Mathematics Teaching in the Foundation Phase 1

F-MAT 221

Mathematics Teaching in the Foundation Phase 1

F-MAT 221

BACHELOR OF EDUCATION IN FOUNDATION PHASE

TEACHING

MATHEMATICS TEACHING IN THE FOUNDATION PHASE 1

YEAR 2

F-MAT 221

Level 6 Credits 12

CURRICULUM AND LEARNING GUIDE

Copyright SANTS Private Higher Education Institution. Pty. Ltd. PO Box 72328, Lynnwood Ridge, 0040

2020

All rights reserved. Apart from any fair dealing for the purpose of research, criticism or review as permitted under the Copyright Act, no part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, without permission in writing, from SANTS.

MATHEMATICS TEACHING IN THE FOUNDATION PHASE 1 CURRICULUM AND LEARNING GUIDE

BEd (FOUNDATION PHASE TEACHING) i

2020 Edition

Program coordinator Prof Ina Joubert

SANTS Private Higher Education Institution

Discipline coordinator Mrs Linda le Hanie SANTS Private Higher Education Institution Dr Ingrid Sapire University of Witwatersrand

Author(s) Dr Lawan Abdulhamid University of Witwatersrand Mrs Ina Nel SANTS Private Higher Education Institution Mrs Mariaan Bouwer SANTS Private Higher Education Institution

Reviewer Dr Tony Mays University of Pretoria, SAIDE

Language editor Yvonne Thiebaut Consultant Editor: Xseed Education

Technical editor Mrs Judith Brown SANTS Private Higher Education Institution

Graphic artist N/A

Printing BusinessPrint


BEd (FOUNDATION PHASE TEACHING) ii

BACHELOR OF EDUCATION IN FOUNDATION PHASE TEACHING 1. WELCOME TO THE MODULE Dear SANTS student, We welcome you to the Mathematics Teaching in the Foundation Phase 1 (F-MAT 221) module that forms part of the Bachelor of Education in Foundation Phase Teaching programme and wish you success in your studies. The purpose of the Bachelor of Education in Foundation Phase Teaching programme is to offer a curriculum that develops teachers who can acquire and eventually articulate focused knowledge, skills and general principles appropriate for Foundation Phase teaching, as specified in the Revised Policy on the Minimum Requirements for Teacher Education Qualifications (Department of Higher Education and Training, 2015). The Bachelor of Education (BEd) qualification requires that teachers develop a depth of specialised knowledge, practical competencies (skills) and experience in a Foundation Phase context. As part of the BEd qualification, you will need to gain experience in applying what you are learning during a period of Workplace Integrated Learning (WIL). This means you will spend some time teaching Foundation Phase learners in an authentic (real) context. The BEd qualification programme is aligned with the Revised Policy on the Minimum Requirements for Teacher Education Qualifications, in particular Appendix C of the policy that outlines the Basic Competencies of a Beginner Teacher (Department of Higher Education and Training, 2015, Government Gazette, No. 38487, p. 62).

2. OUTCOMES OF THE PROGRAMME At the end of the four-year Bachelor of Education Teaching programme, you must demonstrate the following competencies related to your own academic growth and potential to work with Foundation Phase learners:

Read, write and speak the language in ways that facilitate your own academic learning.

Read, write, and speak the language/s of instruction related to Foundation Phase in ways that facilitate teaching and learning instruction in the classroom.

Demonstrate competence in communicating effectively, in general and in relation to Foundation Phase specialised knowledge in order to mediate and facilitate learning.

Interpret and use basic mathematics and elementary statistics to facilitate your own academic learning and to manage teaching learning and assessment.


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Use information and communications technology (ICT) in daily life and in teaching. Explain the contents and purpose of the national curriculum with particular reference

to Foundation Phase. Demonstrate skill in planning, designing, and implementing learning programmes that

are developmentally appropriate and culturally responsive to Foundation Phase context.

Demonstrate competence in identifying and accommodating diversity in the Foundation Phase classroom, and in the identification of learning and social problems. This includes planning, designing and implementing learning programmes to accommodate diversity.

Demonstrate an understanding of the theoretical and pedagogical fields of study that influence education and teaching, as well as learning decisions and practices.

Demonstrate the ability to function responsibly within an education system, an institution and the community in which an institution is located.

Demonstrate a respect for and commitment to the educator profession. Demonstrate an understanding of:

o The principles underpinning the disciplines for the various learning areas;

o Pedagogical content knowledge of the learning subjects to be taught; o Planning and designing learning opportunities; o Resourcing teaching and learning; and o Reflecting on teaching;

Demonstrate competence in observing, assessing and recording learner progress regularly.

Reflect upon and use assessment results to solve problems and to improve teaching and learning.

Demonstrate competence in selecting, using and adjusting teaching and learning strategies in ways that meet the needs of both learners and context.

Demonstrate competence in managing and administering learning environments and supporting learners in ways that promote social justice ideals.

Conduct yourself responsibly, professionally and ethically in the classroom, the school and the broader community in which the school is located.

Display a positive work ethic that benefits, enhances and develops the status of the teaching profession and of early childhood education more broadly.

3. PROGRAMME STRUCTURE The BEd degree is presented on the National Qualifications Framework (NQF) Exit level 7 with minimum total credits of 498, earned over the four years. The table below shows the curriculum implementation plan of the BEd degree you are studying. It also tells you how many credits each module carries. You will also see at which NQF level the study material has been prepared and which modules you need to pass each year. This four-year


BEd (FOUNDATION PHASE TEACHING) iv

programme has been planned to strengthen the competencies you will need as a beginner teacher.

Outline of modules of the BEd (Foundation Phase Teaching) programme: Module name Code NQF L Credits Module name Code NQF L Credits

YEAR 1 SEMESTER 1 SEMESTER 2

Academic Literacy B-ALI 110 5 10 Critical Literacies for Teachers B-CLT 120 5 10

Fundamental Mathematics B-FMA 110 5 10 Introduction to Mathematics Teaching in the Foundation Phase F-MAT 120 5 10

Computer Literacy B-CLI 110 5 10 Professional Studies in the Foundation Phase 1: Classroom Practice

F-PFS 121 5 10

Education Studies 1: Theories of Child Development B-EDS 111 5 10 Education Studies 2: Theories of

Learning and Teaching B-EDS 122 6 12

Introduction to the Language and Literacy Landscape in the Foundation Phase

F-LLL 110 5 10 Introduction to Life Skills Teaching in the Foundation Phase F-LSK 120 5 10

Language of Conversational Competence: isiXhosa / isiZulu / Sepedi

C-LCX 120 C-LCZ 120 C-LCS 120

5 10

50 52-62 Workplace Integrated Learning Year 1 F-WIL 101 5 18 Sub-total credits for Year 1: 120 - 130


English Home and First Additional Language and Literacy Teaching in the Foundation Phase 1

F-EHF 211 6 15 English Home and First Additional Language and Literacy Teaching in the Foundation Phase 2

F-EHF 222 6 15

Home Language and Literacy Teaching in the Foundation Phase 1: Afrikaans / isiXhosa / isiZulu / Sepedi

F-HLA 211 F-HLX 211 F-HLZ 211 F-HLS 211

6 12 Home Language and Literacy Teaching in the Foundation Phase 2: Afrikaans / isiXhosa / isiZulu / Sepedi


6 12

Professional Studies in the Foundation Phase 2: School and Classroom Management

F-PFS 212 6 12 Professional Studies in the Foundation Phase 3: Social Justice and Current Issues in Education

F-PFS 223 6 12

Education Studies 3: Curriculum, Pedagogy and Assessment

B-EDS 213 6 12 Education Studies 4: History of Education and Education Policies B-EDS 224 6 12

Life Skills Teaching in the Foundation Phase 1: Personal and Social Well-being

F-LSK 211 6 12 Mathematics Teaching in the Foundation Phase 1 F-MAT 221 6 12

English First Additional Language and Literacy Teaching in the Foundation Phase 1

F-FLE 221 6 12

First Additional Language and Literacy Teaching in the Foundation Phase 1: Afrikaans / isiXhosa / isiZulu / Sepedi

F-FLA 221 F-FLX 221 F-FLZ 221 F-FLS 221

6 12

48-63 60-63 Workplace Integrated Learning Year 2 F-WIL 202 6 20 Sub-total credits for Year 2: 128 - 146


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Module name Code NQF L Credits Module name Code NQF L Credits



F-EHF 313 6 15 First Additional Language and Literacy Teaching in the Foundation Phase 2: Afrikaans / isiXhosa / isiZulu / Sepedi


6 12



6 12 English First Additional Language and Literacy Teaching in the Foundation Phase 2

F-FLE 322 6 12

Mathematics Teaching in the Foundation Phase 2 F-MAT 312 6 12 Mathematics Teaching in the

Foundation Phase 3 F-MAT 323 6 12

Life Skills Teaching in the Foundation Phase 2: Physical Education

F-LSK 312 6 12 Life Skills Teaching in the Foundation Phase 3: Creative Arts F-LSK 323 6 12

Education Studies 5: Sociology of Education B-EDS 315 7 14

Professional Studies in the Foundation Phase 4: Teacher Identity and the Profession

F-PFS 324 7 14

50-65 38-50 Workplace Integrated Learning Year 3 F-WIL 303 6 22 Sub-total credits for Year 3: 122 - 125



F-EHF 414 7 14

First Additional Language and Literacy Teaching in the Foundation Phase 3: Afrikaans / isiXhosa / isiZulu / Sepedi


7 14



7 14 English First Additional Language and Literacy Teaching in the Foundation Phase 3

F-FLE 423 7 14

Digital Pedagogies for Teachers B-DPT 420 5 10 Mathematics Teaching in the

Foundation Phase 4 F-MAT 414 7 14

Life Skills Teaching in the Foundation Phase 4: Natural Sciences and Technology

F-LSK 414 7 14 Life Skills Teaching in the Foundation Phase 5: Social Sciences

F-LSK 425 7 14

38-52 28-42 Research in Education B-RED 400 7 22 Workplace Integrated Learning Year 4 F-WIL 404 7 26 Sub-total credits for Year 4: 128 - 128 Total credits for programme: 498 - 529

Language competencies will be assessed during the course of your programme.

The modules in the programme can be divided into four broad types of learning (Department of Higher Education and Training, 2015, pp. 9-11). Each type of learning develops specific knowledge, values and attitudes, competencies and skills to achieve the overall exit level outcomes of the programme.


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The different types of learning are: Fundamental learning, which includes student personal and academic development:

This type of learning involves academic literacy, critical literacies for teachers, fundamental mathematics, computer literacy and digital pedagogies for teachers.

Disciplinary learning:

This learning includes subject matter knowledge and includes the study of education and its foundations and specific specialised subject matter;

Knowledge of the child and how the child grows, develops and learns; Understanding of the processes of teaching and learning and the articulation

between child development and teaching and learning; and Understanding of the historical, socio-political, policy and curriculum contexts of

education particularly in South Africa. Situational learning:

Situational learning refers to knowledge of the varied learning situations of learners. This learning involves specifically learning about the context of the learner. These modules are called professional studies.

Professional Studies focuses on: o The complex context of teachers and teaching and learning in general and

Foundation Phase in particular; o Multi-faceted and multi-layered positions and roles a teacher occupies; and o The relationship between teaching and learning in the context of the school

and classroom and specifically the Foundation Phase classroom.

Pedagogical learning: This learning includes disciplinary general pedagogic learning knowledge referring

to the study of principles, practices and methods of teaching; Pedagogic content knowledge which includes specialised pedagogic content or

subject knowledge which includes how to present concepts, methods, strategies, approaches and rules of a specific discipline when teaching; and

It also includes tools for implementing teaching and learning and assessment in context.


BEd (FOUNDATION PHASE TEACHING) vii

Types of learning and modules in the BEd (Foundation Phase Teaching) programme Types of learning Modules

Fundamental learning Student personal and academic development

Academic Literacy Fundamental Mathematics Computer Literacy Critical Literacies for Teachers Digital Pedagogies for Teachers

Disciplinary learning Education studies

Education Studies 1: Theories of Child Development Education Studies 2: Theories of Learning and Teaching Education Studies 3: Curriculum, Pedagogy and Assessment Education Studies 4: History of Education and Education Policies Education Studies 5: Sociology of Education

Situational learning Professional studies

Professional Studies in the Foundation Phase 1: Classroom Practice Professional Studies in the Foundation Phase 2: School and Classroom Management Professional Studies in the Foundation Phase 3: Social Justice and Current Issues in Education Professional Studies in the Foundation Phase 4: Teacher Identity and the Profession

Pedagogical learning Pedagogy

FOUNDATION PHASE (FP) Introduction to the Language and Literacy Landscape in the FP Introduction to Mathematics Teaching in the FP Mathematics Teaching in the FP 1, 2, 3 and 4 Introduction to Life Skills in the FP Life Skills Teaching in the FP 1: Personal and Social Well-being Life Skills Teaching in the FP 2: Physical Education Life Skills Teaching in the FP 3: Creative Arts Life Skills Teaching in the FP 4: Natural Sciences and Technology Life Skills Teaching in the FP 5: Social Sciences Languages: Five language options: English Home and First Additional Language and Literacy Teaching in the FP 1, 2, 3 and 4 Choose another (additional) language at Home Language level OR First Additional Language level: Afrikaans, isiXhosa, isiZulu, Sepedi Only if Afrikaans is chosen as another language: choose between isiXhosa, isiZulu, Sepedi as Language of Conversational Competence (LoCC) Afrikaans Home Language and Literacy Teaching in the FP 1, 2, 3 and 4 English First Additional Language and Literacy Teaching in the FP 1, 2 and 3


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Types of learning Modules

Choose between isiXhosa, isiZulu, Sepedi as Language of Conversational Competence (LoCC) isiXhosa Home Language and Literacy Teaching in the FP 1, 2, 3 and 4 English First Additional Language and Literacy Teaching in the FP 1, 2 and 3 isiZulu Home Language and Literacy Teaching in the FP 1, 2, 3 and 4 English First Additional Language and Literacy Teaching in the FP 1, 2 and 3 Sepedi Home Language and Literacy Teaching in the FP 1, 2, 3 and 4 English First Additional Language and Literacy Teaching in the FP 1, 2 and 3

We call these four types of learning, the knowledge mix of a module (Department of Higher Education and Training, 2015, p. 11). The level of knowledge for this module is set at level 6 and it carries 12 credits. For every credit you should spend approximately 10 hours mastering the content. You will thus have to spend at least 120 hours studying the F-MAT 221 material and doing the assignments and any assessments. The knowledge mix of this level 6 module with the related credits is as follows:

Disciplinary learning, (Study of education and its foundations, 1 credit and Subject knowledge, 5 credits);

Pedagogical learning, (General pedagogic knowledge, 1 credit and Pedagogic content knowledge, 4 credits); and

Situational learning with 1 credit. 4. PURPOSE OF THIS MODULE Purpose This module will prepare students to understand and teach numbers, operations, and number relations by providing both subject and pedagogical content knowledge. Students will understand how to sequence mathematical knowledge and plan, implement and assess mathematical learning in the Foundation Phase. Learning Outcomes At the end of this module students should be able to:

Plan mathematics learning programmes. Evaluate, select, and implement appropriate methods for teaching numbers,

operations and number relations in the Foundation Phase. Assess mathematical learning. Use assessment for learning and teaching. Plan appropriate mathematics learning environments.


BEd (FOUNDATION PHASE TEACHING) ix

Select and develop appropriate resources for mathematics learning. Identify and support learners with barriers to mathematical learning. Reflect on practice.

Content This module continues to lay the foundation for development of mathematical understanding by introducing students to subject and pedagogical content knowledge in order to understand and teach number, operations, and number relations. The content comprises:

Planning mathematics learning programmes. Curriculum mapping for mathematics teaching and learning in the Foundation

Phase. Number relations, place value, ordering, and rounding off. Operations of whole numbers. Assessment of number relations, place value, ordering, and rounding off. Identification of barriers to mathematics learning. Support for learners with barriers to mathematics learning.

Competencies

Sound subject and pedagogical content knowledge; Plan, implement, and assess mathematical learning in diverse contexts. Identification of mathematical barriers to learning; Use of assessment results to improve teaching and learning; and Reflection of practice.

5. WORKING THROUGH THE CURRICULUM AND LEARNING

GUIDE (CLG) We developed the CLG to help you master the content through a distance education mode. You will not have full time tutoring or support but the Student Orientation Booklet, accessible at MySANTS, offers guidelines for distance learning. Aspects such as plagiarism are also explained in this booklet. Make use of MySANTS as a support system for any academic queries. These guidelines will help you to:

Work consistently throughout the semester; Manage your time efficiently; Complete assignments on time; and Prepare for tests and examinations.


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As you read the CLG, draw on your own experiences and the knowledge you already have. The core text and recommended reading texts included in the CLG will also help you to deepen your understanding of the content and concepts you are working through. In the CLG, you will find a glossary (word list). The wordlist will help you understand difficult concepts by providing the definitions (meaning) of such words. You will also find icons (small pictures). The icons indicate the type of activity you must do. If you do each activity as suggested, you ought to advance and consolidate your understanding of the core concepts in the module. You will find a list of the icons used in this CLG on the next page. Reading and writing activities have been designed to help you make connections with what you already know, master the content and reflect on what you have learnt. Scenarios (situations resembling an authentic (real-life) context) and dialogues provide background to what you are learning. The review / self-assessment questions are based on the learning outcomes. Doing each activity will help you understand the content. Get a book or file in which you complete all your activities. Write full sentences and always use your own words to show your understanding. Working systematically through each activity, according to the estimated time for each activity as provided, will also help prepare you for assessments (assignments and the examination). Try to find other students to work with. It is easier to share ideas and complete activities when working in a study group. Doing so, may help you to master the content more easily. Commentaries appear at the bottom of some activities. Commentaries are not answers but rather a reflection to guide your understanding of the activity and to assist you in knowing whether your own answer is appropriate or not. These commentaries alert you to aspects you need to consider when doing the activity.

WRITING ACTIVITY An activity is designed to help you assess your progress and manage your learning. Sometimes you will have to define, explain, and/or interpret a concept. Scenarios and dialogues are often used to contextualise an activity. They will also help you bridge theory and practice by linking the concept and real life situations. When responding to the activities, use your own words to show your understanding. Do not copy directly from the text of the CLG. At the end of most activities, you will find commentary that aims to guide your thinking and assess how well you have understood the concepts. The activities are numbered for easy reference.


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READING ACTIVITY Reading activities may require you to read additional material not printed in the Curriculum and Learning Guide. These readings will be either the full text or part of a core or recommended journal article. Journal articles will give you an expanded or alternative view on a concept. You might be required to explain the concept from a different perspective or compare what has been stated in the CLG with what you read in the journal article.

STUDY GROUP DISCUSSION All study group discussions or peer activities require preparation BEFORE the discussion. Preparation includes reading and completing activities in writing. Study group discussions are an opportunity for reflection and for you to apply what you have learnt. Sharing your learning experiences may help you to learn with and from each other. Study group discussions can be done in your own study group or at the SANTS academic support sessions.

REFLECTION Reflection means to think deeply or carefully about something. Reflection activities require you to review critically what you have learnt and link this with your personal experiences or what you have observed during Workplace Integrated Learning (WIL).

REVIEW / SELF-ASSESSMENT Often questions are provided at the end of each unit to assist self-assessment. These questions are similar to the type of questions that you may be asked in assignments or examinations.

6. SELF-DIRECTED LEARNING As a distance education student, it is your responsibility to engage with the content and to direct your own learning by managing your time efficiently and effectively.

We designed the following self-directed learning programme template so that you can plan your time carefully and manage your independent learning. The template will also help you to keep to due dates and thus complete the assignments on time. Careful time management and breaking the work up into manageable chunks will help you work through the content without feeling too stressed. Once you have worked through the activities you should be able to contribute to discussions with peers in your own study group or during the non-compulsory student academic support sessions.


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When completing the template, consider the following:

This module is offered in the second semester of your second year of study. The semester is 15 - 20 weeks long. The module carries 12 credits and has been developed for NQF level 6. It

should take you about 120 hours to work through this module. The 120 hours will be spent reading, studying, and completing the activities

in this CLG, as well as the assignments. You will also spend time preparing and writing the examination.

The estimated time to read for and complete each activity has been suggested.

You will need 5 to 10 hours to complete each assignment. This means you will need to budget about 20 hours in total.

You should plan to spend about 10 to 20 hours preparing for the examination in order to be successful.

Plan your studies and keep pace of your progress by completing the template below. It is not divided into specific weeks, but into the number of units in the CLG. Depending on the nature of the content, it is possible to complete two or more units in one week. Sometimes, you may only be able to complete one unit in a week. Use the template as a guide to help you plan and pace yourself as you work through the content, and activities in each unit. Add dates to the template indicating when you plan to start working through a particular unit. In addition, using a SANTS academic calendar will also assist you to pace your learning. There is also space to indicate the due dates (deadlines) of the assessments.

UNIT IN CLG CONTENT IN CLG DATE PLANNED

UNIT 1 NUMBERS AND NUMBER

RELATIONSHIPS

Number relationships, place values, and ordering and comparing numbers

Rounding off whole numbers

UNIT 2 ADDITION AND

SUBTRACTION OF WHOLE NUMBERS

Addition

Subtraction

Addition and subtraction – inverse operations


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UNIT IN CLG CONTENT IN CLG DATE PLANNED

ASSIGNMENT 1

ASSIGNMENT 2

EXAMINATION

7. CORE READING Core readings are an important part of your studies as you need to refer to these text(s) when answering some of the questions in the activities. 1. Broadbent, A. (2004). Understanding place-value – a case study of the base ten

game. Australian Primary Mathematics Classroom, 9(4), pp. 45-46. Available on EBSCO: http://search.ebscohost.com/login.aspx?direct=true&db=eue&AN=15254092&site=ehost-live

2. Department of Basic Education. (2011). Curriculum and Assessment Policy

Statement. Mathematics: Grades 1 – 3. South Africa: Government Printing. Open source: https://www.education.gov.za/Curriculum/CurriculumAssessmentPolicyStatements(CAPS)/CAPSFoundation.aspx

The text(s) for the first core reading is available on EBSCOhost. To access the core reading text(s) use the library tab on MySANTS and click on the EBSCOhost link. 8. RECOMMENDED READING As a distance education student, you cannot only rely on your CLG. We recommend that you also study the following sources so that you have broader insight into the study material: 1. Gifford, S. (2005). Teaching Mathematics 3-5 : Developing Learning in the

Foundation Stage. Maidenhead: McGraw-Hill Education. (Chapter 8). Available on EBSCO: http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=233933&site=ehost-live&ebv=EB&ppid=pp_77


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2. Herholdt, R. & Sapire, I. (2014). An error analysis in the early grades mathematics – a learning opportunity? South African Journal of Childhood Education, 2014(1), pp. 42-60. Open source: http://www.scielo.org.za/scielo.php?script=sci_arttext&pid=S2223-76822014000100004

You can access the recommended reading texts by using the library tab on MySANTS and then click on the EBSCOhost link or using the direct URL as indicated.

9. ASSESSMENT OF THE MODULE The SANTS assessment policy is included in the Student Orientation Booklet and is also available on MySANTS. The policy provides information regarding the types of assessment you will need to do. It includes information about progression rules, perusal of marks, or requests for remarking assessments. In this module, both formative and summative assessments are done over a period of time (continuous assessment). The activities in the Curriculum and Learning Guide (CLG) are varied and are aimed at assisting you with self-directed learning. Reflecting on what you are learning and discussing it in a study group is always helpful through self-assessment. The personal reflection or review is aimed at revision, reinforcement, and self-assessment while informal peer assessment takes place during the group discussions. The following table provides a summary of the assessment for this module: 9.1 Summary of assessment Summary of assessment TYPES OF ASSESSMENT

FORM OF ASSESSMENT WEIGHTING

Formative assessment Two written assignments (100 marks each)

60%

Summative assessment Examination (100 marks) 40% TOTAL 100%

9.2 Self-assessment An activity aimed at self-assessment is included at the end of each unit. Before you complete the self-assessment activity, reflect on what you have learnt in the respective unit. Revise the main concepts and if there is any topic or concept you are unsure about, go back to the relevant unit and revise.


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9.3 Assignments

To support you in your self-directed learning and to keep track of your own progress, we will provide guidelines or the memoranda on MySANTS after the assignments have been marked and returned.

In order to demonstrate that you have gained the knowledge, skills, values, and attitudes described in the learning outcomes of the module, you need to do the following:

Complete and submit each assignment (100 marks) before the due date. Submit both assignments that constitute 60% of your final promotion mark to qualify

for admission to the examination.

The task brief (specific information regarding what to do and how to prepare for each assignment) will be explained in the assignment itself. These assignments are provided at the beginning of the semester together with your CLG for this module. The assignments are also available on MySANTS.

9.4 Semester examination

At the end of the semester, you have the opportunity to sit for a formal summative assessment. This includes the following:

Write a formal examination, out of 100 marks that will constitute 40% of your final promotion mark. Please read the SANTS Assessment Policy that deals with all aspects of the general assessment and the examination policy.

A minimum of 40% in the examination is required to qualify for a supplementary examination.

10. PLAGIARISM WARNING FOR STUDENTS Plagiarism is a form of academic misconduct that can lead to educational or disciplinary action and has severe consequences - in some cases civil or criminal prosecution. You are guilty of plagiarism if you copy from another person’s work (e.g. a book, an article, a website or even another student’s assignment) without acknowledging the source and thereby pretending it is your own work. You would not steal someone’s purse so why steal his/her work or ideas? Submitting any work that you have written but have already used elsewhere (thus not “original”), is also a form of plagiarism (auto-plagiarism). An example is when you submit the same assignment or a part of it for two different modules. Avoiding plagiarism by being academically honest is not difficult. Here is what you should do:

Submit only your own and original work. When using another person’s actual words, sentences or paragraphs, Indicate

exactly which parts are not your own (even if presented in the CLG). You must do


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this by referencing in accordance with the Harvard style - a recognised system specified by SANTS, and you must use quotation marks (“...”).

You must also reference precisely when using another person’s ideas, opinions or theory. You must do so even if you have paraphrased using your own words.

You must acknowledge any information or images that you have downloaded from the Internet by providing the URL link (web address) and the date on which the item was accessed (downloaded).

Never allow any student to use or copy any work from you and then to present it as their own.

Never copy what other students have done to present as your own. Prepare original assignments for each module and do not submit the same work

for another module. Always list any student who contributed to a group assignment. Never submit the

work as if only you worked on the assignment.

The Examination Regulations and Procedures policy contains the following in Section 7.10:

Students may not act in a dishonest way with regard to any test or examination assessment, as well as with regard to the completion and/or submission of any other academic task or assignment. Dishonest conduct includes, among other things, plagiarism, as well as the submission of work by a student for the purpose of assessment, when the work in question is, with the exception of group work as decided by the Academic Committee, the work of somebody else either in full or in part, or where the work is the result of collusion between the student and another person or persons.

All cases of suspected plagiarism will be investigated and if you are found guilty, there are serious consequences. Disciplinary action that may result includes:

You may lose marks for the assignment/activity. Your marks may be reduced by as much as 50%. You may even be given zero.

The module may be cancelled and you will have to enrol again. This is a great waste of time and money.

Your registration for that entire year may be cancelled. That means not all the marks you achieved in all the modules you enrolled for will count anything.

In some cases, prosecutions in courts of law may be instituted. Plagiarism is considered such a serious academic crime that you are required to sign the standard document (Declaration of Original Work) to every assignment that you submit by either using the assignment booklet or electronic submission. The Declaration of Original Work is printed on the cover of the assignment booklets.


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CONTENT

BACHELOR OF EDUCATION IN FOUNDATION PHASE TEACHING II 1. WELCOME TO THE MODULE ................................................................................. II 2. OUTCOMES OF THE PROGRAMME ...................................................................... II 3. PROGRAMME STRUCTURE .................................................................................. III 4. PURPOSE OF THIS MODULE .............................................................................. VIII 5. WORKING THROUGH THE CURRICULUM AND LEARNING GUIDE (CLG) ........ IX 6. SELF-DIRECTED LEARNING ................................................................................. XI 7. CORE READING ................................................................................................... XIII 8. RECOMMENDED READING ................................................................................. XIII 9. ASSESSMENT OF THE MODULE ....................................................................... XIV

9.1 Summary of assessment .................................................................................. xiv 9.2 Self-assessment ............................................................................................... xiv 9.3 Assignments ...................................................................................................... xv 9.4 Semester examination ....................................................................................... xv

10. PLAGIARISM WARNING FOR STUDENTS .......................................................... XV

MATHEMATICS TEACHING IN THE FOUNDATION PHASE 1 1 1. INTRODUCTION ...................................................................................................... 1 2. STRUCTURE AND OUTCOMES OF THIS MODULE .............................................. 2 3. GLOSSARY .............................................................................................................. 2

UNIT 1: NUMBERS AND NUMBER RELATIONSHIPS 3 1. INTRODUCTION ...................................................................................................... 3 2. STRUCTURE AND LEARNING OUTCOMES OF UNIT 1 ........................................ 4

SECTION 1: NUMBER RELATIONSHIPS, PLACE VALUES, AND ORDERING AND COMPARING NUMBERS 5 1. INTRODUCTION ...................................................................................................... 5 2. HOW TO TEACH NUMBER RELATIONSHIPS ........................................................ 6

2.1 Teaching part-part-whole relationships ............................................................... 6 2.1.1 The use of counters ............................................................................... 7 2.1.2 The use of home-made clay................................................................... 7 2.1.3 The use of connecting cubes or squares of coloured paper .................. 8 2.1.4 The use of missing part activities ........................................................... 9

2.2 Teaching “more/less” and “equal to” relationships ............................................ 12 2.2.1 The use of learners .............................................................................. 12 2.2.2 The use of counters ............................................................................. 12 2.2.3 The use of dot cards and counters ....................................................... 13 2.2.4 The use of grids ................................................................................... 14

3. HOW TO TEACH PLACE VALUE........................................................................... 15 3.1 Teaching grouping in tens ................................................................................. 15

3.1.1 The use of real objects ......................................................................... 16


BEd (FOUNDATION PHASE TEACHING) xviii

3.1.2 The use of games ................................................................................ 17 3.1.3 The use of ten-frames .......................................................................... 18 3.1.4 The use of workstations ....................................................................... 19

3.2 Teaching place value ........................................................................................ 21 3.2.1 Concrete level: Place value ................................................................. 21 3.2.2 Semi-concrete level: Place value ......................................................... 26 3.2.3 Abstract level: Place value ................................................................... 28

3.3 How to teach trading rules ................................................................................ 34 3.3.1 Trading rules: ....................................................................................... 34 3.3.2 The use of place-value boards to teach trading rules .......................... 34

4. HOW TO TEACH ORDERING AND COMPARING NUMBERS ............................. 37 4.1 The use of learners ........................................................................................... 37 4.2 The use of counters or objects ......................................................................... 38 4.3 The use of number cards and number charts ................................................... 38 4.4 The use of a number line .................................................................................. 40 4.5 The use of blank number charts ....................................................................... 44 4.6 The use of strips, grids, and written exercises .................................................. 44 4.7 The use of place value diagrams ...................................................................... 46

SECTION 2: ROUNDING OFF WHOLE NUMBERS 47 1. INTRODUCTION .................................................................................................... 47 2. HOW TO TEACH ROUNDING OFF ....................................................................... 48

2.1 Rounding off to the nearest ten ........................................................................ 49 2.1.1 The use of a story ................................................................................ 49 2.1.2 The use of a number line ..................................................................... 50

2.2 Rounding off to the nearest hundred ................................................................ 51 2.2.1 The use of a number line ..................................................................... 51 2.2.2 Completing written exercises ............................................................... 53

UNIT 2: ADDITION AND SUBTRACTION OF WHOLE NUMBERS 56 1. INTRODUCTION .................................................................................................... 56 2. STRUCTURE AND LEARNING OUTCOMES OF UNIT 2 ...................................... 58

SECTION 1: ADDITION 59 1. INTRODUCTION .................................................................................................... 59 2. HOW TO TEACH ADDITION .................................................................................. 59

2.1 Steps for introducing addition ........................................................................... 60 2.2 Methods for teaching addition ........................................................................... 64

2.2.1 Use of concrete and semi-concrete materials ...................................... 64 2.2.2 Introduce number symbols and operational signs ................................ 71 2.2.3 The order of addition ............................................................................ 74 2.2.4 Adding one and zero ............................................................................ 76 2.2.5 Doubles ................................................................................................ 78 2.2.6 Basic addition facts .............................................................................. 78 2.2.7 Repeated addition ................................................................................ 80


BEd (FOUNDATION PHASE TEACHING) xix

2.2.8 Number tracks and number lines ......................................................... 83 2.2.9 Written exercises ................................................................................. 90 2.2.10 Breaking down and building up numbers ............................................. 95 2.2.11 Add by breaking down numbers into place value parts ...................... 100 2.2.12 Vertical addition ................................................................................. 104 2.2.13 Three-digit addition using various algorithms ..................................... 107 2.2.14 Application of addition in word problems ............................................ 110

SECTION 2: SUBTRACTION 111 1. INTRODUCTION .................................................................................................. 111 2. HOW TO TEACH SUBTRACTION ....................................................................... 112

2.1 Steps for introducing subtraction .................................................................... 112 2.2 Apply to real life situations .............................................................................. 115 2.3 Methods for teaching subtraction .................................................................... 116

2.3.1 The use of concrete objects and models ........................................... 116 2.3.2 Teach subtraction as “think addition” ................................................. 118 2.3.3 Written exercises ............................................................................... 119 2.3.4 Subtract by using the breaking down method .................................... 120 2.3.5 Vertical subtraction ............................................................................ 122 2.3.6 Multiple operations: Addition and subtraction .................................... 125

SECTION 3: ADDITION AND SUBTRACTION − INVERSE OPERATIONS 129 1. INTRODUCTION .................................................................................................. 129 2. INVERSE RELATIONSHIPS ................................................................................ 132 3. METHODS FOR TEACHING INVERSE RELATIONSHIPS .................................. 132

3.1 Start with addition ........................................................................................... 132 3.2 Start with subtraction ...................................................................................... 135 3.3 Use fact families ............................................................................................. 135

REFERENCES 139

ADDENDUM A: NUMBER EXPANSION CARDS....................................................... 141 ADDENDUM B: 100S CHART .................................................................................... 142 ADDENDUM C: LESSON PLAN TEMPLATE ............................................................ 143 ACTIVITIES 1 TO 48


BEd (FOUNDATION PHASE TEACHING) 1

1. INTRODUCTION This module follows on from the module: Introduction to Mathematics Teaching in the Foundation Phase (F-MAT 120). It is based on the premise that understanding the concept of numbers, place value, rounding off and basic operations is a requirement for young children to develop further mathematical understanding. This first module of four about mathematics and mathematics teaching in the Foundation Phase will continue to empower you, as a student teacher, to teach mathematics effectively to Foundation Phase learners and to reflect on your practice. The emphasis of this module is on developmental progression as well as the theory and practice of mathematics teaching and learning in the Foundation Phase. This module deals with the following: Unit 1: Numbers and Number Relationships will introduce you to some of the core concepts in the content area of numbers – the starting point of mathematical learning. This unit presents many strategies that will enable you to effectively teach the basic number concepts of number relationships, place values, ordering and comparing numbers as well as rounding off whole numbers. You will be introduced to the subject and pedagogical content knowledge to understand, teach and assess these topics in the Foundation Phase, including identifying conceptual errors and supporting learners with such barriers to learning. Unit 2: In this unit, you will learn about the teaching of addition and subtraction. You will learn about the different types of addition and subtraction problems and strategies to solve these problems. Furthermore, you will learn about the inverse relationship between addition and subtraction. In addition, you will learn HOW to teach and assess Foundation Phase learners to add and subtract fluently and WHY operational fluency is important in the Foundation Phase. You will also learn to identify learners’ possible conceptual errors and address these in the classroom.




2. STRUCTURE AND OUTCOMES OF THIS MODULE This module consists of two units and provides you with the opportunity to work towards the outcomes listed below. 3. GLOSSARY Understanding these terms will assist you when working through this module. Addends - The numbers to be added to get a sum.

Ascending order - Shows an increase in size and value.

Commutative - If changing the order of the operands does not change the result.

Consecutive - Numbers that follow on in sequence from smallest to largest, one at a time (Modlin, 2006).

Descending order - Shows a decrease in size and value.


UNIT 1 Numbers and number

relationships Outcomes: At the end of this unit, you should be able to plan, teach and assess: Number relationships, place

value, and ordering and comparing numbers in lessons that are sequenced, coherent, and appropriate to the Foundation Phase context.

Rounding off whole numbers in lessons that are sequenced, coherent, and appropriate to the Foundation Phase context.

UNIT 2 Addition and subtraction of whole

numbers Outcomes: At the end of this unit, you should be able to plan, teach and assess: Addition and subtraction in

lessons that are sequenced, coherent, and appropriate to the Foundation Phase context.

Identify conceptual errors and support learners with these barriers.



Difference - The result of subtracting one number from another.

Operand - The quantity on which an operation is done.

Ordering of numbers - Arrange quantities or numbers according to specific criteria, e.g. from least to most.

Place value of a digit - Refers to the position of a digit in a number; this position determines the digit’s value.

Product - The answer when numbers are multiplied together.

Quotient - The answer after you divide one number by another: dividend ÷ divisor = quotient.

Rounding off - Means making a number simpler but keeping its value close to what it was.

Sum - The total when numbers are added.

Value of a digit - Refers to the specific value of a digit in a number, i.e. in the number 495 the value of the digit 4 is 400.

1. INTRODUCTION In Unit 1 you will be introduced to some of the core concepts in the Numbers, Operations and Relationships content area of mathematics. We will start by introducing numbers and number relationships concerning number sense and counting. This is the starting point of mathematical teaching and learning. This unit presents many strategies that will enable you to effectively teach the basic number concepts of number relationships, place values, ordering and comparing numbers as well as rounding off whole numbers.

UNIT 1: NUMBERS AND NUMBER RELATIONSHIPS



2. STRUCTURE AND LEARNING OUTCOMES OF UNIT 1 Unit 1 provides you with the opportunity to work towards competence in the areas listed below and consists of the following two sections:

UNIT 1 NUMBERS AND NUMBER

RELATIONSHIPS

SECTION 1 Number relationships, place

values, and ordering and comparing numbers

Learning outcomes: At the end of this section, you should be able to: Explain how learners discover

and explore the different relationships among numbers.

Provide learners with activities that will support their understanding of the place value.

Provide learners with activities to enable them to order and compare numbers.

SECTION 2 Rounding off whole numbers

Learning outcomes: At the end of this section, you should be able to: Explain how whole numbers are

rounded off. Provide learners with activities to

round off whole numbers. Identify possible misconceptions

and assist learners to correct conceptual errors.



1. INTRODUCTION A number might mean different things to different people in different contexts. To a learner, the number “7”, for instance, might mean the following different things: These multiple meanings of “7” show how the concept of “7” can be associated with different situations. As a teacher, you therefore need to know some important concepts regarding the different meanings of numbers. Numbers are related to one another through a variety of number relationships. For example:

part-part-whole (involves seeing numbers as being made of two or more parts); more/less relationships and equal to; relationship with 10; the position or order of the number; and about approximation or rounding off.

In this section, you will learn about number relationships, place value, and ordering and comparing numbers and how to teach these topics to learners. We can only describe, order and compare numbers if we understand the relationship between numbers. All these number relationships lay the foundation for place value, which we will be discussing later in this unit.

SECTION 1: NUMBER RELATIONSHIPS, PLACE VALUES, AND ORDERING AND COMPARING NUMBERS

Seven is one of the digits in my mom’s cell phone number.

7 is the number after 6 and before 8.

I live in 7 Mandela Drive. I am seven years old!



1

90 minutes

Reflect on your understanding of the following number relationships by providing two examples of each: 1. Part-part-whole. 2. The position or order of the number. 3. More and less relationships. First, write down your examples and then compare your answers to the commentary below. Commentary: An example of part-part-whole is: Owen has 6 marbles. Two (2) marbles are red, and the rest are blue. How many blue marbles does Owen have? We are given the whole and we need to find the part. This would involve subtraction, 6 − 2 = 4. An example of the position or order of the number could be: What number comes before 9? What number comes after 7? More or less relationships involve problems such as: What number is 2 less than 6? What number is 4 more than 8?

You will learn more about number relationships and how to teach this topic to learners in the Foundation Phase in this section.

2. HOW TO TEACH NUMBER RELATIONSHIPS To be able to order and compare numbers, learners need to be aware of some number relationships. We therefore start by giving ideas on HOW to teach learners to understand some important number relationships. 2.1 Teaching part-part-whole relationships The most important number relationship learners need to understand is that a number is made up of two or more parts. The ability to think about a number in terms of parts is a major milestone in learners’ understanding of numbers.

In this example, you can see how the whole (of 8) is divided into two parts: 5 objects and 3 objects. This relationship can be seen as: “Five and three is eight” or “Five plus three is eight”. In this case, 8 can be thought of as a set of 5 and 3. It means 5 plus 3 is equal to 8. This relationship can be written as: 5 + 3 = 8.



In the same way, 2 plus 6 is equal to 8; 1 plus 7 is equal to 8, etc. However, some learners find it difficult to understand that the same “whole” can be broken up into different parts and, therefore, need enough practice in part-part-whole relationships (Naudé & Meier, 2014). Different activities that can be used to practice part-part-whole relationships will be shared with you. The correct sequence when teaching any new topic in mathematics, e.g. part-part-whole relationships is from concrete to semi-concrete to abstract. In the paragraphs that follow are ideas (strategies) to follow. 2.1.1 The use of counters

Divide the class into small groups (6 to 8 learners per group). During a teacher-guided activity, work with one group at a time, while other groups are involved in side activities. Let learners work in pairs. Hand out a set of 10 counters (such as beans or bottle tops) and a large piece of scrap paper to each pair. Ask them to count out a certain number of counters (say 8). Now ask each pair to divide the designated quantity into two or more amounts. Then learners must explain how they separated the number into different sets and read the parts out loud (e.g. 8 is 5 and 3, or 8 is 4 and 3 and 1). Let learners also listen to the explanation of the other pairs in the small group. Examples of how learners can break up 8 are shown next.

five and three four and four four and two and two

2.1.2 The use of home-made clay

If you have the ingredients, you can make a large bowl of home-made clay (playdough) to use in class.

Here is the recipe:

Dissolve 1 cup salt in 2 cups hot water. Then add 2 cups cake flour and a tablespoon of cooking oil. Add more flour if it is too runny or too sticky. (Have a small bowl of flour on each

table for sticky fingers!)

In some regions, real clay (soil clay) is available. If available, you can successfully use natural clay in your classroom to teach mathematics! Hand out a fist-size ball of home-made clay to each learner. They use the clay to make different combinations of a specific number (e.g. 6). They can be creative and make nests with eggs. For example:



four eggs and two eggs Make sure they describe their combinations correctly (e.g. one nest with four eggs, another nest with 2 eggs makes 6 eggs altogether).

2.1.3 The use of connecting cubes or squares of coloured paper

Provide learners with one type of concrete aid such as connecting cubes or squares of coloured paper. The task is to see how many different combinations of a specific number they can make using two parts, e.g. make different groups of 8:

If you do not have connecting cubes, use old newspaper print to cut-out small squares. If you can, use two different colours of paper so that learners have two sets of different coloured paper. They then make one group with the one colour and the other group with another colour. This helps them distinguish better between the two groups of numbers. As soon as they can write the number symbols, let them write these down, e.g. 5 and 3.

2

90 minutes

1. Choose one of the activities discussed above to teach part-part-whole relationships. Explain what is meant by the following levels when learners complete the activity:

The concrete level. The semi-concrete level.

five and three

four and four

five and three

four and four



2. Do you have any more ideas of activities that can be used to teach part-part-whole relationships? Write down two of your ideas.

Commentary: During the concrete level, learners need to handle concrete objects (objects they can touch and feel). In the semi-concrete level, they can make drawings and use symbols. It is imperative for learners to start with the concrete level where they can use as many senses as possible, for example, to see, touch and feel. Only after learners have mastered the concept on the concrete level, are they ready to move to the semi-concrete level where they no longer work with the concrete objects but with representations thereof, i.e. drawings and pictures. Refer to F-MAT 120 for a complete explanation of the three levels: concrete, semi-concrete and abstract.

2.1.4 The use of missing part activities

Use concrete material Have empty containers (e.g. yogurt or margarine tubs) available, one for each pair of learners. Hand out a specific number of counters to each pair (e.g. the whole is 7). Tell the learners to turn the tub face down and to “hide” some of the counters under the tub, so no-one can see! One learner place a few of the counters under the tub (e.g. 4) and leaves the rest outside (3). The other learner must now identify the number of hidden counters.

Source: Jess (2009)

If the learner sees the 3 counters outside, she or he knows: “Three and four is seven”. If the learner hesitates, the hidden part must be revealed so that the learner can see and count. Learners must take turns to find the missing parts.

“The whole is seven. How many are hidden under the tub?”



Use semi-concrete material

On a semi-concrete level, introduce missing-part cards to play the same type of games as explained above. You can make your own missing part cards for the numbers, say 4 to 10. Use A4 paper (or cardboard). Cut each A4 sheet into 10 equal strips. You will need at least 60 strips. Fold each strip into three equal parts and then unfold as shown alongside. For each long strip, you need ONE extra loose piece (a third) that can be used to cover one of the parts on the long strip. Therefore, use 2 more A4 sheets, cut in 10 equal strips and fold into three equal parts. Cut each strip into 3 equal pieces by cutting on the folded lines. For your classroom, you will need a set with ALL the possible combinations for each number. HINT: Do your planning first to see how many cards you will need for each number. For example, for the number 6 we need at least the following combinations:

Source: SANTS Archive Combinations of 6

0 6 0 + 6 = 6 1 5 1 + 5 = 6 2 4 2 + 4 = 6 3 3 3 + 3 = 6

We use the number 6 as an example here and show ONE possible combination (six is four and two). Now use one of the loose pieces (third) as a flap to cover the dot set on the right-hand side. Use cello tape or a paper clip to attach the flap so that learners can easily lift the flap and cover the dots again.

Make two dot sets in the middle and right-hand side.

Write the number symbol in the left part.

6



Hand out a set of missing-part cards to each group of learners, i.e. at least 6 cards showing different combinations of the number 6. Learners use the cards as in “covered parts” to determine the part-part-whole relationship (e.g. 6 is three and three; four and two; one and five; etc.). Make sure that your learners develop a firm understanding of how numbers, like 4 and 5 are built up before you expect them to extend their work to numbers from 6 to 12. When doing part-part-whole activities, it is important that learners should say or “read” the parts out loud or draw or write them down in some form. This encourages them to focus their thoughts on the part-part-whole relationship.

3

120 minutes

1. Use cardboard to make your missing part cards for the numbers 5 and 7 (according to the instructions in the above-mentioned example). Use cardboard (you may cut from empty boxes). Take your cards along to the next academic support session or study group. (If you have time, make all the cards for the numbers 4 to 10.)

2. In approximately 80 words, justify the tendency to place more emphasis on learning on a concrete level in the Foundation Phase.

Commentary: Did you remember to do your planning tables first before you started making your missing part cards? If you laminate or use DC-fix to cover the missing part cards, they will be much more durable. Keep your set of missing part cards for future use and for your teaching.

4

180 minutes

In the next academic support session or in your own study group: 1. Demonstrate how you will use the missing part cards to develop learners’ number

concept. 2. Listen to and learn from other students. Make notes of what you learn from other

students or your tutor. Next you will learn how to teach “more/less” and “equal to” relationships.

Cover one part with a loose flap.

6 ? Six is four and …? Think first and then lift the flap to check your answer.



2.2 Teaching “more/less” and “equal to” relationships Learners develop the concepts one more or one less, two more or two less when they practically experience that there is “one without a partner” (in a one-to-one-correspondence relationship, as discussed in F-MAT 120). Also, when they share sweets between friends, they discover that someone has one or two more (or less) than the other. For learners who are at the beginning of understanding numbers, this is a difficult concept. We must make sure that we provide enough practice with concrete apparatus to establish the concepts of one more / two more / one less / two less. Furthermore, learners should be proficient in counting forwards and backwards to understand and describe “more than and less than” relationships. Therefore, ensure your learners get enough practice in counting forwards and backwards. As with all other mathematics concepts, learners should be exposed to a variety of learning experiences to develop their understanding of more/less relationships. Naudé and Meier (2014) recommend that “the more than” and “less than” concepts should be taught together because some learners find the “less than” concept more difficult to understand than the “more than” concept. When addressing the two concepts simultaneously, learners will realise the inverse relationship between numbers. For example, 18 is 2 more than 16; therefore, 16 is 2 less than 18. 2.2.1 The use of learners

Let learners participate in a “one and two more, one and two less” game. Ask one learner to make a group of 5 friends, another to form a group of 4 and so on (until all the learners are in groups of 10 or less. Let the groups stand in rows. Learners must compare the rows of learners and order the groups in ascending (increasing) or descending (decreasing) order. Ask learners questions like:

What is different about the row of five and the row of four? Which row has one more (or two more) than the row of four? Which row has one less (or two less) than the row of six? Which rows have the same number of learners?

2.2.2 The use of counters

Hand out counters (the number you are focusing on for the week, say 6) to each learner. Ask them to put out five counters. Now put out one more. How many are there now? Take away two counters. How many are there now? Continue giving instructions involving one and two more and one and two less. Encourage learners to describe the relationship using



the words “more than” and “less than” or “equal to”. Gradually increase the difficulty level and work with larger numbers. After enough practice with concrete materials, introduce semi-concrete aids together with concrete aids, e.g. dot cards (semi-concrete) and counters (concrete). The semi-concrete representation (dots or pictures) is more abstract than a concrete object, but it is not a fully abstract written form. Semi-concrete representations help move the learners towards abstract methods of recording operations.

2.2.3 The use of dot cards and counters

Let learners work in small groups at a table to make more-than sets. Provide each learner with a few dot cards (between 1 and 8) and 10 counters. Ask learners to individually construct a set of counters that is one more than and later, two more than the set shown on their card.

Use the above activity to reinforce the concept “less than” by reminding learners that the five dots are two less than the seven counters. Also, give learners the opportunity to pack out counters that are “equal to” the number of dots.

Group members must check each other’s counters to verify if their answers are correct. Let stronger learners assist struggling learners to find the answer. To expand on this activity, let one of the learners spread out his or her dot cards in the middle of the table. They should then find another card (from the other learners in the group) that is “two-more-than” or “two-less-than” each of the cards displayed. Learners must also take turns to describe the relationship between two of the displayed dot cards. For example: “Four is two less than six” or “Six is two more than four” or “Two more than four is six.” To increase the difficulty level of this activity, learners can be provided with number symbol cards and counters.

Two more than the dots on the dot cards is seven counters.

Dot card with five dots.

Five counters are equal to five dots.

Dot card with five dots.



Grade 1s must begin with working with numbers from 1 to 10. The same principles for one more / two more / one less / two less for numbers between 1 and 10 must later be extended so that learners understand that 16 is one more than 15 and two less than 18. We need to help them to make this connection by providing many opportunities to discover this concept through concrete experiences.

Grade 2s must work with an increased number range on a more semi-concrete level (using models and drawing pictures).

Grade 3s need to work with larger numbers in an abstract way. For example, using grids.

2.2.4 The use of grids

At Grade 2 and Grade 3 level, learners must be able to establish the relationship between numbers in a much more abstract way and be able to write the number names. Design activities like the following:

Grade 3:

Number Number name in words Number

just before

Number just after

2 more 2 less 3 more

413 Four hundred and thirteen

412 414 415 411 416

501

765

919

5

90 minutes

1. In a mind map, summarise the different activities that can be used to promote learners’ understanding of “more/less” relationships.

2. In 120 words, discuss the sequence in which you will present these activities. Clearly indicate the reason you have decided on this specific sequence.

Commentary: In sequencing the activities, keep the recommended trajectory of learning (concrete to semi-concrete to abstract) in mind.

Two more than the number symbol is seven counters. 5 Number

symbol 5.



We will now focus on how to teach place value. 3. HOW TO TEACH PLACE VALUE Understanding “grouping in tens” as a pre-place value concept is a prerequisite (condition) for understanding place value. It is essential that learners should have a good understanding of the meaning of “one ten” to understand place value. 3.1 Teaching grouping in tens “One ten” is still a difficult concept for most Grade 1 learners to grasp. They therefore need a lot of exposure to make groups of ten with concrete apparatus so that they eventually learn to see that (for example) a group of ten together with a group of five makes 15 (without counting). As learners encounter numbers beyond nine, they begin to see that there are big changes in the way we read and write numbers. Up to nine, we simply use a new digit symbol to give each number a name (e.g. 1 or 2 or 3 ...). After nine, we begin to group in tens (and in multiples of ten) and we place digits in different places to show their value. Now learners need to learn that:

The digits in the numbers 10 to 19 stand for one group of ten (10) and a number of units that is not enough to make another group of ten. That means that 11 is one group of ten and one unit (11 = 10 + 1), 12 is one group of ten and two units (12 = 10 + 2), etc.

The digits in numbers from 20 to 29 stand for two groups of ten (20) and a number of units that are not enough to make another group of ten. That means that 21 = 20 + 1; 22 = 20 + 2; ... and so on.

The digits in numbers from 30 to 39 stand for three groups of ten (30) and a number of units that are not enough to make another group of ten. That means that 31 = 30 + 1; 32 = 30 + 2; ... and so on.

Our number system is a base ten system (refer to B-FMA 110) and the most critical time for developing the understanding of place value occurs during the Foundation Phase when many of the required number concepts are being developed. Counting is the first step towards understanding our number system. Many learners in the Foundation Phase might not have mastered rational counting completely. Counting starts with rote counting, i.e. the ability to know the counting words in order. Rational counting,

Base ten: Refers to our decimal number system which uses 10 as the base and needs ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent numbers.



on the other hand, refers to counting with understanding, i.e. counting objects while using counting words correctly in a one-to-one correspondence as illustrated below. Rote Counting Rational Counting

How to teach rote and rational counting was discussed in the module: Introduction to Mathematics Teaching in the Foundation Phase (F-MAT 120). Rational counting above ten is the beginning of understanding place value. Foundation Phase learners need many counting experiences with concrete objects involving grouping in tens to develop a sound concept of place value. Below are some ideas to teaching grouping in 10s to promote learners’ understanding of the base ten concept. 3.1.1 The use of real objects

Provide many opportunities for learners to develop an understanding of a group of ten and how to group a larger number of objects in groups of ten. These experiences MUST at first be on the concrete level. Collect “groupable”/discrete objects, for example:

counters like small stones, bread tags, bottle tops, etc. (store them in containers like empty margarine tubs or ice-cream containers);

sticks or straws (that can be bundled together in 10s and tied with an elastic band); washing pegs (that can be clipped together in chains of tens); paper-clips (that can be chained together in 10s); and strips of 10 squares and loose squares (teacher-made from cardboard).

Have activities where learners make groups or bundles of ten with a variety of objects, e.g. bundle 10 sticks together with an elastic. When they have a clear understanding of 10, learners can start making up different numbers. Start with multiples of ten (20, 30, 40, etc.), and later, extend this activity to include 2-digit numbers that are not multiples of ten (e.g. 13, 28, 34) so that they gain experience with groups of ten and remaining ones. Interlocking cubes can also be used if these are available. To help learners develop an understanding of a group of ten and remaining ones, use an A4 sheet of paper and 20 counters. Draw a line in the middle of an A4 paper. Hand out 20 counters to each learner. Let them count out 10 counters and put them on one side of their

1 2 3 4 5

Source: The Yellow Peach (2009)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, , , , , , , , , , ,11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,22

, ,2,2 … 21, 2222,



paper. Next, have them put five counters on the other side. Together, count all the counters one-by-one. Now say: “Ten and five is fifteen”. Let learners turn the page around and say: “Five and ten is fifteen”. Repeat with other numbers more than 10 but less than 20. Learners pack out the new number without changing the 10 side of the paper. They work in small groups at their tables. Gradually, increase the number of groups of tens. Hand out a sufficient number of counters for each learner to make as many piles of ten (depending on their level) as they can with their counters. They count their piles in tens and say how many piles (tens) they have. They also indicate the remaining ones. Encourage learners to work out how many they need to make another pile of ten. (Learners can also try to borrow from one another to make as many full piles of ten as possible.) With more experience, learners will be able to count in 20s and in 50s and then in 100s.

3.1.2 The use of games

Let learners use games to practice counting and grouping in tens in a more challenging but fun way. Here is an example:

Learners work in small groups at their tables. Hand out 10 counters (e.g. beans) to each group. Each group must draw a scatter-board like the example shown, using 2 concentric circles (2 circles within one-another).

They write 1s on the outer circle and 10s in the smaller inner circle. Explain the rules and procedures of the game to the learners. Learners take turns to close their eyes and scatter their counters onto the board.

The larger, outer circle represents the units and the inner circle represents the tens. If a counter lands in the small circle (or on the line), the value is ten and if a counter lands on the outer circle (or on the outer line), the value is one. Counters that miss the board don’t count. As a group, they need to work out and write down their score. Who had the highest score? (The score on this board is 23.)

1s

10s

Does not count.



3.1.3 The use of ten-frames

Make your own ten-frames to use in class. Use a clean A4 sheet or cardboard from empty cereal boxes to make ten-frames.

You will need at least 50 ten-frames for your class.

NOTE: For Grade 1s, start with 5-frames (cut a ten-frame in half – length-wise) to establish their understanding of benchmark 5. If you are sure that they have acquired an understanding of groups of numbers that make five, introduce ten frames. Let them first use one ten-frame to make combinations of numbers up to ten. They first fill up the first frame with fives and use the second frame for the remaining numbers (e.g. for 8, they fill 5 on the first frame and 3 on the second frame). Work with numbers within their number range. For Grade 2s, you can work with larger numbers. Hand out more ten-frames (up to 10 ten-frames) and 100 beans (or ten packs of ten counters) to each small group of learners. Learners use their ten-frames to fill the tens (make groups of ten) with beans. You call out a number, and learners fill their ten-frames according to your instruction.

Fold the paper/cardboard in half and in half again. Cut into four strips.

You can make TWO ten-frames from one strip. Fold each strip in half (short side) and cut it.

Fold each strip in half (length), draw a line on the fold.

Use a ruler to measure 2 cm widths on the midline. Use a black marker to draw four vertical lines across the strip. If the last part is much wider than 2 cm, also measure 2 cm and cut off the extra part.

“Get forty-two beans and fill up your ten frames.”

Your completed ten-frame should look like this: Ten-frame



To fill up their ten-frames with 42 beans, learners need to use 5 of their ten-frames: 4 for the forty and 1 for the two extra 2 beans. Let learners also record (write down) the amount, for example:

Also, call out other numbers for learners to fill. Let learners check each other’s attempts to see if they correctly filled their ten-frames. Ten-frames can still be used in Grade 3, especially to support learners who still need to develop their concept of numbers.

3.1.4 The use of workstations

On a more advanced level, you can prepare workstation activities where learners use concrete objects to make groups of tens and ones together with a written exercise to record their answers. The above examples are also suitable activities for workstations. Workstation activities need thorough planning and should be incorporated into your daily lesson plan. An example of the planning of a workstation is shown below. Workstation 1: Units and Tens

Use counting bags (or any other container) with a specific amount of counters, e.g. 34 beans. Learners throw out the content, count it and record it as a number symbol on their form. Then they must group it into as many tens as possible, and the learners record the grouping on the form. Learners return all the counters in the container and move to the next station. Worksheet for Workstation 1:

Source: Kenrick (2011)

Group name: How many beans are in the bottle? Write the amount in words: Group the beans in tens and units TENS UNITS Write the number of beans in symbols:

______tens _____ extras



6

180 minutes

You want to reinforce your Grade 2 learners’ understanding of “grouping in tens” before introducing them to place value. Plan four different workstations to reinforce their understanding of base ten. Your planning should include:

the preparation of the required Learning and Teaching Support Materials (LTSM) for each workstation; and

a worksheet with clear instructions for workstation activity and space to record numbers represented in the activities.

Take your workstation activities to the next student academic support session or discuss with your peers in your study group.

Commentary: When planning your workstations, keep in mind that learners enjoy learning through play and that they learn best when taken through the 3 levels of learning, i.e. from concrete through semi-concrete to abstract. Thus, provide for concrete and semi-concrete activities. Recording the numbers at each work station will promote learning on an abstract level. Activities should be within the prescribed number range for Grade 2 learners. Be creative and develop at least one activity of your own.

7

180 minutes

1. In your own study group or at the next academic support session, set up four workstations with different activities on “grouping in tens” by selecting one activity from each group member.

2. Each group member works through the four workstations and copy and completes the checklist (see below) for each workstation. Note: Comments should not be limited to aspects which have been ticked off under the “No” column. Remember to add suggestions on how to improve the activity if this is required.

3. As a group, discuss your experiences, findings and suggestions for improvement on the activities.

Checklist for Workstation Criteria Yes No Comment

The activity is developmentally appropriate for Grade 2 learners.



The activity will reinforce learners’ understanding of “grouping in tens”.

The worksheet’s instructions are clear.

Appropriate space is provided for the recording of numbers.

Suggestions on how to improve the activity:

4. Each group member should consider the suggestions for improvement of their own

activities. 3.2 Teaching place value Place value is the basis of our entire number system (refer to module B-FMA 110). A place value system is one in which the position of a digit in a number determines its value. In the standard system, called base ten, each place represents ten times the value of the place to its right. You can think of this as making groups of ten of the smaller unit and combining them to make a new unit. Ten ones make up one of the next larger unit, tens. Ten of those units make up one of the next larger unit, hundreds. This pattern continues for greater values (ten hundreds = one thousand, ten thousands = one ten thousand, etc.), and lesser, decimal values (ten tenths = one, ten hundredths = one tenth, etc.). To promote learners’ understanding of this concept, we first start with concrete experiences and then proceed to semi-concrete and abstract experiences. 3.2.1 Concrete level: Place value The use of concrete models for base ten concepts can play a key role in supporting learners to understand the idea of “a ten” as both a single entity and as a set of 10 units (van de Walle, Lovin, Karp & Bay-Williams, 2014). These models should be “put-together-and-take-apart” models so that learners can physically group ten items together as one ten or take them apart to have 10 ones. Furthermore, van de Walle et al. (2014) claim that the model for ten should be ten times larger than the model for a unit and ten times smaller than the model for one hundred. Dienes blocks is an example of a teaching aid that can be effectively used to demonstrate the relationship between ones, tens, hundreds, etc. on a concrete level.



Keep in mind that Foundation Phase learners will be focusing on mastering place value as reflected below: Grades 1 and 2: Ones (units) and tens. (Grade 1: two-digit numbers up to 20; and Grade 2: two-digit numbers up to 99.) Grade 3: Ones (units), tens and hundreds (3-digit numbers up to 999).

8

150 minutes

1. How would you expect a learner to group the ice cream sticks below to reveal the number of ice cream sticks as a base ten numeral?

2. How can learners represent the same number using base ten Dienes blocks? 3. Are the base ten Dienes blocks more effective in showing the representation? Why

do you say so? 4. Use base ten blocks to show the exchange, from 19 to 20, in a concrete display. 5. Draw displays of 17 and 27 using base ten blocks. Which represents the biggest

number? How do you know? Commentary: Group the ice cream sticks into groups of ten. There will be 2 groups of ten and 3 loose units will remain. The number of sticks is 23.

Thousands Hundreds Tens Ones



In the display for 17, there is only one ten and 7 units. In the display for 27, there are two tens and 7 units. Thus, 27 represents the biggest number as this number has more tens.

The importance of using concrete material in the development of learners’ understanding of the base ten number system cannot be emphasised enough, as proven by the research in the case study below. The complete article by Broadbent (2004) can be accessed at: http://search.ebscohost.com/login.aspx?direct=true&db=eue&AN=15254092&site=ehost-live Reading: Place value - A case study of the Base Ten Game (Broadbent, 2004) The initial objectives of this project were to research approaches for improving students’ mathematical learning outcomes in relation to the base ten number system and methods for tracking students’ understanding of numbers within a whole class setting. The research also aimed to explore the strategies that support the learning of each student as he or she progresses to the next stage of understanding the number system.

Becomes 20 blocks – 2 groups of 10 blocks 19 blocks Plus 1 more block

17 blocks 27 blocks

Representation of 23



The research attempted to build upon the work of previous studies that explored ways in which teachers could more actively assist students to develop their understanding of the structure of the number system. The project explored the role of a commonly used teaching activity, referred to in the project as the “base ten game”, in developing learners’ understanding of our number system. The game involves students using a place value board and concrete materials to develop an understanding of the structure of the number system and to learn to operate on numbers using this structure. To play the most basic version of the game, the student rolls two dice, adds the numbers shown, and collects that quantity of pop-sticks to add to their game-board, which is ruled up into place value columns (see the table below). The only rule of the game is that there can be no more than nine items in any one column.

Understanding the place value column, the tenth stick is combined to make a bundle of ten sticks, which is then placed in the tens column. Rubber bands can be used to hold the bundles of sticks together. The only rule of the game is that there can be no more than nine in any one column, that is, in the units column there can be no more than nine sticks, in the tens column there can be no more than nine bundles of ten sticks, in the hundreds column there can be no more than nine bundles of 100 sticks, and so on. The project was undertaken using a model of action research, involving nine teachers.

The teachers that were selected to participate in this study had all previously undertaken extended professional development to incorporate constructivist learning theories in their teaching. The teachers were keen to explore the use of concrete materials, especially the base ten game in relation to student learning. The five project schools represented the diverse range of communities within the independent sector, including an isolated rural school with high numbers of indigenous students, small and large urban schools, and two schools with high numbers of students from a low socio-economic background.

The research took place between March and October 2001 and involved 280 primary students. The research methodology enabled the teacher researchers to focus on improved student learning through changes in teacher practice and allowed concentration on the practical, day-to-day realities of the classroom. Under the broad question, “What are the most effective teaching methods and management structures



that will maximise the learning of the base ten number system in a whole class setting?” each teacher developed their own specific research question.

Throughout the project, the teacher researchers were guided by a Project Officer, Mrs Andrea Broadbent. The teacher researchers came together for professional development and sharing of experiences, received visits to their schools, and kept reflective journals and student work samples. Each teacher reflected on their learning and wrote a report on their learning journey. A final research report, drawing together common elements, was constructed.

The major findings from this project are listed below.

Teachers needed to develop their own knowledge of the base ten number system before they could help students learn. The teachers’ own understanding of the number system improved when they focussed on features of the number system that they wanted their students to learn. By clarifying the desired learning outcomes, the project teachers were better able to identify learners who were having difficulties and plan appropriate learning experiences for these learners. Professional development that focussed on the learners’ conceptual understanding also assisted teachers to develop their own understandings.

Once teachers had developed their own relational understanding of the number system, they were better able to:

o discover what each student already knew about base ten; o diagnose any misconceptions that a student might have developed; o offer learning activities that enabled students to build their knowledge;

and o adapt learning activities to meet the individual learning needs of the

diversity of students in the class. Concrete materials, such as those used in the base ten game, can make a

significant contribution to the development of students’ conceptual and procedural knowledge about the number system across all year levels.

Any single set of concrete material or any single teaching activity highlights only certain aspects of the number system. A deep understanding requires a range of materials and activities, chosen according to the features of the number system that they highlight.

The base ten game is a valuable core activity for students of all year levels who are still trying to make sense of the structure of the number system. The addition of complementary activities that both support the ideas being developed through the base ten game and look at the same ideas differently, will enhance its usefulness.

This project revealed the necessity of explicitly developing links between the concrete materials, the learning activities and the structure of the number system to support the development of a relational understanding of place-value.



9

240 minutes

After reading the case study above answer the following questions: 1. What was the aim of this research project? 2. Why will you consider the findings of this research project in your classroom

practice? 3. Broadbent (2004) suggests that concrete materials should be used when teaching

place value. Discuss how this research supports Piaget’s ideas of learning as referred to in the first Mathematics module: Introduction to Mathematics Teaching in the Foundation Phase and Education Studies 2: Theories of Learning and Teaching.

4. Why should teachers have a sound understanding of our number system and its foundations?

5. Could you use this base ten game in your class as a useful teaching strategy? Provide reasons for your answer.

Commentary: To decide if the findings of this project will be of any significance to you, consider the size of the project that is determined by the number of participants. Also look at the relevance of this research to mathematics in the Foundation Phase. Piaget’s ideas of learning were discussed in the module Introduction to Mathematics Teaching in the Foundation Phase (F-MAT 120) as well as in Education Studies 2: Theories of Learning and Teaching (B-EDS 122) during your first year of studies. Revise these sections before answering Question 3. When revising these sections in the relevant modules, you will realise that a learner in primary school is, according to Piaget at the concrete operational stage. Revisit the section about “Developing your own Mathematical knowledge” in the Introduction to Mathematics Teaching in the Foundation Phase (F-MAT 120), before answering Question 4. Base your decision in Question 5 on the cost and availability of the game and the relevancy of this game for learning in the Foundation Phase. Consider how Foundation Phase learners learn best and what is in the curriculum requirements.

3.2.2 Semi-concrete level: Place value You can use grid (squared/quad) paper to make your 2-D base ten LTSM. Cut-out a square to represent a 100 (H), strips to represent tens (T) and loose squares to represent units (U). Ten small squares should fit onto the tens-strip and ten tens-strips should fit onto the hundreds-square.



Ask learners to represent a number using their strips and squares, e.g. “Show me 23”. Learners can then decide how they want to go about showing the 23. If learners counted out 23 loose units, ask them to place each of the units on a strip of ten squares. If the square strip is full, they can replace their ten loose units for one strip. It is important that learners grasp the idea of “10 units make 1 ten.” Ask learners to show more numbers using the models, for example:

8; 16; 32; etc. 39; 46; 78; etc. 123; 231; etc.

More examples are shown of how numbers can be presented, using hundreds blocks, tens strips and units squares or drawings thereof. These examples are already moving towards the abstract level by placing the models in a place value diagram.

Hundreds-square: Large 10 × 10 squared block to represent 100 (H).

Tens-strips (1 × 10 squares) to represent tens (T).

Small squares to represent units (U).



The number 111 can be presented as: The number 233 can be presented as: The above representation already introduced learners to place value charts (diagrams) that will be discussed later in this section. A place-value chart is a way to make sure digits are in the correct places. An excellent way to see the place-value relationships in a number is to model the number with actual objects (place-value blocks, bundles of craft sticks, etc.), write the digits in the chart, and then write the number in words as well as in normal numerical form and expanded notation. An example will be shown later in this section. Once learners had enough experience in place-value relationships on the concrete and semi-concrete level they can proceed to experience place-value relationships on a more abstract level. The focus should be to develop learners’ number skills to such an extent that they will be able to recognise the place value of digits in given numbers. 3.2.3 Abstract level: Place value The understanding of place value is essential to all later mathematics. Without it, keeping track of greater numbers rapidly becomes impossible. Learners need to know the names of the places for recording ever increasing numbers. Can you imagine trying to write/represent 999 with only ones? A thorough mastery of place value is essential to learning the operations with larger numbers. Understanding place value is the foundation for regrouping (“borrowing” and “carrying”) in addition, subtraction, multiplication, and division. After enough base ten experience on the concrete and semi-concrete level, you can introduce learners to base ten cards (also known as flard cards or number expansion cards). These representations introduce learners to the structure of written numbers using place value.

1. The use of base ten cards to build up and break down numbers

Base ten cards (number expansion cards) allow learners to build up numbers using hundreds, tens and units.



Building up and breaking down numbers using base ten cards involve composing (combining) or decomposing (taking apart) a number in multiples of hundreds, tens and units.

e.g. 300 + 50 + 7 = 357. When the base ten cards are moved over one another (overlapped) so that the zeros don’t show, they will look like this:

It is important to point out to learners that in the number 357, there are:

3 hundreds; 35 tens; or 357 units.

357 = 3 hundreds + 5 tens + 7 units = 300 + 50 + 7. This way of writing numbers is called expanded notation It is easy to make your own sets of base ten cards (number expansion cards) for your classroom! Templates for base ten (number expansion) cards are included in Addendum A at the end of the module. Make full use of that! Here, the layout of a set of base ten cards is shown.

100 10 1

200 20 2

300 30 3

400 40 4

500 50 5

600 60 6

300 50 7

300 50 7

300 50 7



Give each learner a set of cards and ask them to pack them out in sequence. For Grades 1 and 2, start with only the tens and units. For Grade 1, begin with a number between 1 and 20. Grade 2s work within the number range 1 to 99. Use the cards with the multiples of 10 to 90 and the single digit cards 1 to 9. Work within a range of numbers learners is familiar with. Ask learners to build, for example, 27 using their base ten cards. At first, learners are likely to take the numbers 2 and 7 and place them next to each other. Rather than telling them what to do, you can say: “What you have here is 2 and 7. Does 2 and 7 make twenty-seven?” This will encourage the learner to reflect, think and realise that 27 is made up of 20 and 7. The 7 must go on top of the 0 of the 20 to make 27. Once all the learners in the small group composed their number, ask them to unpack or “break up” the number and describe which cards they have used. Build up (compose the number) Break down (decompose the number)

Once learners can work with 2-digit numbers, 3-digit numbers can be introduced, for example:

700 70 7

800 80 8

900 90 9

Base ten cards should be large enough and easy to handle. Draw lines to cut 9 strips (exactly the same width and length). Write the numbers as shown alongside. Cut out the numbers (9 cards with 100 to 900; 9 cards with 10 to 90 and 9 cards with 1 to 9). Give one set to every learner. The long cards are the Hundreds, the medium length cards are the Tens and the short cards are the Units.

7 20 + = 2 7 2 7 = 20 + 7



Build up “357”: Learners must be able to recognise that 27 = 20 + 7 (Grade 2); and 357 = 300 + 50 + 7 (Grade 3). Note: Many learners will not understand place value if we write a number as 357 = 3H + 5T + 7U, because this way of writing is too abstract. They will understand better if we say 357 is three hundreds plus fifty plus seven (300 + 50 + 7). Building up and breaking down numbers using base ten cards show how many hundreds, tens and units there are in a number. When learners can confidently compose and decompose numbers, give more challenging instructions, for example:

Build the number “56”. Then add 20 (change the cards to show the answer). Build the number “374”. Then subtract 70 (change the cards to show the

answer).

10

90 minutes

1. Why is it better to write the number 987 in expanded notation as 987 = 900 + 80 + 7 and not 9H + 8T + 7? Explain in your own words.

2. How can writing a number in expanded notation deepen place value understanding and help Foundation Phase learners to get a better understanding of the difference between the place value and the value of a digit?

Commentary: Refer to the Fundamental Mathematics module (B-FMA 110) to ensure that you know and understand the difference between a value and a place value of a digit.

The value of 9 is 900, while its place value is hundreds. The value of 8 is 80, while its place value is tens. The value of 7 is 7, while its place value is units.

H T U 9 8 7

In the next paragraph, the use of place value diagrams in the Foundation Phase classroom will be discussed.

300 50 7



2. The use of place value diagrams Place value diagrams are more abstract LTSM and should be introduced only when you are sure that learners have sufficiently explored grouping and counting in tens in a practical and concrete way. Draw the place value diagrams on the chalkboard and ask learners to redraw a copy in their books (tens and units for Grade 2 and hundreds, tens and units for Grade 3). Explain to them that a number is made of one or more digits. The position of a digit in a number is very important. A digit's value depends on its position in the number. Show them HOW you write a number in the place value diagram and explain the value of each digit in the diagram.

Hundreds Tens Units

Use place value headings to help learners work out the value of each digit in a number. The number 692, for example, is made up of 3 digits, 6, 9 and 2. The digits 692 can be placed under the place value headings in the following way:

Hundreds Tens Units 6 9 2

Explain: 692 contains 6 hundreds, 9 tens and 2 units and is written 692 = 600 + 90 + 2.

The number 692 is written in words as six hundred and ninety-two. Understanding place value will help learners see that the placement of digits is critical in determining their value. They also learn that the sequence for writing down numbers follow a certain rule.

You also need to teach learners about zero as a placeholder in numbers. This is a critical concept in understanding place values.

3. Zero – The place holder

Although zero (0) has no value, we need to write it down at its specific place in a given number. Let’s explain this using 30 as an example. In 30, the place value of the digit “0” is units and the value is zero. The place value of the digit 3 in this

The PLACE VALUE of the 9 is TENS. The VALUE of the 9 is 90.

The PLACE VALUE of the 2 is UNITS. The VALUE of the 2 is 2.

The PLACE VALUE of the 6 is HUNDREDS. The VALUE of the 6 is 600.



number is tens, and the value is 30. If you don’t write the zero down, the number becomes 3. Consequently, the place value and the value of the digit 3 now change to units and 3, respectively.

The purpose of writing the zero is to hold the units’ place in this example as well as to keep the digit 3 in its correct place, namely tens. Therefore, to reinforce the writing down of zero in a number, learners should be encouraged to represent a number, e.g. 30 in a place value diagram as follows:

tens units 3 0

The above diagram clearly shows that there are 3 tens and 0 units in the number 30.

11

60 minutes

Explain what the impact will be on the place value and value of the digits in the number 4 025 if you do not write down the zero.

What learners should know about the place value of numbers is summarised in the next diagram.

In the next paragraph, you will learn more about trading rules and how to teach this to your learners.

Each digit in a number has its own value. The place where we write the digit in the number determines its value:

8 888

We use the ZERO digit as a placeholder to show places in the number where there is no value:

1 023 1 203 1 230

no hundreds

no tens 8 000 (8 thousands)

8 (8 ones)

no units 800 (8 hundreds)

80 (8 tens)

In our number system we group numbers in groups of tens.

What learners should know about place value:

1



3.3 How to teach trading rules An understanding of place value underlies the understanding of certain “trading rules” that govern place value and enable whole number operations to be done. Trading means to “exchange”. Trading can be done because the value of ten units is equal to one ten (10 units = 1 ten). In the same way, ten tens are equal to one hundred (10 tens = 1 hundred), and ten hundreds are equal to one thousand (10 hundreds = 1 thousand). 3.3.1 Trading rules: The following trading rules are important in the Foundation Phase:

Ten units can be traded for one ten. One ten can be traded for 10 units. Ten 10s can be traded for one hundred. One hundred can be traded for 10 tens.

Do not merely teach these trading rules. Learners first need enough practical experience with concrete apparatus to gain an understanding of these rules. By understanding place value, learners will realise that when they take one from the tens column, they are actually taking one group of ten units. They will also realise that when they add numbers in the units column and arrive at a sum above 9 that the amount carried to the tens column actually represents one or more groups of ten.

3.3.2 The use of place-value boards to teach trading rules Place-value boards are simple boards divided into two (or three) sections to hold units and tens pieces (or units, tens and hundreds pieces for Grade 3). You can make your own! To make your own place value board, use A4 paper (or cardboard) and fold it in half. Colour the left-hand side. Write the words TENS and UNITS on the board.

tens units

Learners need to see that, when given a group of more than 10 objects, the number of objects remains the same no matter how they are arranged. Let learners work in pairs. Each pair receives a container with 50 counters (use bottle tops, sticks or straws or any counters that can be easily grouped together) and a place value board.



Tell the learners to put a number of counters greater than ten on the white side of their boards (U), e.g. 34. Ask the pairs to check each other’s counting. Now ask them to make groups of ten and to move each group of ten to the coloured side (T).

tens units

Ask learners questions like:

How many tens are there? How many loose (extra) counters? How many do you have altogether?

To connect learners’ arrangements to number symbols, you can help them write the number down, for example: “3 tens and 4 units”. Let learners now use their remaining counters (of the 50). Ask them if they have enough counters to make another group of 10 to place on the coloured side. Let learners continue making groups of ten to place on the coloured side until they don’t have any counters left. There should be 5 groups of ten on the “tens” side (50 = 5 tens). Another way to help learners construct an understanding of regrouping and renaming through trading activities, is to use learners’ place value boards and paper squares in different colours representing tens and units. Supply them with paper squares, e.g. 40 grey (units) and 20 black squares (tens). Clearly explain that the grey squares represent units and the black squares represent tens.

Learners should use the black and grey squares to repeat the above activity. They first pack out 34 grey squares on the white side (units). Then make groups of ten squares and

Three groups of ten.

Four units.

The black squares are the tens. If you get 30 black squares, it means you have 30 tens (that means 300).



“trade” each group of ten grey squares for one black square. They place the black squares on the coloured side (tens) to represent 3 tens. Four grey squares remain under “Units”. =

Ensure that learners understand that one black square represents 1 group of ten grey squares. Therefore, 3 black squares are equal to 30 grey squares; 30 squares plus 4 squares are equal to 34 squares. When you are sure that learners understand that the blacks are “tens” and the greys are “units”, you can extend the activity. Let them, for example, add seven more grey squares to the units. Ask: “What happens to the units?” Encourage learners to explain what happens. Remind them that there cannot be more than 9 in their units column (right-hand side). Therefore, they should trade 10 grey ones for 1 black square, which should be placed in the tens column.

Their representation should now reflect 34 + 7 = 41 as follows:

Once learners have a clear understanding what to do when units are added, and more than 9 units are in the units column, ask them what they will do if you need more units than are available in the units column. For example, using the example of having 4 tens and 1

tens units

tens units



unit as reflected in the above place value diagram, ask learners how they can take four units from their place value diagram. Encourage learners to discuss the problem and try it out until they find the solution. Verify if all the learners trade in one black for 10 greys, take away 4 and note that they now have 3 blacks and 7 greys (that is 37).

Learners should try several examples until they can confidently apply the trading rules because a clear understanding thereof is imperative in addition and subtraction calculations.

12

90 minutes

Have you ever learnt about “trading rules” in your mathematics class at school? If not, explain what difference it would have made in your mathematics proficiency (ability) should you have been introduced to it.

OR:

If you were introduced, explain how it assisted you to be proficient in mathematics. Commentary: In your response consider the following:

the base ten number system; understanding place values; and the impact of understanding trading rules on a learner’s proficiency in addition

and subtraction calculations. Learners with a clear understanding of the value of numbers will not find it difficult to compare and order numbers. 4. HOW TO TEACH ORDERING AND COMPARING NUMBERS Learners must learn HOW to describe, order and compare numbers in terms of more, less, equal, approximately equal to, and also first, second, …, last. Here are some practical ideas on HOW to teach ordering and comparing numbers. 4.1 The use of learners Divide learners informally into groups of 6 to 8. Let learners compare themselves in their groups in terms of “Who is the shortest?” / Who is the tallest? / “Who is the youngest?” / “Who is the oldest?” etc. Let learners be creative in comparing various aspects of themselves, e.g. who has the longest neck, biggest feet, etc. Let learners also arrange themselves in their groups from shortest to tallest, etc. Now, extend the activity further by asking the tallest (or shortest, oldest, etc.) learner in each group to come forward and then compare their length, age, etc.

1



Reinforce the activity by asking learners to describe their findings by making drawings (e.g. learners draw a picture of the order of learners in their small group from tallest to shortest, etc.). Encourage them to write a few sentences involving the appropriate vocabulary, e.g. taller than, shorter than, almost the same length. If learners cannot write sentences yet (Grade 1), let them tell each other who/what is taller than, shorter than and almost the same length. 4.2 The use of counters or objects For Grade 1 learners who are still working with small numbers, you must use counters to compare and order numbers. Put out piles with different numbers of counters or objects. Ask learners to count each pile (find the total number of each group) and then order the groups in descending order (from greatest to smallest). They must then describe the comparison with appropriate and correct mathematical language.

NOTE: It becomes impractical to use counters to work with very large numbers. Grade 2s, for instance, work with large numbers (e.g. 200). Counting 200 counters or making piles with such large numbers are not practical at all. Use counters at the beginning of Grade 2 when they still work with smaller numbers. Gradually, increase the difficulty level and introduce semi-concrete LTSM like dot cards (let learners arrange dot cards in order from the most dots to the least dots). Eventually, work with numbers only. 4.3 The use of number cards and number charts Prepare cards with numbers on them between 0 and 50. Hand out a card to each learner. Let learners arrange themselves in groups of 5 according to their cards in numerical order from the smallest to the biggest number.

This pile has 10. It is the greatest (highest) pile.

This pile has 2. It is the smallest (lowest) pile.



Let each group describe their position in the row using the words:

between; greater than; smaller than; smallest; and largest (biggest).

Expand on the above activity by doing the following: Let learners take out their 100s-charts. Each learner must have a 100s-chart. An example is included in Addendum B of this module. More examples are available on the Internet and some can be printed for free. You can make these for your learners. Have it laminated so that it is more durable and usable for several years. Hold up a number, e.g. Ask learners questions such as:

What number is this? (35) Show me 35 on your number chart. How many tens are in this number? How do you know? Can you tell me which number is just before 35? Which number is just after 35? Which number is between 34 and 36? Is 36 greater than or smaller than 35? Is 34 greater than or smaller than 35? How far is 35 from 40? Double 35 is …? Half of 35 is about …?

Let learners use their 100-chart to show you the answers. Repeat with other numbers. Encourage learners to write down the comparisons, e.g. 54 is smaller than 55, or 55 is greater than 54.

For Grade 2 and 3 learners, use the number charts and number ranges appropriate to their grades to do similar activities. It is important to reinforce comparing and ordering activities by letting learners complete written exercises. They should describe the relationships by using words like greater than, smaller (less) than, etc.

2 5 17 35 45

35

A hundred chart: contains the numbers from 1 to 100 in sequential order with ten numbers per row.



13

180 minutes

Summarise each of the practical ideas discussed under Ordering and Comparing Numbers. Note the LTSM to be used as well as the role of the teacher (what the teacher must do) and the role of the learners (what the learners will be doing) in each activity. Use your summary to answer the questions. 1. If you need to teach Ordering and Comparing Numbers to learners in the

Foundation Phase, which sequence will you follow? Explain your answer in an essay of approximately 200 words.

2. Which other ideas can you think of to teach Ordering and Comparing Numbers? Write down your ideas.

A number line is a powerful tool to use to order and compare numbers. 4.4 The use of a number line Number lines can be used effectively to teach learners to order and compare numbers. A large number line must be displayed along the top of the chalkboard or any other suitable spot in your classroom on the wall. The number line must represent the correct number range of your grade. Start off with:

Grade 1 and 2: o From 0 to 10 for counting in ones. o From 0 to 100 for counting in tens.

Grade 3: o From 0 to 100 for counting in tens. o From 0 to 1 000 for counting in 100s.

The number lines must be large enough for all learners to see.



Note the following important aspects about number lines:

0 1 2 3 4 5 6 7 8 9 10 The arrows on a number line signify that the line does not stop there. The numbers being shown on the line also do not stop there. The line and the numbers go on to infinity. You should explain to learners that the numbers on either side of the number line carry on and on. The positive numbers go on, as far as you would like them to go; they go to infinity. The negative numbers (which Foundation Phase learners do not work with according to CAPS (DBE, 2011), though they might begin to hear about them) extend on the other side of the number line, to negative infinity. Learners in the Foundation Phase do not learn about the concept of infinity but you can give them an awareness that the number lines show that the line is just a drawing of the numbers we have chosen to label on, it could go on in either direction. This kind of teaching prepares them for the mathematics they will learn in higher grades. A wall number line from 0 to 20 is shown below.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Learners must draw their own number lines in their books. Assist them to do this. It will be easier for them and more accurate if they can use squared/quad paper to do this. Vary the learning experiences on a number line. The number line below shows the numbers from 10 to 20:

10 11 12 13 14 15 16 17 18 19 20

Smaller numbers on this end; < shows “less than”.

Equal distances between numbers.

Larger numbers on this end; > shows “more than”.

Arrows on both ends show that the numbers carry on and on.



Ask learners questions such as:

Which number is greater: 14 or 12? Why do you say so? Which number is between 16 and 18? Is it exactly in the middle? How do you

know? Is 15 less than 16? How can you be sure? Which numbers are more than 15 but less than 18? Which numbers are less than 13 but more than 10?

Increase the difficulty level for Grades 2s and 3s. They should be able to indicate “missing numbers” on a number line in a variety of ways. Keep the number range for each Grade in mind. Number lines do not have to be labelled in 1s. The scale chosen for a number line can vary; you can label a number line in 2s, 5s, 10s, 20s and so on. When you vary the scale of the labels on a number line, you can still ask learners to identify “missing numbers”. This is a good activity to consolidate number and pattern concepts. You will learn about number patterns in Mathematics Teaching in the Foundation Phase: 4 (F-MAT 424). Let learners write the numbers represented by the letters on each number line under the letters:

0 A B 30 C D 60 E F G 100 As an extension activity, let learners play the “secret number game”. Draw a large number line up to 20 on the chalkboard. Then say: Work out the secret number I am thinking about. For example: My number is:

one more than 6 but less than 8; half of 18; double 8; one bigger than half of 20; or I am between 6 and 10, more than 7, but less than 9.

As they gain confidence, expose them to much more challenging “secrets”! In small groups, learners can now also make up their own secret numbers for their group members to work out. When learners are more advanced, you can ask secret number questions, and learners should answer without using the number line (only using the “mental” number line, i.e. in their minds).



Remember, Grade 3s must work with much larger numbers on the number line. Learners in Grade 3 should also be able to draw their own number lines. Give them enough practice by giving challenging instructions, for example:

Draw your own number line. Start at 575 up to 600. Count in 5s. Mark the points on the number line to show the position of each number. Remember

to spread your points evenly across your number line. Write all the numbers from 575 up to 600 in intervals of 5 underneath the points on

the number line. Remind learners to plan their number line before they start. They should work out how many numbers they need to write altogether from 575 to 600. Let learners use their number line to answer questions like:

Which number on a number line is in the middle of 575 and 585? Is 575 less than 585? How can you be sure? Which number(s) are more than 575 but less than 585? Which number(s) are less than 595 but more than 585? What is the third number on your number line? (Also ask learners to show the

position of the first, middle, fourth, last number, etc.) Make a large numberless number line for your class that can stretch across your chalkboard. Use 3 large equal strips of cardboard pasted together with cello tape to create your number line. Now, you can easily write any numbers on the chalkboard below your number line. Let learners use their number cards and pegs to show their answers, e.g. “Show me 50 on the number line”. If the number line is scaled, ask learners questions involving comparing, ordering, doubling, halving, etc. To do this, learners need to study the scale of the labels used on the number line and find out where to place the number you have asked them to place. Here is an example. Ask the learners where they will place 50 on the number line. (They can place it using a number card as shown.)

0 5 10 65

50



4.5 The use of blank number charts Use blank number boards/charts and let learners play the “Where is the number?” game.

How to play the game (this example is appropriate for Grade 3):

One learner in the group calls out a number between 0 and 100 (e.g. “Show me where 44 should be placed on the board”). The other learners find the place on the chart and explain their reasoning (e.g. “I put my finger in the first row and count down in tens to forty, then I count 4 to the right in the row”). Give each learner in the group a chance to call out a number.

4.6 The use of strips, grids, and written exercises Learners in all grades must be exposed to written exercises. Follow the golden rule: Start with simple exercises at their ability-level and gradually increase the difficulty level of the exercises. REMEMBER: Move from the simple to the complex. For Grade 1s, you can display your number symbol cards in the wrong sequence against your chalkboard. Ask learners first to arrange the number cards in the correct sequence and then to write the order of the numbers in the correct sequence in their books. For example:

Later, when learners’ number ranges are extended, let them complete a grid to show their understanding of number relationships. For example:

1 2 3 4 5 6 7 8 9

Blank number board game: Grade 1: 1 to 10; Grade 2: 0 to 50 and Grade 3: 0 to 100.



Number Before After 3 more than

5 less than

Write number name

Position

14 13 15 17 11 Fourteen Fourteenth 20 31

NOTE: Adapt the grid according to the number ranges appropriate to each grade. For Grade 2, you can design strips like the following. Ask learners to find the wrong number(s) in the sequence and correct the sequence.

134 135 136 136 138 Design written work for Grade 2 learners to find numbers before and after a given number and to fill in the missing numbers. Learners must fill in the missing numbers (on their small chalkboards, books or on a worksheet). They must complete number strips like the following:

121 123 124 127

111 115 119 125 129 Learners can complete more advanced tables when they are ready (e.g. Grade 3). The first example in each table is completed to show what is expected:

Number before

300 less than 30 more than Number after

600 601 307 7 15 45 78 79 440 725 62 471 95 429 145 320 411 300 425 299 300 399 362 109

Grade 3s are on a much more advanced level and must be able to work with numbers on an abstract level. They should be able to arrange a list of numbers in sequence from largest to smallest or smallest to largest.



Written activities like the ones below can be provided: Arrange the following numbers in descending order (greatest to smallest):

300; 700; 100; 500; 900 _______________________________________

236; 263; 336; 632; 362 _______________________________________

692; 269; 962; 296; 926 _______________________________________

Grade 3 learners must also be able to compare numbers and describe the relationship between numbers.

Fill in greater than or smaller (less) than sign between the numbers to make the statements true:

234 ________ 243

328 ________ 238

766 ________ 677

889 ________ 898

NOTE: Grade 3 learners do not have to know the symbols > for greater than and ˂ for less than. These are given for your knowledge. 4.7 The use of place value diagrams A practical way to help Grade 3 learners compare and order two 3-digit numbers is to write the numbers in columns so that the digits with the same place value line up. Once learners have a good grasp of place value and place value diagrams, they can be shown HOW to use the place value diagrams to compare 3-digit numbers. Draw a place value table on the chalkboard and ask learners to copy it in their books. They must write the place value headings first: H; T; U. You write the numbers (e.g. 692 and 629) on the chalkboard and ask: “Which is bigger 692 or 629?” Learners must write the digits in the correct columns and then compare the digits. They must start on the left. See the following example:

Greater than > Less than ˂



H

(Hundreds) T

(Tens) U

(Units) 6 9 2 6 2 9

Explain to learners HOW to reason using the place value columns:

Start at the hundreds column. Both numbers have 6 hundreds. The hundreds are equal.

Now look in the tens column. 692 has 9 in the tens column, and 629 only has 2 in the tens column (9 is greater than 2, in other words 90 is greater than 20).

That means 692 is bigger than 629. Repeat with other numbers within the learners’ number range. Remember: Learners must practice a lot!

14

60 minutes

Use numbers not used above to show HOW you would use different teaching strategies to teach ordering and comparing numbers.

In this section, you have learnt different relationships among numbers’ place values and different strategies to teach learners to order and compare numbers. In the next section, we will look at rounding off whole numbers and how to teach this topic to learners in the Foundation Phase.

1. INTRODUCTION Rounding off numbers is an important part of estimation. Rounding off is used when we do not need completely accurate information. Simpler numbers that are “approximately equal to” or “close to” a given number are used to represent more “complicated” numbers. Rounding off means to replace a number by another value that is approximately equal but has a simpler representation. We use rounding off almost every day in our lives, for example when shopping, paying bills, cooking, etc.

SECTION 2: ROUNDING OFF WHOLE NUMBERS

Remember, Grade 3 learners already work with 3-digit numbers!

1



In this section, you will learn to round off numbers and how to teach learners to round off numbers within the number range for their specific grades. 2. HOW TO TEACH ROUNDING OFF We DO NOT always need accurate information. Sometimes you need a good idea of “how much” but not exactly how much. Sometimes it is impossible to know exactly “how much” but an estimate will give you a good enough idea of what you have to work with. Ask prompting questions to show learners that we do not always need (or we are not always given) accurate information, for example:

How much money will you take to the shop if you have calculated that the goods that you need to buy will cost R48,97?

If you know that there are about 25 000 cars manufactured per year, what does it mean?

If people say that approximately 3 000 people were at the soccer match, does it mean that somebody counted them one-by-one?

If we say that there were about 300 people at the funeral, is that the exact number of people who attended the funeral?

The next activity is designed to get you thinking about the words that are used in the context of using round numbers and rounding off, based on the examples given above.

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120 minutes

1. Can you suggest why the words about and approximately are in bold? 2. Can you find answers to the above questions? Write down your answers. Commentary: These words are in bold because in the context that they are being used they indicate that we are not looking for an exact answer but that we are estimating. In this way, we need to make an educated guess based on our understanding of number and number relationships. The amount of money needed to cover the expenses in the first question is R50. This is a rounded or estimated amount. In the other three questions, the numbers given are not exact, they are estimates. This is indicated by the rounded off number given. We do not think that the person saying “there were 300 people at the funeral” counted the people. But if the person said there were “289 people” at the funeral, this would indicate that the exact number of people were counted.



When you introduce rounding off to learners, it is a good idea to ask questions like the ones above. This will stimulate learners cognitively and lead them to deduce their own generalisations. In this manner, they are generating rules using logic and reason. 2.1 Rounding off to the nearest ten To get to the “rounded” numbers spoken about above, there is a mathematical process called “rounding off”. There is a mathematical rule for how this is done so that everyone will round off in the same way. The rule works for numbers of all sizes. In the Foundation Phase, Grade 3 learners are expected to round off to the nearest 10. The tens are the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90, etc. These numbers are called round numbers of ten. The numbers 11, 46, 23 and 89 (for example) are not round numbers. These numbers represent a count (or a few counts) above or below a certain round 10 number. You need to teach your learners how to round off these numbers to the nearest ten. There are several ways you could do this. The rule is that if the digit on the immediate right is less than 5, round down. If the digit on the immediate right is 5 or more, round up. For example: 64 becomes 60 because 4 is less than 5, (61 to 64 is rounded down to the nearest 10, thus 60); and 68 will become 70 because 8 is more than 5 (65 to 69 is rounded up to the nearest 10, thus 70). Below are some ways in which you can teach rounding off to the nearest ten.

2.1.1 The use of a story

Draw an example like the following on your chalkboard. Make up a story, e.g. “Imagine that it is raining heavily. Thumi is at house number 16 and must run to the nearest friend’s house for shelter. He has friends living at number 10 and number 20. Which house is nearer, number 10 or number 20?” Where will Thumi run to?

10 11 12 13 14 15 16 17 18 19 20

10 20 ?



Introduce the topic by explaining to learners how, why and when to round off numbers. Explain to learners that 16 (Thumi’s position) is “nearer to” 20 (than to 10). Ask more questions that allow the learners to think about rounding up and rounding down. Also, discuss the rule that from 15, you round up to 20, although 15 is in the middle between the two tens. This rule applies to the number in the 5s position between any two tens. Next, let us look at how a number line can be used as a visual tool to teach rounding off.

2.1.2 The use of a number line

A number line is one of the best tools to use to explain rounding off. Use a number line to extend the example of the boy in the rain. Draw a large number line on the chalkboard. Write the numbers 0 to 10 on your number line as shown next.

0 1 2 3 4 5 6 7 8 9 10 Ensure all learners are facing the chalkboard so that they can easily see the number line. You stand in front of the 5, facing the 5 with your back towards the learners. Tell learners that from the 5 onwards numbers lean towards the 10. Demonstrate this by stretching out your arms and lean towards the 10. Take one step to your left to stand at the 4, nearer to the 0 and tell the learners that all the numbers less than 5 lean towards the 0. Now stretch out your arms and lean towards the 0.

Now repeat this exercise. Stand at the 5 and together with the learners stretch out to lean towards the 10. Then move to number 4 at your left and lean towards the 0. If you see that they grasped the concept, use the learners and individually place them at a specific spot on the number line and let the learners lead the way to lean towards the 0 or the 10. Ensure your learners understand that the numbers 0 to 4 is nearer to 0 and will be rounded off to 0. The numbers 6 to 10 is nearer to 10 and will be rounded off to 10. The number 5 is in the middle but is rounded up to 10 (the larger number). Let learners draw their own number lines for rounding off. Let them first master rounding off to the nearest 10 before you proceed to teaching rounding off to the nearest 100.

Example: Round off 2, 5, 7 and 8 to the nearest 10.

Let learners:

Draw a number line showing the units from 1 to ten in their class workbooks.



Plot the points that must be rounded off clearly on the number line. Use the plotted points to assist them to decide which ten is nearer.

Remind learners of the rule for 5!

The rule says if the digit is less than 5, rounding off is done to the previous (lower) 10. If the digit is 5 or more, it is rounded off to the next (higher) 10. Thus, 2 is rounded off to zero while 5, 7 and 8 are rounded off to 10. This can be seen on the number line. Grade 3 learners are not expected to round off to the nearest hundred. The following discussion is for your information to equip you should there be an advanced Grade 3 learner wish to find out about rounding off to the nearest 100. 2.2 Rounding off to the nearest hundred If learners have grasped the concept of rounding off to the nearest 10, gradually increase their understanding of rounding off to the nearest 100. The rule is similar to the rule for rounding to the nearest ten, but it focuses on the tens digit (the digit in the tens place) since we are now looking at 3-digit numbers. When rounding off to the nearest hundred, all numbers with tens digits of less than 5 are rounded down to the previous hundred, while numbers with tens digits of 5 or more are rounded up to the next hundred. 2.2.1 The use of a number line

Round off to the nearest 10: Let your Grade 3 learners, for example, help you to find 232 on the number line.

Ask learners questions like:

Where must we plot 232? (Shown above.) Between which two tens is this number? (230 and 240.) How do you know? (232 is 2 more than 230. Learners might give other correct

explanations. Encourage discussion.) This means that 232 rounded off to the nearest ten is 230.

0 1 2 3 4 5 6 7 8 9 10

230 231 232 233 234 235 236 237 238 239 240



Do the same with 235 (remember the rule), 237 and 238. Round off to the nearest 100: Labelling a number line in tens, shows the position of 232 between 200 and 300:

Is 232 nearer to 200 than to 300? Motivate your answer? Which hundred is nearer: 200 or 300?

Which hundred is nearer? The number line can assist learners to see that 200 is nearer than 300. The digit in the tens place is 3; therefore, 232, rounded off to the nearest hundred, is 200, as shown by the arrow on the number line. Once learners have established HOW to round off 232, the same number line can be used to explore rounding off further. Let learners, for example, round off 267 and 250 to the nearest 100. Rounding off 267 to the nearest 100, ask:

What is the digit in the tens place? (6) What is the rule for rounding off when the digit being rounded is a 6? (Round up)

Rounding off 250 to the nearest 100, ask:

What is the digit in the tens place? (5) What is the rule for rounding off when the digit being rounded is a 5? (Round up) What is 250 rounded off to the nearest hundred? (300)

Here is another example: Round 123, 143, 153 and 173 off to the nearest hundred. First, assist learners to determine the range of the number line by asking: “Between which two hundreds do these numbers lie?” It will assist them to determine where the number line must start and end. Let learners then:

Draw a number line showing the tens from 100 to 200 in their class workbooks. Plot the points that must be rounded off clearly on the number line.

267

232

200 210 220 230 240 250 260 270 280 290 300



Use the plotted points to assist them to decide which hundred is nearer. Draw arrows to indicate which hundred is nearer to each of the given numbers. Write down the rounded off (simplified) number for each given number.

The number line is illustrated below, and an explanation is given.

100 110 120 130 140 150 160 170 180 190 200 The rule says: If the tens digit is less than five, rounding off is done to the previous hundred. Thus, 123 and 143 are rounded off to 100 (round down to the previous nearest hundred). Furthermore, the rule says: If the tens digit is five or more than five, rounding off is done to the next hundred (round up). Thus, 153 and 173 are rounded off to 200 (round up to the next nearest hundred). This is shown by the arrows on the number line. Let learners do written exercises to practice estimation and rounding off. 2.2.2 Completing written exercises

Design worksheets where learners must first estimate the answers by rounding the numbers off to the nearest ten and then calculate the exact answer. Learners must then also compare the exact answer with their estimates. Below are examples of questions that can be used for learners to practice their skills. Example questions on rounding off:

Round off to the nearest 10: o 14 o 165

Round off to the nearest 100: o 79 o 450

First, round off the numbers, estimate the answer, and then calculate the correct answer:

o 173 + 35 = o 253 + 149 =

As an extension you can ask: What is the difference between the estimated and the actual answer? Why is there a difference?

123

143

153

173



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240 minutes

1. Be creative and write your attention grabber story as an introduction to a Grade 3 rounding off lesson. Use a context which will be familiar to Grade 3 learners.

2. Use the SANTS lesson plan template in Addendum D to develop a lesson plan on rounding off for Grade 3 learners. Include the story you have created in no 1 in the introduction part of the lesson presentation. In your lesson plan ensure that you include:

Lesson objectives Pre-knowledge LTSM Lesson presentation (include all three lesson phases i.e. introduction,

development and consolidation) In the lesson presentation clearly indicate what you expect learners to do, what your role will be and how the LTSM will be used to enhance learning.

Now that you came to the end of Unit 2, do the following review activity.

17

240 minutes

During a Grade 3 mathematics activity, a learner used the following number expansion cards to build up the number 138: 1. What does the above representation tell you about this learner’s understanding of

the value of digits in numbers? Identify and briefly discuss the learner’s misconception.

2. Develop activities to remediate the learner’s misconception progressively. Start with an activity on base ten concepts and grouping in tens, which form

the basis of understanding place value. Then develop a place value activity on the concrete and semi-concrete level. Finally, design a worksheet so that this learner can practice how to solve place

value related problems using number expansion cards and place value diagrams. The worksheet should: o Require the breaking down of five different numbers using number

expansion cards. o Require the building up of five different numbers using number

expansion cards.o Represent five different numbers in place value diagrams.

1 3 8



o Be in line with the prescribed number range for Grade 3 learners. 3. In 120 words, discuss why teachers should continuously build learners’

understanding of place value, in other words: Why is a clear understanding of place value imperative in being successful in mathematics?

4. How will you explain to a learner to round off 138 to the nearest 10? Explain all your steps and use labelled diagrams to illustrate.

5. How will you convince a Grade 3 learner that 151 should be rounded up and not rounded down when rounding off to the nearest hundred. Use the following to aid and substantiate your explanation:

a number line; and the rounding off rule.

6. According to Heroldt and Sapire (2014, p. 44), error analysis is interwoven with the teacher’s own content (subject) knowledge and pedagogical content knowledge, as well as the teacher’s knowledge of conceptual development. Do you agree with this statement? Write a paragraph of 150 words to explain why you do or do not agree with this statement.

Commentary: Not all errors that learners make are attributed to reasoning mistakes. Some are simply due to negligence, i.e. careless errors (Herold & Sapire, 2014). However, the mistake made above by the learner does not point towards a “slip” only, but it seems like a misconception (conceptual error) about the place values in numbers. It, therefore, is important to do thorough error analysis, as this will assist you as the teacher to understand the learner’s thinking. This, in turn, will help you to support learners in correcting conceptual errors. Learners’ conceptual errors must be addressed and rectified as soon as they are detected. (You will learn more about the importance of identification of conceptual errors through assessment and how to address these in Module 4 of Mathematics Teaching in the Foundation Phase.)

In this Unit, you learnt how learners discover and explore the different relationships among numbers and how to provide learners with activities that will support their understanding of the place-value and ordering and comparing numbers. You also learnt how to round off numbers and how to provide learners with activities to develop and strengthen this skill. Before you go on, reflect on what you have learnt so far and complete the self-assessment activity. If your answer is UNSURE or NO on any of the criteria, go back to the relevant section to study it again.



18

90 minutes

If your answer is UNSURE or NO on any of the criteria, go back to the relevant section to study it again.

Self-assessment activity: Unit 1

Now that I have worked through this unit, I can:

YES UNSURE NO

Explain how learners discover and explore the different relationships among numbers.

Provide learners with activities that will support their understanding of the place-value.

Provide learners with activities to enable them to order and compare numbers.

Explain how whole numbers are rounded off.

Provide learners with activities to round off whole numbers.

Identify misconceptions and assist learners to correct conceptual errors.

In Unit 2, which is the next unit and the final unit of this module, the focus is on the first two operations, namely addition and subtraction, and the relationship between these operations.

1. INTRODUCTION Before learners are introduced to the basic operations, teachers need to ensure that learners have a good concept of numbers, because the operations all work on numbers. In the Introduction to Mathematics Teaching in the Foundation Phase module (F-MAT 120), you were introduced to number concepts. In Unit 1 of this module, number concepts were extended and reinforced. In this unit, you are introduced to the first two basic operations, addition and subtraction. The operations of multiplication and division are dealt with in the next module of this course.

UNIT 2: ADDITION AND SUBTRACTION OF WHOLE NUMBERS



The basic operations are clearly different although they are related in many ways. Learners will grasp through understanding that addition and subtraction are inverse operations. An understanding of these basic operations will enable learners to grasp the inverse relationship between these operations. For example, it is important for learners to understand that 3 + 6 = 9 and 9 – 6 = 3. When you add a number and later subtract the same number the result is the same number that you started with. Understanding the inverse principle enhances learners’ proficiency in the conceptual understanding of numbers, reasoning, problem solving and computational fluency. In this unit, each operation is first introduced as a concept in Sections 1 and 2 respectively, after which the inverse relationship between these operations will be dealt with in Section 3. The connections between the two operations will be made throughout the unit as connected learning enables a deeper understanding on the part of learners. Mastering basic facts and calculations are based on a clear understanding of these basic operations and their relationship with each other. Although historically, the basic operations such as addition and subtraction (and multiplication and division) were taught separately from each other, this unit advocates an approach where addition and subtraction are taught together. The same would apply to multiplication and division. In this unit, we acknowledge that there are important relationships between the operations.

19

60 minutes

Using the information above, consider these questions: 1. How were you taught basic operations at school? 2. Was that useful/not useful to your understanding of the relationship between these

basic operations? 3. Why do you think the inverse principle is considered important for Foundation Phase

learners to understand? Commentary: Teaching basic operations as separate operations focuses more on algorithms (quick mechanical methods) than on developing fluency and reasoning about the operations. In this way of teaching, an understanding of these basic operations and how they relate to each other is not focused on. Understanding the inverse principle enhances learners’ proficiency in conceptual understanding of numbers, reasoning, problem solving and computational fluency and, hence, is critical to the teaching of basic operations in the Foundation Phase.

Inverse: Means something is the opposite or reverse of something else.

1



2. STRUCTURE AND LEARNING OUTCOMES OF UNIT 2 Unit 2 has three sections and provides you with the opportunity to work towards competence in the areas listed below.

SECTION 3 Addition and Subtraction – Inverse Operations

Learning outcomes: At the end of this section, you should be able to: Explain the inverse relationship between

addition and subtraction. Use knowledge and understanding to teach

learners about addition and subtraction as inverse operations.

UNIT 2 ADDITION AND SUBTRACTION OF

WHOLE NUMBERS

SECTION 1 Addition

Learning outcomes: At the end of this section, you should be able to: Use the correct vocabulary

relating to addition as a basic operation.

Identify the different situations that call for addition.

Demonstrate the use of multiple representations for the conceptual development of the concept of addition.

Teach and assess the use of algorithms for addition.

Discuss the role of using addition in problem-solving.

SECTION 2 Subtraction

Learning outcomes: At the end of this section, you should be able to: Use the correct vocabulary

relating to subtraction as a basic operation.

Identify the different situations that call for subtraction.

Demonstrate the use of multiple representations for the conceptual development of the concept of subtraction.

Teach and assess the use of algorithms for subtraction.

Discuss the role of using subtraction in problem-solving.



1. INTRODUCTION Educators agree that early mathematics learning is sequential, i.e. learners should acquire certain foundational knowledge and skills before proceeding to the next level. For example, as you learned in the introduction module (F-MAT 120), learners must be able to count with one-to-one correspondence before doing addition and subtraction. When teaching whole number operations in the Foundation Phase, we aim at developing calculation proficiency and understanding of the four basic operations. The operations of multiplication and division are dealt with in Module 3 of this course. To develop meaning for the basic operations, teachers must design activities for learners to move through the experiences from the concrete to the semi-concrete to the abstract, linking each of these to the others. When learners initially learn the operations of addition, subtraction, multiplication and division, they begin by performing operations on only two numbers. As they advance in their understanding of operations, they start to operate on more than two numbers. They then realise that this entails operating in the same way, on two numbers at a time, until they have operated on the whole number string in the question. The first operation that will be dealt with in Section 1 is that of “Addition”. 2. HOW TO TEACH ADDITION Addition means to make more, to increase, to find the sum, to calculate the total or to add. We use the “plus” sign (+) to show addition. For example: 6 + 3 = 9. Addition is the inverse (opposite) operation of subtraction. For example: If 6 + 3 = 9 then 9 – 3 = 6. Addition is commutative (refer to B-FMA 110), meaning that the order in which the numbers are added does not matter, i.e. 6 + 3 = 3 + 6.

SECTION 1: ADDITION



The important components you need to teach about ADDITION are:

Grade 1: Adding by counting forwards in 1s, 2s, 5s and 10s. Grade 2: Extend adding by counting forwards, including counting in 3s and 4s. Grade 3: Extend adding by counting forwards, including 20s, 25s, 50s and 100s. Adding by putting together groups of numbers; Repeated addition; and Adding 2-digit and 3-digit numbers within the recommended number range for each

grade, by using: o number lines; o breaking down numbers; and o vertical addition.

The recommended number range for each grade is: o Grade 1: Add to 20; o Grade 2: Add to 99; and o Grade 3: Add to 999.

20

60 minutes

1. What may be a probable reason for a Foundation Phase learner to struggle with addition?

2. How will you remediate this problem? Write down your own ideas. Commentary: To remediate the problem, you should consider the various levels of counting, the various levels and strategies of learning, as well as the different LTSM that can be used to promote counting skills. Revise Unit 2 in F-MAT 120.

We start off by suggesting steps for teaching addition. These progressive steps should be followed, especially in Grade 1 when learners are introduced to whole number operations. 2.1 Steps for introducing addition Before learners will be able to add groups of numbers as a single number, they must be able to count with understanding and have an idea of the value (how-many-ness) of numbers. During the first few weeks of Grade 1, you start with small numbers (e.g. numbers between 0 and 5), especially in cases where learners did not attend Grade R. When you are sure that learners understand the numbers 0 to 5, you must progress to numbers up to 10.

2



You already know that all teaching in the Foundation Phase should follow progressive steps to teach, i.e. from the concrete to the semi-concrete and eventually, to the abstract. You should follow the same steps for addition. On the abstract level, learners learn how to write the operation using number symbols and the operation symbol (+). Let learners then apply the steps in real life situations. To teach, for example, the addition sum 1 + 2 = 3, progress must be as follows:

STEP 3: ABSTRACT – Introduce the number symbols and then the number names formally.

Learners must understand the operational signs and use it in a number sentence to describe equivalence (=). They must also use the number symbols to represent numbers, e.g. 1, 2 and 3 in a number sentence.

One and two is three

STEP 1: CONCRETE – Make sets with real objects (e.g. beans or counters). Learners must make sets with real objects and physically put all the objects together to find the sum/total: Make a verbal statement such as: “You have just made a sum that reads: One plus two equals three”. Informally introduce the operation signs (+ and =).

+ =

STEP 2: SEMI-CONCRETE – Use dot cards. From concrete objects (sets) progress to cards with pictures or dots on them, e.g. Progress to: Informally introduce the operational signs (+ and =) and number symbols (1, 2 and 3).

+ =

One plus two equals three

1 2 3 One plus two equals three.

+ =



Finally, introduce the written number names when learners are developmentally ready to read and write number names.

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90 minutes

Considering the number range for each grade, reflect on the appropriateness of the above steps in each of the grades in Foundation Phase. Commentary: Think about the practical implications of using, for example, counters when involving 2 and 3-digit numbers in addition problems. Think about alternative concrete material that can be used instead of counters should it be necessary to start teaching from the first step in Grade 2 or 3.

You now have been introduced to the sequence of steps to follow when you teach addition. Learners must also be introduced to different methods to assist them in doing whole number operations. They need a variety of interesting ways to practice and reinforce their understanding of addition. The following Grade 1 activity calls for addition up to 5 posing semi-concrete and problem-solving questions.

22

120 minutes

Complete the following to show the expected answers of Grade 1 learners: 1. Write number sentences for the following representations:

a. and equals b. and equals

2. Solve the following by making a drawing and writing a number sentence for each.

a. Nosisi has 6 green marbles and 2 red marbles. How many marbles does she have?

b. Tshepo has 5 blue marbles and 3 green marbles. How many marbles does Tshepo have?

one two three + =

2C id i



3. Design activities, similar to the Grade 1 activity above, for Grade 2 and Grade 3 learners. Note: Keep the number range for the different grades in mind.

Commentary: 1. Write a number sentence using symbols for:

a. and equals 4 + 1 = 5 b. and equals 3 + 2 = 5

2. Solve the following by making a drawing and writing a number sentence. a. Nosisi has 6 green marbles and 2 red marbles.

How many marbles does she have? (8 marbles ; 6 + 2 = 8) b. Tsehpo has 5 blue marbles and 3 green marbles.

How many marbles does Tshepo have? (8 marbles ; 5 + 3 = 8) 3. Your activities should follow progressive steps to teach addition, i.e. from the

concrete to the semi-concrete and eventually to the abstract. They must align with the curriculum for the specific grade, i.e. Grade 1, Grade 2 or Grade 3.

In Grade 1, you will start with pictures of real objects and pictures of shapes, adding with totals up to 10. Learners might also draw dots (semi-concrete) but then they should soon start to write the sums using the mathematical symbols (numbers and the + symbol). Learners need to write addition number sentences to show the calculations they have done from Grade 1. They should not use drawings to record what they have done; drawings can help them to do the calculations, but the recording should be symbolic. This will help them to move from concrete through semi-concrete to abstract representations. In Grade 2 and Grade 3, the number range for calculations is extended, and different concrete and semi-concrete representations such as base ten blocks and base ten number cards (number expansion cards or sometimes also called Flard cards) can be used to help learners extend their adding strategies to cope with the higher number range. Once again, these representations are to help them understand the operation strategies and they should be used to help learners understand what they are doing, but they should not be used for the recording of the calculation. Numeric symbolic records should be done to write up the calculations that have been done.

In the next paragraph, we will give you some ideas on HOW to teach addition using different methods and interesting formats.



2.2 Methods for teaching addition At first, learners must use different informal strategies/methods to do addition. They use, for example, concrete materials and models, number lines, building up and breaking down numbers. Start teaching addition with:

Real objects such as small toys, crayons, leaves, stones, shells, buttons, etc. Sort/group them together and match the same objects together. Count and specify more, less, and equal.

Compare groups that have the same objects and same and different numbers as well as groups that have different objects and same and different numbers of objects, e.g. compare 6 leaves with 5 beads.

From using real objects, you can progress to using cut out shapes such as circles, triangles, etc.

Pictures of real objects and pictures of shapes could follow these (semi-concrete). Lastly, use dots (semi-concrete) and symbols (abstract).

Let’s look at some ideas on HOW to use concrete and semi-concrete materials to teach addition. 2.2.1 Use of concrete and semi-concrete materials To help Foundation Phase learners build a rich understanding of addition, you need to use concrete materials as main tools. Concrete materials and models serve as “thinking tools” to help learners to understand the operation. Their understanding improves if they can relate mathematical facts and symbols to an experience they can visualise (see). Concrete materials serve as a reference for later work, but also for constructing understanding for the basic facts (mental mathematics). The most well-known and easily available concrete materials used by learners are their own fingers. As the numbers get larger than 10, learners use counters and physically move them around to solve problems. From there, learners progress to drawing pictures or models (things that represent objects) to solve their problems on a semi-concrete level. Eventually, learners are ready to work with number symbols on an abstract level. Because addition means putting together various groups of objects, it is important that learners understand part-part-whole relationships. You have already been given some ideas on HOW to teach part-part-whole relationships in Unit 1.



Next, we will give you some more ideas on HOW to help learners combine groups (or sets) of objects to make a new bigger group to find out how many objects they have altogether. In other words, how to work with concrete objects to solve addition problems.

1. Making groups with real objects Hand out a pack of counters (beans, bread tags, bottle tops, and small stones – anything that can be counted) and two A4 pages to each learner. On the one page, draw two large circles (large enough for the counters to fit in). Draw a third circle, large enough to hold the total number of counters that are involved in the specific sum, on the second page. Each learner receives enough counters to represent the total, e.g. to teach 1 + 2 = 3, each learner must have 3 counters. You could tell a story or make up a question that leads to the addition of 1 + 2. For example, say to the learners “One girl is sitting on a bench. Two more girls go sit next to her on the bench. How many girls are sitting on the bench?” Ask the learners to show this using their counters. Ask, “How will you show what numbers you need to add?” Learners should work out that you need to add 1 and 2, and to show this you need to put one counter in one circle and 2 counters in the other circle on the one page. Now ask the question: “How many counters are there altogether?” “Use your counters to show this”. Learners physically take the one counter in the first circle and the two counters in the second circle and put them together in the third circle on the second page. Then learners count their counters and say how many they have. Tell your learners: “You have just made a sum that reads: One plus two is equal to three.” Then write this on the chalkboard using symbols: 1 + 2 = 3 so that learners begin to realise how to record addition symbolically. Enough practice will help learners to grasp the idea of addition by putting together a number of objects in a group to another group of objects to give you the total. Therefore, repeat the above activity by giving learners more story sums in a variety of contexts to solve using counters and different “groups”, for example:

Make nests with play dough. Use empty containers such as tins or boxes. Use a piece of wool to make circles.

Remember to increase the number range gradually.



It cannot be emphasised enough that in the Foundation Phase, enough opportunities to manipulate concrete objects will promote learners’ understanding of an abstract concept such as addition. Therefore, more examples of concrete material, that can be used to develop learners’ addition skills, are suggested. 2. The use of counting frames and other concrete materials It is important that you have a variety of concrete materials available in your class. Grade 1s especially need direct access to counters, counting frames, connecting cubes and other concrete apparatus to manipulate. If you do not have these resources in your school, you will need to make them yourself. Grade 2s and 3s also need to use concrete aids to extend their understanding of addition and how to add in the higher number range specified for those grades. Use pumpkin seeds, beans, bottle tops, or any other concrete objects.

Source: Krebs (2013) Pxfuel.com [n.d.] Consider the possibilities of using recycled materials to make your own concrete apparatus such as a counting frame.

Source: SANTS Archive 3. The use of pictures From letting learners make and join sets with concrete objects, progress to working with pictures that represent real objects, e.g. 1 + 2 = 3.

A homemade abacus (counting frame) made from used polystyrene packaging, wool and beads is a valuable resource to use in your class. Ideally, all learners must have their own abacus. They move the beads on their abacus to help them solve problems involving addition. Learners can make their own abacus during a class project.



In this picture sum, we say: “One butterfly plus two butterflies equals three butterflies”. In this case, we can add the objects together as they are all butterflies. It is important to expose learners to the correct mathematical language during both concrete and semi-concrete activities.

NOTE: If you use pictures for addition or subtraction (or any other operation), use pictures of the SAME THINGS (or the same category), e.g. butterflies, insects, animals, flowers, etc. It is confusing and mathematically incorrect to add or subtract several types of things.

This establishes the principle that only “like terms” can be added or subtracted, i.e. objects that are the same.

For example, it doesn’t make mathematical sense to add 1 bird to 2 bananas or subtract 1 bird from 2 bananas. In both the pictorial representations below, there is NO answer or solution.

It is impossible to answer: “How many of WHAT?” Are we referring to birds or bananas?

From using pictures (semi-concrete), we can progress to using dot cards to show addition in another semi-concrete way. Dots are more abstract than pictures, but they are still semi-concrete as they are visual representations of numbers or quantities; they are not symbols. 4. The use of dot cards Dot cards can be effectively used to help learners to abstract the idea of addition.

+

= ?

− = ?

+ =



For example, the number sentence: 1 + 2 = 3 can be shown in the following semi-concrete way:

Encourage learners to use their informal language to talk about and explain their thoughts orally. They must say what they think and explain HOW they reason. Informally introduce the operation sign “+” to describe the operation.

On a semi-concrete level, learners can make their own drawings to present “the putting together of groups”. 5. The use of learner drawings Give learners many opportunities to use drawings and mixtures of words and symbols to represent what they say and do, as well as HOW they think. Allow and encourage learners to use the symbols and operational signs to show the operation. Do not hold them back if they are ready to advance to this level. 1 + 2 = 3 can be shown in the following way:

Gradually help learners to link their concrete explorations of adding to using mathematical symbols and language correctly. Check that your learners have represented the picture sum using the same kinds of objects. So, in the representation above, 1 apple plus 2 apples equals 3 apples. If the learners had drawn different objects, such as 1 apple + 2 bananas, then the total is not equal to

+ =

= equals:

Make the operational sign cards in the same way you made your dot cards, number symbols and number name cards.

+ plus:



3, as we cannot add bananas to apples. We can only add like objects. In the mathematical terminology we refer to these same objects as “like terms”.

As soon as learners are ready to write “sums” (number sentences) symbolically, allow them to do this; do not hold them to drawings. They need to record mathematics using the correct mathematical symbols – you will see when they are ready. Learners will also enjoy writing the symbols as it is faster and more efficient than drawing pictures! Grade 2 and 3 learners should be more proficient in writing addition number sentences. They work with higher number ranges that will require concrete materials such as base ten blocks to show the addition of bigger numbers. Learners might want to draw pictures of these representations (semi-concrete), but some might feel confident to write the symbolic number sentences immediately. You need to be sensitive to the level of the learners. Different learners might be at various levels, and you must be aware of this. You could also provide drawings (semi-concrete pictorial representations) of base ten blocks, showing the addition of bigger numbers. Number lines (you were introduced to these in Unit 1, Section 1, paragraph 4.4) are another semi-concrete representation that can be used in the teaching of addition, starting from Grade 1 and progressing to higher number range addition in Grades 2 and 3. See the progression from HOW Grade 1 learners solved the same problem with counters (concrete) and then using drawings (semi-concrete) in the following example. Ultimately, learners can write the number sentence. Mary picked up 4 pebbles, and Bongani picked up 5 pebbles. How many pebbles did they pick up altogether? At first, learners use concrete materials such as counters to solve addition problems:

Mary’s pebbles must go in one group: I count them 1, 2, 3, 4. Bongani’s pebbles must go in another group: I count them 1, 2, 3, 4, 5. Four counters and five counters must go in the “altogether” group. If I count them all, they make 9 pebbles altogether. Therefore, the learners picked up 9 pebbles altogether.

+ =



Later, learners progress to detailed representations (drawings) of the situation. They do not have to use counters any longer, they use pictures to represent the situation. They can also refer to drawings that you have made.

23

120 minutes

You have been introduced to several methods for the teaching of addition. These include concrete activities, pictures, dot cards and own drawings done by learners.

1. In what way does the sequence of teaching activities for addition concur with Bruner’s idea of learning that progress from enactive to iconic to symbolic representations as you have learnt in the module: Introduction to Mathematics Teaching in the Foundation Phase (F-MAT 120), and the module Education Studies 2: Theories of Learning and Teaching (B-EDS 122).

2. Why is it important to teach the concept of addition initially at the concrete level? Explain in 10 to 15 lines.

3. Do you think it would change the outcome of learning if you started teaching addition at the abstract level? Give reasons for your answer.

Commentary: The suggested sequence of activities is aligned with Bruner’s idea of learning. Manipulating concrete material is in accordance with Bruner’s enactive or action-based stage. Semi-concrete representations, such as pictures, is in line with Bruner’s iconic or image-based stage. The abstract level, e.g. writing of number sentences is like Bruner’s symbolic or language representation. This would be useful to help learners to make their own conclusions to generalise the concept being taught. They will understand better if they use real objects where more senses are used at first, i.e. touch (feeling), seeing, manipulating (handling), etc. Moving from the concrete to the semi-concrete to the abstract is in line with how learners learn.

Number sentences and operation signs are abstract. Learners need to learn what these numbers and symbols mean when written in a number sentence by doing concrete activities that are linked to the written number sentence. That is why you start by introducing addition to learners in a concrete way first. If we start at the abstract level,

If Mary picked up 4 pebbles and Bongani picked up 5 pebbles, they picked up 9 pebbles altogether.



Foundation Phase learners may not have a conceptual understanding of what the operations and symbols mean. They will also not understand the importance of the equals sign and the idea of balancing the left-hand side with the right-hand side. The actual counting of physical objects establishes the concept of addition and enables learners to understand the role of the plus sign and equals sign when writing a number sentence.

Learners need to learn HOW to talk about and write mathematics correctly. Putting experiences into words helps promote meaningful learning. By using mathematical language, learners verbally express their understanding of what they are doing. In the next section, we show you HOW to introduce operational signs in a concrete and understandable way.

2.2.2 Introduce number symbols and operational signs

Although Grade 2 and especially Grade 3 learners work on a more semi-concrete and abstract level, teachers are cautioned NOT to START working on the abstract level when they move to the higher number range. The move to symbols is often made too quickly, and the use of concrete materials dropped too soon. Materials and representations that help learners develop their operational concept must be used at every level. Using concrete materials must come first at any level and then it must be followed by using visual representations and symbols.

It is crucial to introduce and use correct mathematical language, symbols and operation signs so that learners have the “tools” to say and write what they do in the correct mathematical language. Make sure that learners understand what each of the symbols and signs means so that they can use them appropriately. Help learners to see that the mathematical symbols and operational signs are short ways to write what they do and say.

It is important that learners understand what each of the operational signs mean. For addition, the plus and equals signs are used:

Although number sentences and operational signs are abstract, you need to introduce them to learners by linking them to the concrete work you have done and to the semi-concrete representations you have shown the learners. In this manner, learners’ understanding of number sentences and operational signs will be enhanced. It is imperative that learners understand the concept of “equality” or “equal to” in number sentences.

Add, put together, join, find the total, make more, increase with.

“The same as” or “is equal to”.

+ plus: equals: =



To illustrate the concept of “equal to” or “equality”, a balance or a scale is a powerful tool. If learners in the Foundation Phase struggle with this concept, the equals sign can be conceptualised as a balance (van de Walle, Karp & Bay-Williams, 2010). For the scale to be “in balance”, the left-hand side of the scale must always be equal to the right-hand side thereof.

Teaching from a concrete, to semi-concrete to abstract level is recommended. This is shown next:

Concrete level Semi-concrete level (Use actual objects to demonstrate.) (Use a picture/drawing to explain.) 2 + 3 = 5

Abstract level (Use symbols to illustrate.) On a more advanced level, the following activity can be used to reinforce learners’ understanding of “is equal to”. The activity explained here is appropriate for any grade. You must, however, adapt the numbers and sums to fall within the number range of your grade. The number range used here is appropriate for Grade 2. For a whole class activity, prepare the following number and operational symbol cards:

Operational symbol cards:

Make number cards with random numbers between 0 and 99.

Use the above cards to make sums with the learners. Put number bibs around their necks and let them stand in front to form “sums” (number sentences), e.g.

+ + + = Use cardboard, e.g. from empty cereal boxes. Write operational signs appropriate for addition (+) and equals to (=) on square cards. Attach strings to the cards so that they can be tied around learners’ necks.

5 7 9 10 15 21 24 30 39 41



Let each of the learners tell and explain their role or value in the “sum” (number sentence), for example:

I am worth 15; that means 10 and 5 or 3 hands of 5 fingers: 5, 10, 15.

I am the plus sign. It means I put together John’s value and Fred’s value. I say you must add the two numbers. I am worth 9; that means 5 and 4 (or other ideas). I am the equals sign. I show that the value of John’s number, together with Fred’s number is the same as Cindy’s number. I show that the one side is equal to the other side.

I am the answer. I represent the total of the two numbers added together. If learners choose to change the order of their positions on the left-hand side of the equals sign, they should notice that the total is still 24. This shows the commutative property of numbers, which will be discussed in the Paragraph 2.2.3.

Let learners do a few of these “sums” by changing the numbers AND/OR the order of the numbers. They should also write down the number sentences they have been represented and discussed.

9 + 15 = 24

15 9 = 24 +

Images: Freesvg.org (2013)



24

90 minutes

How does the above activity consolidate learners’ understanding of addition and how to record an addition number sentence? Commentary: First, learners are actively involved and link number and operation symbol (+ and =) cards to themselves. This is a fun and active way in which the learners can participate in doing addition.

As they start to tell stories, learners move towards abstracting the addition stories about which they are talking. Writing down their number sentences consolidates the abstract sums they have done.

2.2.3 The order of addition The commutative property of addition means that changing the order of the addends (the numbers to be added) does not affect the sum (the answer). Learners, therefore, must realise that the same two numbers have the same sum, no matter which comes first. In other words, numbers can be added in any order – the result will be the same. To help learners understand the commutative property of addition (i.e. 2 + 3 = 5 and 3 + 2 = 5), you must do activities with concrete material like the one explained next. Do the following practically: Take 5 counters and call two learners. Put 2 counters in Mary’s hand and 3 in John’s hand. Ask: “How many do we have altogether?” Say: “Two and three equals five”. Let learners repeat it after you.

Ask the two learners to keep the counters in their hands, close their hands, and let Mary and John swap places. Let learners open their hands and count how many counters John has in his hand (3). How many does Mary have? (2). Say: “Three and two equals five”. Let learners repeat it after you.

Mary’s hand John’s hand

2



The two learners can swap places again and again, repeating 3 + 2 = 5 and 2 + 3 = 5.

Then write the number sentences on the board: 2 + 3 = 5 and 3 + 2 = 5. (Therefore, 2 + 3 = 3 + 2). Let learners then work with their own counters. By making groups with different coloured counters, learners realise that two white and three black counters give the same number of counters as three white counters and two black counters do.

Use a variety of combinations so that learners clearly get the idea that the order of the numbers that are added does not affect the answer.

Remember to reinforce this concept by doing similar exercises in other situations, for example:

Place several counters in two separate containers, let learners add and record the total. Then let them swap around their two containers, and again add and record the total.

Let learners draw a specific number of dots in two separate groups (circles) on two separate pages. Let them then add and record the total, interchange their pages and again add and record the total.

Let them talk about their findings. Write both the number sentences on the chalkboard, for example, 3 + 6 = 9 and 6 + 3 = 9.

Eventually, learners will know that they can add two numbers in any order – the answer will be the same. This is known as the commutative property (sometimes called the commutative law) of addition. (You will learn more about the properties of numbers in the next module: Mathematics Teaching in the Foundation Phase 2 (F-MAT 312).)

John’s hand Mary’s hand

Swop places

Commutative property: Commutative is derived from commute, which means to move around. Thus, the commutative property of addition means numbers can be moved around in a number sentence without impacting the answer. In other words, numbers can be added in any order and the answer will remain the same.



Learners in the Foundation Phase will not learn the name of this property but you as the teacher will know it and will start to see when learners have realised that it holds and that they can use it when they need to or want to.

25

180 minutes

Plan a lesson for learners where you use the concrete material as well as number cards (make your own number cards for this purpose) to teach Foundation Phase learners that numbers can be added in any order:

Include all the components of the lesson plan. Include at least three activities for learners. Show the trajectory of learning clearly, i.e. from the concrete to the abstract.

Commentary: Revise the lesson plan in the module: Introduction to Mathematics Teaching in the Foundation Phase. Ensure that learners are involved throughout the lesson and that the activities and content you choose is relevant to the topic and provide learning from the concrete to the abstract. Take your lesson plan with to the next academic support session or your study group.

26

120 minutes

In your own study group or at the next academic support session: 1. Present your lesson plan in your group. Explain the trajectory of learning. 2. Listen to and learn from other students. Make notes of what you learn from other

students or your tutor. Keep your lesson for teaching during WIL or in your classroom.

2.2.4 Adding one and zero Adding one to a number is visually easy for most learners. Once again, start at the concrete level and have many experiences for your learners with concrete objects to practice adding one before introducing paper-and-pencil activities like dots and symbols, e.g. 1 + 1 = ; 2 + 1 = ; …; 9 + 1 = . Let learners practice enough so that they can conclude that if they add 1, they get the next counting or whole number. Repeated addition of 1 is the same as counting. Such numbers that follow each other in order, without gaps, by adding 1 each time are called “consecutive numbers”.

Adding with zero is quite difficult for young, inexperienced learners to understand because zero is an abstract idea; it cannot be represented using “something” because it is “nothing”.

2



Concrete experiences with adding “nothing” is tricky because it is difficult to visualise adding nothing! But you can work through examples using stories and talk about adding “no more”. Have many concrete experiences so that learners realise that if they add “no more”, they still have the same number. The number of objects stays the same if they add nothing to that number. As a teacher, you should know that zero (0) is called the “identity element” under addition because when adding zero to any number, you will always get the identical number you started with. Here is an example: Take a basket. Put 4 real apples in the basket. Count as you put them in, one, two, three, four. (You may also use a small empty box and four stones, it will serve the same purpose!) Ask learners how many apples are in the basket. Take an “apple” from the air. Ask learners how many apples are in your hand (0). Now add the “nothing” to the basket and ask learners how many apples are in the basket. Let them count: 1; 2; 3; 4. There are still only 4 apples in the basket. Then show them the sum written in a number sentence: 4 + 0 = 4. You can add another apple to have 5 apples in the basket. Add another “nothing” and make the sum, 5 + 0 = 5. Your activities should focus on this pattern:

1 + 0 =

2 + 0 =

3 + 0 =

4 + 0 =

0 + 1 =

0 + 2 =

0 + 3 =

0 + 4 = You might need to revisit the concept of adding zero; you will do this throughout the teaching of addition. Include sums where you add zero in sets of questions that you give to learners. The number bonds of adding zero to numbers up to 10 are considered basic facts that learners need to learn. They need to know these by heart so that they can apply them when doing operations in the higher number ranges. There are other basic facts that learners need to master that will now be discussed below.

Source: The Yellow Peach (2009)



2.2.5 Doubles Doubles are basic facts in which both addends (numbers to be added) are the same number, for example, 4 + 4 or 6 + 6. To double means to add the same number. At first, connect doubles to familiar situations. For example, use two hands to show that 5 + 5 = 10, use an egg carton to show that 3 + 3 = 6, use two egg cartons show that 6 + 6 = 12, etc.

Source: Pxfuel.com [n.d.]

After experiencing doubles with concrete objects, learners must work with drawings, pictures and dots, for example:

Eventually, they must write the number sentence in symbols, for example, 3 + 3 = 6. Doubling, and ideas to teach doubling, is addressed in more detail in the next module when multiplication is covered. 2.2.6 Basic addition facts

The basic addition facts are those that give an answer up to 20 when whole numbers are added together. Refer to the basic facts, addition table in the addendum of your Fundamental Mathematics module. Do you know the basic facts by heart? Ensure that you revise your knowledge of these regularly. To help learners know and work fluently with basic addition facts, you need to drill them using a variety of methods. Using an array with columns and rows is one such method. You should draw a diagram like the following on the chalkboard. Ask learners to copy it in their class workbooks. The arrows must be included. The arrows show the direction of the operation, i.e. from left to right (horizontal) or top to bottom (vertical) and, therefore, the arrows must be placed on the right-hand side and below (not left or on top).

=

Double 3 eggs on one side shows: 3 + 3 = 6. Double 6 eggs on one side: 6 + 6 = 12.



Ask learners to choose a number less than 9 (basic facts) for each block. You can write the numbers on your diagram on the chalkboard. Now ask learners to use their counters to pack the number in each block. Learners must then add horizontally ( ) and vertically ( ) and write the answer next to or under the arrows.

4

8

9

3

Let learners then replace the counters with the appropriate number symbols:

3 1

4

6 2

8

9

3

Proceed by linking this format to the open block (result unknown) format in number sentences, for example: 3 + 1 = and 6 + 2 = Show the learners informally HOW the horizontal format can be written vertically as:

3 and 6 + 1 + 2

Learners need to learn that a number sentence such as 4 + 2 = is the same as:

4 + 2 ___



2.2.7 Repeated addition Addition also involves the joining (putting together) of more than two numbers as a single number. If these numbers are all equal in size, we call this repeated addition. To do repeated addition means to add the same number several times. It is important that we expose learners to repeated addition, as it prepares learners for multiplication. Repeated addition must be taught following the same teaching sequence as any other addition sum. Let learners first use concrete materials, then progress to semi-concrete representations and eventually, complete abstract activities. Start teaching repeated addition to Grade 1 learners on a concrete level through the involvement of their bodies. Learners can, for example, count the number of eyes of a group of 5 learners. 2 4 (2 + 2) 6 (4 + 2) 8 (6 + 2) 10 (8 + 2)

2 + 2 + 2 + 2 + 2 = 10 In the above activity learners repeatedly (5 times) add groups of 2. They could first skip count in 2s from 2 to 10 as they are pointing at each learner in the group and then write the number sentence. In the same way, learners can repeatedly add groups of 5, counting the fingers on one hand as a group of 5. The following number sentence represent the total number of the fingers of 3 learners: 5 + 5 + 5 + 5 + 5 + 5 = 30. Then use concrete objects to reinforce learners’ understanding of repeated addition. Let learners use concrete examples to show groups of a specific number, for example: “Show me 4 groups of 2”.

Learners must also write down the number sentence: 2 + 2 + 2 + 2 = 8 (At a later stage in Grade 2 and 3, help learners to link the repeated addition format to the multiplication format: 2 + 2 + 2 + 2 = 4 × 2 = 8).

After lots of concrete experiences, learners can progress to using pictures, dots or their own drawings (semi-concrete level) to represent repeated addition. To practice repeated addition on the semi-concrete level, draw (for example) a few smiley faces on the chalkboard. Ask: “How many eyes?”

Images: Freesvg.org (2015)



Let the learners count the eyes. Allow them to count the eyes one by one but encourage them to do mental addition by counting in multiples of two. Let learners also write down the number sentence for the total number of eyes, i.e. 2 + 2 + 2 + 2 + 2 = 10 (skip counting). On their own, let learners make drawings and write the number sentences under the drawings to show, for example:

the wheels of three tricycles; [3 + 3 + 3] the wheels of four cars; or [4 + 4 + 4 + 4] the wheels of three lorries (with 8 wheels each). [8 + 8 + 8]

For Grade 3 learners, you must teach repeated addition of higher numbers. Always try to involve concrete objects, even for Grade 3 learners, when you introduce a new concept. An example involving concrete objects in a money-related problem is given. Collect a number of empty cold drink tins. Place them where all learners can see:

Ask learners: Learners can use counters (concrete) or make drawings (semi-concrete) to find the answer, for example:

+ + +++++ + R7 each

9 tins

“How much will we pay for these 9 cans of cold drink if each can cost R7?”

Images: Needpix.com [n.d.]

Images: Lam (2016)



Learners must also write the number sentence: 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 = 63. At a later stage, when multiplication is introduced to Grade 2 and 3 learners, you will show them how to write the addition number sentence as a multiplication number sentence, i.e. 9 × 7 = 63, which means there are 9 groups of 7. At a more advanced level, learners can be involved in a telephone game. This is a more challenging activity than the one explained previously, but it is a fun way to practice repeated addition on an abstract level. Let 6 to 8 learners sit in a circle. The group decides on a number that must be added each time (e.g. 4). One learner starts the message by whispering a number (any number between 1 and 10) into the ear of the learner on his or her left side. This learner listens carefully, adds the number mentally, and whispers the answer to the learner on his or her left.

The next learner must add the number and whisper the answer, in turn, to the next learner. The last learner must say the total aloud.

27

90 minutes

Several strategies for working with addition have been introduced in the last few pages. Draw a mind map to summarise the last four strategies. Then reflect on these:

How many of them do you remember doing at school? What ideas do you remember most clearly? What does this suggest about the way in which you should teach addition? Do you think learners in the Foundation Phase should do vertical addition? Why

or why not?

Six, add four …

Six

2

Images: Clker.com (2018)



Commentary: You might not clearly remember the strategies used for addition when you were at school. If you remember some successful strategies not mentioned here, write them down. Most people remember strategies best if they were actively involved. Revise the part on “actively involving your learners” in the module: Introduction to Mathematics Teaching in the Foundation Phase (F-MAT 120) and keep this in mind when you plan for teaching. Vertical addition in the Foundation Phase has been debated by Mathematics teachers for a long time. Though there are advantages, one of the disadvantages that must be guarded against is that learners learn algorithms (short methods/recipes) too early without a complete understanding of the base ten number system, trading rules, etc. Guard against this when you teach Foundation Phase learners.

In the next paragraph, we will show you HOW to use number tracks and number lines for addition. 2.2.8 Number tracks and number lines You cannot expect learners to understand and use a number line from the first day in Grade 1. They should gradually progress form physically experiencing the increase and decrease of numbers to efficiently using number lines with large numbers by Grade 3. Learners’ first exposure to number lines should be where they practically experience the increase and decrease of equal segments. Let’s give you some ideas of how to use the number line for addition. Step 1: Increase and decrease of the numbers in equal segments

At the beginning of Grade 1, let the learners pretend to be frogs! They need to jump in a straight line and try to make sure that each jump is the same distance. This helps them to physically experience the equal segments of a number line. They also count while they are jumping.

One Two Three

Images: Clipartkey (2019)



Step 2: Number tracks

Before introducing and using number lines in your class, start with number tracks; they are more concrete and understandable, especially to Grade 1 learners. Number tracks on the ground You can easily make an outdoor number track yourself. Draw large numbered squares on the playground or use your white chalk to draw a number track on the corridor floor. Remember to draw equal segments! Let learners jump forwards and backwards in the squares while counting. Also let them physically do some addition sums, e.g. “Stand on 1. Add 2. Add another 3. Where are you now?” (In the same way, subtraction and a mixture of addition and subtraction can be practiced.) Number tracks on paper After enough practice, you can make number tracks on paper like the following example:

1 2 3 4 5 6 7 8 9 10

Use the number tracks to play games where learners place counters on the number to count forwards and backwards. Give instructions like:

You start on 4 and jump one forward. Where will you land? You start on 7 and jump one backwards. Where will you land? Place 7 counters on your number track. How many more counters do you think

you need to get to ten? Let learners first estimate and then put out counters to check their estimation.

Gradually, extend the number track to 20 and beyond. Once learners understand how to use the number track to represent their actions, you can encourage them to use written methods to show their adding (or subtracting) on the number track, e.g. “You said you started with your counter on 2, then you jumped three (steps) forward and landed on 5. Can we write a ‘sum’ (number sentence) for this?” We can show it like this: 2 + 3 = 5.



1 2 3 4 5 6 7 8 9 10 The number track can also be used to play games with dice and counters, e.g. throw the die, and count on (or add the number on the die to the number of the current place). Learners move the counter and stand on the new place. They throw the die until they reach 10. After this, learners are ready to work on a number line. Step 3: Number lines

The number line is a geometric “model” of all real numbers. Unlike counters, which model only counting, the number line models measurement, which is why it must start with zero. One reason for using the number line in teaching is that learners need to see arithmetic in both contexts: counting and measuring. Another important reason we use the number line is so that our learners understand how the basic operations of addition and subtraction work and to understand what the answer means. Learners make a natural progression from counting to basic addition, but there’s a key moment when they realise that they don’t have to count all the way from one each time. Take for example 3 + 4 = ?. Learners start by counting on their fingers from one to three, then they count four more to get to seven. This is a natural stage of the concrete stage of development. Soon they’ll realise that they don’t have to start at one each time and start at three and then count on to seven. Number lines are an excellent way to accelerate this development. Number lines provide a mental strategy for addition and subtraction. Research has shown that number lines are important because they promote good mental arithmetic strategies. In addition to a number line being a useful pedagogical tool, it is as much a mathematical concept as functions. In the Intermediate Phase, number lines will be important when learners start working with whole numbers to eventually proceed to algebra. It is important that Foundation Phase learners learn how to use a number line efficiently. Foundation Phase teachers are reminded of the importance of displaying a large number line along the top of the chalkboard (or any other suitable visible spot in your classroom). Ideally, learners must all have number lines at their desks. Keep in mind that number lines are more useful in solving addition and subtraction type problems and not necessarily for sharing and grouping problems.

Start here



See how the number line is used to illustrate the adding up of two numbers: 2 + 3 = 5 The learner must use both index fingers for activities on a number line. The learners should put the first index finger on the first number and add (move on) with the second index finger to get to the answer.

28

120 minutes

1. Work through the following set of learner activities. Draw a picture and write a number sentence for each:

a. Khwezi has 9 sweets. Tom has 3 sweets. How many sweets do they have altogether?

b. Grace has 8 sweets. Tuli has 6 sweets. How many sweets do they have altogether?

2. Fill in the numbers on the number line and then write a number sentence for each.

Each number line starts at zero.

a.

b.

3. Why is the number line important to teach addition at the Foundation Phase? Commentary: 1. a. Khwezi has 9 sweets. Tom has 3 sweets. They have 12 sweets altogether.

( ) (9 + 3 = 12) b. Grace has 8 sweets. Tuli has 6 sweets. They have 14 sweets altogether.

( ) (8 + 6 = 14)

The answer is 5. 3 2

0 1 2 3 4 5 6 7 8 10 0 1 2 3 4 5 6 7 8 9 10



2. Fill in the numbers on the number line and then write a number sentence for each. Each number line starts at zero. a. 8 + 5 = 13 b. 6 + 5 = 11

3. One reason to use the number line in teaching is that learners need to see arithmetic in both contexts: counting and measuring. Another important reason we use the number line is so that our learners understand how the basic operations of addition and subtraction work and to understand how the answers to addition and subtraction questions can be located numerically in relation to the numbers being operated on. Number lines are an effective way to accelerate the development from counting to basic addition, especially when number lines that are not labelled in 1s are used, and when number lines in higher number ranges are used. They provide a semi-concrete representation that can be visualised and assist mental addition and subtraction. They also consolidate number concept – locating numbers on a number line requires learners to apply their knowledge of the relative sizes of numbers.

The most abstract level would be when learners do not have any form of representation and begin to work mentally. This is spoken about next using the idea of a “mental number line”. Step 4: The mental number line

Learners eventually develop a mental number line that enables them to solve problems mentally. This mental image of a number line allows learners to sequence numbers and to move flexibly between them. After enough practice on the real number line, you can let learners practice their mental skills using the “mental number line”. This is an advanced activity and challenges learners to develop their number sense on a more advanced level. To practice using the mental number line, have learners participate in a sequence of questions, e.g.:

Teacher: I have 23, what will I get when I add 7? Learner: 30 Teacher: Yes! And now that I have 30, what must I give away so that I am left

with 15? Learner: 15 Teacher: Great. And what will I get by adding another 20?

Carry on with more challenging questions and gradually increase the number range. Let learners answer “Up and Down the Number Line” questions to practice their skills of making compensations using addition and subtraction to calculate an operation mentally.



This is quite advanced thinking, and you should help learners work through the thinking behind making the decisions to add and subtract in the strategies (methods) shown below. There is more than one way to do mental calculations. Let us look at the following strategies:

Completing tens: o I have 16. What do I get if I add 4? (16 + 4 = 20) o What must I add to 26 to get 30? (26 + 4 = 30) o If I have 33, how much do I need to add to get to the next ten? (33 + 7 = 40)

To be able to answer these questions learners must know the number bonds of ten, e.g.: 6 + 4 = 10; 3 + 7 = 10; 2 + 8 = 10; etc.

Bridging tens/hundreds:

o I have 16. What do I get if I add 8? Think: 16 + 4 = 20; 20 + 4 = 24. 16 + 4 + 4 = 24. I get 24.

o What must I add to 27 to get to 34? Think: 27 + 3 = 30; 30 + 4 = 34. I need 3 + 4 = 7 more.

o I have 198. What do I get if I add 24? Think: 198 + 2 = 200; 200 + 22 = 222. o What must I add to 294 to get 340? Think: 294 + 6 = 300; 300 + 40 = 340.

Filling up tens/hundreds:

(Learners first break down the number to be added.) o What do I get if I add 26 and 18?

Think: 26 + 4 + 18 – 4 = 30 + 18 – 4 = 30 + 14 = 44. o What do I get if I add 275 and 35?

Think 275 + 25 + 35 – 25; then, 300 + 35 – 25 = 300; 300 + 10 = 310.

Adding multiples of tens and hundreds: (Learners must realise that if 3 + 2 = 5, then 30 + 20 = 50 and 300 + 200 = 500.) o I have 40, what will I get if I add 20? Think: 40 + 20 = 60. o I have 70, what will I get if I add 30? Think: 70 + 30 = 100. o What must I add to 200 to get 600? Think: 200 + 400 = 600.

Adding to multiples of tens and hundreds:

o I have 16, what will I get if I add 34? Think: 16 + 4 + 30 = 20 + 30 = 50. o What must I add to 27 to get 60? Think: 27 + 3 + 30 = 60. I need 33 more. o What do I get if I add 13 to 227? Think: 227 + 3 + 10 = 230 + 10 = 240.



29

180 minutes

Study the different strategies discussed above for mental calculations. 1. Which strategy do you use most often when you need to do mental mathematics?

Why do you prefer that specific strategy? 2. First, write down the pre-knowledge required by a learner before you can teach the

“filling up tens” strategy to the learner. Then, use your example to show how you would explain the “filling up tens” strategy to a Grade 2 learner.

3. Draw up your mental test consisting of 10 questions to strengthen Grade 3 learners’ competence to bridge hundreds. Also, draw up the memorandum for this test.

Commentary: Always keep the number range in mind when you set mental tests for your learners. The strategies above can help learners do mental calculations and eventually develop their own strategies for mental calculations. Remember to provide enough opportunities for learners to practice their mental calculation strategies.

Mental number line addition can be used as a mental mathematics activity. Number each question, and let learners write down their answers so that the mental mathematics can be assessed afterwards (by themselves, peers or by you). Here is an example provided for you (this activity could be done in 5 to 10 minutes at the start of a mathematics lesson): Mental maths activity: Answer

1. What is 10 more than 750? 760


3. What is 10 less than 750? 740








Addition exercises must always be followed up by written work. Written work must, however, suit the ability level of your grade. In the following paragraphs, some guidelines on HOW to plan for written work on addition is provided.



2.2.9 Written exercises A variety of interesting written activities must be part of mathematics lessons for all grades (even Grade 1). The following activity only tests to see if learners can calculate the sum correctly and does not test learners’ understanding of the process of addition:

Calculate the following: 12 + 1 =

33 + 16 =

114 + 21 =

Although the numbers are all different, they are all the same type of exercise (i.e. the result/answer is unknown). In exercises like these, learners do not get the opportunity to consolidate concepts by practising a variety of addition exercises. When you design written activities for addition, you must have a variety of sums. Follow the guidelines given below. When developing written activities for learners, change the position of the unknown and provide varied types of problems.

Consider the following possibilities and order: Result unknown: 15 + 25 = Start unknown: + 25 = 40 Change unknown: 15 + = 40

Repeated addition: 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = Multiple addition: 4 + 5 + 6 + 7 = Multiple operations (addition and subtraction): 40 + 10 – 5 – 3 + 2 = In the following cases, it is sometimes useful to use a part-part-whole model to identify parts in an addition number sentence.

Result unknown: e.g. 15 + 25 = Start unknown: e.g. + 25 = 35 Change unknown: e.g. 15 + = 40

Whole

Part Part

Result unknown: 15 + 25 = (Whole)

15 (Part) 25 (Part)

Answer / result unknown.



Start unknown: + 25 = 40 40 (Whole)

(Part) 25 (Part)

Change unknown: 15 + = 40 40 (Whole)

15 (Part) (Part) Written exercises should also provide for individualised learning and progression. Below are some practical guidelines to maximise learning, ensure individualised learning and progression with addition. Practical guidelines for progression and individualised learning

Number range: Start with small numbers and gradually increase the number range. To individualise learning, increase or decrease the size of the numbers to match your learners’ grade and ability level.

Position of the unknown: To progress from simple to more complex, start by using smaller numbers with exercises where the result is unknown. For example: 5 + 2 = . To increase the difficulty level, change the position of the unknown, and on an advanced level, work with larger numbers.

Multiple operations: At first, provide activities that focus on one operation (e.g. addition only). Then, increase the number range and give mixed addition exercises. On an advanced level, use multiple operations (e.g. addition, subtraction and multiplication). A few practical examples are provided next.

1. The use of worksheets For Grade 1s, design worksheets like the ones shown here. Learners can use their number lines to complete the following operations:

Activity 1

6 + 10 =

+ 4 = 15

(Have at least 10 examples)

“What” plus 4 will give 15?

Change the position of the unknown (the empty block) to challenge learners. We can make the problem more difficult by putting the unknown at the beginning.



Activity 2

6 + 4 + 2 = + 7 + 6 = 21

7 + 4 + = 16 (Have at least 10 examples)

Activity 3

(Have at least 5 examples) HINT: Have concrete material available for learners who might still need this as learning aids to assist them to solve the problems.

2. The use of work cards On a more advanced level (e.g. Grade 2), activities like the following can be presented. Design a card with pictures and numbers.

6 7 8 9 10 15 27 30 31

48 50 52 64 75 83 90 96 99

Explain to learners that each of the pictures has a different number value, e.g. the flower’s value is 6; the car’s value is 31.

Design work cards like the following and place them on learners’ desks:

9 + 11

+ 3 + 2

Add more numbers and vary the position of the unknown.

Change the format.



Replace the picture with the real value and do the calculation.

Three examples were shown here. Have at least ten examples on varying levels at each table for learners to complete. The more they practice, the more competent they will become in solving similar addition problems.

Design work cards on varying levels to provide for differentiated learning experiences for your learners. Divide your class into groups according to their developmental level. For learners who struggle with addition, use smaller numbers and let them only add two numbers. At first, have the unknown at the end (6 + 7 = ). When your learners are confident with this type of calculation, proceed to slightly more challenging activities in which the position of the unknown is changed (e.g. 6 + = 13). For advanced learners, use larger numbers and let them add three or more numbers and give them problems with multiple operations (addition and subtraction) to solve. There are a variety of interesting ways to let learners practice addition. Ideas for spider web diagrams are provided next. 3. The use of spider web diagrams Spider web diagrams are interesting ways to help learners practice and reinforce their skills in addition. These formats should not be introduced too soon for Grade 1 learners, who should mostly work with concrete objects, but they are interesting alternatives for Grade 2 and 3 learners. You can make your own spider web diagrams, by following the procedure below. You need 3 circles of varying sizes. Any circular object, e.g. tin or lid in large, medium or small, can be used to draw the circles. Use the large circular object to draw the outer circle, then the next size for the middle circle and the smallest shape

+ =

+ = +

+ =



to draw the inner circle. Use a ruler to draw the sector lines of the circles. You must write the instruction (e.g. + 10 or + 13; see spider web examples below) in the inner circle and then fill in the numbers in the middle circle. Draw at least 6 to 8 different examples in your learners’ workbooks. If you have photocopying facilities, you can draw many different spider webs on an A4 page and have it photocopied for your learners to complete.

NOTE: Where the numbers are written on the outer circle, learners need to find the number to put in the inner circle that will give the total in the outer circle. For example, 10 + = 21; then = 11 because 10 + 11 = 21.

30

90 minutes

Reflect on the varying types of written activities that have been suggested by writing a paragraph on how you would use each of them and when. Share your ideas with a peer in your own study group or during the next student academic support session.

+ 13

21

20

12 15

7

9

14

10

+ 10

21

20

12 15

7

9

14

10

3

You can also change the format a little and fill in some of the numbers in the outer circle.

Learners complete the spider-web diagrams according to the instructions given in the inner circle. They calculate and write the answer in the blocks on the outer circle.



The next method that will be discussed is “Building Up” and “Breaking Down” of numbers. In Unit 1, we already explained the “Building Up” and “Breaking Down” of numbers using spray cards (number expansion cards). This requires advanced mathematical skills and challenges learners to think about numbers in creative ways. To complete such activities, require higher-order cognitive thinking skills. 2.2.10 Breaking down and building up numbers Learners must explore a variety of ways to “Build up” and “Break down” numbers within their number range. Grade 1 learners must build up and break down numbers up to 20, using concrete objects at first. As explained in Unit 1, learners must be able to use addition to make different combinations of the same number using concrete materials. However, Grade 1 learners must also be able to break down and build up numbers on a semi-concrete as well as an abstract level within their number range and level of development. A number can be broken down into smaller parts, without considering place values of digits or it can be broken down into place value parts. Let’s look at, for example, the number 16. Sixteen can be broken down into smaller parts without considering the place value of digits, for example: 16 = 5 + 1 + 3 + 7 (16 is broken up in 4 parts). Sixteen can also be built up by putting these parts (5, 1, 3, and 7) together, for example: 5 + 1 + 3 + 7 = 16. There are many possibilities!

31

180 minutes

1. Write down 3 different examples in each case of how 16 can be broken down into: two parts; three parts; and four parts (do not copy the above example).

2. Write down 3 different examples in each case of how 16 can be built up from: two parts; and three parts.

3. What did you learn about the number 16 from doing this activity? 4. What can learners learn from breaking down and building up numbers? Discuss your responses with your own study group or during the next student academic support session.

Sixteen broken down into place value parts will be presented as: 16 = 10 + 6. In the above example, the number has been taken apart into tens and units. Grade 3 learners work with 3-digit numbers and should, therefore, be able to break down numbers into hundreds, tens and units, e.g. 312 can be broken down into 300 (3 hundreds) +



10 (1 ten) + 2 (2 units) and is written as: 312 = 300 + 10 + 2. A place value diagram is a useful tool to use as a visual aid:

In the building up method, a number is built up by putting together hundreds, tens and units, e.g. 300 + 10 + 2 can be built up to 312 and is written as 300 + 10 + 2 = 312.

As learners gain confidence in solving number problems, they will increasingly use techniques that involve breaking down, rearranging and building up numbers. These learners are now ready to work with numbers in a more abstract sense.

Interesting ways to “break down” numbers into three or more smaller parts without considering the place value of digits are provided next.

1. The use of shapes

The following examples are appropriate for Grade 2 learners. The diagrams link to the idea of parts making up a whole (breaking down and building up numbers) diagrammatically. These questions require learners to find parts or the whole, depending on what has been left out of the diagram.

Let learners choose a number and see how many ways they can find to break it up, for example:

Change the format by writing some of the outside numbers and letting learners fill in the missing numbers.

Eventually show learners how to link this format to the “empty box” (unknown) written format: 14 = 4 + 3 +

On a more advanced level, use other shapes with more sides to encourage learners to break numbers into more parts, for example:

20 = 5 + 1 + 6 + 2 + 4 +

H T U

3 1 2

14 4 3

14

14

14 4 3

7



6

2

5

For Grade 3s you can change and extend the shape format of the addition calculations by trying out different shapes and other interesting formats. See the arrows example below. Three possibilities of how 94 can be broken down into a different number of addends are shown:

50 + 44 (2 parts) 40 + 50 + 4 (3 parts) 20 + 20 + 20 + 20 + 10 + 4 (6 parts)

Let your learners find more ways to break down a number, e.g. 94, into a different number of addends.

32

180 minutes

1. Redraw and complete the above arrow diagram by showing how learners can break down 94 in 4, 5, 7, 8 and 9 parts, respectively.

How many varying ways can learners find to break down 94 in 2, 3, 4, 5, 6, 7, 8, and 9 parts respectively? Write down at least two different ways in each case.

What can learners learn from this activity? 2. On an A4 paper, design an interesting activity to enable Grade 2 learners to break

down numbers: Use any shape or format different from those given above. Provide clear instructions for learners to complete the activity. Show an example of how you expect learners to complete the activity. Share your responses with a fellow students in your own study group or at the

next academic support session.

4

1

Addend: Any of the numbers added together.

94

20



2. The use of magic squares to build up numbers Magic squares are an excellent way to reinforce and practice the number you had been working on during the week. At first, your learners might find it difficult, but after some practice, they will benefit a lot from completing them. Some learners might find magic squares quite challenging because in most magic squares the requirement is that the rows, columns and even diagonals must add up to the same total. Therefore, it not only challenges a learner’s addition skills but also practices their problem-solving skills. However, learners enjoy this a lot!! Start with simple magic squares. The following magic square has 9 blocks. For Grade 2 learners, you might have to draw blank magic squares in learners’ workbooks (at least 6). Grade 3s will most probably be able to copy a blank example from the chalkboard. Blank example:

Completed example:

5 5 2

1 4 7

6 3 3

You need to indicate the number for each magic square (use the number you are focusing on for the week, within the learners’ range, say 12). Learners then need to

Magic square: Refers to a square that contains a square number of blocks (e.g. 2 × 2 = 4 or 3 × 3 = 9 blocks) and the total of the rows, columns and diagonals are the same.

12

In this magic square, fill in numbers in each block so that the numbers in each row and each column all add up to 12.

In this magic square, see how the rows, columns as well as the diagonal rows all add up to 12.

12

This is a diagonal.



fill each of the squares so that each row and each column adds up to 12. For advanced learners, you can ask them to try to also add up the diagonal rows to the required number. Also, give learners the opportunity to choose their own numbers and complete their own magic squares. You can ask them to swap their examples with a peer to check each other’s calculations. 3. The use of games You should never underestimate the value of games or activities that actively involve learners when you are teaching important concepts. In this game, learners must not only break down their numbers into smaller parts but particularly into place value parts. One such game is “DRAW LUCKY NUMBERS”. For this game, you need to write numbers on small pieces of paper and put them in an empty box. Do this beforehand. Let each learner draw 5 “lucky numbers”. Now, learners write down their 5 numbers. They break them down into tens and units, e.g. 46 = 40 + 6. (For advanced learners, work with larger numbers, like 3-digit numbers.) Ask them to add a number (say 8) to each of their lucky numbers and break them down again, e.g. 46 + 8 = 54 or 54 = 50 + 4. Encourage learners to discuss and write down their calculations using the appropriate signs and symbols.

33

90 minutes

How does building up and breaking down numbers according to place value of digits consolidate learners’ understanding of place value? Commentary: It gives them an opportunity to talk about units, tens and hundreds; the place values learners work with in the Foundation Phase. Furthermore, they will realise that the value of digits is linked to their place value. The breaking down (or building up) method of addition is a critical method for learners in the Foundation Phase. This method requires learners to break down (or build up) the numbers into units, tens and hundreds and to add them horizontally (next to each other).

Guidelines on HOW to teach learners to add by breaking down numbers into place value parts will be discussed next.

3



2.2.11 Add by breaking down numbers into place value parts Expose learners to varying ways of breaking down numbers for addition. Start with units and tens and progress to units, tens and hundreds once learners grasped this method properly.

1. Progress from informal to formal methods Grade 2 learners are expected to add 2-digit numbers up to 99. The breaking down addition method can be introduced informally as explained next. To add 16 to 12 (16 + 12), the numbers can be broken down in tens and units and make addition easier. Start on a concrete level, using counters in two different colours or Dienes blocks and place value tables. Learners should break up the two numbers into tens and units and use counters to represent the numbers in the place value table. Remind them that, one black counter is equal to 10 grey counters, as explained during the base ten activities completed earlier (see Unit 1, Section 1, Paragraph 3).

16: 12:

After breaking down the numbers using counters, learners can draw the representation in their class workbooks. Let them count the number of tens and units in the place value table and write it down as shown next.

tens units

16 = 10 + 6 and 12 = 10 + 2 28 = 20 + 8 Therefore: 16 + 12 = 28

Tens: 10 10 20 + =

Units: 6 2 8

16 12 28



More formally, the addition sum could be written as follow: 16 + 12 = = (10 + 6) + (10 + 2) = (10 + 10) + (6 + 2) = 20 + 8 = 28 Did you notice the introduction of brackets to pair up and group numbers? Explain to learners that mathematicians use brackets when there is a long string of numbers to make the order in which they are working on numbers clear. In Grade 3, learners must be ready to do more abstract classroom activities using the “building up” or “breaking down” strategies. Two examples of addition activities are given below. Example 1: Adding three-digit and two-digit numbers Do the following example on the board. While you do the working, explain to the learners how you add the hundreds to the hundreds, the tens to the tens and the units to the units. Explain to the learners how you are using the brackets to pair up and group numbers so that you make it clear which numbers will be worked on and in what order. 524 + 82 = = (500 + 20 + 4) + (80 + 2) = 500 + (20 + 80) + (4 + 2) = (500 + 100) + 6 = 600 + 6 = 606 Do another example on your own using the same method: 626 + 32 = (658) Example 2: Adding three-digit and three-digit numbers Do the following example on the board. While you do the working, explain to the learners how you add the hundreds to the hundreds, the tens to the tens and the units to the units.



Explain the use of brackets again.

323 + 436 = = (300 + 20 + 3) + (400 + 30 + 6) = (300 + 400) + (20 + 30) + (3 + 6) = 700 + 50 + 9 = 759 Do another example on your own using the same method: 626 + 142 = (768) NOTE: You should always encourage your learners to estimate their answers before performing the actual calculation. Estimating is the ability to make reasonable guesses about a quantity. In the Foundation Phase, learners deal with estimations informally. It is, however, important that Foundation Phase learners gain experience with estimation and comparing whether their estimate is larger or smaller than the actual count. They must also be able to look at a group of up to 20 objects and have a good sense of whether there are about 5, 10, 15 or 20 objects.

34

90 minutes

1. Calculate the following using building up or breaking down strategies:

a. 524 + 123 = (647) b. 475 + 312 = (787) c. 724 + 121 = (845)

2. Discuss why the strategies of building up and breaking down are important for

Foundation Phase learners? 3. Explain why mental mathematics is one of the most important tools for learning

mathematics? Commentary: All these calculations must follow the method described in Example 1 and 2 above. a. 524 + 123 = (500 + 20 + 4) + (100 + 20 + 3) = (500 + 100) + (20 + 20) + (4 + 3) = 600 + 40 + 7 = 647 b. Similarly, 475 + 312 = 787 c. Similarly, 724 + 121 = 845 Encourage your learners to show all their steps and to use brackets for grouping.



The strategies of building up and breaking down are quite abstract, and so it is crucial that learners have mastered more concrete strategies of addition before being introduced to these more advanced strategies. By building up or breaking down, learners are being exposed to the higher-order cognitive levels of Synthesis and Assimilation in Blooms Taxonomy. These strategies involve the conceptual understanding of number bonds. Using these strategies for addition will establish foundations for problem-solving techniques. Being able to do calculations in your head is an important life skill and an important part of mathematics. Mental mathematics is also a critical component of the Curriculum and Assessment Policy Statements (CAPS) for mathematics (DBE, 2011). The CAPS document lists the number bonds and multiplication table facts that Foundation Phase learners are expected to know and recall for each grade. However, to improve mental calculations, learners need to be taught the most efficient strategies explicitly. Mental mathematics encourages and strengthens accuracy and speed. It involves conceptual understanding and problem-solving. It is a useful skill to use to estimate the answer to a problem. Learners also do not have to rely on a calculator for computing the answers.

A repeated addition problem can also be solved using the breaking down method. Here is an example:

There are 12 trees. There are 6 rows. Thus, we must add 12, 6 times.

Number of trees = 12 + 12 + 12 + 12 + 12 + 12 Which can be broken down into: 10 + 10 + 10 + 10 + 10 + 10 = 60 and 2 + 2 + 2 + 2 + 2 + 2 = 12 Then, 60 + 12 = 72 Therefore, Samuel will plant 72 trees. 2. Introduce vertical methods As Grade 3 learners start to work with 100s, they might find the vertical layout (where they line up digits with the same place value underneath each other) a useful way of helping them keeping track of what they are doing. This is not officially part

Samuel plants 6 rows of trees with 12 trees in each row.

How many trees does he plant?



of the CAPS curriculum as a strategy, but CAPS suggests that learners should be encouraged to use many different algorithms (DBE, 2011). The vertical algorithm is a standard, accepted algorithm, internationally. You should teach your learners how to use it effectively, helping them to understand how the vertical columns help them to layout the number according to place value, and to work efficiently in these columns. This will add to the range of strategies that they have at their fingertips. You should not introduce this strategy until learners are ready to work with numbers and talk about them according to the place values of the digits that make up those numbers. At that stage, they are ready to use the vertical algorithm. Once learners have had lots of practice with different ways of recording their answers in a horizontal format, show them how to write the broken-down numbers underneath each other, e.g. 16 + 12. The horizontal algorithms you have done consolidate learners’ ability to talk about the place values of the digits in the numbers they are working with.

10 + 6 10 + 6 + 10 + 2 + 10 + 2 OR = 20 + 8 20 + 8 = 28 = 28

NOTE: Learners’ informal vertical methods act as an important stepping stone that will help them to understand and use the shorter vertical method that will be introduced later.

16 + 12 28

Although some Foundation Phase learners might not be ready for formal algorithms, some teachers feel that their learners are ready to use the vertical column method for addition. We will show you how to introduce learners to the more formal methods in the next paragraph.

2.2.12 Vertical addition Vertical addition is the standard algorithm for addition. It is called vertical addition because the numbers are written underneath each other, and you add in vertical (from the top to the bottom) columns. Throughout this unit, we have emphasised the teaching approach starting from the concrete to the semi-concrete to the abstract. To learn to use the vertical addition method with understanding, learners need to connect the steps that are used to solve the calculation problems with concrete materials to the steps that are used in the symbolic



(abstract) solution of the problem. We will, therefore, show you HOW to introduce the vertical method in a concrete way. Use base ten blocks to demonstrate how to work with tens and units and record the working using the vertical algorithm. Start with 2-digit addition before you move on to 3-digit numbers. Two examples are shown below that you can use to model the way you would teach this strategy. For example, use base ten blocks to show how to do the calculation for 23 + 15 = ___ Lay out the two numbers using base ten blocks.

Show the learners that there are:

2 tens and 3 units in 23 1 ten and 5 units in 15

Record these numbers vertically underneath each other. Show how when you regroup the tens and the units, you get 3 tens and 8 units, which is 38.

Record the answer in the vertical algorithm. Give learners lots of practice with different numbers where trading or regrouping is not involved. Let them also choose their own numbers to add using this format. It is then important for you to demonstrate using concrete material how the vertical method works, in case of trading or regrouping. Remember that learners will need a lot of concrete experiences to link with the written algorithm. For example, use base ten blocks to show how to do the calculation for 48 + 25 = ___

23 + 15 _____

23 + 15 38



Lay out the two numbers using base ten blocks, as before.

Show the learners that there are: 4 tens and 8 units in 48 2 tens and 5 units in 25 Record these numbers vertically underneath each other. Show how when you regroup the tens and the units, you get 6 tens and 13 units, so you need to regroup to make a new ten.

Use the base ten blocks to exchange: 10 units must be exchanged for 1 ten. Exchanging and regrouping lead to the “carrying” digit you learned about when you were at school. You write this as an extra 1 ten above the tens digit in the algorithm:

Learners must understand where the “carried” digit comes from. So, the total (the answer) is:

7 tens and 3 units, which is 73.

48 + 25 ____

48 + 25 ____

481

48 + 25 73

1



Record the answer in the vertical algorithm. Can you see how important it is for learners to have a clear understanding of place value and groups of tens to do vertical addition? When learners have had a LOT of concrete experiences with adding involving trading and regrouping, the formal vertical method may be introduced. Never force learners to use the formal calculation methods. Only once learners have acquired enough practice inventing their own strategies and methods, can you introduce the vertical addition algorithms. Make sure that your learners know how, why and when to use the vertical column method.

35

60 minutes

How did you learn to add when you were at school? Which method of addition do you prefer? Why?

You will now be shown HOW to add 3-digit numbers using various algorithms. 2.2.13 Three-digit addition using various algorithms You will teach learners many different algorithms to use when adding. Most of these will involve recording numbers, either vertically or horizontally. In vertical addition, calculations always start from the right-hand side. In other words, you first add the units, then the tens, then the hundreds, etc. In horizontal addition, learners will break down the numbers using place value. This can be done in many ways, but place value generally is the guide, and you will break numbers down into hundreds, tens and units. Always start with examples where there is no carrying over (regrouping of using place value) necessary. Then move to examples where regrouping is only necessary in the tens, before you let learners do examples where regrouping is necessary in the tens and hundreds. Here are some examples. These examples show how you can use base ten blocks (concrete) to demonstrate addition strategies and make links between vertical and horizontal strategies when you record calculations you have done. The numeric record of the calculation is done differently depending on what numeric strategy you want to use. The horizontal strategy demonstrated here is that of adding by breaking down the second number only.

This strategy involves adding 3-digit numbers to 3-digit numbers: keeping the first number whole and breaking down the second number and then adding in stages.

3



First example:

323 + 136 = ......... Note: The brackets are used in the calculation strategy. = 323 + (100 + 30 + 6) = (323 + 100) + 30 + 6 = (423 + 30) + 6 = 453 + 6 = 459

Following is an illustration of the algorithm using base ten blocks:

You could also link this to the vertical algorithm strategy: Record the numbers using vertical columns to line up the hundreds, tens and units. Add the digits in the columns, starting from the units.

In the units column, there is a total of 9. In the tens column, there is a total of 5. In the hundreds column, there is a total of 4.

No regrouping was needed; hence, no exchange or carrying was needed.

More examples (see below). While you work through them, you should question the learners about why they

are grouping numbers in the way they suggest.

First, add the hundred of the second addend to the first addend.

Then add the tens to what you have.

Now add the ones to what you have.

First, lay out the base ten representation of the two numbers to be added.

Then regroup to show the addition of the numbers in the different places.

323 + 136 459



141 + 345 = __ = 141 + (300 + 40 + 5) = (141 + 300) + 40 + 5 = (441 + 40) + 5 = 481 + 5 = 486

324 + 125 = __ = 324 + (100 + 20 + 5) = (324 + 100) + 20 + 5 = (424 + 20) + 5 = 444 + 5 = 449

177 + 122 = __ = 177 + (100 + 20 + 2) = (177 + 100) + 20 + 2 = (277 + 20) + 2 = 297 + 2 = 299

141 + 345 486

324 + 125 449

177 + 122 299

Ultimately, having worked through several examples in the class and seeing the links to the base ten blocks, learners should understand the way in which the algorithms work for three-digit numbers – where there is regrouping (or not). They should be able to explain to you what they are doing, using the language of place value.

To summarise, you need to keep the following guidelines in mind when using the vertical and horizontal algorithm. To add using the vertical algorithm:

1. Write the two numbers you are adding underneath each other with the place values aligned in vertical columns.

2. Start with the units digits. Add the units vertically (from top to bottom). Write the answer in the units (U) column in the horizontal bar. If you get more than 10 units when you add the two units digits, regroup, exchange and carry a ten to the tens column.

3. Then add the tens digits. Write the answer in the tens (T) column in the horizontal bar. If you get more than 10 tens when you add the two tens digits, regroup, exchange and carry a hundred to the hundreds column.

4. Add the hundreds digits. Write the answer in the hundreds (H) column in the horizontal bar.

To add using the horizontal algorithm: 1. Write out the two numbers you are adding, breaking down the numbers into

hundreds, tens and units. 2. Regroup the numbers so that you add the hundreds to each other, the tens together

and the units together. 3. Look at the answers you find for each group. 4. If you get more than 10 units when you add the two units digits, regroup, exchange

and carry a ten to the tens group. 5. If you get more than 10 tens when you add the two tens digits, regroup, exchange

and carry a hundred to the hundreds column. 6. Add all the grouped totals to find the final answer.

These skills can only be learnt through LOTS and LOTS of practice! Once learners have mastered the skills to add numbers, let them use these skills to solve word problems. When



learners solve word problems, they should be allowed to use the strategy of their choice to do the working for the calculations needed to find the solution to the problem. 2.2.14 Application of addition in word problems Learners must be taught HOW to apply their addition skills to real life problems and be able to solve word problems. Here is an example:

To find out how many sheep Mr Zo and his neighbour have altogether, learners must add 223 to 191 using the horizontal or vertical algorithm. Horizontal algorithm: 223 + 191 = (200 + 20 + 3) + (100 + 90 + 1) = (200 + 100) + (20 + 90) + (3 + 1) = 300 + 110 + 4 = (300 + 100) + 10 + 4 = 400 + 10 + 4 = 414 Vertical algorithm: 1223 + 191 414 Note that word problems or “story sums” must always be answered in words. Thus, the answer is: Mr Zo and his neighbour have 414 sheep altogether.

36

120 minutes

Prepare yourself thoroughly so that you can actively participate in the group discussion during the next student academic support session or in your own study group. In your group, discuss the following:

The relevant vocabulary relating to the operation of addition and how you will introduce these to Foundation Phase learners;

Mr Zo had 223 sheep. His neighbour has 191 sheep. How many sheep do they have altogether?

3

Note the small “1” showing gcarrying over (trading) of ten y g (tens = 1 hundred.

adind.d



The importance of introducing algorithms for addition through multiple representations with specific reference to how this will impact on learners’ understanding of addition; and

The use of different algorithms for addition in the Foundation Phase and the appropriateness thereof.

Present your own ideas and make notes of what you learn from others during the group discussion. If required, go back to the previous section to substantiate your arguments.

In the next section, you will learn how to teach subtraction at the concrete, semi-concrete and abstract levels. As with addition, you will be taken through the steps of HOW to teach subtraction at the Foundation Phase level. As you work through the theory and examples of how to teach subtraction, remember what you have learned about the teaching of addition and think about ways in which these two are related. The connectedness between the operations is always present, even if you do not focus directly on it. Ultimately, you will draw together the two operations by highlighting the inverse relationship between them.

1. INTRODUCTION Subtraction is the process of taking something away. Subtraction can be thought of as making a number smaller by taking away a part of the number. When you have a number of objects, and you subtract (take away) an item from it, the number becomes smaller or decreases. Subtraction is also a way in which we can find out the difference between two numbers that we are comparing. If we subtract one number from the other number, we can find out which number is bigger and by how much. Many applications of subtraction can be found in real life. We need to understand how to subtract to engage with society effectively, as we use subtraction when dealing with money, cooking, travel and time, among many other daily experiences. In this section, we will look at how to teach subtraction to Foundation Phase learners. This will include the steps and methods to teach subtraction. Even though subtraction is being dealt with in a separate section to addition, we encourage you throughout this section to look for the relationships and links between addition and subtraction rather than viewing them as two distinct number operations. The connection between addition and subtraction should be taught to learners to enhance their understanding of both operations.

SECTION 2: SUBTRACTION



2. HOW TO TEACH SUBTRACTION Subtraction means to make less, to decrease or to find the difference.

Subtraction involves the removal of a part of a number and retaining the rest, resulting in a decrease of the number.

We use the “minus” sign (–) to show subtraction. For example: 9 – 3 = 6 or 9 – 6 = 3.

Subtraction is NOT commutative, i.e. 9 – 3 is NOT equal to 3 – 9; therefore, the order in which the numbers are subtracted is extremely important.

What are the important components you need to teach about SUBTRACTION?

Count backwards in 1s, 2s, 5s and 10s (Grade 2) and 20s, 25s, 50s and 100s (Grade 3).

Subtract 2-digit numbers by using: o the breaking down method; and o vertical subtraction.

Mental Mathematics for subtraction includes counting backwards, basic subtraction facts, mental subtraction.

The recommended number range for each grade is: o Grade 1: Subtract from numbers up to 20; o Grade 2: Subtract from numbers up to 99; o Grade 3: Subtract from numbers up to 999.

The same progressive steps for introducing addition must be followed for subtraction. To refresh your memory, let’s follow these steps to introduce subtraction. 2.1 Steps for introducing subtraction Start with small numbers, between 0 and 5, then progress to numbers between 0 and 10 and finally, extend subtraction to the full number range required for the specific grade according to CAPS. For the next example, use five counters to introduce the subtraction at a concrete level: 5 – 3 = 2. Step 1: The use of counters (concrete level)

Hand out five counters to your learners. Let them count the counters and say how many counters there are. Then ask them to count out and take away three counters.



Learners must then say how many counters are left.

Make a verbal statement such as: “You have five counters, you take away three and there are two left over.” We can also say: “Five minus three equals two”. At first, learners might need to work concretely and handle real objects, but as they gain confidence and experience, they will want to find “shortcuts” to represent and record their solutions. Some learners find drawings easier to work with than counters. Step 2: The use of drawings (semi-concrete level)

On the semi-concrete-level, learners can make drawings to show: 5 – 3 = 2:

Give learners many opportunities to use drawings or mixtures of words and symbols to represent what they do and say and how they think. Learners should always be encouraged to use the correct mathematical language to express their thoughts and methods of solutions.

37

90 minutes

Read the following extract from Baroody (2010) and then reflect on the idea of teaching addition and subtraction in a connected way to promote associative learning (learning through association).

Knowledge of addition combinations has long been thought to facilitate the learning of subtraction combinations (e.g. 8 – 5 = ? can be answered by thinking

number left take away

5 3 2

Draw five counters. Take away (scratch out/delete) three. Two are left.

3Learning through association: is how you connect one idea to another to remember it.



5 + ? = 8). Indeed, it follows from Siegler's (1987) model that an associative facilitating effect should make the correct answer the most common response to a subtraction combination, even in the earliest phase of mental-subtraction development.

How does knowledge of addition strategies facilitate the learning of subtraction combinations?

Were you always aware of this association when you were taught subtraction at school?

Commentary: Addition and subtraction are inversely related. If learners have a good understanding of addition strategies, they can reverse their thinking of addition to solve subtraction problems. Learners need to understand the concept of addition at a concrete level before they can start to undo the addition and think of a difference rather than a sum. If addition and subtraction strategies are not taught in a connected way, learners will not be able to see the relationship between the two operations. They will simply solve each number sentence separately without seeing the association, for example, 5 + 2 = ? will not be associated with the number sentence 7 – 5 = ?

Step 3: Introduce number symbols and operational signs (abstract level)

When you see that learners understand what they are doing, and they are confident with the concept of subtraction, introduce the appropriate operational signs and number symbols. Learners need to know how to record subtraction abstractly, using symbols. We use the special mathematical sign to write “take away” or “minus”.

Also, make sure that learners know the meaning and function of the equals sign (=). Remind learners about the balancing of the scale idea that was used in Section 1 with Addition. The left-hand side of the number sentence must give the same answer as the right-hand side of the number sentence. By using the equals sign, you are implying that you have checked that the left-hand side is equal to the right-hand side.

minus Take away; subtract; make less or find the difference between.

= Equals; one side of the number sentence has the same number value as the other side: 5 – 3 = 2.



Let learners use their number symbol cards and the operational sign cards to form the sum (number sentence) below.

Let them also read the sentence: “Five minus (or take away) three equals two”. This helps you to see if they understand the abstract way of working mathematically. If they struggle with the abstraction, it is best to go back to using concrete objects or pictures of objects (semi-concrete). To extend the activity, also let them reverse the order of the number to be subtracted, i.e.:

To reinforce the relationship between subtraction and addition, learners can use number symbol cards and operational sign cards to show the inverse: Writing down the number names is introduced later when learners can read and write. 2.2 Apply to real life situations Introduce story sums so that they apply their understanding of subtraction to real life situations. For the sum: 5 – 3 = 2, a story sum like the following can be presented:

Always present the number sentences for the story sums, for example: Let learners determine the value of the to make the number sentence TRUE.

5 3 = 2

5 2 3 =

Five learners are playing on the playground. Three of them return to class. How many learners are left on the playground?

5 − 3 =

2 3 = 5 +



On a more advanced level, show learners a number sentence like the following and ask them to make up their own story:

Ask learners questions like: What word problems can we solve using numbers written like this (using subtraction)? Can you make up your own story for the number sentence? Let learners make up stories to show their understanding of the number sentence. For example: “I baked 20 cupcakes for my party. My friends ate 8. I have 12 left.” OR “I have R8. I need R20 to buy a book, so I need R12 more.” You will now learn about many practical ideas on HOW to teach subtraction using different methods. You will notice that they are very closely related to the ways in which the teaching of addition was presented in the previous section. 2.3 Methods for teaching subtraction Throughout the year, learners need a variety of ways to practice and reinforce their understanding of subtraction. In the next paragraphs, different methods and formats suitable for subtractions will be given. The same methods used to teach addition can also be used to teach subtraction. We will now give you some practical ideas to build on. 2.3.1 The use of concrete objects and models

1. The use of fingers and counters Let learners use their fingers to show combinations of 5 and 10. When working with 10, let learners first show all their fingers (10). Then ask them to take away a specific number, for example, “Take away 5 fingers. How many fingers are left?” Do this with all the possible combinations of ten. Reinforce the concepts of subtracting (take away, decrease, minus) as well as, “How many are left?” (The decreased total.)

“Ten fingers. Take away two. How many are left?”

Now hand out ten counters. Let learners show different combinations of ten with their counters. Give learners instructions like: “Put all the counters in your one hand (e.g. left hand). How many do you have altogether?” (10).

Take two counters with your other hand and put it behind your back. You took two away from ten. How many do you have left in your left hand? Do this with all the possible combinations of ten (or any other number).

20 − 8 = 12



2. The use of connecting cubes To build learners’ understanding of subtraction as “finding the difference”, let learners use connecting cubes to compare sets of quantities and the difference between them. In the example below, eight cubes are used. Have learners make two bars with their eight cubes. Discuss the difference between the two bars to generate the third number.

difference For example, if learners made a bar of 5 and a bar of 3 cubes, the difference is 2. It is written as 5 – 3 = 2. Encourage learners to make a drawing of their cubes showing the difference and then to use the appropriate number symbols and signs to represent this in a number sentence. (From the example we can also see that five is “greater than” or “more than” two.) 3. The use of counting frames and other concrete LTSM The abacus (counting frame) can also be used with success to help learners understand the concept of subtraction. Let them move the beads on each row to show different combinations of ten. Or let learners follow your instructions: “You have ten beads in a row. Take away one (move one to the other side). How many are left?” Remember to include activities to subtract one and subtract zero. Once learners mastered adding one and zero (0), they find it easy to learn the related subtraction facts involving zero and one. Once again, learners should notice that zero is also the identity element for subtraction since zero subtracted from any number always results in the same number that you started with. This property was already investigated with addition.



38

30 minutes

Reflect on the similarity between the methods for teaching subtraction and addition. What is the same? What is different?

2.3.2 Teach subtraction as “think addition” For each addition basic fact, there is a related subtraction fact. “Think addition” is the most important thinking strategy for learning and recalling subtraction facts. Encourage learners to recognise, think about and use the relationship between addition and subtraction facts. Do you still remember that subtraction is the inverse operation of addition? The part-part-whole model can be used to think about the relationship between addition and subtraction. The two parts that make up the whole can be found by “thinking subtraction”. This works because of the inverse relationship between addition and subtraction.

Whole Part Part

Consider the following:

Start unknown: + 25 = 40 40 – 25 =

40 (Whole)

(Part) 25 (Part)

Change unknown: 15 + = 40 40 – 15 =

40 (Whole) 15 (Part) (Part)

Learners can find the answers to subtraction facts by thinking about missing addends, for example: 14 − 6 = Think addition:

3

6 + = 14 6 + 8 = 14 So, 14 − 6 = 8



Even if you do not teach addition and subtraction at the same time, the relationship between addition and subtraction must always be pointed out to learners. 2.3.3 Written exercises When designing written activities for learners, change the position of the unknown and provide varied types of problems. Consider the following possibilities for subtraction:

Result unknown: 40 – 25 = Start unknown: – 25 = 15 Change unknown: 40 – = 15 Repeated subtraction: 40 – 5 – 5 – 5 – 5 – 5 – 5 – 5 = Multiple subtraction: 40 – 9 – 5 – 4 – 2 = Multiple operations (addition and subtraction): 40 + 10 – 5 – 3 + 2 =

As with addition, mental mathematics activities are particularly good practice for learners and should be done regularly. Here is an example of such an activity. You could design many activities like this one to give your learners many opportunities to become fluent in doing subtraction.

Calculate the following: Answer

1. 73 – 10 = 63

2. 173 – 10 = 163

3. 86 – 10 = 76

4. 286 – 10 = 276

5. 71 – 10 = 61

6. 571 – 100 = 471

7. 587 – 100 = 487

8. 587 – 300 = 287

9. 587 – 500 = 87

10. 587 – 87 = 500



39

60 minutes

What mental strategies are being consolidated in the mental mathematics activity above? How could you vary the mental maths table to consolidate other subtraction skills? Commentary: Subtraction of 10 and subtraction of 100 are being consolidated. You could design other mental maths activities to consolidate subtraction of multiples of 10 or 100, or subtraction resulting in a round number, and so on.

Examples of other mental maths tables to consolidate other skills: Subtraction of multiples of 10 and 100:

Subtraction that results in a round number:

Calculate the following: Answer Calculate the following: Answer

1. 73 – 30 = 43 1. 73 – 3 = 70

2. 173 – 30 = 143 2. 173 – 73 = 100

3. 86 – 30 = 56 3. 86 – 6 = 80

4. 286 – 30 = 256 4. 286 – 86 = 200

5. 71 – 30 = 41 5. 71 – 1 = 70

6. 571 – 200 = 371 6. 571 – 71 = 500

7. 587 – 200 = 387 7. 587 – 87 = 500

8. 687 – 200 = 487 8. 545 – 45 = 500

9. 487 – 200 = 287 9. 529 – 29 = 500

10. 597 – 200 = 377 10. 507 – 7 = 500 The exposure to more types of different subtraction problems will deepen your learners’ understanding of subtraction. 2.3.4 Subtract by using the breaking down method As with addition, learners must be able to use the breaking down method for subtraction. Let learners explore different ways of building up and breaking down numbers. This is beyond the scope of the Grade 1 curriculum, but examples for Grade 2 and Grade 3 using

3



both the horizontal and vertical breaking down methods for subtraction are provided below, to show the links between these methods. Example for Grade 2: Subtraction using the breaking down method: 16 12

16 – 12 = 10 + 6 – 10 – 2

Subtract units: 6 – 2 = 4 Subtract tens: 10 – 10 = 0_ Then: 16 – 12 = 4

Writing this in vertical format, prepares learners for the formal vertical algorithm/method.

10 + 6 – 10 − 2 0 + 4 = 4

Examples for Grade 3: Example 1: Subtract using the breaking down method without regrouping:

Horizontal method: 476 − 343 = 400 + 70 + 60 – (300 + 40 + 3) = 400 + 70 + 60 – 300 − 40 − 3 = 400 − 300 + 70 − 40 + 6 − 3 = 100 + 30 + 3 = 133 Vertical method: Subtract units 6 − 3 = 3 400 + 70 + 6 Subtract tens 70 − 40 = 30 OR − 300 − 40 − 3 Subtract hundreds 400 − 300 = 100 100 + 30 + 3 = 133 Then 476 − 343 = 133 Example 2: Subtract using the breaking down method with regrouping:

We cannot subtract 9 units from 5 units because we do NOT have enough units. Break down 765 into 700 + 50 + 15 (break down 1 ten in 10 units)

Calculate 476 – 343

Calculate 765 – 549



Then: 15 − 9 = 6 and 50 − 40 = 10 and 700 − 500 = 200 This means 765 − 549 = 216 Example 3: Subtract by breaking down the second number. Do the following example on the board. In this example, you first take away the hundreds, then the tens and then the units. 889 − 137 = = 889 − (100 + 30 + 7) = (889 − 100) − (30 + 7) = (789 − 30) − 7 = 759 − 7 = 752

40

30 minutes

Reflect on the way that you were taught to subtract when you were at school. Was it similar to the methods that have just been discussed? Do you remember only the steps or also the reasoning behind? How important is it to know the reasoning behind?

When learners fully understand the horizontal breaking down method, they are ready to progress to the vertical subtraction method. 2.3.5 Vertical subtraction Vertical subtraction is a method of subtraction where the digits are, like with the vertical addition method, written underneath each other and you subtract in vertical (from the top to the bottom) columns. Place value is again important, and it is a good idea to write both numbers in a place value table before you start. Just like with the addition algorithm, you must link the vertical algorithm to concrete demonstrations that will help learners understand the written method by following the concrete demonstration. When you start with vertical subtraction, let learners first subtract a few one-digit numbers, then two-digit numbers, then three-digit numbers. You might move fast from one to two-digit numbers, but when you get to three-digit numbers, you will have to spend more time to make sure that your learners are with you every step of the way!

4



Start with subtraction calculations where no regrouping, trading or “borrowing” needs to be done. Then do subtraction with regrouping of tens, then hundreds. Examples appropriate for Grade 3: Subtracting a one-digit number

Subtracting a two-digit number

Subtracting a three-digit number

498 − 5 498 − 65

498 −265

Here is an example showing base ten blocks and linking it to the vertical algorithm, where regrouping is necessary: 323 – 115 = __

Lay out the base ten blocks to show the number 323 = 300 + 20 + 3: Record the two numbers in the vertical algorithm.

Start working in the units column: I want to subtract (take away) 5 units from 3 units – I can’t do it. I, therefore, need to exchange (do you still remember the trading rules discussed in Unit 1 – revise this section if you are not sure): Lay out the base ten blocks to show the number 323 as 300 + 10 + 10 + 3:

323 − 115



Exchange the one ten for ten units, so that you can do the subtraction in the units column:

The working to show this is shown in the algorithm box. The display you now have is 300 + 10 + 13, and you can subtract the units:

13 units minus 5 units gives you 8 units. The tens and hundreds remain.

You now need to take away 1 ten from 1 ten, which gives you 0 tens, the units and hundreds remain.

323 − 115 ____

3231 1

323 − 115 8

323 1 1

323 − 115 08

323 1 1



Finally, you take away 1 hundred from 3 hundreds, which gives you 2 hundreds, the units remain.

Before you start to subtract, make sure that the place value columns are vertically aligned. In other words, the units digits are in one column underneath each other, the tens digits are in one column underneath each other, the hundreds digits are in one column underneath each other, etc. In vertical subtraction, like with vertical addition, calculations always start from the right-hand side. In other words, you first subtract the units digits, then the tens digits, then the hundreds digits, etc. If you allow learners to work with base ten blocks while they record numeric calculations, they will understand what is involved in the exchanging required to enable subtraction when the exchange is needed. This will help with whichever algorithm they use: vertical or horizontal. 2.3.6 Multiple operations: Addition and subtraction For written work, multiple operations including addition and subtraction must adhere to certain rules. Refer to B-FMA 110 for a discussion on the order of operations. Order of operation rules: 1. Brackets keep numbers together, and the operation within the brackets should always

be done first. 2. Complete the addition and subtraction in the order it appears from left to right. [You will learn again about the order of operations in Mathematics Teaching in the Foundation Phase 2 (F-MAT 312).] Look at the following examples to see HOW the rules are followed. Do brackets first: 6 + 3 – (5 + 3) = means 6 + 3 – 8 =

323 − 115 208

323 1 1



Then complete the addition and subtraction in the order it appears: 6 + 3 – 8 = 9 – 8 = 1 If there are no brackets, complete the operations (addition and subtraction) in order from left to right: 6 + 3 – 5 + 3 = means 6 + 3 – 5 + 3 = 7 Did you notice that with the brackets the answer is different?

41

60 minutes

Apply the order of operation rules to prepare a memorandum for the following Grade 2 test.

Test: Addition and Subtraction Grade 2

Complete the following: a. 17 – 3 + 5 + 2 = b. 17 – (3 + 5 + 2) = c. 10 + 4 – 5 – 2 = d. 10 + 4 – (5 – 2) = e. 20 – 10 – 8 + 2 = f. (20 – 10) – (8 + 2) = g. 7 + 4 – 2 – 9 = h. 7 + (4 – 2) – 9 =

Commentary: Using the order of operation rules, work from left to right if there are no brackets. If there are brackets, then calculate the brackets first before working from left to right.

a. 17 – 3 + 5 + 2 = 21 b. 17 – (3 + 5 + 2) = 17 – 10 = 7 c. 10 + 4 – 5 – 2 = 7 d. 10 + 4 – (5 – 2) = 14 – 3 = 11 e. 20 – 10 – 8 + 2 = 4 f. (20 – 10) – (8 + 2) = 10 – 10 = 0 g. 7 + 4 – 2 – 9 = 0 h. 7 + (4 – 2) – 9 = 7 + 2 – 9 = 0

You need to be sure that as a teacher you fully understand and can apply the correct order of operations when doing calculations with multiple operations!



Do not underestimate the use of a number line when solving problems involving both addition and subtraction. Draw, for example, a 0 to 20 number line on the chalkboard and show learners how it can be used to solve problems such as 9 + 2 – 4 + 5 or 20 – 6 + 3 – 5 – 2.

Also, expose learners to solving word problems involving addition and subtraction. For example:

Mr Thlapi had 250 sheep. He sold 80 sheep to his neighbour. He then bought another 30 sheep. How many sheep does Mr Thlapi now have?

To facilitate the solving of word problems, it is suggested that you scaffold the story sum by asking questions such as:

1. What is the question? (E.g. How many sheep does Mr Thlapi now have?) 2. What are the numbers involved in the problem? (250, 80 and 30) 3. What are the operations that need to be performed? Or, “What should I do with the

given numbers to solve the problem?” (80 should be subtracted from 250; sheep are taken away when sold. Then 30 should be added to the answer (170) when sheep are bought.)

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120 minutes

The following two word problems were given to Grade 3 learners to solve:

a) Mary has R180. She buys a book for R78. How much change does she get? b) Sihle had 120 marbles. During a game, he lost 25 marbles. In the next game, he

won another 65 marbles. How many marbles does he have now?

1. Describe how you will support the learners to solve each of the two problems. In your description:

Reflect how you will facilitate the solving of each problem through scaffolding. Show how a number line can be used to solve each problem. Write a number sentence to show the solution to each problem.

2. Discuss how you will provide additional support to a learner who still struggles to solve the second problem.

3. How will you change the two word problems to be appropriate for Grade 2 learners?

Commentary: The main thing to remember when you find learners having difficulty doing addition or subtraction is to help them to make the connection between what is required of them to solve the problem. For example: Do I need to find a total amount by combining? (Should



I add?) Do I need to find a difference between two given amounts? (Should I subtract?) How will I perform the required operation? What processes will I use?

First problem: Ask: What is the question? (How much change does she get?) What is the operation? (Subtraction). What are the numbers? (R180 and R78). Show how a number line could be used to solve this problem. 78

Thinking addition: By using counting-on on the number line, we first need to look for ways to work with multiples of 10 or 100. Starting at 78, to find the next multiple of 10, which is 80, we add 2, and then to get us to the next hundred, which is 100, we add 20. Now, to get to R180, we need to add another 80. Subtraction by counting backwards: Start at 180 and count backwards in tens up to 110. Seven times ten equals 70. You still need to subtract 8 (78 – 70 = 8). Subtract 8 from 110 get you to 102. Mary will get: R180 – R70 – R 8 = R102 change. Second problem: Ask: What is the question? (How many marbles does Sihle have?) What is the operation? (Subtraction and addition). What are the numbers? (120, 25 and 65) Show how a number line could be used to solve this problem. 95

+2 +80 +20

100 80 180 -70 -8

Thinking addition.

Counting backwards.

0 30 60 90 120 150 160 1. Subtraction

2. Addition



Sihle has 120 marbles, therefore, start at 120 on the number line. He lost 25 marbles, which means his marbles will become less. Therefore, 25 should be subtracted from 120. Do this by counting backwards in tens up to 100. Two times ten equals 20. You still need to subtract 5. Subtracting 5 from 100 get you to 95. When Sihle won 65 marbles, it implies that his marbles will increase. Therefore, 65 should be added to 95. Now, start at 95 to find the next multiple of 10, which is 100, we need to add 5. Add the remaining 60 by counting forwards in 10s. This will bring you to 160.

Sihle has: 120 – 25 + 65 = 160 marbles. The learner who struggles with solving the problem on the semi-concrete level (number line) needs to be taken back to the concrete level. Dienes blocks can be used to support the learner to promote the learner’s understanding. In your discussion, you should indicate how you will use the Dienes blocks. Consider the recommended number range for Grade 2 (subtract from numbers up to 99) and change the problems accordingly.

In the next section, you will be studying the inverse relationship between addition and subtraction more closely and think about how to use this in your teaching of these additive operations.

1. INTRODUCTION Addition and subtraction are often taught as two different operations. In this way of teaching, the focus is often on teaching algorithms and not on the understanding of these basic operations and how they relate to each other. Young learners come to school with some understanding of addition as an act of putting things together (join) and subtraction as separating or taking away something. When given two sets of tasks such as 5 + 8 = 13 and 13 – 5 = ?, it is often not easy for young learners to recognise that they could use the addition equation provided to

determine the answer to the subtraction task 13 – 5 = □. Thus, the problem can be solved

by translating it into an addition expression 5 + □ = 13. The relationship between addition and subtraction is not obvious to young learners. This relationship links to the part-part-whole model that you have read about earlier in this module. Relational thinking about addition and subtraction enables learners to draw on their knowledge of one operation to solve a problem relating to the other operation. Mastering the inverse relationship between addition and subtraction result in learners being able to

SECTION 3: ADDITION AND SUBTRACTION − INVERSE OPERATIONS



quickly and accurately produce a solution to a task without reliance on unitary counting to produce the answer. This shows their fluency of operational skills.

Teaching addition and subtraction separately hides the internal connectedness of these basic operations. In this section of work, addition and subtraction will be treated together to emphasise the inverse relationship of these operations. There will be times when you teach the two operations separately, but if you show learners the connections between them, you will help them to understand these two operations better.

We will now explore how the inverse relationship between addition and subtraction can be taught to Foundation Phase learners. Extract:

“Children's relational knowledge of addition and subtraction (Baroody, 2010)

Knowledge of addition combinations has long been thought to facilitate the learning of subtraction combinations (e.g. 8 – 5 = ? can be answered by thinking 5 + ? = 8). Indeed, it follows from Siegler's (1987) model that an associative facilitating effect should make the correct answer the most common response to a subtraction combination, even in the earliest phase of mental-subtraction development. Children in the initial or early phase of development were examined in 2 studies. Study 1 involved 25 kindergartners and 15 first graders in a gifted program. Study 2 involved 21 first graders in a regular program. Participants were presented with pairs of items, such as 4 + 5 = 9 and 9 – 4 = ?, and asked if the first item helped them to answer the second. Many participants, particularly the less developmentally advanced ones, did not recognise they could use a related addition equation to determine a difference. Study 2 participants were also administered a subtraction timed test. Contrary to Siegler's model, developmentally less advanced children responded with the correct difference infrequently on nearly all items, and even developmentally advanced children did so on more difficult items. The results of both studies are consistent with earlier findings that suggested the complementary relation between addition and subtraction is not obvious to children. They further indicate that an understanding of the complementary relation is not an all-or-nothing phenomenon. It often develops first with subtraction combinations related to the addition doubles, apparently because such addition combinations are memorised relatively early. Ready facility with related addition combinations might make it more likely that children will connect their knowledge of subtraction to their existing intuitive knowledge of part-whole relations. This process could also account for why Study 2 participants were able to master subtraction complements without computational practice.”

Baroody, A. (2010). Children's relational knowledge of addition and subtraction. Cognition and Instruction, 17(2), pp. 137-175.



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120 minutes

1. What was the purpose of this study? 2. What were the findings of the study? 3. In the two studies mentioned, why do you believe that many participants,

particularly the less developmentally advanced ones, did not recognise that they could use a related addition equation to determine a difference?

4. How would you ensure that your learners understand the relationship between the pair of items: 4 + 5 = 9 and 9 – 4 = ?

Commentary: The purpose of the study was to find out if the learning of addition and subtraction as inverse operations is linked to a learner’s developmental stage. It is interesting to recall at this point the previous aspects of development discussed in this module. Think back to Piaget and the concept of reversibility. Learners in the Foundation Phase are still discovering and acquiring the concept of reversibility. Therefore, learning that subtraction “undoes” addition is intricately linked to the concept of reversibility. The findings of this study support this claim that the link between addition and subtraction is not obvious to learners. The acquisition of the knowledge is linked to both the learner’s developmental stage as well as direct teaching from a teacher. The learners did not recognise the relationship, as they did not have a conceptual understanding of addition and subtraction. They could have been exposed to these types of expressions too early before the operations of addition and subtraction had been concretised with sufficient practice with concrete objects. If these operations had been introduced formally on an abstract level, there would be no connection between the “+” and “−” symbols. The learners would simply see them as symbols and not really understand the role that each symbol plays in each of the expressions. It is critical to expose Foundation Phase learners to many concrete activities involving addition and subtraction by counting physical objects. Once the learners have a meaningful understanding of the operations and have had enough practice to verbally express, using mathematical language, what each expression represents, they can proceed to the next levels of semi-concrete and abstract activities. In these activities, learners can discover how the two expressions are related by using numbers and symbols. If learners are not exposed to the varying levels of abstraction, they will never be able to assimilate that the equation involving subtraction is just another form of writing the equation involving addition.



2. INVERSE RELATIONSHIPS A number fact is made up of three numbers. These three numbers can be used to create other number facts. Knowing one fact can help learners with other facts. Look at the number facts that can be created with the numbers 3, 4, and 7.

Addition Facts Subtraction Facts

3 + 4 = 7 7 − 3 = 4

4 + 3 = 7 7 − 4 = 3

In general, learners find it more difficult to understand and grasp subtraction facts than addition facts. A way to assist learners in this is, for example, if a learner knows that 6 + 9 = 15, and he or she sees the subtraction sentence 15 – 9 = __, the learner can think, 9 and “what” are 15? Or, in other words: What can I add to 9 to get 15? Learners should be encouraged to think of the related addition fact when encountering an unknown subtraction fact. Learners often find themselves either counting on or counting back to solve subtraction, and that is inefficient. If learners learn the important inverse relationship between addition and subtraction, subtraction facts will become much easier. As you work with learners, use questions that encourage this strategy of the inverse relationship between addition and subtraction. 3. METHODS FOR TEACHING INVERSE RELATIONSHIPS The keywords that you need to include in your teaching of inverses relating to addition and subtraction are: Add, Sum, Subtract, Difference, Undo, and Inverse Operations. 3.1 Start with addition

3 + 4 = 7 and 7 − 4 = 3 (Tell the story as you model with unifix cubes. Then write the equations.)



Now, let’s look at the inverse of this story.

First, we added, then we “undid” our adding when we subtracted. Now, we have the same number we started with. We can show this also using an open number line. (Retell the story as you record the hops on the number line.)

“Put together”

Maria has 3 carrots.

Her mom gives her 4 more carrots.

Maria has 7 carrots.

Her mom asks her to give 4 carrots back.

“Take away” How many carrots are left?

3 + 4 = 7

Maria has 7 carrots altogether.

Maria has 3 carrots left.

3 4

7

How many carrots does she have now?

3

7 − 4 = 3

4

7



So, when we add we can undo subtraction, and when we subtract, we can undo addition. Mathematicians say it this way: Subtraction is the inverse operation of addition. That means they are opposites; they “undo” each other.

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60 minutes

Using the number sentences 5 + 3 = 8 and 8 – 3 = 5, do the following: 1. Write your related word problems (stories) for these number sentences. 2. Show how you would teach’ your learners the inverse relationship between the two

given number sentences, making use of concrete and semi-concrete aids.

Commentary: You could make up any stories to fit the number sentences. Use a context known to learners for this. In your explanation, remember to remind learners that addition and subtraction undo each other. On the concrete level, you can use Unifix cubes or any other appropriate concrete objects. Learners should be allowed to work with real objects, count them together and take them away, according to the question. On the semi-concrete level, represent this on a number line. Although this will help learners to understand the inverse relation on a more abstract level, the plotted numbers on the number line are still allowing learners to count on or visualise the operation they are doing.

3 + 4 = 7

7 – 4 = 37 − 4 = 3

3 + 4 = 7

3

3

7

7

+4

−4



3.2 Start with subtraction

This time we’re going to subtract first and see what we get when we add back the subtracted amount. Consider the following example:

Jane has 9 bracelets. She gave 5 bracelets to her friend. How many does she have now? Now, we start by taking away instead of putting together as in the first example: 9 – 5 = 4; Jane has 4 bracelets left. Let’s see what happens if her friend gives the 5 bracelets back: 4 + 5 = 9 First, we subtracted 5 from 9, and we got 4. Then we added the 5 returned bracelets to the remaining 4, and we got 9 again. We added to undo the subtraction. So, the inverse operation is true even when you start with subtraction. If you add back the number you just subtracted, you’ll get back to the number you started with.

45

60 minutes

Illustrate the inverse relation by drawing Unifix cube and number line representations to match the stories and number sentences above. Start with subtraction this time to show how addition “undoes” subtraction.

Commentary: You should draw these representations clearly, like the ones given in the first part (start with addition) of the discussion.

Another way of teaching addition and subtraction as inverse operations is to use the idea of a fact family.

3.3 Use fact families

Explain that a fact family is not a real family, but that the facts are related like people are related; therefore, they have been given the name family. First, write two addition facts on the chalkboard, e.g. 3 + 2 = 5 and 2 + 3 = 5. Ask your learners: “Do you see anything the same about the two facts?” Prompt them to realise that the numbers in the two number sentences are the same. Tell them: “Now, I am going to write down two related facts.” Write 5 – 2 = 3 and 5 – 3 = 2 on the chalkboard. Ask, “What is the same about the four facts?” Prompt learners



to use the same numbers. “Do you see anything different about these four facts?” Prompt learners to respond that the new facts are subtraction facts and the largest number comes first in both facts, while the first two facts are addition facts, and the largest number comes last in both facts. Tell learners that these four facts make up a fact family.

Take time to discuss other fact families with the learners. Then, write 3 + 3 = 6 on the chalkboard. Ask if it has another addition fact. Elicit (point out) that it does not because 3 + 3 turned around would be 3 + 3. Ask if any learner can tell you what the related subtraction fact might be (6 – 3 = 3). Discuss other doubles and why there is only one addition and one subtraction fact in fact families that have doubles.

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60 minutes

Do you think the idea of a “fact family” is a worthwhile number concept to learn about? Motivate your thinking by explaining how knowing one fact (e.g. 4 + 2 = 6) helps/does not help learners to find the rest of the facts in the family.

Now that you came to the end of Unit 2, do the following review activity.

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120 minutes

Read the following statement and complete the questions:

To promote learners’ understanding of addition and subtraction as inverse operations, it is recommended that addition and subtraction should be taught in a connected way to promote learning through association.

1. Use the context-free problems, 322 + 489 = and 811 − 489 = to illustrate how you will teach addition and subtraction in a connected way to promote Grade 3 learners’ understanding of the inverse relationship through associative learning. In your illustration, clearly show how you will use multiple representations (concrete, semi-concrete and abstract) for the development of appropriate concepts and algorithms for addition and subtraction.

2. Explain how this approach will not only help learners to understand addition and subtraction but also help them to recognise the relationship between the two operations. Also, indicate the significance of understanding this inverse relationship in subtraction calculations.

4



3. Describe how you will revise the relevant vocabulary related to addition and subtraction during this approach.

4. One of your learners did the following calculation: 811 – 489 and got 478 as an

answer. Identify the misconception in the subtraction calculation. Explain how you will support the learner to correct his/her conceptual error.

Commentary: In the last question, you may have noted that the learner subtracted the units of the first number from the second number. Also, the tens from the first number was subtracted from the second number. This is a conceptual error, as the second number needs to be subtracted from the first number. Working with subtraction as expanded notation and then with the short method (algorithm) in the vertical format can assist the learner to identify the mistake and correct the conceptual error.

In this Unit, you learnt how to use the correct vocabulary for addition and subtraction and how to identify the different situations for addition and subtraction, respectively. The unit also focused on using multiple representations for conceptual development of addition and subtraction and how to use the algorithms for each. The role of the two operations in problem-solving was also discussed, as well as the inverse relationship between addition and subtraction and how to teach this to learners. Reflect on what you have learnt by completing the self-assessment activity. If your answer is UNSURE or NO on any of the concepts, go back to the relevant sections to study it again.

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90 minutes

If your answer is UNSURE or NO on any of the criteria, go back to the relevant section to study it again.

Self-assessment activity: Unit 2

Now that I have worked through this unit, I can:

YES UNSURE NO

Explain why it is important to use the correct vocabulary relating to addition and subtraction.

Identify the different situations for teaching addition and subtraction, respectively.



Demonstrate the use of multiple representations for the conceptual development of the concept of addition and subtraction, respectively.

Teach and assess the use of algorithms for addition and subtraction, respectively.

Discuss the role of problem-solving using addition and subtraction.

Explain the inverse relationship between addition and subtraction.

Use knowledge and understanding to teach learners about addition and subtraction as inverse operations.

Identify probable misconceptions and assist learners to correct conceptual errors.

You have come to the end of this module. We trust that you will use the knowledge that you have acquired from this module to get you started to become a responsive and effective mathematics teacher. Furthermore, that you will also create engaging environments that will promote meaningful and deep learning for your learners. The focus of the next module will be multiplicative reasoning, fractions, multiple operations and mental mathematics. We wish you all the best in your journey to become an inspiring and successful mathematics teacher!



REFERENCES Baroody, A. (2010). Children's relational knowledge of addition and subtraction. Cognition and Instruction, 17(2), pp. 137-175. Broadbent, A. (2004). Understanding place-value – a case study of the base ten game. Australian Primary Mathematics Classroom, 9(4), pp. 45-46. Clipartkey. (2019). Clipartkey. [Online], Available at: https://www.clipartkey.com/view/ihwRmTR_cute-frog-clipart/ [accessed, 18 May 2020]. Clker.com. (2018). Clker.com. [Online], Available at: http://www.clker.com/clipart-622964.html# [accessed, 18 May 2020]. Department of Basic Education. (2011). Curriculum and Assessment Policy Statement. Mathematics: Grades 1 – 3. Government Printing: South Africa. Department of Higher Education and Training. (2015). Revised policy on the minimum requirements for teacher education qualifications. Government Gazette, 596 (38487). Freesvg.org. (2013). Freesvg.org. [Online], Available at: https://freesvg.org/boy-face-cartoon-vector [accessed, 18 May 2020]. Freesvg.org. (2015). Freesvg.org. [Online], Available at: https://freesvg.org/smiling-face-of-a-child-vector-drawing [accessed, 18 May 2020]. Gifford, S. (2005). Teaching Mathematics 3-5 : Developing Learning in the Foundation Stage. Maidenhead: McGraw-Hill Education. (Chapter 8). Herholdt, R. & Sapire, I. (2014). An error analysis in the early grades mathematics – a learning opportunity? South African Journal of Childhood Education, 2014(1), pp. 42-60. Jess. (2006). flickr. [Online], Available at: https://www.flickr.com/photos/49744078@N00/274772850 [accessed, 16 May 2020]. Kenrick, N. (2011). flickr. [Online], Available at: https://www.flickr.com/photos/33363480@N05/6326252915 [accessed, 15 May 2020]. Krebs, D. (2013). flickr. [Online], Available at:https://www.flickr.com/photos/mrsdkrebs/9718252755 [accessed, 18 May 2020].



Lam, W. (2016). Flickr. [Online], Available at: https://www.flickr.com/photos/85567416@N03/26899145485 [accessed, 18 May 2020]. Modlin, D. (2006). Modlin Dictionaries: Maths Grade 7-9. Modlin e-Learning (Pty) Ltd. Naudé, M. & Meier, C. (Eds.). (2014). Teaching foundation phase mathematics. Pretoria: Van Schaik. Needpix.com. [n.d.]. Needpix.com. [Online], Available at: https://www.needpix.com/photo/837546/smiley-face-smile-happy-free-pictures-free-photos-free-images-royalty-free [accessed, 18 May 2020]. Pxfuel.com. [n.d.]. Pxfuel.com. [Online], Available at: https://www.pxfuel.com/en/free-photo-odljo [accessed, 18 May 2020]. Skelton, C. (2014). Teaching strategies for mental mathematics (Foundation Phase). In: Proceedings of the 20th annual national congress of the Association for Mathematics Education of South Africa (AMESA). [Online], Available at: http://www.amesa.org.za/AMESA2014/Proceedings/index.html [accessed, 2 April 2020]. The Yellow Peace. (2009). flickr. [Online], Available at: https://www.flickr.com/photos/36328518@N07/3384100342 [accessed, 15 May 2020]. Van de Walle, J.A., Lovin, L.H., Karp, K.S. & Bay-Williams J.M. (2014). Teaching student-centered mathematics: Developmentally appropriate instruction for Grades Pre-K-2. Boston: Pearson. Van de Walle, J.A., Karp, K.S. & Bay-Williams J.M. (2010). Elementary and middle school mathematics: Teaching developmentally. 7th ed. Boston: Pearson. Webstockreview.net. [n.d.]. Webstockreview.net. [Online], Available at: https://webstockreview.net/image/back-clipart-hand/244992.html [accessed, 18 May 2020].



ADDENDUM A: NUMBER EXPANSION CARDS

0 0

9 5 0 0 0

8 0 0 0 0 0

7 4 9 3 6 9

6 0 0 0 0 0

5 3 8 0 0 0

4 0 0 2 5 8

3 2 7 0 0 0

2 0 0 0 0 0

1 1 6 1 4 7



ADDENDUM B: 100S CHART

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100



ADDENDUM C: LESSON PLAN TEMPLATE

NAME:

STUDENT NO.

1. SUBJECT e.g. English HL

1.2 DATE

y y y y m m d d 2 0 - -

1.3 GRADE (Mark the grade you will be teaching with an X)

R 1 2 3

2. KNOWLEDGE/CONTENT AREA e.g. Phonics

3. THEME e.g. Healthy living, My body etc.

4. TYPE of LESSON / LESSON FOCUS e.g. Outdoor lesson, group work, class work etc.

PLEASE NOTE THAT THIS LESSON PLANNING TEMPLATE IS AVAILABLE IN ELECTRONIC FORMAT ON MySANTS

5. NCS AIMS/General aims (tick boxes) Learners are able to:

Identify and solve problems and make decisions using critical and creative thinking. Work effectively with others as members of a team, group, organisation and community. Organise and manage themselves and their activities responsibly and effectively. Collect, analyse, organise and critically evaluate information. Communicate effectively using visual, symbolic and/or language skills in various modes. Use science and technology effectively and critically showing responsibility towards the

environment and the health of others. Demonstrate an understanding of the world as a set of related systems by recognising that

problem-solving contexts do not exist in isolation. 6. SUMMARY OF THE CONTENT TO COVER IN THIS LESSON (Briefly summarise the content that you will be presenting in this lesson.)

7. LESSON OBJECTIVE(S): 7.1 PRE-KNOWLEDGE (Write down learners’ existing knowledge, skills and values.) At the start of this lesson the learners should already know… and can do…

SANTS Private Higher Education Institution GRADES R, 1, 2 and 3 LESSON PLANNING FORM



7.2 CONCEPTS and NEW KNOWLEDGE (Write down the new knowledge, skills and values that you are going to teach taking INTEGRATION into consideration.)

Language: English (HL/FAL)

Mathematics Life Skills

7.3 LESSON OBJECTIVES (In your own words, write the lesson objectives based on the general and specific aims from CAPS.) By the end of the lesson the learners should be able to… 7.4 FUTURE LEARNING (Briefly describe what the learners will learn in the lesson that follows this one) 7.5 DIFFERENTIATION (Briefly describe how you will present this lesson taking the following aspects into consideration)

Learner support (Indicate what measures are in place for learners who grasped concepts quickly. How will you challenge them and keep them from getting bored?)

Enrichment activities (Indicate what measures are in place for learners who struggle to grasp the concepts. How will you support them and keep them from getting negative and frustrated?)

Concerns (e.g. Loadshedding – won’t be able to listen to audio book. Will have to read story instead, using instruments for sound effects.)

8. LESSON PHASES: 8.1 INTRODUCTION OF THE LESSON (Give a detailed description of how you plan to begin your lesson by explaining: you will greet the learners, set the atmosphere for the lesson, awaken the learners’ prior knowledge, and create a link between what they already know to the new knowledge that you will be presenting. Also explain how the THEME you selected in 3 above will help you do this.):

8.1.1 Time allocated: 8.1.2 LTSM: (Describe the resources and media you will be using in the introduction phase of the lesson)



8.2 DEVELOPMENT – PRESENTING THE NEW KNOWLEDGE (Give a detailed description of WHAT content you will be presenting (selected in 7.2), HOW you will present it, and WHAT ACTIVITIES THE LEARNERS WILL BE DOING.): 8.3 CONSOLIDATION (Give a detailed description of how you plan to end the lesson by explaining how you will consolidate the new knowledge, incorporate assessment of the objectives and wrap up. If applicable, mention here any HOMEWORK/FUNWORK that you will give the learners.):

8.2.1 Time allocated: 8.2.2 LTSM: (Describe the resources and media you will be using in the development phase of the lesson) 8.3.1 Time allocated: 8.3.2 LTSM: (Describe the resources and media you will be using in the consolidation phase of the lesson)

9. ASSESSMENT At the end of the lesson, I will assess whether the learners have achieved the objectives in the following ways (tick the appropriate blocks): 9.1 FORMS OF ASSESSMENT:

Written work (drawings, painting etc.) Demonstrations (performing actions, experiments etc.) Performances (answers questions, making a speech, presenting a poem, reading aloud, role play, dialogue) Models (artwork, constructions, collages etc.)

Assessment strategy Assessor Assessment instrument

Observation Listening Reading Interpreting Reviewing Questioning Writing

Teacher Self Peer

Checklist Assessment scale Analytical rubric Holistic rubric

10. REFLECTION Briefly reflect on your lesson by discussing its strengths (what went well), its weaknesses (what did not work), what did you find challenging, if the lesson objectives were met and what would you improve if you had to teach this lesson again. Use the following questions to guide your reflection:

Describe aspects of your lesson that worked really well. Which areas of your lesson did not go according to plan? Explain why you think this may have happened. Look again at your lesson objectives. Did you meet them? Why/why not? What did you learn about the learners in your class today? What was your most challenging moment in this lesson and why? How will you respond next time? To what extent were the learners productively engaged in the learning process? Discuss. If you had the opportunity to teach this lesson again to this same group of learners, what would you do differently? Why? What evidence/ feedback do you have that the learners achieved an understanding of the lesson objective(s)?

11. REFERENCE LIST (List all the text books, workbooks, documents such as the CAPS document, websites etc. that you used to prepare this lesson.)

Mathematics Teaching in the Foundation Phase 1

Documents

Transcript of Mathematics Teaching in the Foundation Phase 1