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MATHEMATICS STUDY GUIDE 2/2 Grade 6

Mathematics Study guide 2/2

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Grade 6

Study Guide 2/2 G06 ~ Mathematics

Contents LESSON ELEMENTS.....................................................................................................................................1 YEAR PLAN....................................................................................................................................................2 FACT SHEET...................................................................................................................................................6 UNIT 3..........................................................................................................................................................19 LESSON 17: MASS..................................................................................................................................... 20 Units of mass....................................................................................................................................................21 Measuring mass...............................................................................................................................................21 ACTIVITY 47.............................................................................................................................................25 Comparing and ordering mass..................................................................................................................27 ACTIVITY 48.............................................................................................................................................28 Converting between measuring units....................................................................................................30 Rounding mass................................................................................................................................................32 ACTIVITY 49.............................................................................................................................................33 LESSON 18: WHOLE NUMBERS............................................................................................................ 35 Counting, ordering, comparing, representing and place value of 9-digit whole numbers..............................................................................................................................................................35 ACTIVITY 50.............................................................................................................................................35 LESSON 19: WHOLE NUMBERS – ADDITION AND SUBTRACTION............................................ 37 Adding and subtracting 6-digit whole numbers...............................................................................37 ACTIVITY 51.............................................................................................................................................39 Calculators.........................................................................................................................................................42 ACTIVITY 52.............................................................................................................................................43 LESSON 20: VIEWS OF OBJECTS........................................................................................................... 45 ACTIVITY 53.............................................................................................................................................47 LESSON 21: PROPERTIES OF 2D SHAPES.......................................................................................... 51 Rectangles and parallelograms.................................................................................................................54 Properties of a circle.....................................................................................................................................55 ACTIVITY 54.............................................................................................................................................59 LESSON 22: TRANSFORMATIONS........................................................................................................ 61 Using transformations to describe patterns.......................................................................................63 Using symmetry to describe patterns....................................................................................................64 ACTIVITY 55.............................................................................................................................................65 LESSON 23: TEMPERATURE.................................................................................................................. 70 Measuring instruments................................................................................................................................71 ACTIVITY 56.............................................................................................................................................73

Study Guide 2/2 G06 ~ Mathematics

LESSON 24: PERCENTAGES.................................................................................................................... 76 Equivalent forms of percentages.............................................................................................................78 ACTIVITY 57.............................................................................................................................................81 Arranging percentages, fractions and decimal fractions...............................................................84 ACTIVITY 58.............................................................................................................................................84 Percentage of a number...............................................................................................................................85 ACTIVITY 59.............................................................................................................................................87 LESSON 25: DATA HANDLING............................................................................................................... 90 ACTIVITY 60.............................................................................................................................................91 Analysing and interpreting data...............................................................................................................93 ACTIVITY 61.............................................................................................................................................96

LESSON 26: NUMERIC PATTERNS........................................................................................................ 99 Determining the rule.....................................................................................................................................99 ACTIVITY 62..........................................................................................................................................100 Flow diagrams with two rules...............................................................................................................102 ACTIVITY 63..........................................................................................................................................103 LESSON 27: LENGTH..............................................................................................................................106 Units of length (or distance)...................................................................................................................106 Measuring instruments.............................................................................................................................106 Comparing and ordering length............................................................................................................108 ACTIVITY 64..........................................................................................................................................108 Converting between measuring units.................................................................................................111 Rounding length...........................................................................................................................................115 ACTIVITY 65..........................................................................................................................................115 UNIT 4........................................................................................................................................................118

LESSON 28: WHOLE NUMBERS..........................................................................................................119 Reading and saying 9-digit whole numbers.....................................................................................119 ACTIVITY 66..........................................................................................................................................119 Place value of 9-digit whole numbers.................................................................................................120 ACTIVITY 67..........................................................................................................................................121 Comparing and ordering 9-digit whole numbers..........................................................................122 ACTIVITY 68..........................................................................................................................................122 LESSON 29: WHOLE NUMBERS – MULTIPLICATION....................................................................125 Properties of multiplication....................................................................................................................125 Quick ways of doing multiplication.....................................................................................................127 Multiples, factors and prime factors....................................................................................................128 ACTIVITY 69..........................................................................................................................................129 Calculation techniques..............................................................................................................................130 ACTIVITY 70..........................................................................................................................................132 © Optimi

Study Guide 2/2 G06 ~ Mathematics

Problem-solving...........................................................................................................................................133 ACTIVITY 71..........................................................................................................................................133 LESSON 30: COMMON FRACTIONS....................................................................................................135 Simplifying and ordering fractions......................................................................................................135 ACTIVITY 72..........................................................................................................................................137 Equivalents of fractions, decimal fractions and percentages...................................................139 ACTIVITY 73..........................................................................................................................................141 Calculations with fractions......................................................................................................................143 ACTIVITY 74..........................................................................................................................................145 LESSON 31: PROPERTIES OF 3D OBJECTS......................................................................................147 Properties of 3D objects...........................................................................................................................149 ACTIVITY 75..........................................................................................................................................151 Nets and models of 3D objects...............................................................................................................155 ACTIVITY 76..........................................................................................................................................157 LESSON 32: PERIMETER, AREA AND VOLUME..............................................................................161 Perimeter and area.....................................................................................................................................161 Measuring perimeter with a ruler and tape measure..................................................................163 Diagonals.........................................................................................................................................................165 Area of a rectangle.......................................................................................................................................166 Relation between the perimeter and area of rectangles and squares..................................167 ACTIVITY 77..........................................................................................................................................169 Volume..............................................................................................................................................................172 ACTIVITY 78..........................................................................................................................................173 ACTIVITY 79..........................................................................................................................................177 Relation between the area and volume of a rectangular prism...............................................179 LESSON 33: THE HISTORY OF MEASUREMENT.............................................................................181 How length, mass and volume were measure in the past..........................................................181 How time was measured in the past...................................................................................................183 ACTIVITY 80..........................................................................................................................................184 LESSON 34: WHOLE NUMBERS – DIVISION....................................................................................185 Rules of divisibility......................................................................................................................................185 Using multiplication to do division – clue board...........................................................................186 Long division.................................................................................................................................................187 ACTIVITY 81..........................................................................................................................................188 Ratio and rate................................................................................................................................................189 ACTIVITY 82..........................................................................................................................................190 LESSON 35: NUMBER SENTENCES....................................................................................................192 ACTIVITY 83..........................................................................................................................................194

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Study Guide 2/2 G06 ~ Mathematics

LESSON 36: TRANSFORMATIONS......................................................................................................196 Enlargement and reduction....................................................................................................................197 ACTIVITY 84..........................................................................................................................................199 LESSON 37: POSITION AND DIRECTION.........................................................................................205 Using alpha-numeric grid references to determine location on a grid/map.....................205 Giving directions..........................................................................................................................................207 ACTIVITY 85..........................................................................................................................................208 LESSON 38: PROBABILITY...................................................................................................................212 Identifying possible outcomes...............................................................................................................213 Counting and comparing the regularity of actual outcomes.....................................................213 ACTIVITY 86..........................................................................................................................................215 IMAGE REFERENCES..............................................................................................................................217

Study Guide 2/2 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

Remember or revise

Take note! or

Tips

Important

Research

Self-assessment

Study

Activity

New concept or

Did you know?

definition

LESSON ELEMENT

Study Guide 2/2 G06 ~ Mathematics

Unit

UNIT 3 This unit covers eleven lessons (lessons 17 to 27).

UNIT 3

TOPIC Mental maths: Use Train Your Brain Maths Grade 6 Lesson 17 Mass

Lesson 18 Whole numbers: Counting, ordering, comparing and representing, and place value of digits (9-digit whole numbers) Lesson 19 Whole numbers: Addition and subtraction (6-digit whole numbers) Lesson 20 Viewing objects

Lesson 21 Properties of 2D shapes Lesson 22 Transformations Lesson 23 Temperature Lesson 24 Percentages

Lesson 25 Data handling

Lesson 26 Numeric patterns Lesson 27 Length

Revision: Use the CAMI program

Unit

Study Guide 2/2 G06 ~ Mathematics

LESSON 17: MASS Do you still remember what mass is, from Grade 5? Mass measures the quantity of matter (particles) in an object. Mass is measured in gram (g), kilogram (kg) and ton (t). In Grade 6, we only work with gram and kilogram. We sometimes refer to weight when talking about mass, but they are not the same thing. What is the difference between mass and weight? Mass

Weight

• Mass measures the quantity of matter (particles) in an object. • Mass always stays the same, regardless of where it is measured. • The standard unit is kilogram (kg).

• The weight of an object is determined by gravity. • Weight can change, depending on where it is measured. • The standard unit is Newton (N).

This picture will help you understand the difference. An astronaut’s mass is 90 kg, but his weight here on earth and on the moon, is different. On earth, the astronaut’s mass is 900 N, while on the moon it is merely 15,24 N!

Calculating weight is complicated, but you do not have to do that yet. You will learn more about mass and weight in Natural Sciences, later on. © Optimi

Study Guide 2/2 G06 ~ Mathematics

Unit

Units of mass The standard unit of mass is kilogram (kg). We can also measure objects in gram (g) and ton (t).

In the same way that we convert between millilitre, litre and kilolitre, we can also convert between gram, kilogram and ton. × 1 000

ton (t)

× 1 000

kilogram (kg)

÷ 1 000

gram (g)

÷ 1 000

To convert mass from gram to kilogram, you multiply by 1 000. To convert mass from kilogram to gram, you divide by 1 000. That means

1 ton = 1 000 kg

1 kg = 1 000 g

1 g = 0,001 kg

We will look at converting between units in more detail later on.

Measuring mass You already know that we use scales to measure mass. In Grades 4 and 5, you learned about the different scales we use to measure mass and most were analogue scales.

Unit

Study Guide 2/2 G06 ~ Mathematics

Analogue scales Each of these scales are used to measure the mass of something specific. Can you guess what it is?

_________________________ _________________________ _________________________ _________________________

_________________________ _________________________ _________________________ _________________________ Reading mass on an analogue scale To determine the amount of liquid in a measuring cup, we look at the cup’s graduation markings. With an analogue scale it works the same way – look at the graduation markings. Let us study an example. Example 1

Looking at the graduation markings on a kitchen scale, we can count 10 markings for each numbered interval. (Count the markings from the long marking at 0 to 1 – there are 10 markings, with the long marking at 1 being the tenth marking.) © Optimi

Study Guide 2/2 G06 ~ Mathematics

Unit

But what does it mean? Every numbered interval is 1 kg. Every kilogram consists of 10 markings, which means that every marking is a tenth of a kilogram, therefore 0,10 kg, or 0,1 kg. If we take the reading on the scale on the previous page at eye level, the red arm indicates the mass of the potatoes as 3,8 kg. (Can you count the markings?) How do we take the reading on a scale? •

•

Remember, zeros that follow on digits after the comma can be left out.

Determine the intervals on the scale.

We determined that every numbered interval (the long markings) = 1 kg.

Stand in front of the scale, at eye level, and read the measurement.

3,8 kg

Determine the value of each interval.

There are 10 markings between each kg. So every unnumbered interval is 1 kg = 1 000 g, therefore, 1 000 g ÷ 10 = 100 g

How much grams will that be? We have already determined that every short marking = 0,1 kg.

How do we convert kg to g? We multiply by 1 000, therefore 0,1 kg × 1 000 = 100 g. Therefore, the mass of the potatoes in the image is 3,8 kg, therefore: 3,8 kg × 1 000 = 3 800 g.

Later on, we will look at conversion between units in more detail.

We can also represent a scale’s round disc on a number line. The numbered markings represent intervals of 1 kg, with 10 spaces between each kilogram. So every numbered interval: 1 kg = 1 000 g, therefore 1 000 g ÷ 10 = 100 g.

Every interval is 100 g. 23

1 kg

Unit

Study Guide 2/2 G06 ~ Mathematics

Example 2 Benji weighs himself on a bathroom scale. He stands still, with his feet even and flat. He looks down at the scale reading, how much does he weigh?

• • •

Determine the intervals on the scale.

Every numbered interval represents 5 kg.

Look down and read the measurement.

The red arm indicates the second marking after 20 kg, therefore, Benji weighs 22 kg.

Determine the value of each interval.

There are 5 markings between each 5 kg, therefore every unnumbered interval is: 5 kg ÷ 5 marks = 1 kg.

Digital scales These days we use digital scales more than analogue scales. It works electronically and has a battery. In Grade 5, you revised reading time on an analogue watch and a digital watch. Analogue scales and digital scales work in the same way. Can you guess what these digital scales are used for?

_________________________

_________________________ _________________________ 24

_________________________ _________________________

Study Guide 2/2 G06 ~ Mathematics

Unit

Instead of determining intervals as we do with an analogue scale, a digital scale gives the reading in numbers, so we must literally just read the measurement. Always make sure of the unit. A kitchen scale can measure in grams or kilograms, for example.

55,5 kg

330 g

ACTIVITY 47 1.

In what unit will these items be measured? Use the abbreviations g, kg and t. 1.1 1.3

1.2 1.4

Unit

Study Guide 2/2 G06 ~ Mathematics

1.5 1.6

Read the mass on the scales. Give the reading in the unit on the scale, and then convert it to gram or kilogram.

2.1

2.4

2.2

2.5

2.3

2.6

Study Guide 2/2 G06 ~ Mathematics

Unit

2.7

2.8 95.85 g

How much do the objects weigh? Give your answers in gram. Look carefully at the unit on the scale. 3.1

3.2

3.3

Do you know how much you weigh? Use a bathroom scale if you do not know. Write your weight in kilogram and convert it to gram. I weigh _______ kg. It is _________ g.

Comparing and ordering mass Before we look at converting units of mass in more detail, you must practise comparing and ordering mass. 1 kilogram = 1 kg or 1 000 g

half a kilogram = __  21 kg or 0,5 kg or 500 g

1 kg or 0,25 kg or 250 g quarter of a kilogram =  __ 4

three-quarters of a kilogram = __  43 kg or 0,75 kg or 750 g

Unit

Study Guide 2/2 G06 ~ Mathematics

What is the mass of 1 ℓ water?

What is the mass of 1 mℓ water?

1 mℓ water weighs 1 g

1 ℓ water weighs 1 kg

Here, we converted the amount of water to weight. BUT as a rule it is not possible to convert between different units; water is an exception..

ACTIVITY 48 1. 2.

Explain the difference between mass and weight in your own words. You and your mom buy these items in a shop: A bag of 3 apples 2 ℓ cold drink A 750 mℓ bottle of tomato sauce

Arrange the items from heaviest to lightest. Estimate the mass of the objects.

3.1

3.3

3.2

3.4

A bread 2 kg bag of flour

Study Guide 2/2 G06 ~ Mathematics

Unit

Bonus: Arrange the objects in question 3 in descending order, based on your answers. 4.

Use the correct symbol (>, < or =) to indicate the relationship between the masses.

4.5

__  24 kg _______ 400 g 15 ___  20  kg _______ 850 g 75 _____  100    kg _______ 760 g ___  35 25  kg _______ 1 200 g ___  15 50  kg _______ 400 g

5.1 5.2 5.3 5.4 5.5

30 mℓ 2 300 mℓ 1 550 mℓ 18 458 mℓ 5 mℓ

4.1 4.2 4.3 4.4

6. 7.

Each of these amounts of water is stored in its own container. How much does each container of water weigh? Give your answers in kg.

What is the difference between an analogue scale and a digital scale?

Test your knowledge of conversion between units, before we discuss it in more detail. 7.1 7.2 7.3 7.4 7.5

How many grams are in a kilogram? How many kilograms are in a gram? How many grams are in 3,7 kg? How many grams are in 100 kg? How many kilograms are in 49 350 g?

Unit

Study Guide 2/2 G06 ~ Mathematics

Converting between units of measurement Earlier in the unit, we looked at converting between different mass units.

Converting something means changing it from one thing to another.

× 1 000

ton (t)

× 1 000

kilogram (kg)

÷ 1 000

gram (g)

÷ 1 000

But wait, there is an easier way to convert between units without using a calculator. Can you still remember how, from Grade 5? • •

When we convert from kilogram to gram, we multiply by 1 000. The comma moves three place values to the right (because there are three zeros in 1 000).

× 1 000

When we convert from gram to kilogram, we divide by 1 000. The comma moves three place values to the left (because there are three zeros in 1 000).

÷ 1 000

1 kg = 1 000 g

Let us look at some examples. Example 1 Herlei 2,75 kg na gram. • • •

From which unit to which unit am I converting? Must I multiply or divide?

I am converting from kg to g.

I must convert from a larger unit to a smaller unit, so I must multiply.

Where does the comma move The comma moves to the right. to?

Study Guide 2/2 G06 ~ Mathematics

Unit

To convert to gram, we multiply 2,75 kg by 1 000. You can make a drawing for yourself each time you have to convert. × 1 000

2,75 kg × 1 000 =2 ,7 5 0 = 2 750 g

The comma moves three places to the right – 2 750, We do not write down the comma, because there is nothing after the comma.

You are doing the multiplication sum by moving the comma.

Ask yourself: Does my answer make sense? Yes, because 1 kg is 1 000 g, therefore 2,5 kg will be more than 2 000 g. Example 2 Convert 35__  14 kg to gram. • •

•

From which unit to which unit am I converting? Must I multiply or divide?

I am converting from kg to g.

I must convert from a larger unit to a smaller unit, so I must multiply.

Where does the comma move The comma moves to the right. to?

First, convert the fraction to decimals: 35__  41 kg = 35,25 kg. Now you can do the calculation. × 1 000

35,25 kg × 1 000 =3 5,2 5 0 = 35 250 g

Ask yourself: Does my answer make sense? Yes, because 1 kg is 1 000 g, therefore 35,25 kg will be more than 35 000 g.

Unit

Study Guide 2/2 G06 ~ Mathematics

Example 3 Convert 83 562 g to kilogram.

From which unit to which unit am I converting?

•

Must I multiply or divide?

• • kg

I am converting from g to kg.

I must convert from a smaller unit to a larger unit, so I must divide.

Where does the comma move The comma moves to the left. to? g

÷ 1 000

83 562 g ÷ 1 000 =8 3 5 6 2,

= 83,562 kg = 83 kg and 562 g

Remember the imaginary comma that comes after a whole number.

Ask yourself: Does my answer make sense? Yes, because 1 000 g is 1 kg, therefore, 83 562 g will be more than 83 kg.

Rounding mass Remember this when rounding from gram to kilogram: First, round to the nearest 1 000 g and then convert to kg. Rounding mass and rounding decimal numbers are done in the same way. (Revise lesson 3 and lesson 15 if you cannot remember how.) Example 1 Round 26 574 g to the: Nearest kilogram 26 574 g ≈ 27 000 g 27 000 g ≈ 27 kg

Nearest 100 g 26 574 g ≈ 26 600 g

Nearest 10 g 26 574 g ≈ 26 570 g

Nearest 5 g 26 574 g ≈ 26 575 g

Study Guide 2/2 G06 ~ Mathematics

Example 2 Round 32,98 g to the nearest 5 g.

32,89 g ≈ 33 g 33 g ≈ 35 g

Unit

First, round off to the nearest g and then to the nearest 5 g.

ACTIVITY 49 1.

Convert the masses to kg. 1.1 1.2 1.3

15 g 226 g

1.4 75 g 1.5 999 g

1 235 g

1.6 12 548 g

Convert the masses to g.

Complete the table by rounding the masses.

2.1 2.2 2.3

The nearest kilogram 227 g ≈ ______________ ≈ ______________ 784 g ≈ ______________ ≈ ______________ 988 g ≈ ______________ ≈ ______________ 3 506 g ≈ ______________ ≈ ______________ 11 487 g ≈ ______________ ≈ ______________

2.4 1,236 kg 2.5 45,02 kg 2.6 12,057 kg

The nearest 100 g The nearest 10 g

4.1

1  __  kg

4.3

25 _____  100    kg

__  34 kg

The nearest 5 g

227 g ≈ ____________ 227 g ≈ __________ 227 g ≈ ___________ 784 g ≈ ____________ 784 g ≈ ___________ 784 g ≈ ___________ 988 g ≈ ____________ 988 g ≈ _________ 988 g ≈ ___________ 3 506 g ≈___________

11 487 g ≈ _________

Convert to decimal fractions. 4.2

0,125 kg 4,260 kg 45,3 kg

3 506 g ≈ __________ 3 506 g ≈ _________ 11 487 g ≈ ________ 11 487 g ≈ _______

1 4.4  __ 4  kg

5 4.5  __ 5  kg

Unit

Study Guide 2/2 G06 ~ Mathematics

You are visiting your gran for the school holidays. She is the matron at the retirement home and must decide what groceries to buy for the retirement home’s kitchen. It is cheaper to buy in bulk. Help your gran to convert the amounts to kilogram. 5.1 5.2 5.3 5.4 5.5

6. 7.

5.6

4 × 250 g rice 4 × 225 g sugar 1 × 187 g baking powder 3 × 125 g coffee 3 × 750 g flour

Remember to always give the answer to a word problem as a sentence.

How much will the groceries weigh altogether? Leave your answer in kg.

Your mom buys a 5 kg bag of rice. The bag accidentally tears and 370 g of rice spills out. How much rice is left in the bag? Shane’s textbooks weigh 550 g each. His school bag can take contents of 3 kg. How many textbooks will Shane be able to carry in his bag at a time?

SELF-ASSESSMENT Do you understand the work? Colour the faces to show what you can do. MASS Requirements

Can I do it?

I know what the difference is between mass and weight. I know if what measuring units mass is measured. I know what measuring instruments to use to measure the mass of different objects. I can read the mass of an object on an analogue scale. I can read the mass of an object on a digital scale. I can round mass to the nearest kilogram, 100 g, 10 g and 5 g. I can order and compare mass. I can convert between kilogram and gram.

Study Guide 2/2 G06 ~ Mathematics

Unit

I can do calculations of mass. I can solve problems of mass in context.

LESSON 18: WHOLE NUMBERS Counting, ordering, comparing, representing and place value of 9-digit whole numbers Do you still remember the work from terms 1 and 2 about whole numbers? Revise lessons 1 and 9, as well as lesson 3, that covered adding and subtracting. Let’s quickly look at a place value table again.

An example of a 9-digit number: 368 016 021. Group 3 Millions

Group 2 Thousands

Group 1

Hundred millions

Ten millions

Millions

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Units

HM 3

TM 6

M 8

HT 0

TT 1

TH 6

H 0

T 2

U 1

368 016 021 is three hundred and sixty-eight million sixteen thousand and twenty-one.

ACTIVITY 50 1. 2.

We sometimes refer to half a million. Write the number in digits. Write the numbers in words. 2.1 2.2

South Africa’s population currently grows by 750 420 people per year. In 2020, South Africa had a population of 59 263 077. 35