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MATHEMATICS STUDY GUIDE 1/2 Grade 6
Mathematics Study guide 1/2
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D Botha M Vos
2106-E-MAM-SG01
Í5&È-E-MAM-SG01NÎ
Grade 6
Study Guide 1/2 G06 ~ Mathematics
Contents LESSON ELEMENTS.....................................................................................................................................1 YEAR PLAN....................................................................................................................................................2 FACT SHEET...................................................................................................................................................6 UNIT 1.......................................................................................................................................................... 19 LESSON 1: WHOLE NUMBERS............................................................................................................... 20 Counting in whole numbers.......................................................................................................................21 Prime numbers................................................................................................................................................23 ACTIVITY 1...............................................................................................................................................25 Place value.........................................................................................................................................................28 ACTIVITY 2...............................................................................................................................................32 Ordering whole numbers............................................................................................................................36 ACTIVITY 3...............................................................................................................................................39 LESSON 2: NUMBER SENTENCES......................................................................................................... 41 Word problems................................................................................................................................................41 Ways to solve number sentences.............................................................................................................42 ACTIVITY 4...............................................................................................................................................44 Equivalents........................................................................................................................................................45 Inverse operations.........................................................................................................................................46 Commutative property.................................................................................................................................48 Associative property.....................................................................................................................................48 ACTIVITY 5...............................................................................................................................................49 LESSON 3: WHOLE NUMBERS............................................................................................................... 52 Steps when adding and subtracting.......................................................................................................52 Rounding............................................................................................................................................................53 ACTIVITY 6...............................................................................................................................................53 Methods of addition and subtraction.....................................................................................................56 ACTIVITY 7...............................................................................................................................................57 ACTIVITY 8...............................................................................................................................................59 Calculators.........................................................................................................................................................60 ACTIVITY 9...............................................................................................................................................63 LESSON 4: COMMON FRACTIONS........................................................................................................ 65 Describe and order fractions.....................................................................................................................65 Equivalent forms.............................................................................................................................................66 Tenths and hundredths................................................................................................................................72 ACTIVITY 10.............................................................................................................................................73 Calculations with fractions.........................................................................................................................76 ACTIVITY 11.............................................................................................................................................79
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Study Guide 1/2 G06 ~ Mathematics
LESSON 5: TIME........................................................................................................................................ 81 Time on clocks.................................................................................................................................................81 ACTIVITY 12.............................................................................................................................................84 Calculating with time....................................................................................................................................85 ACTIVITY 13.............................................................................................................................................87 Time on a calendar.........................................................................................................................................88 ACTIVITY 14.............................................................................................................................................89 Time zones.........................................................................................................................................................90 ACTIVITY 15.............................................................................................................................................93
LESSON 6: PROPERTIES OF 2D SHAPES............................................................................................ 95 Types of sides...................................................................................................................................................95 ACTIVITY 16.............................................................................................................................................96 Number of sides of polygons.....................................................................................................................96 Length of sides.................................................................................................................................................99 The size of the angles.................................................................................................................................100 ACTIVITY 17..........................................................................................................................................101 LESSON 7: DATA HANDLING................................................................................................................104 ACTIVITY 18..........................................................................................................................................106 ACTIVITY 19..........................................................................................................................................111
LESSON 8: NUMBER PATTERNS (NUMERIC PATTERNS)............................................................115 Investigate and extend patterns............................................................................................................115 Input and output values............................................................................................................................116 Multiplication and division as inverse operations........................................................................117 ACTIVITY 20..........................................................................................................................................117 Order of multiplication..............................................................................................................................119 Multiply with multiples of 10 and 100...............................................................................................120 ACTIVITY 21..........................................................................................................................................122 UNIT 2........................................................................................................................................................127
LESSON 9: WHOLE NUMBERS ............................................................................................................128 ACTIVITY 22..........................................................................................................................................129 ACTIVITY 23..........................................................................................................................................130 ACTIVITY 24..........................................................................................................................................132 LESSON 10: WHOLE NUMBERS – MULTIPLICATION....................................................................134 ACTIVITY 25..........................................................................................................................................135 Estimation.......................................................................................................................................................136 ACTIVITY 26..........................................................................................................................................137 Distributive property of multiplication..............................................................................................138 ACTIVITY 27..........................................................................................................................................139 Factors to multiply......................................................................................................................................140 © Optimi
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Study Guide 1/2 G06 ~ Mathematics
ACTIVITY 28..........................................................................................................................................142 ACTIVITY 29..........................................................................................................................................144 Vertical column method of multiplication........................................................................................145 ACTIVITY 30..........................................................................................................................................146 Ratio and rate................................................................................................................................................146 ACTIVITY 31..........................................................................................................................................149 Calculators......................................................................................................................................................150 ACTIVITY 32..........................................................................................................................................150 LESSON 11: PROPERTIES OF 3D OBJECTS......................................................................................152 Properties of 3D objects...........................................................................................................................152 Grouping 3D objects...................................................................................................................................153 Objects with flat faces only......................................................................................................................154 ACTIVITY 33..........................................................................................................................................156 ACTIVITY 34..........................................................................................................................................161 Nets and models of 3D objects...............................................................................................................162 ACTIVITY 35..........................................................................................................................................166 LESSON 12: GEOMETRIC PATTERNS................................................................................................168 Repeating patterns......................................................................................................................................169 Patterns that increase................................................................................................................................169 Patterns that decrease...............................................................................................................................170 ACTIVITY 36..........................................................................................................................................171 Patterns with a constant difference.....................................................................................................172 Patterns without a constant difference..............................................................................................174 ACTIVITY 37..........................................................................................................................................175 LESSON 13: SYMMETRY.......................................................................................................................179 ACTIVITY 38..........................................................................................................................................180 ACTIVITY 39..........................................................................................................................................184 LESSON 14: WHOLE NUMBERS – DIVISION....................................................................................186 Rules of divisibility......................................................................................................................................186 Use multiplication to divide....................................................................................................................187 ACTIVITY 40..........................................................................................................................................188 Long division.................................................................................................................................................189 ACTIVITY 41..........................................................................................................................................192 LESSON 15: DECIMAL FRACTIONS....................................................................................................193 Recognising decimal fractions...............................................................................................................194 Place value of decimal fractions............................................................................................................197 ACTIVITY 42..........................................................................................................................................198 Counting in decimal fractions................................................................................................................199 Comparing and arranging decimal fractions...................................................................................200 ACTIVITY 43..........................................................................................................................................201
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Rounding decimal numbers....................................................................................................................202 Adding and subtracting decimal fractions........................................................................................203 Multiplying decimal fractions by 10 and 100..................................................................................204 ACTIVITY 44..........................................................................................................................................205
LESSON 16: CAPACITY/VOLUME.......................................................................................................207 What is the difference between capacity and volume?...............................................................207 Measuring capacity.....................................................................................................................................207 ACTIVITY 45..........................................................................................................................................209 Converting between units of measurement.....................................................................................209 Kilolitre............................................................................................................................................................210 ACTIVITY 46..........................................................................................................................................212 IMAGE REFERENCES..............................................................................................................................215
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LESSON ELEMENTS The guide containts various lesson elements. Each element is important for the learning process. ICON
LESSON ELEMENT
ICON
Think for yourself
Remember or revise
Take note! or
Tips
Important
Research
Self-assessment
Study
Activity
New concept or
Did you know?
definition
LESSON ELEMENT
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Study Guide 1/2 G06 ~ Mathematics
YEAR PLAN UNIT 1 DATE STARTED
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 Number patterns (numeric patterns) Revision: Use the CAMI program
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DATE COMPLETED
Study Guide 1/2 G06 ~ Mathematics
UNIT 2 DATE STARTED
TOPIC
DATE COMPLETED
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
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UNIT 3 DATE STARTED
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 © Optimi
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Study Guide 1/2 G06 ~ Mathematics
UNIT 4 DATE STARTED
TOPIC
DATE COMPLETED
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
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Study Guide 1/2 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
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
Of
Times
Goes into
Increase by a factor of Multiply by
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And
Out of
Product of 6
Ratio of
Twice
Study Guide 1/2 G06 ~ Mathematics
Useful definitions Term
Definition Example A factor is a number that can be Factor divided into another number without Factors of 24 = 1; 2; 3; 4; 6; 8; 12; 24 a remainder. The first 12 multiples of 6 = 6; 12; 18; Multiple Times tables 24; 30; 36; 42; 48; 54; 60; 66; 72 Prime Numbers with only two factors – The first 10 prime numbers = 2; 3; 5; number 1 and the number itself. 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
0
1
2
3
4
5
6
7
8
9
10
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.)
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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.
Th HTU 6 858
Look at the Hundreds column.
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. © Optimi
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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.
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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 1. 2. 3.
Step 1
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. Write the answer of step 2 on top of the denominator: __ 16 . 3
Decimal fractions
Equivalents They all have the same value 1 whole
a quarter one half
=
1 __ 1
=
1,0
=
100%
=
1 __ 2
=
0,5
=
50%
=
three quarters =
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1 __ 4
=
3 __ 4
=
10
0,25 = 0,75 =
25% 75%
Study Guide 1/2 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 number.
Time
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
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Digital time
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Study Guide 1/2 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)
millimetre (mm)
centimetre (cm)
÷ 1 000
÷ 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
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gram (g)
÷ 1 000
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Study Guide 1/2 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
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2D shapes Triangle
Square
3 straight sides 3 angles Rectangle
4 straight sides of equal length 4 right angles (90°)
2 long straight sides of equal length 2 shorter sides of equal length 4 right angles (90°) Pentagon
No angles No straight side (a curved side)
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
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Circle
Hexagon
Octagon
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Study Guide 1/2 G06 ~ Mathematics
3D objects Objects with only curved surfaces Sphere
Objects with flat and curved surfaces Cone Cylinder
Objects with only flat surfaces Prisms Triangular prism Square prism
Pyramids Triangular pyramid Square pyramid OR tetrahedron
Rectangular prism
Pentagonal prism
Rectangular pyramid
Pentagonal pyramid
Hexagonal prism
Octagonal prism
Hexagonal pyramid
Octagonal pyramid
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Types of angles Name
Right angle
Acute angle
Obtuse angle
Straight angle
Reflex angle
Full rotation (or revolution)
Description
Example
An angle of 90° A quarter of a rotation An angle smaller than a right angle (smaller than 90°) An angle bigger than a right angle (bigger than 90°) 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
figu
Translation
re
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figu
n
ctio
refle
re
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Study Guide 1/2 G06 ~ Mathematics
Rotation
Enlargement/reduction
figu
re
rot a
tio
n
enla
rgem
figu
re
ent
point of rotation
Data handling Tally table / Frequency table
Write down all the shoe sizes from small to large.
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
|||| |
6
|||
3
|||| |||
4
||||
5
8 Total
The median is the data item in the middle of the data set. Here it is shoe size 3.
5
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
represents 6 teddy bears. 17
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Study Guide 1/2 G06 ~ Mathematics
Bar graph Grade 5 learners' maths test results 30 25 20 15 10 5 0
A
B
C
Double bar graph
Pie chart
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Study Guide 1/2 G06 ~ Mathematics
Unit
1
UNIT 1 This unit covers eight lessons (lessons 1 to 8).
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 Properties of 2D shapes Lesson 7 Data handling
Lesson 8 Number patterns (numeric patterns) Revision: Use the CAMI program
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Unit
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Study Guide 1/2 G06 ~ Mathematics
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: Zulu
French
Spanish
1
kunye
un
uno
kubili
deux
dos
3
kuthathu
trois
tres
kune
quatre
cuatro
5
kuhlanu
cinq
cinco
isithupha
six
seis
isikhombisa
sept
siete
isishiyagalombili
huit
ocho
isishiyagalolunye
neuf
nueve
ishumi
dix
diez
2 4 6 7 8 9
10
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 26% 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.
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Unit
This is how you write the numbers 1 to 10 in Mandarin: 1
yī
6
liù
3
sān
8
bā
5
wŭ
10
2
èr
4
7
sì
1
qī
9
jiŭ
shí
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
12
+6
18
24
+6
+6
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
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Study Guide 1/2 G06 ~ Mathematics
You may start with any number and count on in 6s: +6
218
+6
224
+6
+6
230
236
242
You may start with any number and count back in 6s: –6
–6
812
806
–6
800
Are the numbers represented above multiples of 6?
+6
248
–6
794
Explain your answer:
788
Circle: Yes / No
Complete the following by filling in the missing numbers in the blocks:
14 639
14 664
14 689
2 695 2 678
14 739
14 764
2 627 2 610 2 593
We can also indicate multiples of 6 on a number line:
This number line indicates multiples of 6 from 30 to 60:
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14 814
14 839
Study Guide 1/2 G06 ~ Mathematics
Unit
1
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.
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? ________________________________________________________________________
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Study Guide 1/2 G06 ~ Mathematics
Colour all the prime numbers up to 100 on the number chart. 1
2
3
4
5
6
7
8
9
10
29
30
11
12
13
14
15
16
17
18
19
31
32
33
34
35
36
37
38
39
21 41 51 61 71 81 91
22 42 52 62 72 82 92
23 43 53 63 73 83 93
24 44 54 64 74 84 94
25
26
45
46
55
56
65
66
75
76
85
86
95
96
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
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
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. © Optimi
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Study Guide 1/2 G06 ~ Mathematics
Unit
1
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 numbes 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.
4.
Complete:
4.1
5 845
25
5 920
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Study Guide 1/2 G06 ~ Mathematics
4.2
16 160
16 340 5.
6. 7.
Complete:
9 068; 8 968; _____________; _____________; _____________; _____________; _____________; 8 368; _____________; _____________; _____________; _____________
What is the sixth multiple of 7? ___________ And the eighth multiple of 12? ___________ Follow the instructions carefully on the number chart. 1
2
3
4
11
12
13
31
32
33
52
53
54
73
74
75
93
94
95
21
22
36
37
38
56
57
77
46
64
65
66
67
85
86
87
82
83
92
35
45
63
91
17
44
62
81
16
43
34
84
8
15
25
61
72
7
24
42
71
6
23
41
51
14
5
26
55
76
96
27
47
97
18
9
10
19
20
39
40
58
59
60
78
79
98
99
28
29
48
49
68
88
30
50
69
70
89
90
80
100
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. © Optimi
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Study Guide 1/2 G06 ~ Mathematics
8.
Unit
1
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? __________
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. 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.
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Unit
1
Study Guide 1/2 G06 ~ Mathematics
Requirements
Can I do it?
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: Hundred Thousands
Ten Thousands
Thousands
Hundreds
Tens
Units
HT
TT
T
H
T
U
6
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.
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Study Guide 1/2 G06 ~ Mathematics
Unit
1
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?
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 806 042. 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. 29
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Unit
1
Study Guide 1/2 G06 ~ Mathematics
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
Hundred Thousands 4
Ten Thousands 5
Group 1
Thousands
Hundreds
Tens
Units
6
8
7
9
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
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Study Guide 1/2 G06 ~ Mathematics
Unit
1
Write the numbers in words:
678 412 is _______________________________________________________________________________________________ 590 478 is _______________________________________________________________________________________________ 600 809 is _______________________________________________________________________________________________ 470 200 is _______________________________________________________________________________________________ Let’s write these words in numbers:
Thousands Hundreds
8 Tens 5 Thousands 14 Tens
5
0
Take another look at 14 Tens – how must it be represented?
Tens 8 0 14
Units 0 0 0
OOPS!
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 looks like this:
8 Tens 5 Thousands 14 Tens
Thousands Hundreds 5
0 1
Tens 8 0 4
Units 0 0 0
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 1
Tens 4
Units 0
Always make sure in which column you must write the digits.
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Unit
1
Study Guide 1/2 G06 ~ Mathematics
Write the following numbers as digits in the table.
Hundred Ten Thousands Hundreds Thousands Thousands
Tens
Units
5 Ten Thousands 5 Thousands
6T + 7U + 3TT
150 + 3 + 3 200 15 Units 22 Tens
54 Thousands
83TT + 32H + 14TH
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.
ACTIVITY 2 1.
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
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Write the numbers in the place value table.
Hundred Ten Thousands Hundreds Thousands Thousands
14 306
1
5 322
136 003 432 486 899 360
150 000 + 20 000
5U + 6TT + 9H + 6TH 845 Thousands
6 ×1 000 + 5 × 1
15 × 100 + 62 × 10
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4
3
Tens
Units
0
6
Study Guide 1/2 G06 ~ Mathematics
Unit
Hundred Ten Thousands Hundreds Thousands Thousands
Tens
1
Units
12 × 10 + 6 × 10 000 1.10 + 34 × 0 + 54 × 10 000
2.
Follow the example and write the numbers in expanded notation. Example 17 689 = 1TT + 7TH + 6H + 8T + 9U = 1 × 10 000 + 7 × 1 000 + 6 × 100 + 8 × 10 + 9 × 1 = 10 000 + 7 000 + 600 + 80 + 9
2.1
54 890
2.2
601 469
2.3
671 999
2.4
300 005
2.5
919 010
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Unit
1
3.
4.
Study Guide 1/2 G06 ~ Mathematics
Write the numbers in words.
Example 89 457 = Eighty-nine thousand four hundred and fifty-seven
3.1
391 903 = _______________________________________________________________________________
3.2
49 300 = _______________________________________________________________________________
3.3
909 480 = _______________________________________________________________________________
3.4
896 045 = _______________________________________________________________________________
3.5
605 160 = _______________________________________________________________________________
How many Units are there in: 4.1
1 Thousand _____________________________________________________________________________
4.3
60 Tens _________________________________________________________________________________
4.2 5.
How many Thousands are there in: 5.1
1 Ten Thousand ________________________________________________________________________
5.3
4 Hundred Thousands + 19 Ten Thousands __________________________________________
5.2
6.
5.4
190 000 _________________________________________________________________________________ 60 Ten Thousands _____________________________________________________________________
Answer these questions about the number 918 090. 6.1
How many Units are in this number? _________________________________________________
6.3
What is the value of the 9 on the right? ______________________________________________
6.2
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1 Hundred Thousand __________________________________________________________________
What is the value of the 9 on the left? ________________________________________________
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Study Guide 1/2 G06 ~ Mathematics
6.4 7.
6.5
Unit
1
If you added 1 000 to the number, what would the answer be? ___________________
What are the two zeros in the number for? __________________________________________
Solve the riddles. 7.1
I am a number. If you multiplied me by 10, I would be 120 960. What am I?
7.2
I am a number. If you subtracted 400, I would be 96 840. What am I?
7.3
Three times my value would equal 3 less than 999 999. What am I?
7.4
I am the number 167 798. How would you change me to 166 798?
7.5
I am the number 908 908. How would you change me to a number with 0 in my Thousands place value?
SELF-ASSESSMENT Do you understand the work? Colour the faces to show what you can do. PLACE VALUE OF WHOLE NUMBERS Requirements
Can I do it?
I can write numbers in expanded notation in three different ways. I can write numbers of up to 6-digit whole numbers in words. I can convert numbers written in words into digits. I can write numbers in a place value table.
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