4.1 Learning outcomes
Explanation of the learning outcomes required in Grade 7, 8 and 9:
Learning Outcome 1
Learning
Learning
Learning
5
3
4
2
Numbers, calculations and relationships
4.2 Levels of difficulty
The number of stars next to the sums indicates the level of difficulty. This indication is given in the study guide, facilitator’s guide, tests and examination papers.
Level 1 * Knowledge of basic theory and procedures is required before any calculations can be done. Studying theory is required. 25% of the question paper.
Level 2 ** Applying routine multistep procedures in various contexts. Continuous practice will enable learners to master this section of the question paper. 45% of the question paper.
Level 3 *** Applying knowledge and procedures in a variety of contexts. Complex calculations based on a combination of acquired knowledge. Questions are asked indirectly with the emphasis on comprehension. 20% of the question paper.
Level 4 **** Applying relevant knowledge to unfamiliar, nonroutine questions. The most difficult part of the question paper – testing the learners’ ability to apply all the knowledge they have gained. 10% of the question paper.
Level 5 ***** Advanced
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 Number systems
Exponents
Construction of geometric figures
Geometry of 2-D figures
Geometry of straight lines
Term 2 Fractions
Number patterns and relationships
Perimeter, area and volume
Term 3 Algebra Graphs
Transformations
Ratio and rate
Finances
Term 4 Statistics
Probability
Revision
Year Plan
Grade 7 Mathematics
6. Subject advisor’s information
Contact the subject matter expert for more information or support.
Exercise
Exercise
Exercise
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
and high order arguments
Euclidean Geometry
No stipulated path to follow
Similarities and differences between presentations
Need conceptual and holistic approaches to problems
Problems in different sections
Still in curriculum and study guide
Mostly aimed at practical and everyday situations. High order of thinking
Often regarded as acceleration of the curriculum. Not included in tests and exam papers.
Unit 1:Number systems
Content in this unit:
Where do number systems come from?
Natural numbers
Whole numbers(Counting numbers)
Integers
Fractions(Rationalnumbers)
Roots andsurds(Irrationalnumbers)
Properties of number systems(associative, commutative anddistributive laws)
Exercise1:Where do number systems come from?
There isproof that the Ishango people of the Democratic Republic of the Congo (DRC) made marks on bones to count their cattle or family members.Theoldest bone found to confirm these assumptions is approximately 20000 years old. The number system that we use today is the Hindu-Arabic system and it was developed more than 1000 years ago by Hindu-Arabic mathematicians. The Egyptian number system consists of symbols.Examples of the two number systems are:
1.1*Write down the Egyptian equivalent for the number 3516. Answer:
1.2*Write down the Egyptian equivalent for the number 2182. Answer:
1.3***Write the Hindu-Arabic number for
Answer: 1 million+ 2 thousands+ 2 tens + 3 units 100223
1.4*Study the number 234654365123987341236687.
Write down the number that is 10000 more than the given number. Answer: 234654356123987341246 687
1.5***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.
EnglishAdditionSubtractionMultiplicationDivision AfrikaansOptellingAftrekkingVermenigvuldigingDeling SepediGo hlakantšha Go tloṧaGo atišaGo arola
Exercise2: Natural numbers
Natural numbers are the numbersfrom 1 to infinity. The symbol usedis 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:
The symbol for natural numbers is N.
The letter A represents natural numbers. As the unit progresses, the other number systems will be explained.
Each digit in natural numbers has its own meaning.
Example:
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
the table by identifying the place value of the 7
2.2** Write the number from the description given.
Number
4 Hundred Thousands 400 000 2.2.1* 6 Tens 2.2.2* 12 Thousands 2.2.3** 876 Millions 2.2.4* 9 Units 2.2.5** 47 Hundreds 2.2.6*** 639 Ten Thousands 2.2.7**** 32
2.2.4*
2.2.5**
2.2.6***
2.3*** Add 10 to all the following values. Write your answers in the table.
2.3.1*
2.3.2*
2.3.3**
2.3.4***
2.6**
2.4.3** 78 493 Add 9 000. 87 493 2.4.4** 12 121 212 Subtract 1 million. 11 121 212 2.4.5**** 875 462 867 Add four hundred thousand. 875 862 867
2.4.6**
Break down the natural numbers as shown in the example.
2.4.3** 78 493 Add 9 000.
2.4.4**
2.4.6** 444 444 Add 5 000.
2.7**
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}
2
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:
2.8** Place a < or > symbol between the values in the table.
Use your answer and write down the correct answer for 17 hundred plus 25 hundred.
When doing addition and subtraction of natural numbers, your answer must have enough steps to show that you did not use a calculator.
2.12**** Determine the sum of 17
Pay attention to the zeros. The line is the same as the = sign
Pay attention to the zeros.
Thousands and 25 Ten Thousands.
Calculations with natural numbers can only be one of the following: plus minus multiplication division
Examples of calculations withnaturalnumbers:
Use the prescribed methods and determine the answers tothe following.You are not allowed to use a calculator.
2.13**2345 + 999
Answer 2345 + 999 = 2345 +999 3 344
2.14**2345 –999 Answer 2345 –999 = 2345 –999 1 346 No calculators. Write down all the calculations to show that you did not use a calculator.
2.15**2345 x 99
Answer 2345 99 = 2345 99 21105 + 211050 232 155
Remember the zero
2.16***2345 9with the long divisionmethod.
Answer
Two numbers beloweach other without a +, –, x or means nothing. In this case it is a minus(–) and you must write it down.
2.17***2345 99 with the long divisionmethod.
Answer
2.18****2345 999 with the long divisionmethod.
Answer
RewritingEnglish sentences asmathematical sentences(number sentences) is essential to solvingword problems. The following termsfor calculations may be used. Study them and use them.
Symbol Meaning Other Englishwords
Multiply Product of Divide Quotient of – Subtract Difference between + Add 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.
If you cannot give the answers quickly…write the numbers below one other and add them.
2.20**Calculate the product of 12 and 56.
Marks are allocated for:
1 for the Maths sentence 1 for the 72 + 600 1 for the answer
2.21**What is the quotient if 56 is divided by 12?Give the remainder.
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.
2.22***A printingcompany prints 1 436 books per day.How many books did they print in July 2013 if they did not work on Saturdays and Sundays?
Answer
There were23working days inJuly 2013. 1436 books per day. 1436x23for the month
2.23****A yachtsails inone direction. On the first day it sails 275 km and then 35 kilometres further each day. How far will the yacht be from the harbour at the end of the third day? Day 1= 275 km Day 2
Harbour
3
Answer
First day=275 km
Second day275 + 35 km = 310 km
Third day275 + 35 + 35 km= 345 km
Total at end of third day = 275 + 310 + 345 = 930 km
2.24*** Juan drove 1 035 metres to a shop. John drove 2 285 metres to the same shop. How much further did John drive?
Answer Juan 1 035
John 2 285
Difference = 2 285 – 1 035 = 2 285 – 1 035 1 250 metres
2.25 Sandra has to make 34 skirts for girls participating in a school play. She buys 2 rolls of material of 44 metres each. Each skirt requires 3 metres of material.
2.25.1*** Show all your calculations and determine how much material she still needs to buy. You are not allowed to use a calculator.
Answer 34 girls each 3 metres gives in total 34 x 3 = 102 metres She bought 88 metres of material
2.25.2*** To make the skirts, a single piece of 3 metres of material is required. The material can’t be joined. Explain how many rolls of material must be bought and how many metres will remain on each roll after 34 pieces of 3 metres have been cut.
Rounding off natural numbers must be done in Tens, Hundreds and Thousands. Examples
Rounding off to
Rounding off to 100. (It means that the number must be divided by 100 without leaving a remainder.)
Rounding off to 1 000. (It means that the number must be divided by 1 000 without leaving a remainder.)
Look at the Units.
When less than 5, round off downwards.
When 5 or more, round off upwards.
Look at the Tens and Units (last two digits).
When less than 50, round off downwards.
When 50 or more, round off upwards
Look at the last three digits (Hundreds, Tens and Units).
When less than 500, round off downwards.
When 500 or more, round off upwards.
2.26** Complete the table by rounding off the numbers. Read the headings of the table.
7 766 10 7 770 6 is more than 5
7 766 100 7 800 66 is more than 50
7 719 1 000 8 000 719 is more than 500
7 371 10 7 370 1 is less than 5
890 100 900 90 is more than 50
1 million 1 000 1 000 000 Look at the last 3 digits. It is not necessary to round off
Round off the following numbers to the nearest 5.
Example:
(It means that 5 must be divided into the value without a remainder. Remember, the Units that are a 1 or a 2 must be rounded off downwards and Units that are 3 and 4, upwards.)
62 rounded off to the nearest 5 is 60.
63 rounded off to the nearest 5 is 65.
84 rounded off to the nearest 5 is 85.
When you round off in Mathematics, the following sign is used: For clarity, the number you round off to is written at the end of the answer.
For example:
26 30 to the nearest 10
3 456 3 500 to the nearest 100
2.27*** Round off 123 456 to the nearest 5. Answer 123 456
2.28**** Round off 74 to the nearest 7. Answer
Multiples of 7 are 63; 70; 77;… Every number can be divided by 7. Between 70 and 77 the middle value is between 73 and 74. 71; 72; 73; 74; 75; 76; 77
74 rounded off to the nearest 7 will be 77. Round off downwards Round off upwards
2.29*** Shaun buys a pair of trousers for R243 and a jacket for R679.
2.29.1* Round off each to the nearest R10 and give the total amount he paid for the trousers and the jacket.
Answer: R243 R240 to the nearest R10 R679 R680 to the nearest R10 In total: 920 R 680 240
First round off and then add.
2.29.2** What will the amount be if Shaun first adds the amounts before rounding off?
Answer: R243 + R679 R920 R922
First added and then rounded off.
2.29.3*** What is the difference between the answers in the previous two questions? Do you think one will always get to the same answers?
Answer:
The answer is the same in both cases. It seems that there is no difference between the two methods.
Order of operations
When questions are asked that include a combination of adding, subtracting, multiplying or division, it is important to remember the rules regarding the order of operations.
Remember there are priorities for +, –, and
This sequence applies up to Grade 12. Study this now and you will never have to study it again.
The order of operations:
1. Brackets (…)
2. Exponents
3. Multiplication and division from left to right
4. Addition and subtraction from left to right
Simplify the following by doing only onecalculationfor each step:
Brackets first. A bracket disappears when the answer is written down.
Remember:Once the calculation in the bracket has been completed, the bracket disappears.
Simplify the following expression by doing one calculation per steponly.
Do one calculation for each step. It will teach you the preferenceand the use of the = sign. Remember that everything must be identical before and after the = sign.
The assignment is to do one calculation for each step. Do not do more than one!
x and have the same priority. The first sign must be executed first.
Be aware
All the number 2s can be confusing. Work systematically … one calculation for each step.
2.36** )15(6328
Answer:
463x28)15(6328
2.37**
Answer:
2.38**0203
Answer: 0 00 0x20 0203
2.39**(6–2+7) +(3 –2 + 12)
Answer: (6–2+7) +(3 –2 + 12)
=(4 +7)+(3 –2 + 12) =11+(1 + 12)
= 11 + 13 =24
It is easy to multiply by 0. The answer stays zero.
(0)4 = 0 and not 4 6 x 0 = 0 and not 6
Factors
2.40***Simplify by doing only one calculationfor each step.
Answer:
Treat the division line as two large brackets.
Simplify the numerator on its own and the denominator on its own. Thendivide the numerator by the denominator. Facilitator’s
This is a general term which means these arenumbers that divideexactly into a bigger number, with an integer as an answer. Therefore 10 will be a factor of 20 because 10 can bedivided into 20.
Prime factors
A prime number is a number that has exactly two factors,namely the number 1 and the number itself.
This means that only two numbers can be divided into a prime number without a remainder. There are aninfinite number of prime numbers, but no formula exits to determine the number of prime numbers.
The set of prime numbers is= {2; 3; 5; 7; 11; 13; 17;19…}
The number 1 1 is neithera prime number nora compositenumber.
Compositefactors
Numbers that can be divided by more than 2 factorsare called compositenumbers. For example,24 is compositebecause it can be divided by 1, 2, 3, 4, 6, 8, 12 and 24.
Compositenumbershave three or more factors.
Example:Factors of24 = {1;2;3;4;6;8;12;24}
={the number 1} + {prime numbers}+ {compositenumbers} ={1} + {2;3} + {4;6;8;12;24}
4 is written as 2 x 2
12 is written as 2 x 2 x 3
24 is written as 2 x 2 x 2 x 2 x 3
Prime numbers that are multiplied are called composite factors.
Write each number given as sets of 1, prime numbers and compositefactors. Use the example given.
2.41*12
Answer
Factors of 12= {1;2;3;4;6;12} = {1} + {2;3} + {4;6;12}
Factors of 30 = {1; 2; 3; 5; 6; 10; 15; 30} = {1} + {2; 3; 5} + {6; 10; 15; 30}
2.43** 36
Answer
Factors of 36 = {1; 2; 3; 4; 6; 9; 12; 18; 36} = {1} + {2; 3} + {4; 6; 9; 12; 18; 36}
2.44** 35 Answer
Factors of 35 = {1; 5; 7; 35} = {1} + {5; 7} + {35}
Use the ladder method to determine prime factors.
=2
2 x 2 x 3 x 3 x 5
Divide the number by the first prime number that can be divided into that number. Continue dividing by the same number until you end up with a remainder. Then use the next prime number.
numbers = {2; 3; 5; 7; 11; 13 ...}
Write the following numbers as the product of their prime numbers. Use the ladder method. This means the same as writing the number as the product of prime factors. 2.50* 24
the largest prime number that can be
The largest prime number is 11
First learn the following: Multiples. Numbers into which the given number can be divided. Example: multiples of 6 are {6; 12; 18; 24; 30 …}
Factors. These are numbers that can be divided in the given numbers:
Example: factors of 12 is {1; 2; 3; 4; 6; 12}
Example:
Factors of 8 = {1; 2; 4; 8}
Multiples of 8 = {8; 16; 24; 32 …}
Determine the multiple and factors of the following: 2.53* 6
Answer
Factors of 6 = {1; 2; 3; 6}
Multiples of 6 = {6; 12; 18; 24 …} 2.54* 12
Answer:
Factors of 12 = {1; 2; 3; 4; 6; 12}
Multiples of 12 = {12; 24; 36 …} 2.55* 18
Answer:
Factors of 18 = {1; 2; 3; 6; 9; 18}
Multiples of 18 = {18; 36; 54 …} 2.56* 24
Answer:
Factors of 24 = {1; 2; 3; 4; 6; 8; 12; 24}
Multiples of 24 = {24; 48; 72 …}
Multiples are large numbers.
Factors are small numbers.
Determine the LCM (lowest common multiple) and HCF(highest common factor) of two or more integers.
Example 1: Determine the HCF of 6 and 8
Factors of6 = {1; 2; 3; 6}
Factors of 8 = {1; 2; 4; 8}
Common factors = {1;2} HCF = 2
(Highest common factor.This is the biggest factor that can divide intobothnumbers)
Example 2:
Multiples of 6 = {6;12;18; 24 …}
Multiples of 8 = {8;16; 24;32; 40…}
LCM= 24
(Lowestcommon multiple.The smallest number that 6 and 8 can divide into.)
2.57**Determine the HCF of 16 and 24.
Answer:
Factors of 16 = {1; 2; 4; 8; 16}
Factors of 24= {1; 2; 3; 4; 6; 8; 12; 24} HCF = 8
2.58**Determine the HCFof 20 and 30.
Answer:
Factors of20 = {1; 2; 4; 5; 10; 20}
Factors of 30= {1; 2; 3; 5; 6; 10; 15; 30} HCF = 10
2.59***Determine the LCMof9 and 12.
Answer:
Multiples of 9 = {9;18; 27; 36 …}
Multiples of 12 = {12; 24; 36;48 ..}
LCM =36
2.60***Determine the LCMof5, 6, and 15.
Answer:
Multiples of5= {5;10; 15; 20;25; 30 ..}
Multiples of 6 = {6; 12; 18; 24; 30; 36...}
Multiples of 15 = {15; 30; 45...}
LCM =30
2.61***Determine the LCMof 7,9and 21.
Answer:
HCF means highest common factor.
LCM means lowest common multiple.
LCD means lowest common denominator.
Exercise 3: Counting numbers
The zero is added. Remember undefined is 0 2 0 2 0
3.1**(2 + 0 –1)+(3 –0)
Answer: (2 + 0 –1)+(3 –0) =(2 –1)+(3 –0) =1 +(3 –0) =1 + 3 = 4
3.2**22 x 0+4 –0 +(10 –0)
Answer:
22 x 0 + 4 –0 +(10 –0) =22 x 0 + 4 –0 + 10 = 0 + 4 –0 + 10 = 4 –0 + 10 = 4 + 10 = 14
Multiples of9 ={9 ;18;27; 36; 45; 54; 63; 72…}
Multiples of7 = {7; 14; 21; 28; 35; 42; 49; 56; 63; 70…}
Multiples of 21 = {21; 42; 63 ...}
LCM =63
Counting numbers B include the natural numbers and zero.
B = {0; 1; 2; 3; 4; 5; 6 …}
Natural numbers
A = {1; 2; 3; 4; 5; 6 …}
The symbol used for counting numbers is N0.
Do not be afraid of
Remember that you are not allowed to divide by zero. The numerator does not matter; it is the denominator that can neverbe zero.
3.9****The temperature in Sutherland was -9°C on Saturday and -5°C on Sunday. What was the change in temperature from Saturday to Sunday? Answer
Study the temperature on the number line.
Saturday Sunday
From -9 to -5 is 4 units to the right. The change is then positive 4°C.
3.10****If the temperature inSutherland was-5°C on Saturday and -9 °C on Sunday, what was the change in temperature from Saturday to Sunday? Answer
Study the temperature on the number line. -10-9-8-7-6-5-4-3-2-101
From -5 to -9 is 4 units left. The change is negative -4°C.
