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Mathematics
Facilitator’s guide
Grade 6
2106-E-MAM-FG01
CAPS aligned
D Botha M Vos
LESSON ELEMENTS
The guide containts various lesson elements. Each element is important for the learning process.
ELEMENT
Think for yourself
Tips
Research
Remember or revise
Take note! or Important
Self-assessment
Study Activity
New concept or definition Did you know?
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)
SAMPLE
• 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.
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.
SAMPLE
Guidelines for time allocation per topic
UNIT 1
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 3 Whole numbers: Addition and subtraction (5-digit whole numbers)
into
Mental maths: Use Train Your Brain Maths Grade 6
(divided into 10 minutes per day) Lesson 9
Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers)
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)
UNIT 4
Mental maths: Use Train Your Brain Maths Grade 6
28
Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers)
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%.
Always refer to the assessment plan for the formal assessments that must be completed per term. (This excludes the activities and investigations in the study guide.)
SAMPLE
YEAR PLAN
UNIT 1
TOPIC DATE
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
SAMPLE
Lesson 7
Data handling
Lesson 8
Number patterns (numeric patterns)
Revision: Use the CAMI program
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
TOPIC
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
SAMPLE
Revision: Use the CAMI program
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
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.
Useful definitions
Factor
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
Multiple Times tables
Prime number Numbers with only two factors –1 and the number itself
The first 12 multiples of 6 = 6; 12; 18; 24; 30; 36; 42; 48; 54; 60; 66; 72
The first 10 prime numbers = 2; 3; 5; 7; 11; 13; 17; 19; 23; 29
Prime factor Factors that are also prime numbers. The prime factors of 24 are 2 and 3.
Rounding
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.
Look at the Tens column.
Th HTU 6 858 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.)
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.
HTU
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)
SAMPLE
(a × b) × c = a × (b × c) 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.
(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.
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
1. Divide the numerator with the denominator. Here it is 16 ÷ 3 = 5 rem. 1.
2. Write down the whole number. Here it is 5.
3. Write the rest on the numerator, next to the whole number: 5 1 3 .
How to convert a mixed number to an improper fraction 16 = 5 1 3 3
× =15 + Step 3 = Step 2 Step 1
1. Multiply the whole number with the denominator. In the example it is 5 × 3 = 15
2. Add the answer of step 1 to the numerator. Here it is 15 + 1 = 16
3. Write the answer of step 2 on top of the denominator: 16 3
Decimal fractions Equivalents
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 number.
Time
SAMPLE
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
Analogue time
NOTE
Leap year: Leap year is every 4th year. There are 366 days in a leap year. e.g. 2020; 2024; 2028; ...
Digital time
Length
Units: kilometre (km), metre (m), centimetre (cm) and milimetre (mm).
mm = 1 cm
cm = 1 m 1 000 m = 1 km
(km)
Mass
Units: ton (t), kilogram (kg) and gram (g)
SAMPLE
Volume
Units: kilolitre (kℓ), litre (ℓ) and millilitre (mℓ).
1 000 mℓ = 1 ℓ 1 000 ℓ = 1 kℓ
× 1 000
1 000
1 000 × 1 000 millilitre (mℓ)
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
2D shapes
3 straight sides 3 angles
2 opposite long straight sides of equal length
2 opposite shorter sides of equal length
4 straight sides of equal length 4 right angles (90°)
4 right angles (90°) Circle No angles No straight side (a curved surface)
NOTE:
The Castle of Good Hope in Cape Town is a pentagonshaped building
5 straight sides and 5 angles
straight sides and 7 angles
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
Types of angles
Name
Right angle
An angle of 90° A quarter of a rotation
Acute angle
An angle smaller than a right angle (smaller than 90°)
Obtuse angle An angle bigger than a right angle (bigger than 90° and smaller than 180°)
Straight angle
Reflex angle
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°)
Full rotation (or revolution)
A full (complete) rotation Transformations Reflection (Mirror image)
Rotation
Data handling
Tally table / Frequency table
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.
Frequency indicates the answer of the count. It shows how many units of data there are. Write down all the shoe sizes in ascending order.
The median is the data item in the middle of the data set. Here it is shoe size 3.
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 represents 6 teddy bears.
Bar graph
Grade 5 learners' maths test results
SAMPLE
Double bar graph
Pie chart
UNIT 1
This unit covers eight lessons (lessons 1 to 8). UNIT 1
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)
numbers: Addition and subtraction (5-digit whole numbers)
(divided into 10 minutes per day)
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:
1 kunye un uno
2 kubili deux dos
3 kuthathu trois tres
4 kune quatre cuatro
SAMPLE
5 kuhlanu cinq cinco
6 isithupha six seis
7 isikhombisa sept siete
8 isishiyagalombili huit ocho
9 isishiyagalolunye neuf nueve
10 ishumi dix diez
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.
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:
Watch this video to learn to say the numbers in Mandarin: bit.ly/3hZJFIJ.
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).
6 12 18 24 30 36
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:
You may start with any number and count back in 6s:
Are the numbers represented above multiples of 6?
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.
Complete the following by filling in the missing numbers in the blocks: + 25 + 25
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:
Indicate the multiples of 7 between 30 and 100 on this number line:
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
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.
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?
• Omvattende verduidelikings van konsepte in eenvoudige taal.
• Praktiese, alledaagse voorbeelde met visuele voorstellings en diagramme wat leerders help om konsepte te bemeester.
• Leerders werk teen hul eie pas.
• Aktiwiteite wat leerders se toepassing van kennis en hul redeneervermoë uitdaag.
• Die fasiliteerdersgids bevat stapvirstapbewerkings en antwoorde.
