Curriculum



PROVINCE OF THE

EASTERN CAPE

EDUCATION

DIRECTORATE: CURRICULUM FET PROGRAMMES

LESSON PLANS

TERM 2

MATHEMATICS

FOREWORD

The following Grade 10, 11 and 12 Lesson Plans were developed by Subject Advisors from 09 March – 13 March 2009. Teachers are requested to look at them, modify them where necessary to suit their contexts and resources. It must be remembered that Lesson Plans are working documents, and any comments to improve the lesson plans in this document will be appreciated. Teachers are urged to use this document with the following departmental policy documents: Subject Statement; LPG 2008; SAG 2008; Examination Guidelines 2009 and Provincial CASS Policy/ Guidelines.

Lesson planning is the duty of each and every individual teacher but it helps when teachers sometimes plan together as a group. This interaction not only helps teachers to understand how to apply the Learning Outcomes (LOs) and Assessment Standards (ASs) but also builds up the confidence of the teachers in handling the content using new teaching strategies.

It must please be noted that in order to help teachers who teach across grades and subjects, an attempt has been made to standardise lesson plan templates and thus the new template might not resemble the templates used in each subject during the NCS training. However, all the essential elements of a lesson plan have been retained. This change has been made to assist teachers and lighten their administrative load.

Please note that these lesson plans are to be used only as a guide to complete the requirements of the Curriculum Statements and the work schedules and teachers are encouraged to develop their own learner activities to supplement and/or substitute some of the activities given here (depending on the school environment, number and type of learners in your class, the resources available to your learners, etc).

Do not forget to build in the tasks for the Programme of Assessment into your Lesson Plans.

Strengthen your efforts by supporting each other in clusters and share ideas. Good Luck with your endeavors to improve Teaching, Learning and Assessment.

TERM 2

LESSON PLAN: 1 FOR TERM 2

|Subject: MATHEMATICS Grade 12 |

|Lesson Plan: Factorization - Third degree polynomials Number of Activities 3 |

|Duration: 4h30 Week 11 Date |

|Context: Mathematical : Factorization of 3rd degree polynomials |

|Link with previous lesson: Products of binomials, factorization of quadratic equations, solving quadratic equations |

|KNOWLEDGE (K): Factorization of third degree polynomials SKILLS (S): Factorize, identify and solve 3rd degree polynomials, application, and calculation VALUES (V): Appreciation and sharing ideas. |

|Learning Outcome 1: |Learning Outcome 2: |Learning Outcome 3: Space, Shape and |Learning Outcome 4: Data Handling and |

|Number and Number Relationships |Functions and Algebra |Measurement The learner is able to describe, |Probability |

|When solving problems, the learner is able to recognise, |The learner is able to investigate, analyse,describe and |represent, analyse and explain properties of |The learner is able to collect, organise, |

|describe, represent and work confidently with numbers and |represent a wide range of functions and solve related problems.|shapes in 2-dimensional and 3-dimensional |analyse and interpret data to establish |

|their relationships to estimate, calculate and check | |space with justification. |statistical and probability models to solve |

|solutions. | | |related problems. |

|12.1.2Demonstrate an | |12.2.1(a)Demonstrate the | |12.3.3Use a two dimensional Cartesian | | |

|understanding of the definition | |ability to work with | |co-ordinate system to derive and | | |

|of a logarithm and any laws | |various types of functions| |apply:the equation of a circle (any | | |

|needed to solve real-life | |and relations including | |centre);the equation of a tangent to a | | |

|problems | |the inverses listed in the| |circle given a point on the circle. | | |

| | |following Assessment | | | | |

| | |Standard.(b)Demonstrate | | | | |

| | |knowledge of the formal | | | | |

| | |definition of a function | | | | |

|Activity 1 |Educator give l earners |Learners are given |Discussion, question and |Class work home work |Work sheets, calculator | |

|Factors, products and equations |worksheets on the revision of |worksheets on the revision|answer |Memo | | |

| |previous grades work.The |of previous grades work. | |Educator, individual | | |

| |following work need to be |The following work need to| | | | |

| |revised: |be revised: | | | | |

| |What is a polynomial? |What is a polynomial? | | | | |

| |Products of binomials and |Products of binomials and | | | | |

| |trinomials. |trinomials. | | | | |

| |The factorization of quadratic |The factorization of | | | | |

| |equations |quadratic equations | | | | |

| |Solving quadratic equations |Solving quadratic | | | | |

| |Structuring a quadratic equation|equations | | | | |

| |if roots are known. |Structuring a quadratic | | | | |

| | |equation if roots are | | | | |

| | |known. | | | | |

|Activity 2 |Teacher demonstrates and give |Learners given worksheets|Discussion, question and |Class work home work |Work sheets, calculator | |

|Factor Theorem |learners worksheets on examples |on examples of different |answer |Memo | | |

| |of different methods to |methods to factorize 3rd | |Educator, individual | | |

| |factorize 3rd degree |degree polynomials. | | | | |

| |polynomials. |Taking out a common factor| | | | |

| |Taking out a common factor |Two special types of | | | | |

| |Two special types of |factorization for cubes. | | | | |

| |factorization for cubes. |Grouping | | | | |

| |Grouping |The factor theorem | | | | |

| |The factor theorem |e.g. Factorize f(x)= 3x3 | | | | |

| |e.g. Factorize f(x)= 3x3 + x2 |+ x2 -12x -4 | | | | |

| |-12x -4 | | | | | |

|Activity 3 |Educator demonstrate to learners|Learners need to apply |Discussion, question and |Class work home work |Work sheets, calculator | |

|Solve 3rd degree equations |the need to apply their |their knowledge to solve |answer |Memo | | |

| |knowledge to solve third degree |third degree equations. | |Educator, individual | | |

| |equations. |e.g. Solve the following | | | | |

| |e.g. Solve the following |equations | | | | |

| |equations |x3 -2x2 -31x +21 =0 | | | | |

| |x3 -2x2 -31x +21 =0 |Learners are given mixed | | | | |

| | |equation to solve. | | | | |

|Expanded Opportunities Different examples and remedial work Use of different equations |

LESSON PLAN: 2 for Term 2

|Subject: Mathematics Grade 12 Lesson |

|Plan: Calculus Number of Activities 3 Duration: 13H 30 Min |

|Week 13-15 / Date |

|Context: Mathematical: real life situations |

|Link with previous lesson: Average rate of change ,Instantaneous rate of change and Limits ,Gradient of a curve at a point on a curve and First principles |

|KNOWLEDGE (K): Rules of differentiation Equations of tangents to graphs. Sketch graphs of cubic functions using differentiation SKILLS (S): Solve problems |

|Learning Outcome 1: |Learning Outcome 2: |Learning Outcome 3: Space, Shape and Measurement|Learning Outcome 4: Data |

|Number and Number Relationships |Functions and Algebra |The learner is able to describe, represent, |Handling and Probability |

|When solving problems, the learner is able to recognise, describe, |The learner is able to investigate, analyse,describe and represent a |analyse and explain properties of shapes in |The learner is able to collect, |

|represent and work confidently with numbers and their relationships |wide range of functions and solve related problems. |2-dimensional and 3-dimensional space with |organise, analyse and interpret |

|to estimate, calculate and check solutions. | |justification. |data to establish statistical |

| | | |and probability models to solve |

| | | |related problems. |

|12.1.2Demonstrate an understanding of the definition of a | |12.2.1(a)Demonstrate the ability to work with various types of| |12.3.3Use a two dimensional Cartesian | | | |

|logarithm and any laws needed to solve real-life problems | |functions and relations including the inverses listed in the | |co-ordinate system to derive and | | | |

| | |following Assessment Standard.(b)Demonstrate knowledge of the | |apply:the equation of a circle (any | | | |

| | |formal definition of a function | |centre);the equation of a tangent to a | | | |

| | | | |circle given a point on the circle. | | | |

|12.1.3a)Identify and solve problems involving number | |12.2.2a)Investigate and generate graphs of the inverse | |12.3.4(a)Use the compound angle | | | |

|patterns, including but not limited to arithmetic and | |relations of functions, in particular the inverses of:y = ax +| |identities to generalise the effect on | | | |

|geometric sequences and series. (b)Correctly interpret | |q y = ax2y = ax ; a > 0(b) Determine which inverses are | |the co-ordinates of a point (x ; y) | | | |

|sigma notation.(c)Prove and correctly select the formula | |functions and how the domain of the original function needs to| |after rotation about the origin through | | | |

|for and calculate the sum of series, | |be restricted so that the inverse is also a function. | |an angle (.\(b)Demonstrate the knowledge| | | |

| | | | |that rigid transformations | | | |

|12.1.4(a)Calculate the value of n in the formula A = P(1| |12.2.3 Identify characteristics as listed below and hence use | |12.3.5Derive and use the following | | | |

|( i)n b)Apply knowledge of geometric series to solving | |applicable characteristics to sketch graphs of the inverses of| |compound angle identities: | | | |

|annuity, bond and sinking fund problems, with or without | |the functions listed above:(a) domain and range;(b)intercepts | | | | | |

|the use of the formulae: | |with the axes;(c)turning points, minima and maxima; | | | | | |

| | |(d)asymptotes;(e)shape and symmetry;(f)average gradient | | | | | |

| | |(average rate of change); intervals on which the function | | | | | |

| | |increases/decreases. | | | | | |

| | | | | | | | |

|12.1.5Critically analyse investment and loan options and | |12.2.4 Factorise third degree polynomials (including examples | |12.3.6 Solve problems in two and three | | | |

|make informed decisions as to the best option(s) (including| |which require the factor theorem) | |dimensions by constructing and | | | |

|pyramid and micro-lenders’ schemes). | | | |interpreting geometric and trigonometric| | | |

| | | | |models | | | |

|12.1.6Solve non-routine, unseenproblems. | |(12.2.7 a)Investigate and use instantaneous rate of change of |√ | | | | |

| | |a variable when interpreting models ofsituations:demonstrating| | | | | |

| | |an intuitive understanding of the limit concept in the context| | | | | |

| | |of approximating the rate of change or gradient at a | | | | | |

| | |point;establishing the derivatives of the following functions | | | | | |

| | |from first principles: (c)Determine the equations of tangents | | | | | |

| | |to graphs.(d)Generate sketch graphs of cubic functions using | | | | | |

| | |differentiation to determine the stationary points (maxima, | | | | | |

| | |minima and points of inflection) and the factor theorem and | | | | | |

| | |other techniques to determine the intercepts with the | | | | | |

| | |x-axis.(e) Solve practical problems involving optimisation | | | | | |

| | |and rates of change. | | | | | |

| | |12.2.8 Solve linear programming problems by optimising a | | | | | |

| | |function in two variables, subject to one or more linear | | | | | |

| | |constraints, by establishing optima by means of a search line | | | | | |

| | |and further comparing the gradients of the objective function | | | | | |

| | |and linear constraint boundary lines. | | | | | |

| |Teachers Activities |Learners Activities |Teaching Methods |Assessment |Resources |Date Completed |

|Activity 1 |Educator give learners worksheets where they use the |Learners given worksheet where they use the following |Discussion |Class work ,Memo |calculator, | |

| |following rules of differentiation: |rules of differentiation: | |Educator, individual |worksheets | |

|Rules of |[pic] |[pic] | | | | |

|differentiation |[pic] |[pic] | | | | |

| |Differentiate by using the power rule (If[pic], then [pic]) |Differentiate by using the power rule (If[pic], then | | | | |

| |Examples: |[pic]) | | | | |

| |Differentiate |Examples: | | | | |

| |(a) [pic] |Differentiate | | | | |

| |(b) [pic] |(a) [pic] | | | | |

| |(c) [pic] |(b) [pic] | | | | |

| |(d) [pic] |(c) [pic] | | | | |

| | |(d) [pic] | | | | |

|Activity 2 |Educators give worksheets to learners to determine |Learners are given worksheets to determine equations of|Group work , |Class work , home work, |calculator, | |

|Equations of tangents |equations of tangents to graphs. |tangents to graphs. |Discussion, |memo rubric |worksheets | |

|to graphs. Sketch |By determining the equation of a cubic function from a given|They need to generate sketch graphs of cubic functions |Investigative |Educator, individual | | |

|graphs of cubic |graph. |using differentiation to determine the stationary points | | | | |

|functions using |Using the second derivative or any other means to determine |(maxima, minima and points of inflection) and the factor | | | | |

|differentiation |a point of inflection where applicable. |theorem and other techniques to determine the intercepts | | | | |

| |Discuss the nature of stationary points including local |with the x-axis. learners are expected to be able to | | | | |

| |maximum, local minimum and points of inflection. |interpret cubic functions | | | | |

| |Integration with transformation. |By determining the equation of a cubic function from a | | | | |

| | |given graph. | | | | |

| | |Using the second derivative or any other means to | | | | |

| | |determine a point of inflection where applicable. | | | | |

| | |Discuss the nature of stationary points including local | | | | |

| | |maximum, local minimum and points of inflection. | | | | |

| | |Integration with transformation. | | | | |

| | |Candidates are expected to interpret the graph of the | | | | |

| | |derivative of a function. | | | | |

|Activity 3 |Educator demonstrate to learners to solve practical |Learners solve practical problems involving optimisation |Group work , |Class work , home work, |calculator, | |

|Solve practical |problems involving optimisation and rates of change. |and rates of change. |Discussion |Memo rubric, Educator, |worksheets | |

|problems involving | | | |individual | | |

|optimisation and rates| |E.g. If the price for a chocolate bar is p(x) cents where| | | | |

|of change | |p(x) = 100 – x/10, then x thousand chocolate bars will be| | | | |

| | |sold in a certain area. | | | | |

| | |Find an expression for the total income from the sale of | | | | |

| | |x thousand chocolate bars. | | | | |

| | |Find the value of x that leads to maximum income. | | | | |

| | |Find the maximum income. | | | | |

|Expanded Opportunities Different examples and remedial work Use of different equations |

LESSON PLAN: 3 for Term 2

|Subject: Mathematics Grade 12 |

|Lesson Plan: Data Handling Number of Activities 3 |

|Duration: 4H 30 Min Week 16- 17 / Date |

|Context: Data Handling |

|Link with previous lesson: Revision of grade 10-11 Data Handling |

|KNOWLEDGE (K): Measures of central tendency and dispersion in univariate numerical data and bivariate numerical data |

|SKILLS (S): Measuring, calculating, drawing, interpretation VALUES (V): appreciation |

|Learning Outcome 1: |Learning Outcome 2: |Learning Outcome 3: Space, Shape and |Learning Outcome 4: Data Handling and |

|Number and Number Relationships |Functions and Algebra |Measurement The learner is able to describe, |Probability |

|When solving problems, the learner is able to recognise, |The learner is able to investigate, analyse,describe and |represent, analyse and explain properties of |The learner is able to collect, |

|describe, represent and work confidently with numbers and their |represent a wide range of functions and solve related problems.|shapes in 2-dimensional and 3-dimensional |organise, analyse and interpret data to |

|relationships to estimate, calculate and check solutions. | |space with justification. |establish statistical and probability |

| | | |models to solve related problems. |

|12.1.2Demonstrate an understanding of the definition of| |12.2.1(a)Demonstrate the ability to work with various | |12.3.3Use a two dimensional Cartesian | |11.4.1 |√ |

|a logarithm and any laws needed to solve real-life | |types of functions and relations including the inverses | |co-ordinate system to derive and | |Calculate and represent| |

|problems | |listed in the following Assessment | |apply:the equation of a circle (any | |measures of central | |

| | |Standard.(b)Demonstrate knowledge of the formal | |centre);the equation of a tangent to a| |tendency and dispersion| |

| | |definition of a function | |circle given a point on the circle. | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

| | | | | | | | |

|12.1.3a)Identify and solve problems involving number | |12.2.2a)Investigate and generate graphs of the inverse | |12.3.4(a)Use the compound angle | |Represent bivariate |√ |

|patterns, including but not limited to arithmetic and | |relations of functions, in particular the inverses of:y =| |identities to generalise the effect on| |numerical data as a | |

|geometric sequences and series. (b)Correctly interpret | |ax + q y = ax2y = ax ; a > 0(b) Determine which | |the co-ordinates of a point (x ; y) | |scatter plot and | |

|sigma notation.(c)Prove and correctly select the | |inverses are functions and how the domain of the original| |after rotation about the origin | |suggest intuitively | |

|formula for and calculate the sum of series, | |function needs to be restricted so that the inverse is | |through an angle (.\(b)Demonstrate the| |whether a linear, | |

| | |also a function. | |knowledge that rigid transformations | |quadratic or | |

| | | | | | |exponential function | |

| | | | | | |would best fit the data| |

| | | | | | |(problems should | |

| | | | | | |include issues related | |

| | | | | | |to health | |

| | | | | | | | |

| | | | | | | | |

|12.1.4(a)Calculate the value of n in the formula A =| |12.2.3 Identify characteristics as listed below and hence| |12.3.5Derive and use the following | | | |

|P(1 ( i)n b)Apply knowledge of geometric series to | |use applicable characteristics to sketch graphs of the | |compound angle identities: | | | |

|solving annuity, bond and sinking fund problems, with | |inverses of the functions listed above:(a) domain and | | | | | |

|or without the use of the formulae: | |range;(b)intercepts with the axes;(c)turning points, | | | | | |

| | |minima and maxima; (d)asymptotes;(e)shape and | | | | | |

| | |symmetry;(f)average gradient (average rate of change); | | | | | |

| | |intervals on which the function increases/decreases. | | | | | |

|12.1.5Critically analyse investment and loan options | |12.2.4 Factorise third degree polynomials (including | |12.3.6 Solve problems in two and three| | | |

|and make informed decisions as to the best option(s) | |examples which require the factor theorem) | |dimensions by constructing and | | | |

|(including pyramid and micro-lenders’ schemes). | | | |interpreting geometric and | | | |

| | | | |trigonometric models | | | |

|12.1.6Solve non-routine, unseenproblems. | |(12.2.7 a)Investigate and use instantaneous rate of | | | | | |

| | |change of a variable when interpreting models | | | | | |

| | |ofsituations:demonstrating an intuitive understanding of | | | | | |

| | |the limit concept in the context of approximating the | | | | | |

| | |rate of change or gradient at a point;establishing the | | | | | |

| | |derivatives of the following functions from first | | | | | |

| | |principles: (c)Determine the equations of tangents to | | | | | |

| | |graphs.(d) Generate sketch graphs of cubic functions | | | | | |

| | |using differentiation to determine the stationary points | | | | | |

| | |(maxima, minima and points of inflection) and the factor | | | | | |

| | |theorem and other techniques to determine the intercepts | | | | | |

| | |with the x-axis.(e) Solve practical problems involving | | | | | |

| | |optimisation and rates of change. | | | | | |

| | |12.2.8 Solve linear programming problems by optimising a | | | | | |

| | |function in two variables, subject to one or more linear | | | | | |

| | |constraints, by establishing optima by means of a search | | | | | |

| | |line and further comparing the gradients of the objective| | | | | |

| | |function and linear constraint boundary lines. | | | | | |

| |Teachers Activities |Learners Activities |Teaching Methods |Assessment |Resources |Date Completed |

|Activity 1 |Educator give learners |Learners given worksheets |Group work, question and |Class work |Calculator, exemplars, | |

|Measures of central tendency and |worksheets to calculate and |to calculate and represent|answer, |Memo |worksheet | |

|dispersion in univariate |represent measures of central |measures of central | |Educator, Peer | | |

|numerical data |tendency and dispersion in |tendency and dispersion in| | | | |

| |univariate numerical data |univariate numerical data | | | | |

| | |by: | | | | |

| | |five number summary | | | | |

| | |(maximum, minimum and | | | | |

| | |quartiles); | | | | |

| | |box and whisker diagrams; | | | | |

|Activity 2 |Educator demonstrate to learners|Learners to draw and |Group work, question and |Class work |Calculator, exemplars, | |

|Measures of central tendency and |how to draw and interpret |interpret ogives and also |answer, |Memo |worksheet | |

|dispersion in univariate |ogives and also to |to | |Educator, Peer | | |

|numerical data |calculate variance and standard |calculate variance and | | | | |

| |deviation (use of the calculator|standard deviation (use of| | | | |

| |is advised). Interpret standard|the calculator is | | | | |

| |deviations for normal |advised). Interpret | | | | |

| |distributions. |standard deviations for | | | | |

| | |normal distributions. | | | | |

|Activity 3 |Educator give work sheets given |Work sheets given to |Group work, question and |Class work |Calculator, exemplars, | |

|Represent bivariate numerical |to learners to represent |learners to represent |answer, |Memo |worksheet | |

|data |bivariate numerical data as a |bivariate numerical data | |Educator, Peer | | |

| |scatter plot and suggest |as a scatter plot and | | | | |

| |intuitively whether a linear, |suggest intuitively | | | | |

| |quadratic or exponential |whether a linear, | | | | |

| |function would best fit the data|quadratic or exponential | | | | |

| |(problems should include issues |function would best fit | | | | |

| |related to health, social, |the data (problems should | | | | |

| |economic, cultural, political |include issues related to | | | | |

| |and environmental issues). |health, social, economic, | | | | |

| | |cultural, political and | | | | |

| | |environmental issues). | | | | |

|Expanded Opportunities Different examples and remedial work Use of different equations, Additional question papers given |

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