Intermediate Phase Grade 6 Study Guide Mathematics by Impaq

Intermediate Phase Grade 6 • Study Guide 1/2

Mathematics CAPS IEB Mathematics

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

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946486

Study Guide 1/2

2106-E-MAM-SG01

Mathematics Study guide 1/2 Grade 6

2106-E-MAM-SG01

9 781990

946486

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D Botha M Vos

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 © Optimi

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 v

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 © Optimi

Study Guide 1/2 G06 ~ Mathematics

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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Study Guide 1/2 G06 ~ Mathematics

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

LESSON ELEMENT

ICON

Think for yourself

LESSON ELEMENT

Remember or revise

Take note! or

Tips

Important

Research

Self-assessment

Study

Activity

New concept or

Did you know?

definition

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

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

Study Guide 1/2 G06 ~ Mathematics

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

DATE COMPLETED

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 5

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

Times

Goes into

Increase by a factor of Multiply by

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

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.)

Study Guide 1/2 G06 ~ Mathematics

Round to the nearest 100

Numbers smaller than 5 are rounded to the previous 100 (end with a 0). Numbers ending in 5, 6, 7, 8 and 9 are rounded to the next 100 (end with a 0). Example: Round 6 858 to the nearest 100. 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 10 + 8 = 8 + 10 18 = 18

a×b=b×a 12 × 3 = 3 × 12 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

Study Guide 1/2 G06 ~ Mathematics

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

Common fractions

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

4(7 + 2 + 3) = (4 × 7) + (4 × 2) + (4 × 3) = 28 + 8 + 12 = 48 47 × 45 = 47 × (40 + 5) → Break down the number = 47 × 40 + (47 × 5) → Distributive property = 1 880 + 235 = 2 115

Numerator

1 4

Denominator

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

Study Guide 1/2 G06 ~ Mathematics

Important: Before you compare or add fractions, you must always make the denominators the same. It means that you determine the Least Common Denominator (LCD). When you multiply the denominator by a number, you must also multiply the numerator by the same number. How to convert an improper fraction to a mixed number 16 3 1. 2. 3.

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

How to convert a mixed number to an improper fraction =

Step 3

16 1 + Step 2 = 5 × =15 3 3 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

1 __ 1

1,0

100%

one half

1 __ 2

0,5

50%

a quarter three quarters

= =

1 __ 4

3 __ 4

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

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

Digital time

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

gram (g)

÷ 1 000

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 13

Study Guide 1/2 G06 ~ Mathematics

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 surface)

5 straight sides and 5 angles Heptagon

6 straight sides and 6 angles

7 straight sides and 7 angles

8 straight sides and 8 angles

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

Circle

Hexagon

Octagon

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

Study Guide 1/2 G06 ~ Mathematics

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° and smaller than 180°) An angle of 180° Half a rotation

An angle bigger than a straight angle but smaller than a full rotation (bigger than 180° but smaller than 360°) 360° A full (complete) rotation

Transformations Reflection (Mirror image)

figu

Translation (Glide image)

ctio

refle

figu

tran

slat

ion

Study Guide 1/2 G06 ~ Mathematics

Rotation

Enlargement/reduction

figu

rot a

tio

enla

rgem

figu

ent

point of rotation

Data handling Tally table / Frequency table

Write down all the shoe sizes in ascending order.

Shoe size

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

Tally

2 3

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

Frequency

|||| |

|||

|||| |||

||||

8 Total

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

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

Pictograph Day

Teddy bears

Day 1 Day 2

Day 3 Every

represents 6 teddy bears. 17

Study Guide 1/2 G06 ~ Mathematics

Bar graph Grade 5 learners' maths test results

30 25 20 15 10 5 0

Double bar graph

Pie chart

Study Guide 1/2 G06 ~ Mathematics

Unit

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

Unit

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

kunye

uno

kubili

deux

dos

kuthathu

trois

tres

kune

quatre

cuatro

kuhlanu

cinq

cinco

isithupha

six

seis

isikhombisa

sept

siete

isishiyagalombili

huit

ocho

isishiyagalolunye

neuf

nueve

ishumi

dix

diez

2 4 6 7 8 9

It’s not that easy! Watch these videos on the internet to help you with your pronunciation: Zulu: bit.ly/3iaN0VI French: bit.ly/31d0OsE Spanish: bit.ly/2Nq61W1

Did you know? The most common language spoken as a first language by South Africans is isiZulu. Approximately 23% of all South Africans’ home language is isiZulu. 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.

Study Guide 1/2 G06 ~ Mathematics

Unit

This is how you write the numbers 1 to 10 in Mandarin: 1

yī

liù

sān

bā

wŭ

èr

sì

qī

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

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

Unit

Study Guide 1/2 G06 ~ Mathematics

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

218

224

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?

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:

14 814

14 839

Study Guide 1/2 G06 ~ Mathematics

Unit

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? ________________________________________________________________________ 23

Unit

Study Guide 1/2 G06 ~ Mathematics

Colour all the prime numbers up to 100 on the number chart. 1

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

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

Study Guide 1/2 G06 ~ Mathematics

Unit

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.

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

Write 10 multiples of the 8 × table. Start at 64.

Complete: 4.1

5 845

5 920

Unit

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

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

Study Guide 1/2 G06 ~ Mathematics

Unit

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.

Unit

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

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.

Study Guide 1/2 G06 ~ Mathematics

Unit

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 806 043. 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. 29

Unit

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

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

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

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

Say the number again: four hundred and fifty-six thousand eight hundred and seventy-nine

Let’s practise saying these large numbers:

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