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Mathematics
Study guide 2/2
Grade 6
2106-E-MAM-SG02
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.
Subtraction
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
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
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
Acute angle An angle smaller than a right angle (smaller than 90°)
Obtuse angle
Straight angle
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°) Full rotation (or revolution)
A full (complete) rotation Transformations
(Mirror image)
image)
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 3
This unit covers eleven lessons (lessons 17 to 27).
UNIT 3
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
LESSON 17: MASS
Do you still remember what mass is, from Grade 5?
Mass measures the quantity of matter (particles) in an object. Mass is measured in gram (g), kilogram (kg) and ton (t). In Grade 6, we only work with gram and kilogram.
We sometimes refer to weight when talking about mass, but they are not the same thing.
What is the difference between mass and weight?
Mass
• Mass measures the quantity of matter (particles) in an object.
• Mass always stays the same, regardless of where it is measured.
• The standard unit is kilogram (kg).
SAMPLE
Weight
• The weight of an object is determined by gravity.
• Weight can change, depending on where it is measured.
• The standard unit is Newton (N).
This picture will help you understand the difference. An astronaut’s mass is 90 kg, but his weight here on earth and on the moon, is different. On earth, the astronaut’s mass is 900 N, while on the moon it is merely 15,24 N!
Calculating weight is complicated, but you do not have to do that yet. You will learn more about mass and weight in Natural Sciences, later on.
Units of mass
The standard unit of mass is kilogram (kg). We can also measure objects in gram (g) and ton (t).
In the same way that we convert between millilitre, litre and kilolitre, we can also convert between gram, kilogram and ton.
To convert mass from gram to kilogram, you multiply by 1 000. To convert mass from kilogram to gram, you divide by 1 000.
That means 1 ton = 1 000 kg 1 kg = 1 000 g 1 g = 0,001 kg
We will look at converting between units in more detail later on.
Measuring mass
SAMPLE
You already know that we use scales to measure mass. In Grades 4 and 5, you learned about the different scales we use to measure mass and most were analogue scales.
Analogue scales
Each of these scales are used to measure the mass of something specific. Can you guess what it is?
Reading mass on an analogue scale
To determine the amount of liquid in a measuring cup, we look at the cup’s graduation markings. With an analogue scale it works the same way – look at the graduation markings.
Let us study an example.
Example 1
SAMPLE
Looking at the graduation markings on a kitchen scale, we can count 10 markings for each numbered interval. (Count the markings from the long marking at 0 to 1 – there are 10 markings, with the long marking at 1 being the tenth marking.)
But what does it mean?
Every numbered interval is 1 kg. Every kilogram consists of 10 markings, which means that every marking is a tenth of a kilogram, therefore 0,10 kg, or 0,1 kg.
If we take the reading on the scale on the previous page at eye level, the red arm indicates the mass of the potatoes as 3,8 kg. (Can you count the markings?)
How do we take the reading on a scale?
• Determine the intervals on the scale.
• Determine the value of each interval.
Remember, zeros that follow on digits after the comma can be left out.
We determined that every numbered interval (the long markings) = 1 kg.
There are 10 markings between each kg. So every unnumbered interval is 1 kg = 1 000 g, therefore, 1 000 g ÷ 10 = 100 g
• Stand in front of the scale, at eye level, and read the measurement. 3,8 kg
How much grams will that be?
We have already determined that every short marking = 0,1 kg.
How do we convert kg to g? We multiply by 1 000, therefore 0,1 kg × 1 000 = 100 g.
SAMPLE
Therefore, the mass of the potatoes in the image is 3,8 kg, therefore: 3,8 kg × 1 000 = 3 800 g.
Later on, we will look at conversion between units in more detail.
We can also represent a scale’s round disc on a number line. The numbered markings represent intervals of 1 kg, with 10 spaces between each kilogram.
So every numbered interval: 1 kg = 1 000 g, therefore 1 000 g ÷ 10 = 100 g. 0 1 kg
Every interval is 100 g.
Example 2
Benji weighs himself on a bathroom scale. He stands still, with his feet even and flat. He looks down at the scale reading, how much does he weigh?
• Determine the intervals on the scale. Every numbered interval represents 5 kg.
• Determine the value of each interval.
• Look down and read the measurement.
Digital scales
SAMPLE
There are 5 markings between each 5 kg, therefore every unnumbered interval is: 5 kg ÷ 5 marks = 1 kg.
The red arm indicates the second marking after 20 kg, therefore, Benji weighs 22 kg.
These days we use digital scales more than analogue scales. It works electronically and has a battery. In Grade 5, you revised reading time on an analogue watch and a digital watch. Analogue scales and digital scales work in the same way.
Can you guess what these digital scales are used for?
Instead of determining intervals as we do with an analogue scale, a digital scale gives the reading in numbers, so we must literally just read the measurement.
Always make sure of the unit. A kitchen scale can measure in grams or kilograms, for example.
ACTIVITY 47
1. In what unit will these items be measured? Use the abbreviations g, kg and t.
2. Read the mass on the scales. Give the reading in the unit on the scale, and then convert it to gram or kilogram.
3. How much do the objects weigh? Give your answers in gram. Look carefully at the unit on the scale.
I weigh _______ kg. It is _________ g.
SAMPLE
Comparing and ordering mass
4. Do you know how much you weigh? Use a bathroom scale if you do not know. Write your weight in kilogram and convert it to gram.
Before we look at converting units of mass in more detail, you must practise comparing and ordering mass.
1 kilogram = 1 kg or 1 000 g half a kilogram = 1 2 kg or 0,5 kg or 500 g
quarter of a kilogram = 1 4 kg or 0,25 kg or 250 g
three-quarters of a kilogram = 3 4 kg or 0,75 kg or 750 g g kg kg
What is the mass of 1 mℓ water?
1 mℓ water weighs 1 g
What is the mass of 1 ℓ water?
1 ℓ water weighs 1 kg
Here, we converted the amount of water to weight. BUT as a rule it is not possible to convert between different units; water is an exception..
ACTIVITY 48
1. Explain the difference between mass and weight in your own words.
2. You and your mom buy these items in a shop:
A bag of 3 apples
A bread
2 ℓ cold drink 2 kg bag of flour
A 750 mℓ bottle of tomato sauce
Arrange the items from heaviest to lightest.
3. Estimate the mass of the objects.
Bonus: Arrange the objects in question 3 in descending order, based on your answers.
4. Use the correct symbol (>, < or =) to indicate the relationship between the masses.
4.1 2 4 kg _______ 400 g
4.2 15 20 kg _______ 850 g
4.3 75 100 kg _______ 760 g
4.4 35 25 kg _______ 1 200 g
4.5 15 50 kg _______ 400 g
5. Each of these amounts of water is stored in its own container. How much does each container of water weigh? Give your answers in kg.
5.1 30 mℓ
5.2 2 300 mℓ
5.3 1 550 mℓ
5.4 18 458 mℓ
5.5 5 mℓ
6. What is the difference between an analogue scale and a digital scale?
7. Test your knowledge of conversion between units, before we discuss it in more detail.
7.1 How many grams are in a kilogram?
7.2 How many kilograms are in a gram?
7.3 How many grams are in 3,7 kg?
7.4 How many grams are in 100 kg?
SAMPLE
7.5 How many kilograms are in 49 350 g?
Converting between units of measurement
Earlier in the unit, we looked at converting between different mass units.
Converting something means changing it from one thing to another.
(t) kilogram (kg) × 1 000
1 000
1 000 × 1 000 gram (g)
But wait, there is an easier way to convert between units without using a calculator. Can you still remember how, from Grade 5?
• When we convert from gram to kilogram, we divide by 1 000. The comma moves three place values to the left (because there are three zeros in 1 000).
SAMPLE
• When we convert from kilogram to gram, we multiply by 1 000. The comma moves three place values to the right (because there are three zeros in 1 000). kg ÷ 1 000 × 1 000 g
1 kg = 1 000 g
Let us look at some examples.
Example 1
Convert 2,75 kg to gram.
• From which unit to which unit am I converting?
• Must I multiply or divide?
I am converting from kg to g.
I must convert from a larger unit to a smaller unit, so I must multiply
• Where does the comma move to? The comma moves to the right
• 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.
