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Intermediate Phase Grade 6 • Facilitator’s Guide

Mathematics CAPS Mathematics

• Comprehensive explanations of concepts in simple language. everyday examples with visuals and diagrams to help • Practical, master concepts. • Learners work at their own pace. • Activities that test learners’ knowledge application and reasoning. • The facilitator’s guide contains step-by-step calculations and answers. • Use in school or at home.

home classroom college workplace

9 781990

Facilitator’s Guide

2106-E-MAM-FG01

946462

Mathematics Facilitator’s guide Grade 6

2106-E-MAM-FG01

9 781990

946462

CAPS aligned

D Botha M Vos

Facilitator’s Guide G06 ~ Mathematics

Contents LESSON ELEMENTS.....................................................................................................................................................1 INTRODUCTION...........................................................................................................................................................2 YEAR PLAN....................................................................................................................................................................9 FACT SHEET................................................................................................................................................................13 UNIT 1..........................................................................................................................................................................26 LESSON 1: WHOLE NUMBERS...............................................................................................................................27 Counting in whole numbers...................................................................................................................................... 28 Prime numbers................................................................................................................................................................ 30 ACTIVITY 1.............................................................................................................................................................. 33 Place value......................................................................................................................................................................... 36 ACTIVITY 2.............................................................................................................................................................. 41 Ordering whole numbers............................................................................................................................................ 44 ACTIVITY 3.............................................................................................................................................................. 47 LESSON 2: NUMBER SENTENCES.........................................................................................................................49 Word problems................................................................................................................................................................ 49 Ways to solve number sentences............................................................................................................................. 50 ACTIVITY 4.............................................................................................................................................................. 51 Equivalents........................................................................................................................................................................ 53 Inverse operations......................................................................................................................................................... 55 Commutative property................................................................................................................................................. 56 Associative property..................................................................................................................................................... 56 ACTIVITY 5.............................................................................................................................................................. 57 LESSON 3: WHOLE NUMBERS...............................................................................................................................60 Steps when adding and subtracting....................................................................................................................... 60 Rounding............................................................................................................................................................................ 61 ACTIVITY 6.............................................................................................................................................................. 62 Methods of addition and subtraction.................................................................................................................... 65 ACTIVITY 7.............................................................................................................................................................. 66 ACTIVITY 8.............................................................................................................................................................. 68 Calculators......................................................................................................................................................................... 69 ACTIVITY 9.............................................................................................................................................................. 71 LESSON 4: COMMON FRACTIONS........................................................................................................................73 Describe and order fractions..................................................................................................................................... 73 Equivalent forms............................................................................................................................................................. 74 Tenths and hundredths................................................................................................................................................ 79 ACTIVITY 10............................................................................................................................................................ 80 Calculations with fractions......................................................................................................................................... 83 ACTIVITY 11............................................................................................................................................................ 86 LESSON 5: TIME........................................................................................................................................................88 Time on clocks................................................................................................................................................................. 88 i

Facilitator’s Guide G06 ~ Mathematics

ACTIVITY 12............................................................................................................................................................ 91 Calculating with time.................................................................................................................................................... 93 ACTIVITY 13............................................................................................................................................................ 94 Time on a calendar........................................................................................................................................................ 96 ACTIVITY 14............................................................................................................................................................ 97 Time zones........................................................................................................................................................................ 97 ACTIVITY 15..........................................................................................................................................................101

LESSON 6: PROPERTIES OF 2D SHAPES......................................................................................................... 103 Types of sides.................................................................................................................................................................104 ACTIVITY 16..........................................................................................................................................................104 Number of sides of polygons...................................................................................................................................105 Length of sides...............................................................................................................................................................107 The size of the angles..................................................................................................................................................108 ACTIVITY 17..........................................................................................................................................................109 LESSON 7: DATA HANDLING............................................................................................................................... 112 ACTIVITY 18..........................................................................................................................................................114 ACTIVITY 19..........................................................................................................................................................120 LESSON 8: NUMBER PATTERNS (NUMERIC PATTERNS)........................................................................... 125 Investigate and extend patterns.............................................................................................................................125 Input and output values............................................................................................................................................126 Multiplication and division as inverse operations.........................................................................................127 ACTIVITY 20..........................................................................................................................................................127 Order of multiplication..............................................................................................................................................129 Multiply with multiples of 10 and 100................................................................................................................130 ACTIVITY 21..........................................................................................................................................................131 UNIT 2....................................................................................................................................................................... 135

LESSON 9: WHOLE NUMBERS ........................................................................................................................... 136 ACTIVITY 22..........................................................................................................................................................137 ACTIVITY 23..........................................................................................................................................................138 ACTIVITY 24..........................................................................................................................................................140 LESSON 10: WHOLE NUMBERS – MULTIPLICATION................................................................................... 143 ACTIVITY 25..........................................................................................................................................................144 Estimation.......................................................................................................................................................................145 ACTIVITY 26..........................................................................................................................................................146 Distributive property of multiplication..............................................................................................................147 ACTIVITY 27..........................................................................................................................................................148 Factors to multiply.......................................................................................................................................................150 ACTIVITY 28..........................................................................................................................................................151 ACTIVITY 29..........................................................................................................................................................153 Vertical column method of multiplication.........................................................................................................155 ACTIVITY 30..........................................................................................................................................................156 Ratio and rate.................................................................................................................................................................156

Facilitator’s Guide G06 ~ Mathematics

ACTIVITY 31..........................................................................................................................................................159 Calculators.......................................................................................................................................................................160 ACTIVITY 32..........................................................................................................................................................161

LESSON 11: PROPERTIES OF 3D OBJECTS..................................................................................................... 163 Properties of 3D objects............................................................................................................................................163 Grouping 3D objects....................................................................................................................................................164 Objects with flat faces only.......................................................................................................................................165 ACTIVITY 33..........................................................................................................................................................167 ACTIVITY 34..........................................................................................................................................................172 Nets and models of 3D objects...............................................................................................................................173 ACTIVITY 35..........................................................................................................................................................176 LESSON 12: GEOMETRIC PATTERNS............................................................................................................... 178 Repeating patterns......................................................................................................................................................179 Patterns that increase.................................................................................................................................................180 Patterns that decrease................................................................................................................................................181 ACTIVITY 36..........................................................................................................................................................183 Patterns with a constant difference.....................................................................................................................185 Patterns without a constant difference...............................................................................................................187 ACTIVITY 37..........................................................................................................................................................188 LESSON 13: SYMMETRY...................................................................................................................................... 191 ACTIVITY 38..........................................................................................................................................................193 ACTIVITY 39..........................................................................................................................................................196 LESSON 14: WHOLE NUMBERS – DIVISION................................................................................................... 198 Rules of divisibility......................................................................................................................................................198 Using multiplication to divide.................................................................................................................................199 ACTIVITY 40..........................................................................................................................................................200 Long division..................................................................................................................................................................202 ACTIVITY 41..........................................................................................................................................................203 LESSON 15: DECIMAL FRACTIONS................................................................................................................... 206 Recognising decimal fractions................................................................................................................................207 Place value of decimal fractions.............................................................................................................................210 ACTIVITY 42..........................................................................................................................................................211 Counting in decimal fractions.................................................................................................................................212 Comparing and arranging decimal fractions....................................................................................................213 ACTIVITY 43..........................................................................................................................................................213 Rounding decimal numbers.....................................................................................................................................215 Adding and subtracting decimal fractions.........................................................................................................216 Multiplying decimal fractions by 10 and 100...................................................................................................217 ACTIVITY 44..........................................................................................................................................................217 LESSON 16: CAPACITY/VOLUME...................................................................................................................... 220 What is the difference between capacity and volume?................................................................................220 Measuring capacity......................................................................................................................................................220 iii

Facilitator’s Guide G06 ~ Mathematics

ACTIVITY 45..........................................................................................................................................................222 Converting between units of measurement......................................................................................................222 Kilolitre.............................................................................................................................................................................223 ACTIVITY 46..........................................................................................................................................................225

UNIT 3....................................................................................................................................................................... 228

LESSON 17: MASS.................................................................................................................................................. 229 Units of mass..................................................................................................................................................................230 Measuring mass............................................................................................................................................................231 ACTIVITY 47..........................................................................................................................................................234 Compare and order mass..........................................................................................................................................236 ACTIVITY 48..........................................................................................................................................................237 Convert between units of measurement............................................................................................................239 Rounding mass..............................................................................................................................................................241 ACTIVITY 49..........................................................................................................................................................242 LESSON 18: WHOLE NUMBERS......................................................................................................................... 244 Counting, ordering, comparing, representing and place value of 9-digit whole numbers...........245 ACTIVITY 50..........................................................................................................................................................245 LESSON 19: WHOLE NUMBERS – ADDITION AND SUBTRACTION......................................................... 247 Addition and subtraction of 6-digit whole numbers.....................................................................................247 ACTIVITY 51..........................................................................................................................................................249 Calculators.......................................................................................................................................................................254 ACTIVITY 52..........................................................................................................................................................256 LESSON 20: VIEWS OF OBJECTS........................................................................................................................ 258 ACTIVITY 53..........................................................................................................................................................261 LESSON 21: PROPERTIES OF 2D SHAPES....................................................................................................... 266 Rectangles and parallelograms..............................................................................................................................269 Properties of a circle...................................................................................................................................................270 ACTIVITY 54..........................................................................................................................................................273 LESSON 22: TRANSFORMATIONS..................................................................................................................... 276 Use transformations to describe patterns.........................................................................................................278 Use symmetry to describe patterns.....................................................................................................................279 ACTIVITY 55..........................................................................................................................................................280 LESSON 23: TEMPERATURE............................................................................................................................... 285 Measuring instruments.............................................................................................................................................286 ACTIVITY 56..........................................................................................................................................................288 LESSON 24: PERCENTAGES................................................................................................................................. 292 Equivalents of percentages......................................................................................................................................293 ACTIVITY 57..........................................................................................................................................................296 Ordering percentages, fractions and decimal fractions...............................................................................299

Facilitator’s Guide G06 ~ Mathematics

ACTIVITY 58..........................................................................................................................................................299 Percentage of a number.............................................................................................................................................301 ACTIVITY 59..........................................................................................................................................................303

LESSON 25: DATA HANDLING............................................................................................................................ 307 ACTIVITY 60..........................................................................................................................................................309 Analysing and interpreting data............................................................................................................................313 ACTIVITY 61..........................................................................................................................................................315 LESSON 26: NUMERIC PATTERNS..................................................................................................................... 319 Determining the rule..................................................................................................................................................319 ACTIVITY 62..........................................................................................................................................................320 Flow diagrams with two rules................................................................................................................................322 ACTIVITY 63..........................................................................................................................................................323 LESSON 27: LENGTH............................................................................................................................................. 326 Units of length (or distance)....................................................................................................................................326 Measuring instruments.............................................................................................................................................326 Comparing and ordering length.............................................................................................................................328 ACTIVITY 64..........................................................................................................................................................328 Converting between units of measurement......................................................................................................331 Rounding length............................................................................................................................................................334 ACTIVITY 65..........................................................................................................................................................335 UNIT 4....................................................................................................................................................................... 338

LESSON 28: WHOLE NUMBERS......................................................................................................................... 339 Reading and saying 9-digit whole numbers.....................................................................................................339 ACTIVITY 66..........................................................................................................................................................339 Place value of 9-digit whole numbers.................................................................................................................341 ACTIVITY 67..........................................................................................................................................................342 Comparing and ordering 9-digit whole numbers...........................................................................................344 ACTIVITY 68..........................................................................................................................................................344 LESSON 29: WHOLE NUMBERS – MULTIPLICATION................................................................................... 347 Properties of multiplication.....................................................................................................................................347 Quick ways of doing multiplication......................................................................................................................349 Multiples, factors and prime factors....................................................................................................................350 ACTIVITY 69..........................................................................................................................................................351 Calculation techniques...............................................................................................................................................352 ACTIVITY 70..........................................................................................................................................................354 Problem-solving............................................................................................................................................................356 ACTIVITY 71..........................................................................................................................................................356 LESSON 30: COMMON FRACTIONS................................................................................................................... 359 Simplifying and ordering fractions.......................................................................................................................360 ACTIVITY 72..........................................................................................................................................................361 Equivalents of fractions, decimal fractions and percentages....................................................................365 v

Facilitator’s Guide G06 ~ Mathematics

ACTIVITY 73..........................................................................................................................................................367 Calculations with fractions.......................................................................................................................................372 ACTIVITY 74..........................................................................................................................................................374

LESSON 31: PROPERTIES OF 3D OBJECTS..................................................................................................... 377 Properties of 3D objects............................................................................................................................................379 ACTIVITY 75..........................................................................................................................................................381 Nets and models of 3D objects...............................................................................................................................384 ACTIVITY 76..........................................................................................................................................................386 LESSON 32: PERIMETER, AREA AND VOLUME............................................................................................. 390 Perimeter and area......................................................................................................................................................390 Measuring perimeter with a ruler and tape measure..................................................................................393 Diagonals..........................................................................................................................................................................394 Area of a rectangle.......................................................................................................................................................395 Relation between the perimeter and area of rectangles and squares...................................................396 ACTIVITY 77..........................................................................................................................................................398 Volume..............................................................................................................................................................................401 ACTIVITY 78..........................................................................................................................................................402 ACTIVITY 79..........................................................................................................................................................406 Relation between the perimeter and volume of a rectangular prism....................................................409 LESSON 33: THE HISTORY OF MEASUREMENT............................................................................................ 412 How length, mass and volume were measured in the past........................................................................412 How time was measured in the past....................................................................................................................414 ACTIVITY 80..........................................................................................................................................................415 LESSON 34: WHOLE NUMBERS – DIVISION................................................................................................... 416 Rules of divisibility......................................................................................................................................................416 Using multiplication to do division – clue board............................................................................................418 ACTIVITY 81..........................................................................................................................................................419 Ratio and rate.................................................................................................................................................................421 ACTIVITY 82..........................................................................................................................................................422 LESSON 35: NUMBER SENTENCES................................................................................................................... 425 ACTIVITY 83..........................................................................................................................................................427 LESSON 36: TRANSFORMATIONS..................................................................................................................... 431 Enlargement and reduction.....................................................................................................................................432 ACTIVITY 84..........................................................................................................................................................434 LESSON 37: LOCATION AND DIRECTION....................................................................................................... 440 Using alpha-numeric grid references to determine location on a grid/map ....................................440 Giving directions...........................................................................................................................................................442 ACTIVITY 85..........................................................................................................................................................443 LESSON 38: PROBABILITY.................................................................................................................................. 447 Identifying possible outcomes................................................................................................................................448

Facilitator’s Guide G06 ~ Mathematics

Counting and comparing the regularity of actual outcomes.....................................................................449 ACTIVITY 86..........................................................................................................................................................450

IMAGE REFERENCES............................................................................................................................................. 452

vii

Facilitator’s Guide G06 ~ Mathematics

LESSON ELEMENTS The guide containts various lesson elements. Each element is important for the learning process.

ICON

LESSON ELEMENT

ICON

Think for yourself

LESSON ELEMENT

Remember or revise

Take note! or

Tips

Important

Research

Self-assessment

Study

Activity

New concept or

Did you know?

definition

Facilitator’s Guide G06 ~ Mathematics

INTRODUCTION Grade 6 learners are now in the third year in the new phase in their schooling career, and they are already familiar with the concept of self-study. The study guide and facilitator’s guide will help learners and facilitators to navigate the process and lay the foundation for future academic success. However, the facilitator’s role in supporting learners remains crucial since learners must gain self-confidence and develop a love for maths. The study guide has a friendly and informal tone to involve learners and make the subject accessible and interesting. It contains theory, activities and research elements. It is important to complete all the activities in the study guide to help learners understand and apply new knowledge. There is an individual self-assessment after each lesson. Use this to determine whether a learner still requires help with a particular lesson or concept, and do revision or new exercises as soon as possible to ensure learners master the concept. The self-assessment can also be used to plan enrichment activities that can be completed once learners have mastered the concepts in each lesson. With the help of the facilitator’s guide, the facilitator can support learners so the theory of mathematics is well established at the end of each lesson. Many of the lessons have a practical component and there are recommended for these in the guide. Both guides are divided into four units. Aim to complete one unit per term.

Do 10 minutes of mental arithmetic every day. Use the supplementary Train Your Brain Maths Grade 6 product. The CAMI programme (www.camiweb.com) is recommended for revision and additional practise.

The products include: • A facilitator’s guide • Two study guides (units 1 and 2 are in study guide 1/2 and units 3 and 4 are in study guide 2/2) • Train Your Brain Maths Grade 6 (for mental arithmetic)

The learners need a book to complete the activities in. It must preferably be a black hard cover book with a grid (quad ruled) instead of lines. Writing in blocks helps learners to space numbers correctly and to write numbers neatly underneath each other, like with the vertical column method, for example.

TIME MANAGEMENT

The time allocation per topic is a guideline and may be adapted according to the learners’ pace. Some topics are covered extensively but have shorter time allocation, for example, whole numbers in term 1. The learners studied whole numbers in Grade 4, and only do revision in Grade 5. However, the topic is covered comprehensively to ensure the information is readily available, should the facilitator identify gaps in the learners’ knowledge while doing revision. If this is the case, use the opportunity to repeat the work. Adjust the time allocation based on the learners’ mastery of the topic. © Optimi

Facilitator’s Guide G06 ~ Mathematics

From term 2, learners are referred to previous lessons. The time allocation includes the revision to be completed before doing the activity.

It is important to not continue with the next lesson or topic before the learners have a thorough understanding of the current topic, even if you exceed the recommended time. Continually adjust the time allocation to address the learners’ needs.

It is, however, important to keep in mind that the required lessons must be completed before tests or exams can be attempted. Learners must spend six hours per week on maths. This does not include all activities, assessments, and examinations. If learners are working at a slower pace, make the necessary adjustments to allow them to complete all the work in time.

Facilitator’s Guide G06 ~ Mathematics

Guidelines for time allocation per topic UNIT 1 TOPIC

TIME AND NOTES

Mental maths: Use Train Your Brain Maths Grade 6

Lesson 1 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (3-digit whole numbers) Lesson 2 Number sentences

Lesson 3 Whole numbers: Addition and subtraction (5-digit whole numbers) Lesson 4 Common fractions

8 hours (divided into 10 minutes per day) 2 hours 3 hours 7 hours 10 hours

Lesson 5 Time

4 hours

Lesson 6 Properties of 2D shapes

8 hours

Lesson 7 Data handling

10 hours

Lesson 8 Number patterns (numeric patterns)

4 hours

Revision: Use the CAMI program

4 hours

TOTAL

60 hours

Facilitator’s Guide G06 ~ Mathematics

UNIT 2 TOPIC

TIME AND NOTES

Mental maths: Use Train Your Brain Maths Grade 6

Lesson 9 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers)

7 hours (divided into 10 minutes per day) 1 hour

Lesson 10 Whole numbers: Multiplication (4-digit whole numbers by 2-digit whole numbers)

5 hours

Lesson 12 Geometric patterns

6 hours

Lesson 11 Properties of 3D objects

Lesson 13 Symmetry

Lesson 14 Whole numbers: Division (4-digit whole numbers by 2-digit whole numbers) Lesson 15 Decimal fractions

5 hours

2 hours 8 hours 10 hours

Lesson 16 Capacity/Volume

5 hours

Revision: Use the CAMI program

5 hours

Assessment – examinations of all subjects

6 hours

TOTAL

60 hours

Facilitator’s Guide G06 ~ Mathematics

UNIT 3 TOPIC

TIME AND NOTES

Mental maths: Use Train Your Brain Maths Grade 6 Lesson 17 Mass

Lesson 18 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers) Lesson 19 Whole numbers: Addition and subtraction (6-digit whole numbers) Lesson 20 Viewing objects

8 hours (divided into 10 minutes per day) 5 hours 1 hour 8 hours 3 hours

Lesson 21 Properties of 2D shapes

4 hours

Lesson 22 Transformations

3 hours

Lesson 23 Temperature

1 hour

Lesson 24 Percentages

5 hours

Lesson 25 Data handling

9 hours

Lesson 26 Numeric patterns

5 hours

Lesson 27 Length

5 hours

Revision: Use the CAMI program

3 hours

TOTAL

60 hours

Facilitator’s Guide G06 ~ Mathematics

UNIT 4 TOPIC

TIME AND NOTES

Mental maths: Use Train Your Brain Maths Grade 6

Lesson 28 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers)

7 hours (divided into 10 minutes per day) 1 hour

Lesson 29 Whole numbers: Multiplication (4-digit whole numbers by 3-digit whole numbers)

5 hours

Lesson 31 Properties of 3D objects

5 hours

Lesson 30 Common fractions

Lesson 32 Perimeter, area and volume

5 hours

7 hours

Lesson 33 The history of measurement

1 hour

Lesson 34 Whole numbers: Division (4-digit whole numbers by 3-digit whole numbers)

7 hours

Lesson 36 Transformations

3 hours

Lesson 35 Number sentences

Lesson 37 Position and direction

3 hours

2 hours

Lesson 38 Probability

2 hours

Revision: Use the CAMI program

6 hours

Assessment – examinations of all subjects

6 hours

TOTAL

60 hours

Facilitator’s Guide G06 ~ Mathematics

Units 1 and 2 are in study guide 1/2 and units 3 and 4 are in study guide 2/2.

ASSESSMENT REQUIREMENTS Formal assessment tasks and tests are part of the year-long formal assessment programme. Refer to the portfolio book or my.Impaq for the requirements. Formal assessment tasks and tests count 75% of the final mark and the November examination counts 25%.

The formal assessment programme indicates which components must be completed in each term (this includes the activities and investigations in the study guide.) Term (Unit)

Number of formal assessments per term/unit

Formal assessment

• •

1 × assignment 1 × test

• •

1 × test 1 × project

4 • •

• • • • •

1 × test 1 × June examination 1 × assignment 1 × research 1 × November examination

TOTAL

Total % contribution to the final mark

The June examination covers all the work in terms 1 and 2 (units 1 and 2). The November examination covers all the work in terms 3 and 4 (units 3 and 4).

100

Facilitator’s Guide G06 ~ Mathematics

YEAR PLAN UNIT 1 DATE STARTED

TOPIC

DATE COMPLETED

Mental maths: Use Train Your Brain Maths Grade 6 Lesson 1 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (3-digit whole numbers) Lesson 2 Number sentences

Lesson 3 Whole numbers: Addition and subtraction (5-digit whole numbers) Lesson 4 Common fractions Lesson 5 Time Lesson 6 Properties of 2D shapes Lesson 7 Data handling Lesson 8 Number patterns (numeric patterns) Revision: Use the CAMI program

Facilitator’s Guide G06 ~ Mathematics

UNIT 2 DATE STARTED

TOPIC

Mental maths: Use Train Your Brain Maths Grade 6 Lesson 9 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers) Lesson 10 Whole numbers: Multiplication (4-digit whole numbers by 2-digit whole numbers) Lesson 11 Properties of 3D objects Lesson 12 Geometric patterns Lesson 13 Symmetry

Lesson 14 Whole numbers: Division (4-digit whole numbers by 2-digit whole numbers) Lesson 15 Decimal fractions Lesson 16 Capacity/Volume Revision: Use the CAMI program

DATE COMPLETED

Facilitator’s Guide G06 ~ Mathematics

UNIT 3 DATE STARTED

TOPIC

DATE COMPLETED

Mental maths: Use Train Your Brain Maths Grade 6 Lesson 17 Mass

Lesson 18 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers)

Lesson 19 Whole numbers: Addition and subtraction (6-digit whole numbers) Lesson 20 Viewing objects

Lesson 21 Properties of 2D shapes Lesson 22 Transformations Lesson 23 Temperature Lesson 24 Percentages

Lesson 25 Data handling

Lesson 26 Numeric patterns Lesson 27 Length

Revision: Use the CAMI program 11

Facilitator’s Guide G06 ~ Mathematics

UNIT 4 DATE STARTED

TOPIC Mental maths: Use Train Your Brain Maths Grade 6

Lesson 28 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers) Lesson 29 Whole numbers: Multiplication (4-digit whole numbers by 3-digit whole numbers) Lesson 30 Common fractions

Lesson 31 Properties of 3D objects

Lesson 32 Perimeter, area and volume

Lesson 33 The history of measurement

Lesson 34 Whole numbers: Division (4-digit whole numbers by 3-digit whole numbers) Lesson 35 Number sentences Lesson 36 Transformations

Lesson 37 Position and direction Lesson 38 Probability

Revision: Use the CAMI program © Optimi

DATE COMPLETED

Facilitator’s Guide G06 ~ Mathematics

FACT SHEET Whole numbers Whole numbers are numbers without fractions or decimals. Whole numbers are always positive and never negative. Remember: 0 is also a whole number. Even numbers

• •

All numbers that are divisible by 2 without a remainder. Even number end with 2, 4, 6, 8 or 0.

• •

Alternative words used for operations

Uneven numbers

All numbers that are not divisible by 2 without a remainder. Odd numbers end with 1, 3, 5, 7 or 9.

Addition Add

And

More than

Add to

Sum of

Plus

Increase by

In total

Subtraction

Decrease by

Left

Less than

Less

Take away

Altogether

Deduct

Left over

Fewer than

Division

Divide

Halve

Difference between/of

Total of

Minus

Percent

Divide equally

How many times

Quotient

Half of

Per

Multiplication

Split into

Double

Times

Goes into

Increase by a factor of Multiply by

Out of

Product of

Ratio of

Twice

Facilitator’s Guide G06 ~ Mathematics

Useful definitions Term

Definition

Factor Multiple

Example

A factor is a number that can be divided into another number with a remainder.

Factors of 24 = 1; 2; 3; 4; 6; 8; 12; 24

Numbers with only two factors – 1 and the number itself

The first 10 prime numbers = 2; 3; 5; 7; 11; 13; 17; 19; 23; 29

Times tables

Prime number Prime factor

Rounding

Factors that are also prime numbers.

The first 12 multiples of 6 = 6; 12; 18; 24; 30; 36; 42; 48; 54; 60; 66; 72 The prime factors of 24 are 2 and 3.

Round to the nearest 5

Numbers ending in 3, 4, 5, 6 and 7 are rounded to 5. Numbers ending in 1 and 2 are rounded to the previous 10 (end with a 0). Numbers ending in 8 and 9 are rounded to the next 10 (end with a 0). Round to the nearest 10

Numbers smaller than 5 are rounded to the previous 10 (end with a 0). Numbers ending in 5, 6, 7, 8 and 9 are rounded to the next 10 (end with a 0). Example: Round 6 858 to the nearest 10.

Th HTU 6 858

Look at the Tens column.

The ones column must help you decide (8 is greater than 5, therefore the number is rounded to the next ten and ends with a 0 the answer is 6 860.)

Facilitator’s Guide G06 ~ Mathematics

Round to the nearest 100 Numbers smaller than 5 are rounded to the previous 100 (end with a 0). Numbers ending in 5, 6, 7, 8 and 9 are rounded to the next 100 (end with a 0). Example: Round 6 858 to the nearest 100.

Look at the Hundreds column.

Th HTU 6 858

The tens column must help you decide (5 and greater are rounded to the next 100 and end with a 0 the answer is 6 900.)

Apply the same method when you round to 1 000. (The Hundreds column must help you decide.)

Properties/Laws Commutative law: Numbers may be added or multiplied together in any order. a + b = b + a a×b=b×a 10 + 8 = 8 + 10 12 × 3 = 3 × 12 18 = 18 36 = 36

Associative law: When you add or multiply numbers together, it does not matter how the numbers are grouped. (a + b) + c = a + (b + c)

13 + 24 + 22 = 13 + 24 + 22 (13 + 24) + 22 = 13 + (24 + 22)

(a × b) × c = a × (b × c)

(37) + 22 = 13 + (46) 59 = 59

3×4×2=3×2×4 (3 × 4) × 2 = 3 × (2 × 4) (12) × 2 = 3 × (8) 24 = 24

Distributive law: The distributive law of multiplication means that you can break down one or all of the numbers in a multiplication sum, multiply them separately and add the products together.

Facilitator’s Guide G06 ~ Mathematics

Regular method

(5)(8) = 40 (4)(12) = 48 47 × 45 = 2 115

Common fractions

Distributive method 5(6 + 2) = (5 × 6) + (5 × 2) = 30 + 10 = 40

4(7 + 2 + 3) = (4 × 7) + (4 × 2) + (4 × 3) = 28 + 8 + 12 = 48

47 × 45 = 47 × (40 + 5) → Break down the number = 47 × 40 + (47 × 5) → Distributive property = 1 880 + 235 = 2 115

Numerator

1 4

Denominator

The top number (numerator) counts how many of the bottom number (denominator) there are. You can remember the difference by seeing that the denominator is down below. The denominator determines what we name the fraction, for example, quarters, eighths, etc. You may use a fractions wall to compare fractions.

Facilitator’s Guide G06 ~ Mathematics

Important: Before you compare or add fractions, you must always make the denominators the same. It means that you determine the Least Common Denominator (LCD). When you multiply the denominator by a number, you must also multiply the numerator by the same number. How to convert an improper fraction to a mixed number

16 3 1. 2. 3.

Divide the numerator with the denominator. Here it is 16 ÷ 3 = 5 rem. 1. Write down the whole number. Here it is 5. 1 Write the rest on the numerator, next to the whole number: 5__ . 3

How to convert a mixed number to an improper fraction =

Step 3

16 1 + Step 2 = 5 × =15 3 3 Step 1

1. 2. 3.

Multiply the whole number with the denominator. In the example it is 5 × 3 = 15. Add the answer of step 1 to the numerator. Here it is 15 + 1 = 16. 16 Write the answer of step 2 on top of the denominator: __ 3 .

Decimal fractions

Equivalents They all have the same value 1 whole

__ 11

1,0

100%

__ 12

0,5

50%

a quarter

three quarters

one half

__ 14

__ 34

0,25 0,75

= =

25% 75%

Facilitator’s Guide G06 ~ Mathematics

Rules of divisibility • • • • • • • • • • •

No number is divisible by 0. We say it is undefined. Any number is divisible by 1.

A number is divisible by 2 if the last digit is an even number (2, 4, 6, 8, or 0). A number is divisible by 3 if the sum of the digits is a multiple of 3. A number is divisible by 4 if the last two digits is a multiple of 4. A number is divisible by 5 if the last digit is a 5 or a 0. A number is divisible by 6 if it is divisible by 2 and 3.

A number is divisible by 7 if the three steps work (see below).

A number is divisible by 8 if the last three digits are divisible by 8. A number is divisible by 9 if the sum of the digits is divisible by 9. A number is divisible by 10 if the number ends with 0.

Divisibility by 7

Step 1 Double the last digit in the number.

Step 2 Subtract this number from the remaining digits.

Step 3 If the new number is 0 or a number divisible by 7, the original number is also divisible by 7. If the number is still too large to quickly see if it is divisible by 7, repeat step 1 and 2 with the new

Time

number.

Analogue time

1 minute = 60 seconds 1 hour = 60 minutes 24 hours = 1 day 7 days = 1 week 4 weeks = 1 month 12 months = 1 year 10 years = 1 decade 100 years = 1 century

NOTE Leap year: Leap year is every 4th year. There are 366 days in a leap year. e.g. 2020; 2024; 2028; ...

Digital time

Facilitator’s Guide G06 ~ Mathematics

Length Units: kilometre (km), metre (m), centimetre (cm) and milimetre (mm). 10 mm = 1 cm

100 cm = 1 m

1 000 m = 1 km

× 1 000 × 1 000

kilometre (km)

× 10

× 100

metre (m)

÷ 1 000

millimetre (mm)

centimetre (cm)

÷ 10

÷ 100 ÷ 1 000

Mass Units: ton (t), kilogram (kg) and gram (g)

1 000 g = 1 kg

1 000 kg = 1 ton

× 1 000

ton (t)

× 1 000

kilogram (kg)

÷ 1 000

gram (g)

÷ 1 000

Facilitator’s Guide G06 ~ Mathematics

Volume Units: kilolitre (kℓ), litre (ℓ) and millilitre (mℓ).

1 000 mℓ = 1 ℓ

1 000 ℓ = 1 kℓ

× 1 000

kilolitre (kℓ)

× 1 000

litre (ℓ)

÷ 1 000

millilitre (mℓ)

÷ 1 000

Conversions: Multiply or divide by 10, 100 and 1 000 Example 1 Convert 4 000 g to kilogram. 4 000 g ÷ 1 000 = 4 kg

Picture a comma at the end of the number.

When you divide by 1 000, move the imaginary comma three place values to the left (because there are three zeros in 1 000).

4 0 0 0 , g ÷ 1 0 0 0 = 4 , 0 0 0 kg

Example 2 Convert 4 kg to gram.

4 kg × 1 000 = 4 000 g

When you multiply by 1 000, move the imaginary comma three place values to the right (because there are three zeros in 1 000).

4 , 0 0 0 kg × 1 0 0 0 = 4 0 0 0 g

Facilitator’s Guide G06 ~ Mathematics

2D shapes Triangle

Square

3 straight sides

4 straight sides of equal length

Rectangle

Circle

3 angles

4 right angles (90°)

2 long straight sides of equal length 2 shorter sides of equal length

No angles

4 right angles (90°) Pentagon

No straight side (a curved surface)

5 straight sides and 5 angles Heptagon

6 straight sides and 6 angles

7 straight sides and 7 angles

8 straight sides and 8 angles

NOTE: The Castle of Good Hope in Cape Town is a pentagonshaped building

Hexagon

Octagon

Facilitator’s Guide G06 ~ Mathematics

3D objects Objects with only curved surfaces

Objects with flat and curved surfaces

Sphere

Cone

Cylinder

Objects with only flat surfaces Prisms

Pyramids

Triangular prism

Square prism

Triangular pyramid OR tetrahedron

Square pyramid

Rectangular prism

Pentagonal prism

Rectangular pyramid

Pentagonal pyramid

Hexagonal prism

Octagonal prism

Hexagonal pyramid

Octagonal pyramid

Facilitator’s Guide G06 ~ Mathematics

Types of angles Name

Description

Example

An angle of 90° A quarter of a rotation

Right angle

An angle smaller than a right angle (smaller than 90°)

Acute angle

Obtuse angle

Straight angle

Reflex angle

Full rotation (or revolution)

An angle bigger than a right angle (bigger than 90° and smaller than 180°) An angle of 180° Half a rotation An angle bigger than a straight angle but smaller than a full rotation (bigger than 180° but smaller than 360°) 360° A full (complete) rotation

Transformations Reflection (Mirror image)

figu

Translation (Glide image)

figu

ctio

refle

tran

slat

ion

Facilitator’s Guide G06 ~ Mathematics

Rotation

Enlargement/reduction

figu

rot a

tio

enla

rgem

figu

ent

point of rotation

Data handling Tally table / Frequency table

Write down all the shoe sizes in ascending order.

Shoe size

Make a mark for each data unit. Every fifth mark goes across the group of four marks to make a group of five. It is easier to count data in groups of five.

Tally

2 3

Frequency indicates the answer of the count. It shows how many units of data there are.

Frequency

|||| |

|||

|||| |||

||||

8 Total

The median is the data item in the middle of the data set. Here it is shoe size 3.

22 The mode is the data item with the most tallies. The mode of the data set is: Shoe size 3

Pictograph Day

Teddy bears

Day 1 Day 2 Day 3 Every © Optimi

represents 6 teddy bears. 24

Facilitator’s Guide G06 ~ Mathematics

Bar graph Grade 5 learners' maths test results

30 25 20 15 10 5 0

Double bar graph

Pie chart

Facilitator’s Guide G06 ~ Mathematics

Unit

UNIT 1 This unit covers eight lessons (lessons 1 to 8).

UNIT 1

TOPIC

TIME AND NOTES

Mental maths: Use Train Your Brain Maths Grade 6

Lesson 1 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (3-digit whole numbers) Lesson 2 Number sentences

Lesson 3 Whole numbers: Addition and subtraction (5-digit whole numbers) Lesson 4 Common fractions

8 hours (divided into 10 minutes per day) 2 hours 3 hours 7 hours 10 hours

Lesson 5 Time

4 hours

Lesson 6 Properties of 2D shapes

8 hours

Lesson 7 Data handling

10 hours

Lesson 8 Number patterns (numeric patterns)

4 hours

Revision: Use the CAMI program

4 hours

TOTAL

60 hours

Unit

Facilitator’s Guide G06 ~ Mathematics

LESSON 1: WHOLE NUMBERS Grade 6 is the last year in the Intermediate Phase and learners must master the concepts before moving on to the Senior Phase. The number range increases to 9-digit whole numbers (in term 4) and place value is an important concept that learners must have a thorough grasp of to enable them to work with larger numbers.

In Term 1, revision only includes the number range up to 6-digit whole numbers as dealt with at the end of Grade 5. Pay special attention to learners’ understanding of place value up to hundred thousands, as well as of the properties of numbers and operations, especially in the first part of term 1. In this lesson, learners will do this with whole numbers: • count • order • compare • represent • indicate place value (up to 6-digit whole numbers in term 1)

Counting comes as naturally as breathing and blinking – we do it consciously and subconsciously. You are used to counting because we count all day long. You probably count down the minutes to break or the end of the school day. Let’s learn how to count to 10 in three languages: Zulu

French

Spanish

kunye

uno

kubili

deux

dos

kuthathu

trois

tres

kune

quatre

cuatro

kuhlanu

cinq

cinco

isithupha

six

seis

isikhombisa

sept

siete

isishiyagalombili

huit

ocho

isishiyagalolunye

neuf

nueve

ishumi

dix

diez

1 2 4 5 7 8 9

It’s not that easy! Watch these videos on the internet to help you with your pronunciation: Zulu: bit.ly/3iaN0VI French: bit.ly/31d0OsE Spanish: bit.ly/2Nq61W1 Did you know? The most common language spoken as a first language by South Africans is isiZulu. Approximately 23% of all South Africans’ home language is isiZulu. © Optimi

Facilitator’s Guide G06 ~ Mathematics

Unit

French is also spoken in many African countries, with Spanish and English competing for second place on the list of most spoken languages on earth.

In your opinion, which language is the most spoken in the world? It is Mandarin – a dialect of Chinese, which means it is a form of Chinese. This is how you write the numbers 1 to 10 in Mandarin: 1

yī

liù

sān

bā

wŭ

èr

sì

qī

jiŭ

Watch this video to learn to say the numbers in Mandarin: bit.ly/3hZJFIJ.

shí

Divide the learners into three groups and let them help one another to learn counting in the three languages. Encourage learners to learn to count in different languages at home, e.g., isiXhosa, Sepedi, German, Russian, etc., and then teach the class. It could be a fun activity for learners to research the languages spoken in different countries. For example, let the learners each choose a country and do research about the language (or languages) they speak and how to count from 1 to 10 in that language.

Counting in whole numbers You probably still remember these words: count backwards, count back, count forwards, count on, skip counting (this is counting by 2s, 3s, 4s, 5s, 25s, 30s, etc.) – these are different ways of counting. Skip counting is also a way of practising times tables.

The 6 × table is skip counting in 6s (you add 6 each time).

Unit

Facilitator’s Guide G06 ~ Mathematics

We counted in sixes – we call this multiples of 6. The numbers above can be divided exactly by 6. It means when we divide the number by 6, there is no remainder. Example: 24 ÷ 6 = 4

Therefore, 24 is a multiple of 6. Revise all the times tables. Let the learners count together as a class. Do this a few times and then timetest them. Do this every day to help learners memorise the times tables. The concept of multiples must also be mastered. The time test can comprise individual or combined times tables.

Show the learners examples of numbers that are not multiples and explain why (as below). Let the learners test a few numbers on their own. Is 46 a multiple of 6?

Circle: Yes / No

Explain your answer: 46 ÷ 6 = 7,666666666, which means there is a remainder. Therefore 46 is not a multiple of 6. You may start with any number and count on in 6s:

218

224

230

You may start with any number and count back in 6s:

–6

812

–6

806

236

242

–6

800

Are the numbers represented above multiples of 6?

248

–6

794

788

Circle: Yes / No

Explain your answer: In the first representation, the example is 218 ÷ 6 = 36,333333, which has a remainder. In the next representation, the example is 794 ÷ 6 = 132,3333, which also has a remainder. Therefore, the representations are not multiples of 6.

Facilitator’s Guide G06 ~ Mathematics

Unit

Complete the following by filling in the missing numbers in the blocks: + 25

14 639

+17

2 712

+ 25

2 695

14 714

14 689

14 664

14 739

-17

2 661

2 678

2 644

2 627

14 764

2 610

2 593

14 789

2 576

14 814

2 559

2 542

14 839

2 525

We can also indicate multiples of 6 on a number line:

This number line indicates multiples of 6 from 30 to 60:

Indicate multiples of 4 from 24 to 60 on the number line: 24

Indicate the multiples of 7 between 30 and 100 on this number line: 35

Prime numbers

The number line does not start at 30 but at the multiple of 7 closest to 30: 5 × 7 = 35.

A prime number is a number with exactly two factors: itself and 1.

Unit

Facilitator’s Guide G06 ~ Mathematics

What are factors again? A factor is a number that divides exactly into another number without a remainder. Or, as you learned in Grade 5, a factor is a number that multiplies with another number. For example 2 × 3 = 6

2 and 3 are factors of 6. And 2 and 3 is a factor pair of 6.

For example, the factors of 12 are: 1; 2; 3; 4; 6; 12.

Can you tell that 5 is not a factor of 12? 12 cannot be divided by 5 without leaving a remainder, therefore 5 is not a factor of 12. The set of the first ten prime numbers are:

{2; 3; 5; 7; 11; 13; 17; 19; 23; 29}

Remember: 1 is not a prime number, because it has only itself as a factor. Why are even numbers not prime numbers? Any even number has 2 as a factor. An even number has 2, itself and 1 as factors, therefore an even number cannot be a prime number. Then why is 2 a prime number? It is the first even number and only has two factors, namely 1 and 2. (The number 1 and the number 2 itself.) Colour all the prime numbers up to 100 on the number chart. 2

52 72

73 83

25 45 55 65 75 85 95

19 29

79 89

100

Make it a daily habit to ask the learners mental maths as they enter the classroom, such as: • • • •

What is 6 × 5; 7 × 8; 3 × 12, etc.? Is 5 a factor of 30? What are the first three multiples of 12? Is 24 divisible by 6?

Facilitator’s Guide G06 ~ Mathematics

Unit

You can alternate this exercise with questions about work covered the previous day or week, or add in a few riddles, such as: I’m a number. When multiplied by 3, I’m 168. What am I? Ask the learners to explain why these numbers are not prime numbers: 28; 51; 57; 69; 77; 87 and 93. Factor pairs are the easiest way to explain why numbers are not prime: • The factor pairs of 28: 1; 2; 4; 7; 14; 28 • The factor pairs of 51: 1; 3; 17; 51 • The factor pairs of 57: 1; 3; 19; 57 • The factor pairs of 69: 1; 3; 23; 69 • The factor pairs of 77: 1; 7; 11; 77 • The factor pairs of 87: 1; 3; 29; 87 • The factor pairs of 93: 1; 3; 31; 93

Also, practise listing all the factor pairs of these numbers: 24; 36; 48; 56 • Factor pairs of 24: 1; 2; 3; 4; 6; 8; 12; 24 • Factor pairs of 36: 1; 2; 3; 4; 6; 9; 12; 18; 36 • Factor pairs of 48: 1; 2; 3; 4; 6; 8; 12; 16; 24; 48 • Factor pairs of 56: 1; 2; 4; 7; 8; 14; 28; 56 • Natural numbers are number sets that start at 1 and continues to infinity. We write a set of natural numbers like this:

N = {1; 2; 3; 4; 5; 6; ...}

This shortened way of writing, is called the language of mathematics. Mathematics is an exact subject (it means we have to work precisely and accurately), therefore we have to pay attention to certain things. When we ‘speak’ maths, certain signs have special meanings, for example: curly brackets { } semicolon ; three dots ...

set of numbers separate numbers infinite set

Explain to the learners that mathematics is systematic. Get excited about maths so they can share in your enthusiasm. Make mathematics ‘real’ by giving examples of how it is used in everyday life. Emphasise the importance of neatness and that mathematics is written in a particular way. Explain the symbols and signs when you use it and repeat the meaning. Later on, you can ask the learners to explain it, for example: What do three dots mean?

Unit

Facilitator’s Guide G06 ~ Mathematics

If this line, written in mathematical language N = {1; 2; 3; 4; 5; 6; ...} was written in English, it would look like this: The set of natural numbers is equal to the set of numbers 1 and 2 and 3 and 4 and 5 and 6, and to infinity. Or even shorter: The set of natural numbers is from 1 to infinity.

The set of whole numbers is from 0 to infinity. In mathematical language we would write it as follows:

N0 = {0; 1; 2; 3; 4; 5; 6; ...}

Did you notice the tiny 0 below the N? It also serves as a reminder that whole numbers start at 0.

Starting today, work accurately and neatly when doing maths. Remember to use the correct symbols. Since Grade 6 is the last year in the Intermediate Phase and you have already learned a lot of maths, we will use this year to refine techniques, to practise methods and to learn to work neatly and accurately. If you start working neatly and accurately now, it will be much easier in later grades. Do a speed test. (A speed test is revision in between work.)

Skip count on in 5s: 750; 755; 760; 765; 770; 775; 780; 785 Skip count backwards in 3s: 196; 193; 190; 187; 184; 181; 178; 175; 172 6 × 5 × 2 = 60 8 × 7 × 2 = 112 9 × 7 × 10 = 630 4 × 5 × 3 = 60 8 × 6 × 2 = 96 6 × 7 × 3 = 126 11 × 3 × 3 = 99 6 × 3 × 4 = 72

Emphasise the commutative property of multiplication again – numbers can be multiplied in any order.

ACTIVITY 1 Lesson 1’s activities may be completed in the space provided in the study guide, the other activities must be completed in the learners’ workbooks (quad-ruled hard cover book). Learners are encouraged to copy their answers to lesson 1’s activities into their workbooks. © Optimi

Facilitator’s Guide G06 ~ Mathematics

Unit

Give the set of even numbers between 415 and 435. 416; 418; 420; 422; 424; 426; 428; 430; 432; 434

What do you call the set of numbers that starts at 0 and continues to infinity? The set of whole numbers

Write 10 multiples of the 8 × table. Start at 64. 64; 72; 80; 88; 96; 104; 112; 120; 128; 136

Complete:

4.1

(+ 75)

5 845

5 920

6 145

6 070

5 995

15 620

15 800

15 980

16 160

16 880

16 700

16 520

16 340

5 695

5 770

6 220

4.2

5. 6.

(– 180)

Complete: 9 068; 8 968; 8 868; 8 768; 8 668; 8 568; 8 468; 8 368; 8 268; 8 168; 8 068; 7 968 What is the sixth multiple of 7? 42 And the eighth multiple of 12? 96

Unit

Facilitator’s Guide G06 ~ Mathematics

Follow the instructions carefully on the number chart. 1

41 51 61 71 81

11 21

7.1 7.2 7.3 7.4

22 42 52 62 72 82 92

23 43 53 63 73 83 93

24 44 54 64 74 84 94

27 47 57 67 77 87 97

28 48 58 68 78 88 98

49 59 69 79 89 99

20 40 50 60 70 80 90

100

Colour the multiples of 7 yellow. Count in 3s from 45 to 78, and cross out the numbers with a blue pen or pencil. Draw a green triangle on each multiple of 9 between 60 and 100. Which number on the table is yellow, crossed out and a triangle? 63 Is it an even or odd number? Odd number Is it a prime number? No

Study this representation of rows and columns on an Excel worksheet:

Start at cell B21 and count the blue blocks in row 21. Skip count to B4 to determine how many blue blocks there are. Write your ‘skips’ down below.

10 blocks from cell B21 to cell K21. Skip count in 10s – 0; 20; 30; 40; 50; 60; 70; 80; 90; 100; 110; 120; 130; 140; 150; 160; 170; 180. There are 180 blue blocks in total.

Facilitator’s Guide G06 ~ Mathematics

Unit

After every lesson (and some concepts), learners must complete a self-assessment on the lesson. The facilitator may use the evaluation to determine whether the learner needs more assistance with that specific lesson and address it immediately. The facilitator may also use it to plan enrichment. Once the learner has mastered the lesson, further enrichment can be done.

SELF-ASSESSMENT Do the learners understand the work? Let them colour the faces that show what they can do. COUNTING IN WHOLE NUMBERS Requirements

Can learners do it?

I know what a multiple of a number is. I know what a prime number is. I know what a factor of a number is. I can skip count on and backward by different numbers and multiples. I can determine the factor pairs of numbers. I know what natural numbers are. I know what whole numbers are. I can write sets of numbers using the correct symbols.

Place value In Grade 5, we worked with numbers of up to 6-digit whole numbers, for example 695 428.

You learned that the digits we work with (0 to 9), can be used in different places or positions in a number. Every ‘place’ has a value, such as Units, Tens, Hundreds, etc.

Unit

Facilitator’s Guide G06 ~ Mathematics

Study the representation where place values are indicated in the columns:

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Units

It represents the number 695 428. It can be written as six hundred and ninety-five thousand four hundred and twenty-eight. The digit 8 is in the Units column and means 8 × 1 = 8. The digit 2 is in the Tens column and means 2 × 10 = 20. The digit 4 is in the Hundreds column and means 4 × 100 = 400. And so on.

The digit 4 in the number 695 428, is in the Hundreds place. Therefore, the place value of the 4 is equal to 4 Hundreds or 4H. Thus, the value of the digit 4 is in fact not 4, but 400. The digit 9 in the number 695 428, is in the Ten Thousands place. Therefore, the place value of the 9 is equal to 9 Ten Thousands or 9TT. Thus, the value of the digit 9 is in fact not 9, but 90 000. What is the difference between place value and number value? Place value is exactly what it says – the ‘place’ or position of a digit in a number. Your answer must show that you know where (in which position or place) the digit is in the number.

Number value is the value of a digit in a number what value does the digit represent?

Practise place value on the blackboard, or use Dienes blocks, place value cards or flash cards so learners can get used to 6-digit whole numbers. Start with smaller 3-digit numbers and then increase it to 4-, 5- and 6-digit whole numbers. Say a number or write the number on the blackboard. Learners can then use their place value cards to build the number. Then provide instructions, such as increase the number by 100, decrease the number by 50, increase the number by 1 000, and so on. Remind the learners that ‘increase’ means ‘adding’ and ‘decrease’ means ‘subtracting’. The place value cards also teach them about expanded notation. Let them break down the numbers they just made, and write it in expanded notation (the third method). You can then explain the other two methods using the third. © Optimi

Facilitator’s Guide G06 ~ Mathematics

Unit

Expanded notation We can also represent the number in three ways without the use of a table. We call these methods expanded notation (your learned this in Grade 5). This is how we would write the number 695 428 in expanded notation:

Method 1 6 Hundred Thousands + 9 Ten Thousands + 5 Thousands + 4 Hundreds + 2 Tens + 8 Units OR You may abbreviate the words. 6HT + 9TT + 5TH + 4H + 2T + 8U Method 2 6 × 100 000 + 9 × 10 000 + 5 × 1 000 + 4 × 100 + 2 × 10 + 8 × 1 Method 3 600 000 + 90 000 + 5 000 + 400 + 20 + 8 = 695 428 What if a number contains a 0, for example 806 043?

There is a 0 in the Hundreds place and a 0 in the Ten Thousands place. In expanded notation it will look like this:

8HT + 0TT + 6TH + 0H + 4T + 3U

But we do not have to write 0TT and 0H in expanded notation, because you are already demonstrating that you know in which place each digit of the number must go, therefore you can write it as: 8HT + 6TH + 4T + 3U

BUT you cannot leave out the zeros when writing the number 806 043. In this number, the zeros are ‘place’ holders for Ten Thousands and Hundreds. Otherwise the number would be 8 643, which is much smaller. Can you see the difference between 806 043 and 8 643? The zeros in 806 043 are place holders for TT and H.

The digit 0 is very important in a number.

What does the 0 in 100 or in 1 000 mean? I cannot simply leave out the 0, otherwise both numbers will be just 1. The 0 is a place holder. In 100, the 0 is a place holder for Units and the other 0 for Tens. Wow, that was a lot of information! What did we just revise?

Now you: 9 can use digits to write numbers; 9 can use words to write numbers; 9 know in which place each digit in a number is; and 9 can write numbers three ways in expanded notation. 38

Unit

Facilitator’s Guide G06 ~ Mathematics

The numbers are written in groups of three digits with a space in between each group. (This is another way in which mathematics is very precise: numbers are written in groups of three digits to make it easier to read and pronounce.) Group 2

Group 1

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Units

Let’s say the number: four hundred and fifty-six thousand eight hundred and seventy-nine. We divide the number into groups of three digits each, so that it is easier to read: Group 1 Group 2

The last three digits 879 are easy – eight hundred and seventy-nine.

The next group of digits 456 is four hundred and fifty-six thousand. Can you see that the group name is ‘thousand’?

Say the number again: four hundred and fifty-six thousand eight hundred and seventy-nine Let’s practise saying these large numbers:

978 459 in words: nine hundred and seventy-eight thousand four hundred and fifty-nine 608 413 in words: six hundred and eight thousand four hundred and thirteen Write the numbers in words:

678 412 is six hundred and seventy-eight thousand four hundred and twelve 590 478 is five hundred and ninety thousand four hundred and seventy-eight 600 809 is six hundred thousand eight hundred and nine 470 200 is four hundred and seventy thousand two hundred Let’s write these words in numbers:

Thousands

8 Tens

5 Thousands 14 Tens

Hundreds

Tens

Units

0 0

OOPS!

Facilitator’s Guide G06 ~ Mathematics

Unit

We can’t write two digits in the Units column. The extra digit must jump over to the next column (like you learned in Grade 5). Then it will look like this:

Thousands

8 Tens

5 Thousands 14 Tens

Hundreds

Tens

Units

Take note: You cannot write 14 Tens as 14. Why not? There is a difference between 14 Units and 14 Tens. 14 Tens are actually 140. Thousands

Hundreds

Tens

Units

Always make sure in which column you must write the digits. Write the following numbers as digits in the table. HT

Hundred Ten Thousands Hundreds Thousands Thousands 5

5 Ten Thousands 5 Thousands

6T + 7U + 3TT

150 + 3 + 3 200 15 Units 22 Tens

54 Thousands

83TT + 32H + 14TH

Tens

Units

In term 4 we will work with a new group – millions. For now, you must make absolutely sure that you know all the names for the place values up to Hundred Thousands, very well. NOTE: 8

Do a speed test.

Emphasise that 1 × 0 = 0 but 1 + 0 = 1. 11 × 1 = 11 × 0 = 110 × 10 = 110 × 1 = 110 × 0 =

11 0 1 100 110 0

110 × 1 + 0 = 110 + 1 × 0 = 10 × 0 + 110 × 0 = 101 + 1 + 0 × 110 = 1 010 + 10 + 0 + 1 =

110 110 0 102 1 021

3 7

2 2

0 0