is canceling incorrectly. Learners must first factorise and then cancel:

TIPS FOR TEACHERS Lesson 1 It is important to first revise factorisation of algebraic expressions with your learners before tackling this lesson. Make sure that your learners do not cancel incorrectly. _ x 2 + x is canceling incorrectly. Learners must first factorise and then cancel: x 2 1 x(x + 1) (x + 1)(x 1) = x_ x 1 Encourage your learners to make use of the very important results discussed in this lesson. Emphasises the difference between simplifying an expression and solving an equation. Be careful of incorrect use of equal sign with equations. Learners tend to misuse the equal sign as follows: 2 = 5_ x 1 = 2(x 1) = 5 = 2x 2 = 5 Rather encourage them to use the therefore sign when solving equations: 2 = 5_ x 1 2(x 1) = 5 2x 2 = 5 Make sure that learners understand the concept of taking out a negative when factorising an expression: 3x + 12 = 3(x 4) but not 3(x + 4) 3x 12 = 3(x + 4) but not 3(x 4) Lesson 2 When we complete the square we write a quadratic expression in the form a(x p) 2 + q If a < 0 then q is the maximum value of the expression at x = p. If a < 0 and q is less than 0, the expression will always be negative. If a > 0 then q is the minimum value of the expression when x = p. If a > 0 and q is greater than 0, then the expression will always be positive. Lesson 3 Examples should include simple concepts to test understanding and not complicated algebraic manipulation. Encourage learners to memorize powers of 2 up to 2 6, powers of 3 to 3 4 and powers of 5 to 5 3. Stress to them never to divide across + and signs but to factorise. The investigations can be used for portfolio exercises and marked by the rubric at the end of the lesson. 377 LC Maths G11 WB.indb 377 2008/08/09 08:29:25 AM

Rubric for investigations Presentation of work Investigation1 Mathematics Constructions Investigation2 Mathematics Messy and unable to read. Did not understand what mathematics to use. Did not know what to do. Did not see the sequence as the mathematics is incorrect. Very little logical sequence. Only able to perform mathematical operations with help and mistakes were made. Needed help with constructions and not completely accurate. Neat and tidy but not enough. 2 2 2 Needed help with Pythagoras and only one length investigated (no nth term). 2 Correct mathematics with help from the teacher. Constructions correct but help was needed. Investigated different lengths but could not get the nth term. Untidy but correct. Well investigated with few errors. Tidy work and correct. Work done on their own but some concepts not included. Investigated different lengths and obtained the nth term for each sequence. Everything correct. Everything correct. Everything correct and a general term found for any length. TOTAL: 20 Lesson 4 Worksheet 2 can be used for portfolio exercises. Make sure that the learners make the quadratic equation equal to zero. It is a good idea to note the restrictions when they write down the LCD. Yet they should still be encouraged to check solutions. Learners should know the difference between an identity and an equation. When they need to prove a conjecture they must make the left-hand side equal to the right-hand side. Lesson 5 It is a good idea to put an asterisk next to the surd before the learners square so that they remember to check. Drawing the graph by testing points helps the learners to understand the restrictions. Lesson 6 378 Stress to the learners that solving equations by completing the square is like a theorem and should only be used when they are told to. LC Maths G11 WB.indb 378 2008/08/09 08:29:26 AM

They could learn this algorithm in step form where dividing by the coefficient if x squared is golden rule number one. Always make them aware of when there are solutions and when they actually could have factorised (rational solutions). Lesson 7 Alert the learners to read the question carefully and decide how they must produce the answer (decimal place or surd form or, in fact, simple surd form). The nature of roots is no longer in the curriculum but I do think it is still important for the learners to analyse the quadratic formula and decide how they will solve the equation. Lesson 8 Encourage learners to look at the second line and if it is too complicated, create a k substitute. They need to remember to substitute back. They must check b 2 4ac to see if they can factorise. Lesson 10 Allow learners to go straight to the number line and put the critical values on the number line. Otherwise they often make mistakes with the inequality signs. Inequalities with rational expressions (variables in the denominator) are no longer on the syllabus, so if you do them, it will only be for extension. Lesson 11 The routine-type questions such as speed, distance and time are important to help learners formulate equations, but I think it is important that they learn to cope with non-routine questions. Encourage the learners to draw pictures and make tables. Remember the best way to make equations is : Big minus small equals the difference. Lesson 12 13 It is important to teach this section slowly because the learners battle with the signs. Encourage the learners to use their calculators to check answers. Let the learners experiment with the calculator. Lesson 14 When the learners have to prove the left-hand side is the same as the righthand side, it is a good idea to draw a line down the middle so that they do not treat the identity as an equation. The only time they need to change 1 into sin 2 α + cos 2 α is to create factors. Encourage the learners to work inside the brackets. 379 LC Maths G11 WB.indb 379 2008/08/09 08:29:26 AM

Lesson 15 Make sure the learners use brackets after they have reduced the ratios otherwise they get confused with addition and multiplication. When they are doing multiplication encourage them to look for factors to cancel. Even though they can always find the trig ratios of angles on their calculators, it is good training and enhances understanding to reduce. Lesson 16 18 One of the reasons why we no longer teach sec θ cosec θ and cot θ is because the equations were too complicated, so I suggest keeping the factors as simple as possible. Learners can work in pairs and make up equations for their partners to solve. Lesson 19 20 Let the learners work together in this section. They learn to use their calculators and at the same time handle the sequences. Lesson 21 This whole section is best done as portfolio work. Learners could be asked to do projects and research on the Golden Ratio, Pascal s triangle and various sequences of numbers. Lesson 22 This section of work should certainly be done before volume and area because many of these concepts are needed The learners do need to know theorems. Emphasis will be placed on actual angles and not on variables. Encourage learners to reproduce diagrams and sometimes draw their own diagrams. Lesson 23 This section of work should certainly be done before volume and area because many of these concepts are needed The learners do need to know theorems. Emphasis will be placed on actual angles and not variables. Encourage learners to reproduce diagrams and sometimes draw their own diagrams. 380 Lesson 24 Emphasis must be placed on real life type problems. LC Maths G11 WB.indb 380 2008/08/09 08:29:27 AM

This section can be used for portfolio type work learners can be asked to use trigonometry to find heights and angles of objects in their daily lives such as trees, rugby posts, etc. Lesson 25 The best way to explain the more difficult three dimensional shapes is to make the learners construct models. Group work is fun to use in this section. Lesson 26 The best way to explain the more difficult three dimensional shapes is to make the learners construct models. Group work is fun to use in this section. Lesson 27 Inclination and straight line equations are meant to be done in Grade 11, but it is necessary to revise the formulae done in Grade 10. This Lesson basically covers the Grade 10 work. Lesson 28 Learners work nicely in pairs in this section. Always encourage the learners to draw diagrams. Analytical Geometry is about making equations. Learners should not copy out the formula from the formula sheet, they must substitute directly into the formula. Lesson 29 Learners work nicely in pairs in this section. Always encourage the learners to draw diagrams. Analytical Geometry is about making equations. Learners should not copy out the formula from the formula sheet, they must substitute directly into the formula. Lesson 30 Learners work nicely in pairs in this section. Always encourage the learners to draw diagrams. Analytical Geometry is about making equations. Learners should not copy out the formula from the formula sheet, they must substitute directly into the formula. 381 LC Maths G11 WB.indb 381 2008/08/09 08:29:27 AM

Lesson 31 This section will be examined in Paper 3 and is not compulsory. However, it is important that as many children as possible write this paper. When doing geometry it is a good idea to let the learners work together in groups. Geometry is a discipline, so the setting out of the work is very important. Lesson 32 This section will be examined in Paper 3 and is not compulsory. However, it is important that as many children as possible write this paper. When doing geometry it is a good idea to let the learners work together in groups. Geometry is a discipline, so the setting out of the work is very important. Lesson 33 This section will be examined in Paper 3 and is not compulsory. However, it is important that as many learners as possible write this paper. When doing geometry, it is a good idea to let the learners work together in groups. Geometry is a discipline so the setting out of the work is very important. Lesson 35 Graphs are a huge part of the curriculum so we need to do them slowly and carefully. It is a good idea to put the learners in pairs and let them give each other graphs to draw. Lesson 36 This is a good section to do in pairs or in groups. The emphasis must be on translating graphs. Lesson 37 Graphs are a huge part of the curriculum so we need to do them slowly and carefully. It is a good idea to put the learners in pairs and let them give each other graphs to draw. 382 Lesson 38 Graphs are a huge part of the curriculum so we need to do them slowly and carefully. LC Maths G11 WB.indb 382 2008/08/09 08:29:28 AM

It is a good idea to put the learners in pairs and let them give each other graphs to draw. Lesson 39 This is an optional section of the curriculum at present but will become an important section in the future. It is important to start teaching this section. Most countries in the world study probability and it is a very important section needed to prepare learners for tertiary education. Lesson 40 This is an optional section of the curriculum at present but will become an important section in the future. It is important to start teaching this section. Most countries in the world study probability and it is a very important section needed to prepare learners for tertiary education. Lesson 41 Learners need to understand the difference between simple and compound interest as well as the difference between linear and reducing balance depreciation. It is extremely important for learners to know the formulae for simple and compound interest as well as the two formulae for depreciation, even though these formulae may be provided in a formula sheet in the final examinations. It is vital for teachers to link the concepts of linear and exponential functions to these formulae. Learners must be able to efficiently use their calculators when doing the calculations. The use of the CASIO fx-82es calculator is recommended. The use of time lines (see first example In this lesson) is highly recommended, especially when learners are required to answer more complicated questions (as will be seen in the next Lessons). In questions requiring the calculation of the interest rate, learners must ensure that they determine the interest rate as a percentage. Leaving the answer as a decimal is not sufficient. Lesson 42 Learners should understand that the nominal rate (taking different compounding periods into account), and the effective annual rate will yield the same accumulated amount at the end of the investment period. You could demonstrate this by using the idea of two people investing the same amount of money for one year and receiving the same accumulated amount at the end of that year. Person A invests money and the interest is calculated, say, every month. Person B invests money and interest is calculated at the end of the year. The yearly effective rate is clearly greater than the nominal rate, but the accumulated amounts will be the same for both people. However, the quoted rate without compounding cannot yield the same accumulated amount. For example, suppose that James and John each invest R2 000 for one year at 18% per annum. 383 LC Maths G11 WB.indb 383 2008/08/09 08:29:28 AM

The interest rate for James is 18% per annum compounded monthly (nominal). The interest rate for John is the effective annual rate of 19,195618171%. They will both receive the same accumulated amount at the end of the year, even though the interest rates differ. James: A = 2000 ( 1 + _ 0,18 12 ) 12 = R2391,24 (Interest is calculated at the end of each month) John: A = 2000(1 + 0,195618171) 1 = R2391,24 (Interest is calculated at the end of the year) You use the formula 1 + i eff = ( 1 + _ i n nom n ) to convert 18% per annum compounded monthly to the annual effective rate of 19,6%. When converting from the nominal to the effective annual rate using the formula, the calculation is always done for one year, even though the question might be asking for accumulated amounts over more than one year (see example 3). Lesson 43 Learners should be encouraged to use the methods as illustrated in the summary example rather than to remain with the long methods. This will save a lot of time and produce more accurate answers. The effective annual rates should not be rounded off when using them in calculations. Learners should not round off interest rate calculations such as _ 0,13 12 = 0,0108 3. It is more effective to leave the interest rate as _ 0,13 in the calculation. 12 Lesson 44 For more manageable calculations, it is highly recommended that educators encourage learners to use the recommended methods in this lesson. Lesson 45 It is essential for learners to investigate and discover the transformation rules for themselves. Learners must be able to draw images under different transformation rules as well as be able to describe the rule when a given figure is transformed into an image. The use of the (x; y) notation is essential for learners to become familiar with. 384 Lesson 46 It is essential for learners to investigate and discover the transformation rules for themselves. LC Maths G11 WB.indb 384 2008/08/09 08:29:28 AM

Learners must be able to draw images under different transformation rules as well as be able to describe the rule when a given figure is transformed into an image. The use of the (x; y) notation is essential for learners to become familiar with. Lesson 47 It is extremely important for learners to understand how to locate the quartiles in data with odd and even numbers of values. Quartiles can form part of the data set or stand outside the data as averages between two data values. Comparing Box and Whisker plots are important for exam purposes. Lesson 48 It is extremely important for learners to understand the difference when working with ungrouped and grouped data. With ungrouped data, the intervals on the horizontal axis will contain values which are the same. To illustrate this, consider the first example. The values in the interval 24 < x 25 are all 25. With grouped data, the values in these horizontal axis intervals are not the same. Refer to the second example and you will find that marks in the interval 9 < x 19 vary from 10 to 19. This means that for ungrouped data, you read off the value in the interval exactly, whereas with grouped data, you have to read off a value approximately by estimating. There are sophisticated methods of estimating the quartiles more accurately. However, these methods are beyond the scope of the Grade 11 syllabus. Lesson 49 Learners need to be encouraged to master the calculator technique of calculating the standard deviation for given data. The long method is cumbersome and should be discouraged with large sets of data. However, the principle can be examined but with a very small set of data values. Lesson 50 In Grade 11, learners are expected to intuitively draw a line of best fit. Therefore any feasible line of best fit will be accepted. In Grade 12, they use regression to find the true line of best fit. Correlation is explored in more depth in the Grade 12 year. 385 LC Maths G11 WB.indb 385 2008/08/09 08:29:29 AM