture or photograph to view

6. sketching cylinders and cones, attempting to show depth

7. creating nets from everyday packages e.g., a cereal box

8. sketching 3D objects from different views including top, front and side views

9. making and visualising the resulting cut face (plane section) when a 3D object receives a straight cut


describe 3D objects using everyday language and mathematical terminology (Communicating Mathematically)

recognise and describe the use of 3D objects in a variety of contexts e.g., buildings, packaging (Reflecting & Evaluating, Communicating Mathematically)

compare features of 3D objects and two-dimensional shapes (Strategically Thinking & Representing, Reflecting & Evaluating)

compare own drawings of 3D objects with other drawings and photographs of 3D objects (Reflecting & Evaluating)

explore, make and describe the variety of nets that can be used to create particular 3D objects (Strategically Thinking & Representing, Reasoning & Justifying, Communicating Mathematically)

draw 3D objects (hand-drawn or using a computer drawing package), attempting to show depth (Strategically Thinking & Representing)

Background Information The formal names for particular prisms and pyramids are not introduced at this Year Level. Prisms and pyramids are to be treated as classes to group all prisms and all pyramids. Names for particular prisms or pyramids are introduced in Years 5 and 6.

In Geometry a 3D object is called a solid. The 3D object may in fact be hollow but it is still defined as a geometrical solid. Models at this Year Level should include skeletal models.

Language Examples of students’ explanations: ‘Cylinders and prisms have two bases that are parallel to each other. One difference is in the shape of the bases. Cylinders have circular bases while prisms have bases that are polygons.’ ‘Cones and pyramids have one base and one vertex.’ ‘Faces of a pyramid from the base to the vertex are always triangular in shape.’

Keywords to be used: prisms, pyramids, uniform cross-section, polygon, rectangle, triangle, perpendicular, base, vertex, section, solid, face, cylinders, cones, spheres

Resources cubes, prisms, pyramids, cylinders, cones, spheres

Links Two Dimensional Shapes Visual Arts


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LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES SG4.1 Compares, explains and models 3D objects including cylinders, cones and spheres; constructs nets from everyday packages; and describes cross-sections of 3D objects

ACTIVITY SG4.1.1: CAN YOU FIND AN OBJECT LIKE ME?

Part A Put out a pyramid, a prism, a cylinder, a cone and a sphere and see if students can find real objects that have the same shape. You may get disagreement about which are alike.


Is this real object similar to this solid? Justify your answer.

If the real object is not similar to the solid, what is different about it? What is this different shape called?

Part B Provide students with photos or pictures and ask them to identify pyramids, prisms, cylinders, cones and spheres. Students make models of the solids identified, sort and record their classifications into groups by hand drawing the solids identified.

ACTIVITY SG4.1.2: WHO DOES NOT BELONG? Given the picture of solids below, students are asked to determine which solids are prisms, pyramids, cylinders, cones and spheres and which do not belong to any group. Since there are many ways to solve this problem, be ready to encourage lively discussion.


Which solids are cylinders? Justify your answer.

Which solids are cones? Justify your answer.

Which solids are spheres? Justify your answer.

Which solids are pyramids? Justify your answer.

Which solids are prisms? Justify your answer.

Are there other possible answers? Why?

What are similarities, if any, between the solids?

What are differences between the solids?

ACTIVITY SG4.1.3: A WALK AROUND THE SCHOOL COMPOUND Students are asked to take a walk around the school compound and identify 3D shapes they see along the way, for example, cylinders, prisms, pyramids, spheres, and cones as well as shapes that may not have geometric names.

Students draw each shape, label it with its name and describe its shape. Students display their drawings in the classroom.

ACTIVITY SG4.1.4: VIEWS OF SOLIDS Students are given the following pictures of solids to explore.

(i) Which of the following are not views of the

solid above?

(ii) Which of the following drawings could be a view of one of the solids A, B, C, D and E?


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A B C D E Drawings:

ACTIVITY SG4.1.5: NETS OF SOLIDS Students gather boxes such as cereal boxes, eleni boxes and other solids to create and determine their nets.

Students are then asked to draw the nets for the following solids:

ACTIVITY SG4.1.6: STRAIGHT CUTS Students are asked to visualise the resulting cut face (plane section) when each of the solids in Activity 5 above is cut parallel to its base. Enlarged pictures of each solid are shown in Appendix SG1 to better assist students with their thinking and reasoning.

Students are asked to record their answers using drawings and words for each of the solids.

ACTIVITY SG4.1.7: TOP VIEWS Students are provided with pictures of solids (see Appendix SG2). They are asked to determine and draw the shape for each solid if they were looking down from above (i.e., the top view).

Extension. Students are asked to discuss and brainstorm what the views will be for each solid from the front and back. They record their findings using drawings and words.

ASSESSMENT FOR LEARNING INDICATORS SG4.1 Compares, explains and models 3D objects including cylinders, cones and spheres; constructs nets from everyday packages; and describes cross-sections of 3D objects The student, for example:

compares and describes features of prisms, pyramids, cylinders, cones and spheres

identifies and names 3D objects as prisms, pyramids, cylinders, cones and spheres

recognises similarities and differences between prisms, pyramids, cylinders, cones and spheres

identifies 3D objects in the environment and from drawings, photographs or descriptions

makes models of cylinders, cones and spheres given a 3D object, picture or photograph to view

sketches cylinders and cones, attempting to show depth

creates nets from everyday packages e.g., a cereal box

sketches 3D objects from different views including top, front and side views

makes and visualises the resulting cut face (plane section) when a 3D object receives a straight cut


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APPENDIX SG1: SOLIDS FOR ACTIVITY 6: STRAIGHT CUTS


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APPENDIX SG2: SOLIDS FOR ACTIVITY 7: TOP VIEW


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SG4.2a Two Dimensional Space SG4.2a Substrand: Two Dimensional Space Recognises, visualises, classifies, models, describes, names and examines 2D shapes including octagons and hexagons, and describes their features

Key Ideas Attitudes & Values Rearrange, label, compare, describe, build models of, and draw 2D shapes including octagons and hexagons presented in different orientations Group 2D shapes using multiple attributes Use translations and rotations to create tessellating designs Use symmetry and identify symmetry in the environment and tapa designs to create symmetrical patterns and pictures



1. manipulating, comparing and describing features of 2D shapes, including octagons and hexagons.

2. identifying and naming octagons, hexagons, and trapeziums presented in different orientations

e.g.,

3. using measurement to describe features of 2D shapes e.g., sides of a regular pentagon are equal

4. grouping 2D shapes using multiple attributes e.g., those with equal sides and right angles

5. making representations of 2D shapes in different orientations

6. constructing 2D shapes from a variety of materials e.g., cardboard, popsticks and paper

7. making tessellating designs by reflecting (flipping), translating (sliding) and rotating (turning) a 2D shape

8. finding lines of symmetry for a given shape

Working Mathematically Students learn to a. select a shape from a description of its

features (Strategically Thinking & Representing, Communicating Mathematically)

b. describe objects in the environment that can be represented by 2D shapes (Communicating Mathematically, Reflecting & Evaluating)

c. explain why a particular two-dimensional shape has a given name e.g., ‘It has 6 sides.’ (Communicating Mathematically, Reflecting & Evaluating)

d. recognise that a particular shape can be represented in different sizes and orientations (Reflecting & Evaluating)

e. use computer drawing tools to make, or manually cut-out, multiples copies of a shape, to create a tessellating design by pasting and rotating regular shapes (Strategically Thinking & Representing)

f. describe designs in terms of reflecting, translating and rotating (Communicating Mathematically)


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Background Information It is important for students to experience a variety of shapes in order to develop flexible mental images. Students need to be able to recognise shapes presented in different orientations. In addition, they should have experiences identifying both regular and irregular shapes. Regular shapes have all sides equal and all angles equal. Regular Tessellations For a regular polygon to tessellate, the size of

the angle must divide evenly into 360, the number of degrees around a point. That is why the only regular 2D shapes that tessellate are the equilateral triangles, squares and hexagons (see Year 3 SG3.2a) and diagram below.

The vertex shown below is just a corner point where the tessellating shapes meet. For regular

tessellations, the pattern is identical at each vertex. To name a tessellation, go around

a vertex and write down how many sides each polygon has. The regular tessellation shown here has three hexagons meeting at each vertex. Since a hexagon has six sides, then the pattern shown is named a “6.6.6” one. When naming (or coding) tessellations, always start at the polygon with the least number of sides, so "3.12.12", not "12.3.12"

Semi-regular Tessellations If two or more regular polygons are arranged in the same order (clockwise and anticlockwise) about each vertex, a semi-regular tessellation is formed. There are only eight of them. Three of the examples are provided below and the rest are shown in Appendix SG6.

The three semi-regular tessellations shown above, in the order from left to right, are named and coded: 3.3.3.3.6, 3.3.3.4.4 and 3.3.4.3.4. Choose any vertex and see why.

Language It is actually the angles that are the focus for the general naming system used for shapes. A polygon (Greek ‘many angles’) is a closed shape with three or more angles and sides. Quadrilateral is a term used to describe all four-sided figures.

Keywords to be used: polygons, parallelograms, quadrilaterals, octagons, hexagons, pentagons, regular, irregular, opposite sides, orientations, tessellations, reflection, flipping, lines of symmetry, vertex

Resources pictures of geometric shapes, scissors, a variety of cut out regular and irregular shapes

Links Visual Arts Fractions and Decimals Three-dimensional Space


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LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIESSG4.2a Recognises, visualises, classifies, models, describes, names and examines 2D shapes including octagons and hexagons, and describes their features

ACTIVITY SG4.2A.1: WHAT IS A POLYGON? – UP TO OCTAGONS Students working in pairs or individually are given a sheet of geometric shapes (as shown below) and are asked to identify and describe each shape.


How many groups of shapes can you form using the given shapes? (students may cut out the shapes for sorting)

Can you write down a sentence about each group to explain why the shapes are put into that group?

Once students have sorted their shapes, ask for pairs of students to place their groups of shapes on the overhead projector (or tape them onto the blackboard) and then

challenge the class to determine their criteria for forming the group.

ACTIVITY SG4.2A.2: CONSTRUCTING THE 7-PIECE TANGRAM This activity provides students with some insight as to how the Tangram pieces fit together, and to stimulate their interest in forming shapes and exploring patterns using the Tangram pieces.

A Tangram is a Chinese puzzle (see diagram) made up of 5 triangles, 1 square, and 1 parallelogram.

The 7 pieces of a Tangram can be arranged to form different shapes.

Students can construct a 7-piece tangram

from A4 paper by following directions to fold and cut as explained below.

Students are asked to: a. make observations on the pieces

formed and compare how they are related to each other.

b. explore patterns and shapes with Tangram

Students are seated in small groups so they can discuss and record observations after each step. 1. Students should fold their square paper

(e.g., 8.5” x 8.5”) in half along the diagonal. Unfold and cut along the crease. Possible questions include:

What observations can you make about the two pieces you have?

What are the two shapes called?


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How can you prove your observations are correct?

2. Students take one of the halves and fold it in half and cut along the crease. Have students make more observations and be able to support their statements. Possible questions include:




3. Students take the remaining half of paper and lightly crease to find the midpoint of the longest side. Fold so that the vertex or corner (point where two sides of the original square meet to form a right angle) touches that midpoint and cut along the crease. Possible questions include:




5. Students take the four-sided shape (What is it called? [trapeziums]) and fold it in half and cut. Possible questions include:



What relationships do the pieces cut have?

Can you determine the measure of any of the angles?


6. Students fold the base angle that is less than a right angle of one of the trapeziums to the adjacent right base angle and cut on the crease. Possible questions include:



What relationships do the pieces cut have?


7. Students fold the right base angle of the other trapezium to the opposite angle larger than a right angle. Cut on the crease. Students should now have the 7 Tangram pieces. Possible question include:

What observations can you make about the 7 pieces you have?

ACTIVITY SG4.2A.3: MAKING SHAPES USING THE 7-PIECE TANGRAM This activity uses the 7 pieces of the Tangram constructed in Activity 2. Possible question include:

Order the 7 pieces from smallest to largest? What criteria did you use?

Which 2 pieces of the puzzle form a square? Justify your answer.

Which 3 pieces of the puzzle form a square? Justify your answer.

Can you make a square using more than 3 pieces? Explain your answer

Can you form a triangle using 2 pieces? Explain your answer.

Can you make a triangle using 3 pieces? Explain your answer.

Can you make a triangle using 5 pieces? Explain your answer.

Can you make a parallelogram with 2 pieces? Explain your answer.




Can you make a square with all 7 pieces? Explain your answer.

Students record their observations using drawings and explanations.

ACTIVITY SG4.2A.4: AREAS OF THE 7-PIECES OF THE TANGRAM Students are asked to focus on the arrangement of pieces based on area. Using the small triangle as the basic unit of area, what are the areas of each of the other 6 pieces?


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Students form squares with 3 pieces, 4 pieces, 5 pieces, 6 pieces, and all 7 pieces. Possible question include:

What is the area of each square?

Are there multiple solutions?

Are there no solutions for any?

Do you notice any patterns? Students record their observations using drawings and explanations. Possible questions include:

If the length of a side of the original square is 2, what are the lengths of the sides of each of the Tangram pieces?

How do you know? How do you check your answers based on your findings from Activity 3?

Students may observe that the areas of the squares appear to be powers of 2 and that they are unable to make a 6 piece square. When all combinations of 6 pieces are considered, the possible areas are not powers of 2.

ACTIVITY SG4.2A.5: 5-PIECE TANGRAM Students are provided with square papers and are asked to construct a 5-piece tangram (as an extension of Activities 2 – 4) based on their analysis of the 5-pieced tangram below. See an enlarged version in Appendix SG2.

5 piece tangram

Students are asked to construct different

figures using all 5 pieces. Some examples are provided in Appendix SG3.

Students record their observations and insights about the relationships between the different shapes.

Variation. Determine the shape formed by joining the dots in diagram in Appendix SG4.

ACTIVITY SG4.2A.6: LINES OF SYMMETRY Students are given enlarged copies (Appendix SG5) of the following geometric sheet to cut out each shape and for them to determine their lines of symmetry. Students answer the questions:

How many lines of symmetry for each shape? How do you know?

How do you check your answers?

ACTIVITY SG4.2A.7: SEMI-REGULAR TESSELLATIONS Students are provided with copies of 8 semi-regular tessellations shown below (see Appendix SG6) to explore and to answer the following questions:

1. Which regular shapes are used in these semi-regular tessellations?

2. Choosing a vertex point P in each tessellation, how many sides do the shapes about P have?

3. What is the greatest number of equilateral triangles that will fit around any point? Justify your answer.


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ASSESSMENT FOR LEARNING INDICATORS SG4.2a Recognises, visualises, classifies, models, describes and examines 2D shapes including pentagons and parallelograms presented in different orientations and compares and describes special groups of quadrilaterals The student, for example:

manipulates, compares and describes features of 2D shapes, including octagons and hexagons.

identifies and names octagons, hexagons, and trapeziums presented in different orientations

uses measurement to describe features of 2D shapes groups 2D shapes using multiple attributes

makes representations of 2D shapes in different orientations

constructs 2D shapes from a variety of materials

makes tessellating designs by reflecting (flipping), translating (sliding) and rotating (turning) a 2D shape

finds lines of symmetry for a given shape


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Appendix SG2: Five Piece Tangram (Activity 5)


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Appendix SG3: Pictures formed from the 5 Pieces of the 5-Piece Tangram


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Appendix SG4: Joining the Dots – What is the Shape?


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Appendix SG5: Lines of Symmetry


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Appendix SG6: Semi-regular Tessellations


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SG4.2b Two Dimensional Space cont’d SG4.2b Substrand: Two Dimensional Space Recognises, understands, compares and defines angles in turns and classifies angles as equal to, greater than or less than a right angle

Key Ideas Attitudes & Values Recognise and describe 2D shapes using the terms ‘parallel sides’ and ‘right angles’ Describe and sort angles into groups of ‘equal to’, ‘greater than’ or ‘less than’ a right angle Understand and recognise the angle in a turn where one arm is visible Compare and order angles of adjacent sides of shapes in relation to a right angle


Knowledge and Skills Students learn about 1. identifying and naming parallel lines 2. identifying angles with two arms in practical

situations e.g., corners 3. comparing angles using informal means such as

an angle tester 4. drawing angles of various sizes by tracing along

the adjacent sides of shapes and describing the angle drawn

Working Mathematically Students learn to a. identify examples of angles in the environment

and as corners of 2D shapes (Strategically Thinking & Representing, Reflecting & Evaluating)

b. identify angles in 2D shapes and 3D objects (Strategically Thinking & Representing)

c. create simple shapes using computer software or hand drawn shapes involving direction and angles (Strategically Thinking & Representing)

d. explain why a given angle is, greater than, less than or equal to, a right angle (Reasoning & Justifying)

Background Information At this level, students need informal experiences of creating, identifying and describing a range of angles. This will lead to an appreciation of the need for a formal unit to measure angles which is introduced in Year 5. The use of informal terms ‘sharp’ and ‘blunt’ to describe acute and obtuse angles respectively are actually counterproductive in identifying the nature of angles as they focus students. Attention to the external points of the angle rather than the amount of turning between the angle arms. Paper folding is a quick and simple means of generating a wide range of angles for comparison and copying.

A simple angle tester can be created by cutting the radii of two equal circles and sliding the cuts together. Another can be made by joining two narrow straight pieces of card with a split-pin to form the rotatable arms of an angle. Angles are formed when two lines meet at a point called a vertex. An angle is the amount of turning about the vertex needed to bring one line on top of the other.

Language Polygons are named according to the number of angles e.g., pentagons have five angles, hexagons have six angles, and octagons have eight angles.

Keywords to be used: parallel lines, vertex, slope, arms, right angles, greater than, less than, quarter turn

Resources pencils, paper, 2D paper shapes

Links Health & Physical Education Fractions


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LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES SG4.2b Recognises, understands, compares and defines angles in turns and classifies angles as equal to, greater than or less than a right angle

ACTIVITY SG4.2B.1: ANGLES IN PICTURES Students collect a variety of pictures that show various angles e.g., buildings, football fields, aerial views. They identify angles in the pictures as ‘right’ angle, ‘greater than’ or ‘less than’ a right angle, trace them onto overhead transparencies and then describe them.


what strategies did you use to describe your angles?

did you discover anything about the type of angles identified?

Variation: Students measure the angles traced and record their findings.

ACTIVITY SG4.2B.2: CLASSIFYING RIGHT ANGLES Students identify, record and classify angles in their classroom and school environments using the terms ‘right’ angle or ‘greater than’ or ‘less than’ a right angle by estimating the size of each angle and then checking their estimates by using their ‘right’ angle tester.

In pairs, students describe the angles they have classified. Students draw each type of angle and label the vertex and arms.


❚ were some of your estimations closer than others?

❚ why do you think this was?

ACTIVITY SG4.2B.3: PARALLEL LINES Students are asked to identify which of the following pairs of lines are parallel to each other. Students justify their answers.

Students are asked to identify two more

pairs of lines that are parallel to each other and to justify their answers.

ASSESSMENT FOR LEARNING INDICATORS SG4.2b Recognises, understands, compares and defines angles in turns and classifies angles as equal to, greater than or less than a right angle The student, for example:

identifies and naming parallel lines

identifies angles with two arms in practical situations e.g., corners

compares angles using informal means such as an angle tester

draws angles of various sizes by tracing along the adjacent sides of shapes and describes the angle drawn manipulates, compares and describes features


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SG4.3 Position SG4.3 Substrand: Position Describes position using coordinates and uses compass points to give and follow directions

Key Ideas Attitudes & Values Represent position and follow routes using simple maps and grids Determine the directions NE, NW, SE and SW, given one of the directions Use coordinates or compass directions to describe the location of an object on a simple map



1. using a compass to find North and hence East, South and West

2. using an arrow to represent North on a map 3. determining the directions N, S, E and W, given

one of the directions 4. using N, S, E and W to describe the location of

an object on a simple map, given an arrow that represents North e.g., ‘The treasure is east of the cave.’

5. using a compass rose to indicate each of the key directions, e.g.,

6. determining the directions NE, NW, SE and SW,

given one of the directions 7. using NE, NW, SE and SW to describe the

location of an object on a simple map, given a compass rose e.g., The church is north-east of the village malae.

Working Mathematically Students learn to a. use and follow positional and directional language

(Communicating Mathematically) b. discuss the use of compasses in the environment

e.g., map of Samoa (Communicating Mathematically, Reflecting & Evaluating)

c. create a simple map or plan using computer paint, draw and graphics tools or hand draw map of your village (Strategically Thinking & Representing)

d. use simple coordinates in games such as map reading (Strategically Thinking & Representing)

e. interpret and use simple maps found in factual texts and newspapers (Strategically Thinking & Representing,

Background Information Students need to have experiences identifying North from a compass in their own environment and then determining the other three directions, East, West and South. This could be done in the playground before introducing students to using these directions on maps to describe the positions of various places.

Language The four directions NE, NW, SE and SW could be introduced to assist with descriptions of places that lie between N, S, E or W.

Keywords to be used: position, location, direction, coordinates, north, south, east, west, north-east, north-west, south-east, south-west, plot, legend, key, path, route

Resources compass, chalk, blindfold, mystery objects, ice-cube tray, beads, counters, tote trays, geoboards, dot paper, rubber bands, atlases, simple maps, street directories

Links Social Studies Length Two Dimensional Space


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LEARNING EXPERIENCES AND ASSESSMENT OPPORTUNITIES SG4.3 Describes position using coordinates and uses compass points to give and follow directions

ACTIVITY SG4.3.1: BODY TURNS The teacher marks the four major compass directions (N, S, E, and W) on the ground. Students face north. Students are asked to turn left or right in quarter turns and state in which direction they then face.

Students are given north and are then asked to face particular compass directions. Students record on a compass rose.

Students are then asked to face a place in the playground whose direction is halfway between the North and East directions.

Students are introduced to the NE, SE, NW, and SW directions to enable them to describe places that lie between N, S, E and W.

Students are given north and are then asked to face particular compass directions (to be selected from N, S, W, E, NE, SE, NW or SW). Students record on a compass rose.

ACTIVITY SG4.3.2: BIKE TRACK In pairs, students are given grid paper to design a bike track within the school grounds or the local park. Students discuss their layout, such as ensuring the route does not cross itself and provides an entry/exit to the school grounds.

Students draw a grid over their map and are asked to describe their bike tracks using positional language (i.e., N, S, W, E, NE, SE, NW or SW), in relation to other structures or pathways.

Students use a compass rose to indicate directions.Stage2

ACTIVITY SG4.3.3: MAPS Students are given atlases and/or road maps and are asked to locate north and then find other compass points.

Students use a compass rose and use N, S, E or W to describe the location of a point on a map.

Students are asked to find places on a map that are in a given direction from a starting point e.g., find a village which is due north of

Siumu. Students are asked to pose their own questions using directional language.

On a map of Samoa students locate

Falease’ela. They then locate places NE, NW, SW, SE of there. Students describe the location of the places in relation to Falease’ela and record using a compass rose.

ACTIVITY SG4.3.4: USING A COMPASS In small groups in the playground, students use a compass to locate the directions N, S, E, W, NE, NW, SW, or SE. Students mark on the ground a grid with sufficient spaces for each student in the group to have a space of their own.

A leader is chosen and blindfolded to call out compass directions i.e., North, South, East, West, North-West, South-West, North-East, South-East. Students follow the directions, moving one grid space at a time, until they are off the grid and ‘out’. Players must call ‘I’m out’ when they are off the grid. The last student to survive wins and becomes the new leader. Students could experiment with rule changes to add further interest to the game.

ACTIVITY SG4.3.5: VILLAGE DIRECTORIES Students are given a simple map of a town with grid lines superimposed. They find places on the map, given coordinates.

Students give the coordinates of particular places on the map.

Students use a page of a street directory or construct a map of the village in which they live. Students are asked to give the coordinates of:

the place where they live

the school


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the church. Students are asked to state what building or physical feature is shown on the map at certain grid positions e.g., ‘What would we find if we walked to A7?’


how many different ways can you get from one point to another?

what does …….. mean in the key? Where can I find it on the map?

can you describe the location of an object in relation to a landmark?

what coordinates or directions can you use to identify the …… (landmark)?

can you determine the directions N, S, E, W, NE, SE, NW, SW on the map? How did you know? What could you use to check?

ACTIVITY SG4.3.6: TURNING IN RIGHT ANGLES AND HALF RIGHT ANGLES Students are encouraged to discuss compass points e.g., N, S, W, E, NE, NW, SE, SW.

Students could use this knowledge to play ‘Robots’. In pairs, students label grid paper using the same coordinates and a scale.

Student A gives directions while Student B is the robot e.g., Student A says ‘Face East, go forward 3 paces, turn right one right angle, go forward 4 paces and turn two right angles to your left….’. At each instruction Student B tells Student A in which direction they are facing. Student B draws the route onto their grid paper. Students compare routes.


what angle have you turned through?

what direction would you be facing if you turned through one more right angle?

ACTIVITY SG4.2B.7: CONSTRUCT A SIMPLE MAP/PLAN Students construct a simple map/plan of their bedroom, classroom or playground. Students plot coordinates on the map/plan and include a key.


can you construct a simple map or plan using coordinates?

does your key allow you to locate specific objects?

can you draw a path from one point to another on your map/plan?

can you describe how to get from one point to another?

can you use directions (i.e., N, S, E, W, NE, NW, SE, SW) to follow a route on your map?

can you describe the location of an object in relation to another using more than one descriptor?

can you describe the position of ……….. using coordinates?

ASSESSMENT FOR LEARNING INDICATORS SG4.3 Describes position using coordinates and uses compass points to give and follow directions The student, for example:

uses a compass to find North and hence East, South and West

uses an arrow to represent North on a map

determining the directions N, S, E and W, given one of the directions

uses N, S, E and W to describe the location of an object on a simple map, given an arrow that represents North e.g., ‘The treasure is east of the cave.’

uses a compass rose to indicate each of the key directions

determines the directions NE, NW, SE and SW, given one of the directions

uses NE, NW, SE and SW to describe the location of an object on a simple map, given a compass rose e.g., The church is north-east of the village malae.


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GLOSSARY

This glossary provides brief explanations of the meaning of particular terms within the Mathematics 1-8

Syllabus document. It contains those terms that may be new to primary teachers. This is particularly the case for

terms that arise in the Years 7 and 8 Content that is included in the Years 1-6 syllabus. The glossary is not

intended to address all mathematical terminology used in the document. Terms written in italics have their own

alphabetical entry in the glossary.

Addends: the numbers you add together to get a sum Angle: is a measure of a turn. A turn is a movement which may be represented by an arrow as shown: Arc (of a circle): Part of the circumference of a circle.

Average: (see Mean) Axes: the zero-lines on a grid. The x-axis is across. The y-axis is up. Cardinal numbers: numbers used for counting when the order does not matter, such as one, two, three, four, five and so on. Sometimes also called ‘counting numbers’. Capacity: The amount that a container can hold. Categorical data: data that has no numerical ordering, for example, colours, food types, favourite drinks Census: Collection of data from a population (e.g., all Year 5 students) rather than a sample. Charts: are diagrams used to show data in a form (e.g., drawings, pictures, tally marks, and/or numbers) that can be seen at a glance. Chance: something that occurs in an unpredictable way. You can forecast your chances in the long term, but you can never guarantee what will happen next. Class interval: A subdivision of a set of data e.g., students’ heights may be grouped into class intervals of 150 cm – 154 cm, 155 cm – 159 cm. Cluster: A ‘crowding’ of data round a particular score e.g., for the set of scores 7, 8, 19, 19, 19, 20, 20, 21, 21, 36, there is a cluster of scores around the score 20. Column: things placed one below the other. In a table, the entries which are in a line that goes up and down the page. Column graph: A graph that uses separated vertical columns or horizontal bars to represent data. Composite number: A number that has more than two factors e.g., 15 is a composite number because it has factors 1, 3, 5 and 15. Concave quadrilateral: A quadrilateral that contains a reflex angle

e.g.,

Continuous data: Data that can take any value within a given range e.g., the heights in centimetres of the students in a class. Conversion graph: A line graph that can be used to convert from one unit to another e.g., from $A to $US. Coordinates (or co-ordinates): the pair of numbers that tells you the position of a point on a graph. They are usually enclosed with brackets. The order for coordinates is always alphabetical: (horizontal,vertical), (x,y) or (across,up). Cross-section: The shape (plane section) produced when a solid is cut through by a plane, parallel to the base e.g., the cross-section of a cone is a circle

a

r

c


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Cumulative frequency: The total of all frequencies up to and including the frequency for a particular score in a frequency distribution

e.g.,

Score Frequency Cumulative Frequency

10 6 6

20 7 13

30 8 21

40 4 25

The cumulative frequency of the score 30 is 21, since the total of the frequencies up to and including the frequency for 30 is 6+7+8 = 21.

Denominator: The lower number of a fraction that represents the number of equal fractional parts a whole has been divided into. Discrete data: Data that can only take certain values within a given range e.g., the number of students enrolled in a school. Divided bar graph: A graph that uses a single bar divided proportionally into sections to represent the parts of a total

e.g.,

Faalavelave

Clothing

Food

Fares

Enter- tainment

Savings

Divided Bar Graph of Weekly Expenditure

Dot plot: A data display in which scores are indicated by symbols such as dots or crosses drawn above a horizontal axis e.g., Empty number line: An unmarked number line providing a means for students to record their calculation strategies e.g., jump strategies for addition and subtraction.

(Unmarked number line) 46 56 66 76 77 78 79

(Recording of a jump strategy for calculating 46 + 33) Equilateral triangle: A triangle with all sides equal in length. Equivalent fractions: Fractions that can be reduced to the same basic fraction i.e. fractions that have

the same value e.g.,

1

2 2

4 3

6 4

8

Factor: A factor of a given number is a whole number that divides it exactly e.g., 1, 2, 3, 4, 6 and 12 are the factors of 12. Fibonacci numbers: Numbers in the sequence which begins with two ones and in which each subsequent term is given by the sum of the two preceding terms i.e. the numbers 1, 1, 2, 3, 5, 8,…

+1

+1

+10

+1

0

+1

0

+

10


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Figurate numbers: Numbers that can be represented by a geometric pattern of dots e.g., triangular numbers, square numbers, pentagonal numbers.

Fraction notation: Representation of numbers in the form

a

b where a and b are whole numbers and b is

not equal to zero. Frequency distribution (table): A table that lists a set of scores and the frequency of occurrence of each score e.g., frequency distribution table for the set of scores: 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9

Score

Frequency

5 2

6 5

7 4

8 3

9 1 Frequency histogram: A graph of a frequency distribution that uses vertical columns (with no gaps between them) to represent the frequencies of the individual scores e.g., frequency histogram for the data in the example above:

Frequency polygon: A graph of a frequency distribution formed by joining the midpoints of the tops of the columns of a frequency histogram e.g., frequency polygon (with histogram) for the data given in the table above:

Graph: is a kind of map with patterns of crossing lines called grids, designed to show you where things are. Grid: a pattern of lines that cross at right angles that is used to make it easier to set out your work Growing patterns: patterns that involve progression from step to step, in later years, these are called sequences. Histogram: is a form of column graph with vertical bars that touch (no gaps between bars). Histograms are used to display numerical data rather than categorical data. Hefting: The comparison of objects, holding one in each hand, to determine which is heavier or lighter. Improper fraction: A fraction in which the numerator is greater than the denominator. Index (plural: indices): The number expressing the power to which a number or pronumeral is raised

e.g., in the expression 23 , the index is 2.


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Inverse operation: The operation that reverses the effect of the original operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations. Isosceles triangle: A triangle with two sides equal in length.

Jump strategy: An addition or subtraction strategy in which the student places the first

number on an empty number line and then counts forward or backwards firstly by tens and

then by ones to perform a calculation. The number of jumps will reduce with increased

understanding. e.g., 46 + 33

Method 1:

46 56 66 76 77 78 79

Method 2:

46 56 66 76 77 78 79

Line graph: A graph in which information is represented through plotting and joining points with a line or line segments. Meaning can be attached to the points between the plotted points e.g., temperature and population trends may be represented using line graphs. Line symmetry: A figure has line symmetry if one or more lines (‘line of symmetry’ or ‘axis of symmetry’) can be drawn that divide the figure into two mirror images. Linear scale: A scale where equal quantities are represented by equal divisions e.g., ruler, thermometer. Mean (or Average): The total of a set of scores divided by the number of scores

e.g., for the scores 4, 5, 6, 6, 9, 12, the mean is

4 56 6 912

6 7

Median: The middle score when an odd number of scores is arranged in order of size. If there is an even number of scores, the median is the average of the two middle scores

e.g., for the scores 3, 3, 6, 8, 9, the median is 6; for the scores 5, 6, 9, 9, the median is

69

2 7 1

2

Mental facility: The ability to use a variety of strategies to calculate mentally.

Mixed numeral: A number that consists of a whole number part and a fractional part e.g.,

2 12

.

Mode: The score that occurs most often in a set of scores i.e. the score that has the highest frequency. A set of scores may have more than one mode e.g., for the scores 1, 2, 3, 3, 4, 4, 4, 5, the mode is 4; for the scores 3, 5, 5, 5, 6, 6, 6, 7, there are two modes, 5 and 6. Multiple: A number that is the product of a given number and any whole number greater than zero e.g., the multiples of 4 are 4, 8, 12, 16, 20, … Number: one or more numerals placed together represent the size of something (e.g., 45 is the numerals four and five placed together to represent the number forty-five). Number sense: The ability to use an understanding of number concepts and operations in flexible ways to make mathematical judgements and to develop useful strategies for handling numbers and operations. Numeral: a symbol standing for a number. The modern numerals are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Roman numerals are I, V, X, etc Numerator: The upper number of a fraction that represents the number of equal fractional parts. Numerical Data: data that is numerical and can be ordered on a number line. Oblique prism: (See Prism). Ogive (or ‘cumulative frequency polygon’): A graph formed by joining the top right-hand corners of the columns of a cumulative frequency histogram e.g.

+

3

+

30

+1

+1

+1

0

+1

0

+1

0

+1

+1


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Ordered pairs: any pair of things (such as coordinates, for example) in which the order matters. Order of rotational symmetry: The number of times a figure coincides with its original position in turning through one full rotation e.g., an equilateral triangle has rotational symmetry of order three and a square has rotational symmetry of order four. Ordinal numbers: numbers used for putting things in order, such as first, second, third, fourth, fifth and so on. Outlier: A score that lies well outside most of the other scores in a set of data e.g., 25 is an outlier in the set of scores 1, 2, 4, 4, 6, 7, 25. Palindromic numbers: Numbers that are the same if read forward (as for ‘Jump Strategy’) or backwards e.g., 44, 23 532. Parallelogram: A quadrilateral with both pairs of opposite sides parallel. Pascal’s triangle: A triangular array of numbers bordered by 1’s such that the sum of two adjacent numbers is equal to the number between them in the next row.

1 1 1

1 2 1

1 3 3 1

1 4 6 4 1 etc

Pentagonal numbers: Numbers that can be represented by a pentagonal pattern of dots. The first five pentagonal numbers 1, 5, 12, 22 and 35 can be represented by

Perimeter: The distance around the boundary of a two-dimensional shape. Platonic solids: The five regular polyhedra i.e., the five polyhedra whose faces are regular congruent polygons: tetrahedron (4 faces); cube (6 faces); octahedron (8 faces); dodecahedron (12 faces); icosahedron (20 faces). The Platonic solids (with nets):


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Polygon: A two-dimensional shape having three or more straight sides. Polyhedron (plural: polyhedra): A solid in which each face is a polygon. Population: The whole group from which a sample is drawn. Position: The location of an object in relation to oneself or another object. Prime factor: A prime factor of a given number is a prime number that divides it exactly e.g., the prime factors of 42 are 2, 3 and 7. Prime number: A number that has only two factors, itself and one e.g., 3 is a prime number because its only factors are 1 and 3. Prism: A solid comprising two congruent parallel faces (‘bases’) and the (‘lateral’) faces that connect them. The lateral faces are parallelograms. If they are all right-angled (i.e., rectangles) the prism is a ‘right prism’; if they are not all right-angled then the prism is an ‘oblique prism’

e.g., Right prisms Oblique prism

(rectangular prism) (triangular prism)

Pyramid: A solid with any polygon as its base. Its other faces are triangles that meet at a common vertex. Pyramids are named according to their base e.g., a pyramid with a square base is a ‘square pyramid’.

Quadrant: A sector with arc equal to a quarter of a circle (and therefore centre angle 90(sometimes) an arc equal to a quarter of a circle.

Quantitative data: Data that can be counted (discrete data) or measured (continuous data) e.g., the number of students enrolled in a school (discrete); the heights in centimetres of the students in a class (continuous). Range: The difference between the highest and lowest scores in a set of scores e.g., for the scores 5, 7, 8, 9, 10, 11, the range is 11 – 5 = 6. Repeating Patterns: patterns that involve the repeat of a string of elements. Elements can be sounds, actions, objects, numbers or a mix of these. Rhombus: A parallelogram with all sides equal. Rhythmic counting: Counting with emphasis on rhythm e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, … (where the bold numbers are said more loudly). Right angle: an angle which is exactly a quarter of a complete turn Right prism: (see Prism) Row: things placed side by side. In a table, the entries which are in any line across the page Sample: Part of a population chosen so as to give information about the population as a whole. Scalene triangle: A triangle with no two sides equal in length. Scatter diagram: A display consisting of plotted points that represent the relationship between two sets of data e.g., the scatter diagram shows the Mathematics and English test scores of a class of twenty students. Each point on the diagram represents the pair of scores for one student.


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Section: The flat surface obtained by cutting through a solid in any direction e.g., the section shown for a square pyramid is a trapezium.

Sector: Part of a circle bounded by two radii and an arc.

Sector (pie) graph: A data display that uses a circle divided proportionally into sectors to represent the parts of a total. Semicircle: Part (half) of a circle bounded by a diameter and an arc joining the ends of the diameter; or (sometimes) the arc equal to half the circumference of a circle.

In the diagram, both the shaded and unshaded regions are semicircles.

Sequences: number patterns that involve progression from step to step Skip counting: Counting forward or backwards in multiples of a particular number e.g., 3, 6, 9, 12, … . Solid: A three-dimensional object. Square numbers: Numbers that can be represented by a square pattern of dots. The first three square numbers 1, 4, and 9 can be represented by Stem-and-leaf plot: A display that provides simultaneously a rank order of individual scores and the shape of the distribution. The ‘stem’ is used to group the scores and the ‘leaves’ indicate the individual scores within each group.


99

e.g., 0 5 6 9 1 1 2 4 4 2 3 5 7 (Stem-and-leaf plot for the set of data: 9, 6, 12, 14, 14, 11, 5, 23, 25, 27.) A back-to-back stem-and-leaf plot has two sets of data displayed on either side of the common stem. Step graph: A graph that increases or decreases in ‘steps’ rather than being a continuous line e.g.,

Subitising: The skill of immediately recognising the number of objects in a small collection without having to count the objects. Summary statistics: Measures such as mean, mode, median and range used in analysing a set of data. Symbol: a mark written on paper or something else to stand for a number or an idea of any kind. Table: is an arrangement of rows and columns for sorting and storing data. Tally marks: marks made as a way of recording the total Tangram: is a set of puzzle shapes. The standard set of seven tangram pieces is cut from a square as shown below including a 5 piece tangram puzzle:

Tessellate: to make a perfectly interlocking pattern to cover a surface without gaps or overlaps Translation: Sliding of a figure without rotation or changing of its shape or size. Trapezium: A quadrilateral with at least one pair of opposite sides parallel. Travel graph: A graph that represents the relationship between time and distance travelled. Triangular numbers: Numbers that can be represented by a triangular pattern of dots. The first three triangular numbers 1,3, and 6 can be represented by Uniform cross-section: A solid has a uniform cross-section if cross-sections taken parallel to its base are always the same size and shape (cross-sections parallel to the base of prisms are uniform, whereas cross-sections parallel to the base of pyramids are not).

Unit fraction: A fraction that has a numerator of one e.g.,

1

2, 13,14, 15.

Vertex (plural: vertices): A point where two or more sides of a polygon or edges of a solid meet e.g., a square has 4 vertices and a cube has 8 vertices.


100

Visualise: To recreate and manipulate images mentally.

Volume: The amount of space occupied by an object or substance


APPENDICES APPENDIX I ANNUAL PLAN

CALENDAR OF ANNUAL PLAN MATHEMATICS Years 1 – 6 YEAR......

TERM ONE TERM TWO TERM THREE

Number and Operations Patterns and Algebra Data Measurement Space and Geometry

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TERM ONE TERM TWO TERM THREE TERM FOUR Numbers and Operations Patterns /Algebra Data Measurements Space and Geometry

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teachers’ manual 2013 - mesc.gov.ws manuals/year 4 maths.pdf · unit plan ... mathematics year 4...

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