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MATHEMATICS FACILITATOR’S GUIDE

Grade 7

A member of the FUTURELEARN group

Mathematics Facilitator’s guide

1907-E-MAM-FG01

Í3’È-E-MAM-FG01&Î

Grade 7

CAPS aligned

DM Oost

Facilitator’s Guide G07 ~ Mathematics

CONTENTS Guidelines for facilitators 1.

2.

3.

4.1 4.2 5.

6.

General........................................................................................................................ ii Study guide with activities and exercises .................................................................... ii Year mark tasks .......................................................................................................... ii Learning outcomes ..................................................................................................... iii Levels of difficulty ....................................................................................................... iii Year plan .................................................................................................................... iv Subject advisor’s information..................................................................................... vii

Study Guide Memorandum UNIT 1: NUMBER SYSTEMS .............................................................................................. 1 UNIT 2: EXPONENTS ....................................................................................................... 77 UNIT 3: ALGEBRA ............................................................................................................ 94 UNIT 4: NUMBER PATTERNS AND RELATIONSHIPS .................................................. 136 UNIT 5.1: GEOMETRY .................................................................................................... 168 UNIT 5.2: GEOMETRY: AREA, PERIMETER AND VOLUME ......................................... 220 UNIT 6: TRANSFORMATION GEOMETRY .................................................................... 256 UNIT 7: RATIO AND RATE ............................................................................................. 282 UNIT 8: FINANCE ............................................................................................................ 301 UNIT 9: STATISTICS ....................................................................................................... 319 UNIT 10: PROBABILITY .................................................................................................. 354

Note that the page numbers of the Study Guide Memorandum start at p. 1 again.

The table of contents will guide you to easily access the units.

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Facilitator’s Guide G07 ~ Mathematics

CONTENTS Guidelines for facilitators 1.

2.

3.

4.1 4.2 5.

6.

General........................................................................................................................ ii Study guide with activities and exercises .................................................................... ii Year mark tasks .......................................................................................................... ii Learning outcomes ..................................................................................................... iii Levels of difficulty ....................................................................................................... iii Year plan .................................................................................................................... iv Subject advisor’s information..................................................................................... vii

Year plan: Abridged version Term 1 1 1 2 2

Content Unit in study guide Number systems 1 Exponents 2 Geometry 5.1 Number systems: Fractions 1 Number patterns and 4 relationships 2 Perimeter, area and volume 5.2 3 Algebra 3 3 Transformations 6 3 Ratio and rate 7 3 Finance 8 4 Statistics 9 4 Probability 10 Revision November examination covers the year’s work

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Facilitator’s Guide G07 ~ Mathematics

5.

Year plan To comply with the National Curriculum and work schedule, the content set out below has priority at certain stages of the year. The following schedule will be important when setting papers for examinations and tests. The year plan is based on this schedule, but some of the topics repeat and appear a few times during the year. It is very important to teach the curriculum in the same order as the rest of South Africa so that learners can move between schools with ease.

Term 1

Term 2

Term 3

Term 4

• • • • • • • • • • • • • • • •

Number systems Exponents Construction of geometric figures Geometry of 2-D figures Geometry of straight lines Fractions Number patterns and relationships Perimeter, area and volume Algebra Graphs Transformations Ratio and rate Finances Statistics Probability Revision

Year Plan Grade 7 Mathematics Term 1 Unit 1 Number systems LO1

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Exercises 1 2 3 4 5 6 7 8 9 10

Content Where do number systems come from?

Natural numbers Whole numbers (Counting numbers) Integers Fractions (Rational numbers) Roots and surds (Irrational numbers) Properties of calculations Percentages Calculator work Mixed exercises

iv

Lessons/ days 1 2 2 3 4 2 2 2 2 1 [20]

Facilitator’s Guide G07 ~ Mathematics

Unit 2 Exponents L02

1 2 3 4 5

Expanded exponential notation Calculations with exponents Prime factors Calculator work Mixed exercises

Unit 5.1 Geometry LO3

1 2 3 4 5 6 7 8 9

Measure and construct lines Measure and construct angles Different angles Parallel lines Triangle Quadrilaterals Polygons Circles Mixed exercises

The lessons and time allocated are a guideline only.

Term 2 Exercises Content Unit 4 Number patterns LO1

Unit 5.2 Geometry area, perimeter and volume LO3

1 2 3 4 5 6 7

Visual presentation of patterns Number patterns in a sequence Types of number patterns (sequences) Determine rules for general terms Relationships: Input and output diagrams Number coordinates Mixed exercises

1 2 3 4 5 6 7 8 9

Convert units Rectangles Triangles Circles Polygons Volume Surface area Mixed exercises Summary of formulas

Revise the two terms’ work for the June examination.

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1 3 3 2 1 [10] 2 2 2 3 3 3 3 1 1 [20] Lessons or days 1 2 2 2 2 3 1 [13] 1 3 3 3 1 2 2 2 1 [18]

Facilitator’s Guide G07 ~ Mathematics

Term 3

Exercises

Content

Unit 3 Algebra LO2

1 2 3 4 5 6 7 8 9

Variables Basic calculations Algebraic expressions English to Maths Substitution Linear equations Inequality Basic word problems Mixed exercises

Unit 6 Transformation geometry LO3

1

Symmetry

2 3 4 5 6

Reflection Translation Rotation Enlargement and reduction Mixed exercises

Unit 7 Ratio and rate LO1

1 2 3 4 5 6 7

Ratios and equivalent fractions Divide in specific ratios Rate Increase and decrease Distance, speed and time Scale drawings Mixed exercises

Unit 8 Finance LO1

1 2 3 4 5 6 7 8

Calculations with money Money and ratios Percentages Profit, loss and discount VAT Interest Exchange rate Mixed exercises

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Lessons or days 1 3 2 2 3 2 2 2 2 [19] 1 1 1 2 2 2 [9] 2 2 2 2 2 2 1 [13] 1 2 2 2 2 2 2 1 [14]

Facilitator’s Guide G07 ~ Mathematics

Term 4

Exercises

Content

Unit 9 Statistics LO5

1 2 3 4 5 6

Data collection Organisation and summary of data Presentation of data Analysis and calculation of data Interpretation and evaluation of data Mixed exercises

Unit 10 Probability LO5

1 2 3 4 5

Terminology Relative frequency of outcomes The probability scale Probability P(x) of outcome x Mixed exercises

Revision

6.

Use the mixed exercises at the end of each unit for revision. The November Examination covers the entire year’s work.

Subject advisor’s information Contact the subject matter expert for more information or support.

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Lessons or days 1 2 3 3 1 1 [11] 1 2 2 3 1 [9]

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Facilitator’s Guide G07 ~ Mathematics

Unit

1

CONTENTS Unit 1: Number systems ................................................................................................. 3 Exercise 1: Where do number systems come from? ..................................................... 3 Exercise 2: Natural numbers ......................................................................................... 4 Exercise 3: Counting numbers ..................................................................................... 26 Exercise 4: Integers ..................................................................................................... 29 Exercise 5: Fractions (rational numbers) ..................................................................... 33 Exercise 6: Roots (irrational numbers)......................................................................... 52 Exercise 7: Properties and calculations ....................................................................... 58 Exercise 8: Percentages .............................................................................................. 63 Exercise 9: Calculator work ......................................................................................... 68 Exercise 10: Mixed exercises ...................................................................................... 72 Bibliography ................................................................................................................. 76

Simplify means…

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Facilitator’s Guide G07 ~ Mathematics

Unit

Level of difficulty 1 Basic knowledge 25%

*

Level of difficulty 2 Routine procedures 45%

**

Level of difficulty 3 Complex procedures 20%

***

Level of difficulty 4 Problem-solving 10%

****

Level of difficulty 5

**** *

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Assessment analysis • Use mathematical facts and vocabulary • Use the correct formulas • Predict answers and round off values • Theoretical knowledge • Complete known procedures • Apply knowledge and facts with more than one step • Come to conclusions from given information • Basic calculations as learned from examples and exercises • Complex calculations and high order arguments • Euclidean Geometry • No stipulated path to follow • Similarities and differences between presentations • Need conceptual and holistic approaches to problems • Unseen and not-routine problems.

• Problems in different sections • Still in curriculum and study guide • Mostly aimed at practical and everyday situations.

• High order of thinking Advanced Often regarded as acceleration of the curriculum. Not included in tests and exam papers.

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Facilitator’s Guide G07 ~ Mathematics

Unit

1

Unit 1: Number systems Content in this unit: • Where do number systems come from?

• Natural numbers • Whole numbers (Counting numbers) • Integers • Fractions (Rational numbers) • Roots and surds (Irrational numbers) • Properties of number systems (associative, commutative and distributive laws)

Exercise 1: Where do number systems come from?

There is proof that the Ishango people of the Democratic Republic of the Congo (DRC) made marks on bones to count their cattle or family members. The oldest bone found to confirm these assumptions is approximately 20 000 years old.

The number system that we use today is the Hindu-Arabic system and it was developed more than 1 000 years ago by Hindu-Arabic mathematicians.

The Egyptian number system consists of symbols. Examples of the two number systems are: Egyptians

Hindu-Arabic

staff

1

heel

10

rope

100

lotus

1000

pointing finger

10 000

burbot

100 000

astonished man

1000 000

1.1*

Write down the Egyptian equivalent for the number 3 516.

Answer:

1.2*

Write down the Egyptian equivalent for the number 2 182.

Answer:

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Facilitator’s Guide G07 ~ Mathematics

1.3***

1.4*

1.5***

Unit

1

∩∩

Write the Hindu-Arabic number for

Answer: 1 million + 2 thousands + 2 tens + 3 units 100223 Study the number 234 654 365 123 987 341 236 687.

Write down the number that is 10 000 more than the given number.

Answer: 234 654 356 123 987 341 246 687 Investigate: For fun Draw the following table in your answer book and translate the terms addition, subtraction, multiplication and division in any other two languages. + English Afrikaans Sepedi

Addition Optelling Go hlakantšha

– Subtraction Aftrekking Go tloṧa

x Multiplication Vermenigvuldiging Go atiša

÷

Division Deling Go arola

Exercise 2: Natural numbers Natural numbers are the numbers from 1 to infinity. The symbol used is N. If tabulated: N = {1; 2; 3; 4; 5 ...} If we look at all the number systems together, A in this diagram represents natural numbers: E The letter A represents natural numbers. As the unit progresses, the other number systems will be explained.

The symbol for natural numbers is N.

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Facilitator’s Guide G07 ~ Mathematics

Unit

Units

7

Tens

1

Hundreds

Millions

3

Thousands

Ten millions

4

Ten thousands

Hundred millions

2

Hundred thousands

Milliards

Each digit in natural numbers has its own meaning.

Example:

5

9

0

6

8

Milliard and billion denote the same number. Milliard is almost never used in America and Britain, but is often used in Continental Europe.

Decimal system 1 million= 1000 000 = 106 Six zeros 1 billion = 1000 000 000 = 109 Nine zeros 1 trillion =1000 000 000 000 =1012 Twelve zeros 1 quintillion = 1018 Eighteen zeros

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(1000)2

A thousand million

(100 0000)2

(100 0000)3

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Facilitator’s Guide G07 ~ Mathematics

2.1**

Unit

Complete the table by identifying the place value of the 7 .

Example

Number 345 2 7 8

Description Tens

2.1.1*

34 52 7 823

2.1.2*

2 7 82

2.1.3**

5 2 7 8 254

2.1.4**

7 254 164

2.1.5***

952 7 54 123

2.1.6*

1233452 7

2.1.7**

34 52 7 825 455

2.1.8*

7 82

2.1.9****

34 7 84 556 723

Answer: Number 345 2 7 8

Description Tens

2.1.1*

34 52 7 823

Thousands

2.1.2*

2 7 82

Hundreds

2.1.3**

5 2 7 8 254

Ten Thousands

2.1.4**

7 254 164

Millions

2.1.5***

952 7 54 123

Hundred Thousands

2.1.6*

1233452 7

Units

2.1.7**

34 52 7 825 455

Millions

2.1.8*

7 82

Hundreds

2.1.9****

34 7 84 556 723

Hundred Millions

Example

2.2**

Write the number from the description given.

Example 2.2.1* 2.2.2* 2.2.3** 2.2.4* 2.2.5** 2.2.6*** 2.2.7****

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Description 4 Hundred Thousands 6 Tens 12 Thousands 876 Millions 9 Units 47 Hundreds 639 Ten Thousands 32 456 Hundreds

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Number 400 000

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Facilitator’s Guide G07 ~ Mathematics

2.2

Unit

Answer:

Example 2.2.1* 2.2.2* 2.2.3** 2.2.4* 2.2.5** 2.2.6*** 2.2.7****

2.3***

Description 4 Hundred Thousands 6 Tens 12 Thousands 876 Millions 9 Units 47 Hundreds 639 Ten Thousands 32 456 Hundreds

Number 400 000 60 12 000 876 000 000 9 4 700 6 390 000 3 245 600

Add 10 to all the following values. Write your answers in the table.

Number 2.3.1* 65 382 2.3.2* 1 234 2.3.3** 87 592 2.3.4*** 96 795 2.3.5**** 9 328 999

Answer

Answer: Number 2.3.1* 65 382 2.3.2* 1 234 2.3.3** 87 592 2.3.4*** 96 795 2.3.5**** 9 328 999 2.4**

Complete the table by calculating what is asked.

Example 2.4.1** 2.4.2* 2.4.3** 2.4.4** 2.4.5**** 2.4.6**

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Answer 65 392 1 244 87 602 96 805 9 329 009

Number 3 761 676 767 78 493 78 493 12 121 212 875 462 867 444 444

Calculation Add 400.

Add 1 million.

Subtract 50.

Add 9 000.

Subtract 1 million.

Add four hundred thousand.

Add 5 000.

7

Answer 4 161

1

Facilitator’s Guide G07 ~ Mathematics

Unit

1

Answer Number 3 761 676 767 78 493 78 493 12 121 212 875 462 867

2.5**

Calculation Answer Example Add 400.

4 161 2.4.1** Add 1 million.

1 676 767 2.4.2* Subtract 50.

78 443 2.4.3** Add 9 000.

87 493 2.4.4** Subtract 1 million.

11 121 212 2.4.5**** Add four hundred 875 862 867 thousand.

2.4.6** 444 444 Add 5 000.

449 444 Break down the natural numbers as shown in the example.

Number Answer Example 1 234 567 = 1000000 + 200000 + 30 000 + 4 000 + 500 + 60 + 7 2.5.1* 648 2.5.2* 33 333 2.5.3**** 54 545 454 2.5.4** 5 678 2.5.5* 6 2.5.6** 765 432 Answer Number 1 234 567

2.6**

2.7**

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Answer Example = 1 000 000 + 200 000 + 30 000 + 4 000 + 500 + 60 + 7 2.5.1* 648 = 600 + 40 + 8 2.5.2* 33 333 = 30 000 + 3 000 + 300 + 30 + 3 2.5.3**** 54 545 454 = 50 000 000 + 4 000 000 + 500 000 + 40 000 + 5 000 + 400 + 50 + 4 2.5.4** 5 678 = 5000 + 600 + 70 + 8 2.5.5* 6 =6 2.5.6** 765 432 = 700 000 + 60 000 + 5 000 + 400 + 30 + 2 Arrange the following natural numbers from small to big. {2 542; 154; 2 441; 2 523; 2 509} Answer {2 542; 2 523; 2 509; 2 441; 154} = {154; 2 441; 2 509; 2 523; 2 542} Arrange the following natural numbers from big to small. {592; 523; 2 600; 2 699} Answer {592; 523; 2 600; 2 699} = {2 699; 2 600; 592; 523}

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Facilitator’s Guide G07 ~ Mathematics

Unit

2 379 + 6 666 = 2 379 + 6 666 9 045 In Mathematics we use symbols to show greater than or less than.

If you look at the following symbols from left to right, you can say:

>

greater than

<

Example: 234 > 233 345 < 346 2.8**

less than

Place a < or > symbol between the values in the table.

Number 1 < or > Number 2 546 124 12 65 3 232 6 756 437 436 112 233 112 234 4 356 11 111 Answer Number 1 < of > Number 2 546 > 124 12 < 65 3 232 < 6 756 437 > 436 112 233 < 112 234 4 356 < 11 111

When doing addition and subtraction of natural numbers, your answer must have enough steps to show that you did not use a calculator.

Example Make sure you know where the = 9 370 – 6 666 2 379 + 6 666 sign is placed. = 9 379 = 2 379 Know how and – 6 666 + 6 666 where this sign is 2 713 9 045 placed.

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Facilitator’s Guide G07 ~ Mathematics

Unit

1

Simplify the following. Show your calculations. You are not allowed to use a calculator.

2.9.*

17 + 25 Answer 17 + 25 = 17 + 25 42

2.10**

Use your answer and write down the correct answer for 17 hundred plus 25 hundred.

Answer The line is the same as the = sign

1 700 + 2 500 = 1 700 + 2 500 4 200 2.11**

Now use your answer and write down the correct answer for 17 million plus 25 million.

Answer Pay attention to the zeros.

17 000 000 + 25 000 000 = 17 000 000 + 25 000 000 42 000 000 2.12****

Determine the sum of 17 Hundred Thousands and 25 Ten Thousands.

Answer Pay attention to the zeros.

1 700 000 + 250 000 = 170 000 + 250 000 1 950 000

−

Calculations with natural numbers can only be one of the following:

+

plus

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×

minus

multiplication

10

÷ division

Facilitator’s Guide G07 ~ Mathematics

Unit

Examples of calculations with natural numbers:

Short division 25 863 ÷ 8 3 232 rem 7 = 8 ) 25 863

Addition 6765 + 25863

=

6765 + 25863 32628

Long division 25863 ÷ 8

=

3232 8 25863 − 24 18

Multiplication 456 x 18

=

456 x 18 3648 + 4560 8208

− 16 26 − 24 23 − 16 res 7 remainder 7

Subtraction 25863 − 4444

=

25863 − 4444 21419

Use the prescribed methods and determine the answers to the following. You are not allowed to use a calculator.

2.13**

2 345 + 999 Answer 2 345 + 999 = 2 345 + 999 3 344

2.14**

2 345 – 999 Answer 2 345 – 999 = 2 345 – 999 1 346 No calculators. Write down all the calculations to show that you did not use a calculator.

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Facilitator’s Guide G07 ~ Mathematics

2.15**

Unit

2 345 x 99 Answer 2 345 99 =

Remember the zero

2 345 99

21 105 + 211 050 232 155 2.16***

2 345 ÷ 9 with the long division method.

Answer

Two numbers below each other without a +, –, x or ÷ means nothing. In this case it is a minus (–) and you must write it down.

2.17***

2 345 ÷ 99 with the long division method.

Answer

2.18****

2 345 ÷ 999 with the long division method.

Answer

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Facilitator’s Guide G07 ~ Mathematics

Unit

Rewriting English sentences as mathematical sentences (number sentences) is essential to solving word problems. The following terms for calculations may be used. Study them and use them.

Symbol × ÷ – +

Meaning Multiply Divide Subtract Add

Other English words Product of Quotient of Difference between Sum of

Write the following English sentences in mathematical (number) sentences. Simplify them without using a calculator.

2.19**

Determine the sum of 12 and 56.

Answer 12 + 56 = 68

If you cannot give the answers quickly…write the numbers below one other and add them.

or 12 + 56 68

2.20**

Calculate the product of 12 and 56.

Answer 12 x 56 = 12 x 56 72 + 600 672

2.21**





Marks are allocated for: 1 for the Maths sentence 1 for the 72 + 600 1 for the answer



What is the quotient if 56 is divided by 12? Give the remainder.

Answer 56 ÷ 12 = 12

4 56 − 48 remainder 8

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The remainder is asked.

That means that you have to do this question with long division.

In future we will use decimals, but for now the remainder and quotient are both integers.

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