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Mathematics
Study guide 1/2
Grade 6
CAPS aligned
D Botha M Vos
LESSON ELEMENTS
The guide containts various lesson elements. Each element is important for the learning process. ICON
Think for yourself
Tips
Research
Remember or revise
Take note! or Important
Self-assessment
Study Activity
New concept or definition Did you know?
YEAR PLAN
UNIT 1
TOPIC
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
SAMPLE
Properties of 2D shapes
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
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.
Decrease by Left Less than
Deduct Left over Minus
SAMPLE
Difference between/of Less Take away Fewer than Division
Divide Halve Percent
Divide equally How many times Quotient Goes into Out of Ratio of Half of Per Split into Multiplication
Double Of Times
Increase by a factor of Product of Twice Multiply by
Useful definitions Term
Factor
A factor is a number that can be divided into another number without a remainder.
Multiple Times tables
Prime number Numbers with only two factors –1 and the number itself.
Factors of 24 = 1; 2; 3; 4; 6; 8; 12; 24
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.
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) (37) + 22 = 13 + (46) 59 = 59
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.
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.
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
× =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
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
SAMPLE
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).
(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
(ℓ) × 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
angles
2 opposite long straight sides of equal length 2 opposite shorter sides of equal length 4 right angles (90°)
NOTE: The Castle of Good Hope in Cape Town is a pentagonshaped building
5 straight sides and 5 angles
straight sides of equal length 4 right angles (90°)
No angles No straight side (a curved surface)
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
An angle smaller than a right angle (smaller than 90°)
An angle bigger than a right angle (bigger than 90° and smaller than 180°)
An angle of 180° Half a rotation
Reflex angle
An angle bigger than a straight angle but smaller than a full rotation (bigger than 180° but smaller than 360°)
Data handling
Tally table / Frequency table
Write down all the shoe sizes in ascending order.
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.
Pictograph
Day 1
Day 2
Day 3
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
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).
TOPIC
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
SAMPLE
Number patterns (numeric patterns)
Revision: Use the CAMI program
UNIT 1
LESSON 1: WHOLE NUMBERS
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:
kunye un uno
kubili deux dos 3 kuthathu trois tres 4 kune quatre cuatro 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
SAMPLE
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.
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 + 6 + 6 + 6 + 6 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.
Is 46 a multiple of 6?
Explain your answer:
Circle: Yes / No
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:
Complete the following by filling in the missing numbers in the blocks:
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
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.
SAMPLE
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? _________________________________________________________
Then why is 2 a prime number? ________________________________________________________________________
Colour all the prime numbers up to 100 on the number chart.
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 { } set of numbers semicolon ; separate numbers three dots ... infinite set
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.
ACTIVITY 1
1. Give the set of even numbers between 415 and 435.
2. What do you call the set of numbers that starts at 0 and continues to infinity? 3. Write 10 multiples of the 8 × table. Start at 64.
Complete:
6. What is the sixth multiple of 7? ___________ And the eighth multiple of 12? ___________
7. Follow the instructions carefully on the number chart.
7.1 Colour the multiples of 7 yellow.
7.2 Count in 3s from 45 to 78, and cross out the numbers with a blue pen or pencil.
7.3 Draw a green triangle on each multiple of 9 between 60 and 100.
7.4 Which number on the table is yellow, crossed out and a triangle?________ Is it an even or odd number? ____________________ Is it a prime number? ________
8. 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.
SELF-ASSESSMENT
Do you understand the work? Colour the faces to show what you can do.
SAMPLE
COUNTING IN WHOLE NUMBERS
Requirements Can I 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.
Requirements
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.
Study the representation where place values are indicated in the columns:
9 5 4 2 8
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.
Expanded notation
Number value is the value of a digit in a number what value does the digit represent?
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
What if a number contains a 0, for example 806 043?
You may abbreviate the words.
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.
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
SAMPLE
Group 1
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 The last three digits 879 are easy – eight hundred and seventy-nine.
Group 2
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
• 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.
