Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Abuse of notation wikipedia , lookup
Musical notation wikipedia , lookup
Location arithmetic wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
History of mathematical notation wikipedia , lookup
Large numbers wikipedia , lookup
Big O notation wikipedia , lookup
Positional notation wikipedia , lookup
Section 1: Number Skills 1. Introduction This section on number skills lays the foundation for the other sections, as it provides the student with the opportunity to review basic mathematical concepts that will be used in this subject throughout the year. As the student you need to make sure that on completion of this section you are comfortable working with all of its contents. 1.1 Learning outcomes At the end of this section on Number Skills you should be able to competently do the following: Fractions Multiply and divide fractions Add or subtract fractions with the same or different denominators Simplify fractions Decimals Convert a fraction to a decimal by using a calculator Use the correct notation to indicate a recurring decimal Convert a decimal to a fraction Rounding off Moet nog invul Percentages Convert decimals and fractions to percentages and vice versa Do simple increasing and decreasing problem solving by using percentages Ratio and Proportion Compare information in terms of ratios Do simple problem solving by using direct and inverse proportion Sigma notation Determine sums written in sigma notation Write sums in sigma notation Exponents 1 Know and be able to apply all the Laws of Exponents to simplify expressions Logarithms Moet nog invul Start up activity 1.1: What do I know? Pair up with a class mate and complete the following activity within 15 minutes: Try to define each of the concepts listed below and support your definition with an example of each. Then share your findings with the rest of the class. Fractions Decimals Rounding off Percentages Ratio and Proportion Sigma notation Exponents Logarithms 2. Fractions Most of us are accustomed to fractions such as “three over four” 3 , and would 4 interpret this specific fraction as “ three parts of four” or “three divided by four”. This is indeed correct. However, fractions are also known as rational numbers and the set of all rational numbers are indicated with the sign . Definition of a Rational Number A rational number is a number that can be written as a fraction a , where a b and b are integers and b 0 , as division by 0 is not defined. We can also refer to a as the quotient of a and b , where a is called the b numerator and b the denominator or divisor of the quotient. Example 5 20 0 4 3 , , , and are all examples of rational numbers. 7 9 6 1 8 2 To do different operations such as addition, multiplication and division with rational numbers, also called fractions, we need certain properties to guide us in working with these special numbers. 2.1 1. Properties of Fractions a c ac b d bd To multiply fractions, we multiply the numerators with each other and the denominators with each other. Example 3 7 3 7 21 5 2 5 2 10 2. a c a d b d b c To divide fractions, invert the divisor and multiply it with the first fraction. This operation becomes meaningful if we look at the following explanation: a d ad ad a c b c bc bc ad b d c d cd 1 bc d c dc Example 3 7 3 2 6 5 2 5 7 35 Please note that we are NOT doing cross-multiplication here. Crossmultiplication is an operation which is performed ONLY if you are working with equations, not expressions. Recall that an expression is obtained by combining variables and constants by using operations such as addition, subtraction, multiplication, division and roots. This is an example of an 1 3 expression: 2 x x 2 7 4 x . An equation is a statement that two mathematical expressions are equal. These are examples of equations: 7 4 3 and 2x 4 9 . 3. a b ab c c c To add fractions that have the same denominators, you only need to add the numerators. Example 3 6 36 9 5 5 5 5 3 4. a c ad bc b d bd To add fractions with different denominators, find a common denominator and rewrite the terms in the numerator with respect to the new denominator. Then add the new terms in the numerator. This process once again becomes meaningful if we look at the following explanation: a c a d c b ad bc b d bd d b bd Example 3 7 3 2 7 5 6 35 41 5 2 5 2 25 10 10 Please note that we are once again NOT doing cross-multiplication here. Cross-multiplication is an operation which is performed ONLY if you are working with equations, not expressions. 5. ac a bc b Factors that are common in the numerator and the denominator can be cancelled if there is multiplication and/or division between all the factors. Example 3 4 3 5 4 5 6. If a c , then ad bc b d Now this is cross-multiplication, as we are working with an equation, not an expression. Example 3 9 , because 3 15 9 5 5 15 Example Evaluate 9 211 24 300 Solution: When evaluating a sum consisting of fractions for which a common denominator can be found which is smaller than the product of the two denominators as described in property 4 above, the following procedure is used. Each denominator is factored into prime factors (factors that can only divide into 1 and itself). So 24 2 3 3 and 300 2 2 3 5 2 . The Least Common 4 Denominator (LCD) is then determined by forming the product of all the factors that occur in the factorizations, using the highest power of each factor. Therefore the LCD of 24 and 300 is 2 3 3 5 2 600 . Then 9 211 24 300 9 25 211 2 24 25 300 2 225 422 600 600 647 600 3. Decimals It is not always easy and appropriate to work with numbers in fractional form. You would, for instance, be very surprised to see the price of a pair of jeans being advertised as R129, 95 75 . Or a packet of biltong being sold for R . We 100 4 therefore often choose to express fractions as decimals. All fractions (rational numbers) can be expressed as recurring decimals, that is, a decimal in which one or more of the digits are repeated endlessly. A recurring decimal is indicated by placing a dot or a line over the digit or digits that repeat. Here are some examples of recurring decimals: 1 0,333333 0,3 3 157 0,317171717 0,317 495 9 1,285714285714285714 1, 285714 7 1 0,50000 0,50 2 The last example is also called a terminating decimal, as it can be written as a fraction with a denominator that can be expressed as a power of 10. 1 5 0,50000 0,50 2 10 It is customary to omit indicating the recurring decimal when working with terminating decimals. We would, for example, rather write 1 0,5 than 2 1 0,50 . 2 Whenever we encounter a recurring or terminating decimal, we know that it can be expressed as a fraction. To convert a repeating decimal such as x 2,578787878 2,578 to rational form, we complete the following procedure: 5 Step 1: Multiply the decimal number with an appropriate power of 10 so that the comma moves enough spaces to past the first recurrence. In this example we will multiply with 1000 to move the comma until after the first number 8. 1000 x 2578,7878 Step 2: Multiply the decimal number with an appropriate power of 10 so that the comma moves enough spaces to stop just before the first recurrence. In this example we will multiply with 10 to move the comma until just before the first number 7. 10 x 25,787878 Step 3: This step involves subtracting the two equations obtained in steps 1 and 2 from each other in order to eliminate the decimals. 1000 x 2578,787878 10 x 25,787878 990 x 2553,0 Step 4: Divide by 990 on either side of the last equation to obtain the fractional answer. So 2,578 2553 . 990 4. Rounding off Moet dit nog doen. 5. Percentages Percentages form part of our everyday lives. They are often quoted in the media, particularly in connection with money matters: ‘Inflation now stands at 7,5%.’ ‘Ford has given its workforce a 24,2% raise.’ ‘Unemployment in South Africa is about 36%.’ 26% of children watch 15 hours or more television per week. 6 In the sections on fractions and decimals, we learned some very useful ways to express parts of quantities and to compare quantities. In this section we now standardise our comparison of parts of quantities. We will do this by expressing quantities in relation to a standard unit of 100. We use this relationship, called percent, to solve many different types of problems. The word percent means “per hundred” or “for every hundred”. So, if you pay 14 percent VAT on an item, it means that for every R100 you will pay R14 VAT. The symbol % represents a “percent”. Definition of a Percent A percent is a fractional part of a hundred expressed by a percent sign (%). A hundred percent (100%) represents one whole quantity, or one hundred out 100 . 100 of a hundred parts, Using the label “percent” is an easy way to express the relationship of any quantity to 100. When we want to make a comparison or state a problem, the term percent may save time in performing a calculation. But we can't use percentages as such when solving a problem. First of all, we need to convert the percentage to a fraction or a decimal. To change a percentage such as 45% to a numerical equivalent, we need to divide by 100. Then, for every real number a , a% is numerically equal to a 100 or a 45 . So 45% . 100 100 Percentages often help us make comparisons between numbers. Example Suppose a student obtained the following marks in a series of tests: Mathematics English Life Skills 16 25 31 50 30 40 It is difficult to compare the results that the student obtained in each subject, as the tests were not out of the same mark. The student obtained the most marks in English, but was this the best result? In order to compare the marks, the fractions should all be converted to percentages, which will give us results expressed as fractions with the same denominator, namely 100. We go about doing this as follows: 7 16 4 64 64% 25 4 100 31 2 62 62% 50 2 100 30 2,5 75 75% 40 2,5 100 The test results are therefore 64% in Mathematics, 62% in English and 75% in Life Skills. The best result was obtained in Life Skills. To convert a fraction or decimal to a percentage, it is easiest to just multiply with 100. Examples Expressing a decimal as a percentage 0,59 100 59% 1. 1,45 100 145% 2. Expressing a fraction as a percentage 3. 4. 5. 34 100 34% 100 1 103 100 103% 100 1 19 100 1900 52,78% 36 1 36 Expressing a percentage as a decimal and a fraction 6. 5.1 30% 0,30 30 100 Increasing an Amount In real life it is often indicated that products will rise in price with a certain percentage. We could hear on the news that bread, for example, will increase in price with 3% as from next week. What does such a percentage increase mean in cash terms? Example If bread currently costs R 7,50 and increases in price with 3%, how much will it cost after the raise? There are two ways in which we can go about determining the new price of bread. Firstly, we can calculate 3% of R 7,50 and then add it to the original price: 3% of R 7,50 3 R 7,50 100 23c 8 New price R7,50 R0,23 R7,73 Secondly, we can consider the original amount of R 7,50 as 100% of the price. Increasing it by 3% is the same as calculating 103% of the original cost. New cost 103% R7,50 103 R 7,50 100 R 7,73 5.2 Decreasing an Amount Just as prices of items that we buy everyday may rise occasionally, they do sometimes become cheaper. The amount of money that we pay for those items then decreases. Example The price of a movie ticket is R 23,50 . What would the price be after the discount as indicated on the banner is applied? There are two ways in which we can go about determining the new price of a movie ticket. Firstly, we can calculate 5% of R 23,50 and then subtract it from the original price: 5 R 23,50 100 R1,18 R 23,50 R1,18 5% of R 23,50 New price 5% off on all movie tickets R 22,32 Secondly, we can consider the original amount of R 23,50 as 100% of the price. Decreasing it by 5% is the same as calculating 95% of the original price. New cost 95% R 23,50 95 R 23,50 100 R 22,32 6. Ratio and Proportion 6.1 Ratio Definition of Ratio A ratio compares two or more quantities of the same kind, expressed in the same unit. 9 Example At a BMW dealership there are 24 blue BMWs for sale and 15 silver ones. The ratio of blue to silver BMWs can be written as 24 : 15 . Note that the unit in this example is simply “cars” and is not included or indicated in the ratio. A ratio can be simplified by dividing each side of the ratio by the same number. Example Cost price : Selling price 250 : 300 (No units are indicated, but have to be the same - rands on either side) (Divide both sides by 25) 10 : 12 (Divide both sides again – now by 2 – to 5:6 obtained the simplified form) Quantities can be divided in a certain ratio. Example Mr Jones, Mrs Black and Dr Edwards invested in an investment opportunity through ABSA bank and made a total profit of R1 048 238. Mr Jones invested R180 000, Mrs Black invested R315 000 and Dr Edwards invested R135 000. Divide the profit among the investors in the same ratio as their investments. First we need to determine the ratio between the amounts that they invested. Jones : Black : Edwards = 180 000 : 315 000 : 135 000 = 4 : 7 : 3 We get this simplified ratio by dividing each amount by 45 000. The sum of the parts is 4 + 7 + 3 =14. 4 R1 048 238 = R299 496,57 14 7 R1 048 238 = R524 119,00 Mrs Black receives 14 3 R1 048 238 = R224 622,43 Mr Edwards receives 14 Therefore: Mr Jones receives 6.2 Proportion Definition of Direct Proportion Two quantities are in direct proportion if the one quantity increases (decreases) as the other quantity increases (decreases). 10 These two quantities, which we can represent by x and y , are in a functional relationship that can be indicated by: 1. an equation in the form y mx such as y 4 x . Then: y x -2 -8 -1 -4 Here we can clearly see how the y values increase as the 0 0 x values increase. 1 4 2 8 3 12 11 2. a straight line graph going through the origin (0, 0). y 12 2 2 28 2 2 2 4 2 2 21 2 2 2 12 2 -2 -1 1 2 3 x -4 -8 This straight line graph is representative of the equation y 4x . Example 4 kg of potatoes cost R12,50 . How much does 11 kg cost? If 4 kg cost R12,50 , then 4 12,50 R3,13 . 1 kg costs 4 4 Therefore 11 kg of potatoes will cost 11 R3,13 R34,43 Definition of Inverse Proportion Two quantities are in inverse proportion if the one quantity increases as the other quantity decreases. These two quantities, which we can represent by x and y , are also in a functional relationship to each other and can be indicated by: 12 k x 1. an equation in the form xy k or y If y 3 , then we have that x y 3 4 3 1 3 3 Here we can clearly see how the y values decrease 2 3 3 3 from to 3 as the x values increase from 4 to 1 . 1 4 3 Division by 0 is not defined. Undefined 0 3 3 1 3 3 It is also clear that the y values decrease from 3 to 2 4 3 as the x values increase from 1 to 4. 1 3 3 4 x -4 -3 -2 -1 0 1 2 3 4 or 13 2. a graph of a hyperbola with center at (0, 0). y 3 2 2 22 4 2 21 2 2 2 2 12 2 -4 -3 -2 -1 1 2 3 4 x -1 11 4 -2 -3 This hyperbola is representative of the equation y 3 . x Example 9 workers are asked to paint the inside walls of an apartment building and they are given 2 weeks to do it in. How many days will it take to paint the building if 15 workers are doing the job? This is an inverse proportion problem as the number of days that it would take to paint the building will surely be less if more painters are employed to do the job. We will therefore use the equation xy k and first determine k . The values of x and y are: x 9 workers and y 14 days (two weeks) k xy 9 14 126 If the number of workers (x ) increases to 15, we then have that y k x 126 15 8,4 days to paint the inside walls of the apartment building. 14 7. Sigma Notation Summation or sigma notation is basically shorthand for writing sums. The sums that can be written in this way are special. The terms that make up such a sum have to belong to a generated sequence and are written in a specific order. The sign used in sigma notation is derived from the Greek letter , which corresponds to our S for “sum”. If we want to add the terms a1 , a2 , a3 ,, an , sigma notation can be used to n express the sum as a k 1 k a1 a 2 a3 a n . The letter k is called the index of summation and a k is the general term of the sum. The idea is to substitute the values of k into the general term to obtain the sum on the right-hand side of the equation. The index of summation indicates that the first value we should substitute is k 1 and the last value is k n. Example To find the first three terms, the 20th term and the 598th term of the sequence defined by the general term ak 2k 15 , we have to substitute 1 , 2 , 3 , 20 and 598 into the formula representing the general term. Then we get that the first term a1 2(1) 15 13 , the second term a2 2(2) 15 11 , the third term a3 2(3) 15 9 , the twentieth term a20 2(20) 15 25 and the five hundred and ninety eight term a598 2(598) 15 1181 . If the sum of a number of terms is expressed in sigma notation, we can determine the sum as demonstrated in the following three examples. Example 7 Determine the sum k 2 . k 1 7 k 2 12 2 2 3 2 4 2 5 2 6 2 7 2 k 1 1 4 9 16 25 36 49 140 Example 5 Determine the sum 3 k k 1 2 . 15 5 3 k k 1 2 3 3 3 3 3 2 2 2 2 2 1 2 3 4 5 3 1 3 3 3 4 3 16 25 3600 900 400 225 144 1200 1200 1200 1200 1200 5269 1200 Example 6 Determine the sum 5 . k 1 6 5 5 5 5 5 5 5 k 1 5 6 30 This summation means that the value 5 must be added 6 times. To determine how many terms are in a given summation, subtract the bottom 6 index from the top index and add 1 . In the summation 5 the number of k 1 terms will therefore be N 6 1 1 6 terms. In the summation 12 4t the t 3 number of terms will be N 12 3 1 10 . As suggested in the previous sentence, the index of a summation does not have to start at k 1 . It is quite possible to determine the sum of 12 4t , as t 3 illustrated in the next example. Example 12 Determine the sum 4t . t 3 12 4t t 3 4(3) 4(4) 4(5) 4(6) 4(7) 4(8) 4(9) 4(10) 4(11) 4(12) 12 16 20 24 28 32 36 40 44 48 300 Just as you should be able to determine the value of a summation written in sigma notation, you should be able to express a sequence of terms using sigma notation. 16 Example Express 12 2 2 32 4 2 in sigma notation. Then 4 k 2 12 2 2 3 2 4 2 . k 1 Notice that it is not compulsory to use the letter k to represent the index. Any alphabet letter can be used. A certain preference is given to the letters k and n , however. The previous example can thus also be written as 4 n 2 12 2 2 3 2 4 2 , without changing the meaning of the summation or the n 1 value of the sum. Example Write the sum 5 7 9 21 using sigma notation. First we need to determine the general term a k , that represents all of the terms in the sum. It seems as if it is ak 2k 1 , because all the values under the root signs are odd values. Next we need to determine the starting and ending points of the index of summation, k . The starting point has to be k 3 , as 2k 1 2(3) 1 5 . The ending point will be k 11 , because 2k 1 2(11) 1 21 . 11 Then 5 7 9 21 2k 1 . k 3 But there is no unique way to write a sum in sigma notation. The answer we obtained in the previous example is just one way of writing the sum 5 7 9 21 in sigma notation. It all depends on the choice of index, which in turn will influence how the general term is expressed. If we choose to let k start at 4 instead of 3 , we obtain the following notation: 12 5 7 9 21 2k 3 k 4 The index of summation now runs from 4 to 12 and the general term is expressed by ak 2k 3 . The terms obtained by these indices and general 11 term are, however, exactly the same as those represented by 2k 1 . k 3 8. Exponents In this section we recall information regarding integer exponents a m , radicals x and n th roots n a in order to give meaning to an exponent in which the m power is a fraction a n . 17 8.1 Integer Exponents Instead of writing out a product of identical numbers as 3 3 3 3 3 , it can be written in exponential form, as 35 . Definition of an Exponent If a is any real number and n is a positive integer, then the n th power of a is a n a a a a a a . n times The number a is called the base and n the exponent. Power an Exponent Base In order to work with exponents, we need to know the laws according to which they operate. 8.1.1 Laws of Exponents 1. a m a n a mn To multiply powers with the same bases, keep the base and add the exponents; then a 4 a 6 a 46 a10 . Proof: If a is any real number and m and n positive integers, then a m a n (a a a)( a a a) a a a a a a m n m factors n factors m n factors 2. am a mn n a To divide powers with the same bases, keep the base and subtract the exponent in the denominator from the exponent in the numerator; then a6 a 6 4 a 2 . 4 a Proof: If a is any real number and m and n positive integers, then factors m a m a a a a a a a a a mn n a a a a a m n factors n factors 3. a m n a mn 18 To raise a power to a new power, keep the base and multiply the 3 exponents with each other; then a 6 a 63 a18 . 4. ab n a nb n To raise a product to a power, raise each factor of the product to the power; then ab 4 a 4 b 4 . n 5. an a n b b To raise a quotient to a power, raise both the numerator and the 6 a6 a denominator to the power; then 6 . b b 6. a b n b a n To raise a fraction to a negative power, invert the fraction and change a the sign of the exponent; then b 7. 2 2 b . a a n b m b m a n When moving a power from the numerator to the denominator or vice versa, change the sign of the exponent; then a n b m . b m a n There are two more important definitions that follow from the Laws of Exponents, namely: 8. If a 0 is any real number and n is a positive integer, then a 0 1 . Proof: a 0 a 4 a 0 4 a 4 . But this is only possible if a 0 1 . 9. If a 0 is any real number and n is a positive integer, then a n Proof: a n a n a n ( n ) a 0 1 . But this is only possible if a n 1 . an 1 . an Examples Eliminate negative exponents and simplify as far as possible: x 4 x 7 x 4 7 x11 1. 19 2. x 3 x 9 x 3 ( 9 ) x 6 3. y 12 y 125 y 7 5 y 4. 5. c 2 7 5 x 2 1 x6 c 27 c14 5 2 x 2 25 x 2 3 6. 7. 8. h3 h3 h 3 27 3 3 x 5 1 2 5 2 y y x 4a c 7ac 4 a c 7ac 16a c 7ac 9 3 2 6 9 2 2 3 2 18 6 6 6 (16)(7)a18 ac 6 c 6 112a 19 c12 9. 4 s 2 t 3 k 6 12t 4 sk 4 1s 2 sk 3t 4 t 3 k 6 4 s2s4k 4 3t 7 k 6 s6 7 2 3t k 10. x y 3 y 2 x3 z 2 4 5 y 3 y 8 x12 5 x z 2 3 5 y y 8 x12 z 2 x y3 3 y 40 x 60 z 10 x y 3 y 40 x 3 x 60 z 10 y 43 x 57 z 10 8.2 Radicals (Roots) Ek moet dit nog doen. 20 8.3 Rational Exponents Ek moet dit nog doen. 9. Logarithms The study of logarithms is motivated by the need to solve equations such as bx a for x. To be able to express x in terms of a and b in this setting requires a manipulation not yet presented. In fact, a new notation must be introduced in order to solve this problem. In view of this, we define x log b a to mean the same as b x a . Logarithm Notation x log b a means the same as b x a where b 0, b 1, and x is any real number. The equation x log b a is read as “x is the logarithm of a to the base b” or as “x is the logarithm to the base b of a.” Notice that the logarithm is equal to x, and x is the exponent. Thus, a logarithm is an exponent. In this case, x is the exponent to which b must be raised to produce a. The number a results from raising the positive base b to some power x. Consequently, a is always positive in b x a . In turn, this means that a must be positive in x log b a . In other words, log b a is defined only for a 0 . The logarithm of a negative number (or zero) is not defined. Here are some examples of equations written in logarithmic form along with their equivalent exponential form. Logarithmic equation Exponential equation log 3 9 2 log 2 8 3 log 4 1 0 log 10 0,01 2 log e 1 0 log 10 ( 2) undefined 32 9 23 8 40 1 102 0,01 e0 1 10 ? 2 ? there is no possible answer 21 The two most popular bases for logarithms are 10 and e. Logarithms using base 10 are called common logarithms, and “ log 10 ” is written simply “log”. When no base is written, base 10 is assumed. log 10 x is written log x Example Solve the following: (a) log 2 64 y 1 y 243 1 (c) log 49 x 2 (b) log 3 Solutions (a) 2 y 64 y6 (b) 1 3 5 5 3 y 5 3y (c) x 49 x 1 2 1 1 49 7 Assessment Activity 1.1 Solve the following: 1. y log 9 27 2. log b 8 3 4 2 3 4. y log 27 3 1 7 5. log b 128 1 6. log 1 x 4 16 3. log 8 x 22 9.1 The Laws of Logarithms From our knowledge of logarithms we have log 2 8 log 2 16 log 2 2 3 log 2 2 4 3 4 7 log 2 128 log 2 8(16) Thus log 2 8(16) log 2 8 log 2 16 . It turns out that this equation is a special case of the first law of logarithms. Laws of Logarithms If M and N are positive, b 0, and b 1, then (i) log b MN log b M log b N M log b M log b N N (iii) log b ( N k ) k log b N (ii) log b (iv) log b (b x ) x (v) b log x x In addition, log 1 0 because 10 0 1 OR: log b 1 0 because b 0 1 b Example Use the properties of logarithms to expand each expression. x2 x 5 (b) log 32 x (a) log (c) log[ e 3 ( x)] (d) log b 3 2x Solution (a) Use property (ii) log x2 log( x 2) log( x 5) x 5 (b) Use property (iii) log 32 x 2 x log 3 (c) Use property (i) log[ e 3 ( x)] log e 3 log x 3 log e log x (d) Use property (i) and (ii) log b 3 log b 2 x 23 log b 3 log b 2 log b x log b 3 log b 2 log b x Example Use the properties of logarithms to simplify each of the following expressions: (a) log 3 log x (b) log x log 2 1 2 (c) 2 log b ( x 1) log b x Solution (a) Use property (I) log[ 3( x)] (b) Use property (ii) log x 2 (c) Use property (iii) log b ( x 1) log b x 2 1 2 Use property (i) 12 log b [( x 1) x ] 2 Assessment Activity 1.1 Use the properties of logarithms to simplify each of the following expressions: 1. log b x log b x log b 3 2. log x log( x 1) 2 log( x 2) 3. log 4 log 10 2 4. log A 2 log A 1 log B log B 2 5. 2 log A 3 log B log A 2 Learning activity 2.1 [or Portfolio Activity 2.1] 24 Dui vir my aan waar jy dink ek ‘n Learning Activity (oefening) kan invoeg. Group Activity 1.1 Dui vir my aan waar jy dink ek ‘n Group Activity (oefining) kan invoeg. 10. End of section comments This wraps up all the content that you had to acquire before being able to proceed to the next sections. Make sure that you understand and can confidently work with everything that was addressed in this section. We will now proceed to the next section which is on calculator usage. 11. Feedback Hierdie sal ek invoeg as ek die Learning en Group Activities klaar geskryf het. 12. Tracking my progress You have reached the end of this section. Check whether you have achieved the learning outcomes for this section. Learning outcomes I feel confident I still need more practice Multiply and divide fractions Add or subtract fractions with the same or different denominators Convert a decimal fraction to a fraction with a numerator and denominator Convert decimals and fractions to percentages and vice versa Do simple increasing and decreasing problem solving by using percentages Compare information in terms of ratios Do simple problem solving by using 25 direct and inverse proportion Write sums in sigma notation Determine sums written in sigma notation Know and be able to apply all the Laws of Exponents to simplify expressions What did you like best about this section? What did you find most difficult in this section? What do you need to improve on? How will you do this? 26