Technical Mathematics PAT GR 12 2021 Eng

Downloaded from hlayiso.com TECHNICAL MATHEMATICS GUIDELINES FOR PRACTICAL ASSESSMENT TASKS GRADE 12 2021 These guidelines consist of 37 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 2 DBE/PAT 2021 NSC TABLE OF CONTENTS Page 1. INTRODUCTION 3 2. TEACHER GUIDELINES 4 2.1 How to administer the PATs 4 2.2 Moderation of the PATs 4 3. LEARNER GUIDELINES 4 4. EVIDENCE OF MODERATION WITH MARK ALLOCATION AND 5 DECLARATION OF AUTHENTICITY 5. CONCLUSION 6 6. TASKS 7 TASK 1: Complex numbers 7 TASK 2: Euclidean geometry 19 TASK 3: Circles, angles and angular movement 31 Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 3 DBE/PAT 2021 NSC 1. INTRODUCTION The 18 Curriculum and Assessment Policy Statement subjects which contain a practical component all include a practical assessment task (PAT). These subjects are:  AGRICULTURE: Agricultural Management Practices, Agricultural Technology  ARTS: Dance Studies, Design, Dramatic Arts, Music, Visual Arts  SCIENCES: Computer Applications Technology, Information Technology, Technical Sciences, Technical Mathematics  SERVICES: Consumer Studies, Hospitality Studies, Tourism  TECHNOLOGY: Civil Technology, Electrical Technology, Mechanical Technology, Engineering Graphics and Design A practical assessment task (PAT) mark is a compulsory component of the final promotion mark for all candidates offering subjects that have a practical component and counts 25% (100 marks) of the examination mark at the end of the year. The practical assessment task for Technical Mathematics Grade 12 consists of three tasks (one task per term) which should be completed by end of Term 3. The tasks are COMPULSORY for ALL candidates offering Technical Mathematics in Grade 12. The PAT is implemented during the first three terms of the school year. The PAT allow learners to be assessed regularly during the school year and it also allows for the assessment of skills acquired and it applies the science of Mathematics to the technical field where the emphasis is on application. It is therefore important that schools ensure that all learners complete the practical assessment tasks within the stipulated period to ensure that learners are promoted at the end of the school year. The planning and execution of the PAT differs from subject to subject. The tasks should be administered under supervised conditions. Moderation may be done onsite. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 4 DBE/PAT 2021 NSC 2. TEACHER GUIDELINES 2.1 How to administer the PATs  The following documents must be available for all formal tasks: o Task instructions explaining the procedures to be followed o The worksheets which include questions to be answered under examination conditions o The teacher guidelines with task instructions, worksheets and marking guidelines (The teacher guidelines MUST NOT be released to the learners.) o Teachers should compile marking guidelines (memoranda) for the actual results of the task (teachers should do the tasks themselves FIRST)  The tasks must be done individually.  Each learner must record his/her OWN INDIVIDUAL data and observations.  Each learner must be provided with his/her OWN worksheet and answer the questions INDIVIDUALLY under examination conditions.  Only once all the learners are ready to do the task and they are seated and ready to answer questions may teachers hand out a worksheet to each learner. Examination conditions have to be applied.  If it is not possible to do the task and complete the worksheet on the same day, the teacher must collect the learners' tasks. These tasks must be kept at school. 2.2 Moderation of the PATs For moderation the following documents are required in the teacher's file:  Index indicating all tasks with raw and weighted marks  All task instructions  Marking guidelines for all tasks, with ticks and totals  Composite working mark sheet for all learners showing raw and weighted marks  Evidence of internal moderation For moderation the following documents are required in the learner's file:  Index stating all tasks with raw and weighted marks  Answer sheets for all tasks 3. LEARNER GUIDELINES 3.1 This PAT for Grade 12 consists of THREE tasks. 3.2 The PAT contributes 25% towards your final promotion mark for Grade 12. 3.3 All the work in the PAT must be your own work. Group work will NOT be allowed. 3.4 Show ALL calculations clearly and include units. Round off answers to TWO decimal places. Use correct SI units where necessary. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 5 DBE/PAT 2021 NSC 4. EVIDENCE OF MODERATION Learner's name: School: MARK ALLOCATION TASK MAXIMUM WEIGHTING LEARNER'S MODERATED MODERATED MODERATED MARK MARK MARK MARK MARK (TEACHER) (SCHOOL) (DISTRICT) (PROVINCE) 1 40 40 2 30 30 3 30 30 TOTAL 100 100 NAME SIGNATURES DATE DECLARATION OF AUTHENTICITY I hereby declare that the tasks submitted for assessment is my own original work and have not been submitted for assessment or moderation previously. ___________________________________ ____________ SIGNATURE OF LEARNER DATE As far as I know, the above declaration by the candidate is true and I accept that the work offered is his/her own. _________________________________ ____________ SIGNATURE OF TEACHER DATE SCHOOL STAMP Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 6 DBE/PAT 2021 NSC 5. CONCLUSION On completion of the practical assessment task learners should be able to demonstrate their understanding of the industry, enhance their knowledge, skills, values and reasoning abilities as well as establish connections to life outside the classroom and address real-world challenges. The PAT furthermore develops learners' life skills and provides opportunities for learners to engage in their own learning. Copyright reserved

Downloaded from hlayiso.com TECHNICAL MATHEMATICS PRACTICAL ASSESSMENT TASK 1 GRADE 12 2021 SURNAME AND NAME SCHOOL TERM: 1 MARKS: 40 This task consists of 12 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 2 DBE/PAT 2021 NSC TECHNICAL MATHEMATICS TASK 1 TOPIC: COMPLEX NUMBERS AIM: To apply and develop mathematical skills, reasoning and demonstrate an understanding of complex numbers in real-life technical problems A complex number is any number that can be written in the form a + bi where a and b are real numbers, where a is a real part, b is an imaginary part and i is an imaginary unit. INSTRUCTIONS AND INFORMATION 1. This PAT Task 1 worksheet consists of FIVE questions. 2. Answer ALL the questions. 3. Resources required are calculators, grids (provided) and mathematical sets. 4. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 3 DBE/PAT 2021 NSC QUESTION 1 CLASSIFICATION OF COMPLEX NUMBERS AND INTRODUCTION TO COMPLEX CONJUGATES Classify each complex number by placing a value in the appropriate column(s). Solution Marks No. Complex Imaginary Real Part Complex Number Part Conjugate 1. 5 + 2i 1 2. 8 + i 2 3. 3i [3] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 4 DBE/PAT 2021 NSC QUESTION 2 BASIC OPERATIONS AND GRAPHICAL REPRESENTATION OF COMPLEX NUMBERS Complex numbers can be represented on a complex plane, the Argand diagram with the horizontal axis as the real part and the vertical axis as the imaginary part. A Complex number z = a + bi in rectangular form has a, representing the distance along the real axis and b, the distance along the imaginary axis. 2.1 Determine, with the aid of an Argand diagram: 2.1.1 The sum of z1   8  3i and z2  3  5i (3) Solution Marks (3) 2.1.2 The difference between z3  9  7i and z4  6  i (3) Solution Marks (3) Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 5 DBE/PAT 2021 NSC 2.2 Sketch the given complex number and its complex conjugate on the grid provided: 2.2.1 z3  9  7i (3) Solution Marks (3) 2.2.2 z4  6  i (2) Solution Marks (2) 2.3 What can you deduce from QUESTIONS 2.2.1 and 2.2.2 about the complex number and its complex conjugate? (1) Solution Marks (1) [12] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 6 DBE/PAT 2021 NSC QUESTION 3 POLAR FORM AND GRAPHICAL REPRESENTATION OF COMPLEX NUMBERS  A complex number in rectangular form a + bi has polar coordinates r cis   r ( cos θ + i sin θ ) = r  where:  r = a 2 +b2 , r is the modulus b  tan   ,  is the argument a  In a complex plane, by drawing a vector from the origin to the point representing z = a + bi , an angle  in standard position is formed. The point a + bi is r units from the origin.  When converting a complex number from a rectangular form to a polar form:  The quadrant in which the complex number lies should be identified  The required angle is found by using a tangent ratio, the signs of a and b indicate the quadrant in which the angle is found Convert to polar form and represent on an Argand diagram the following rectangular complex numbers: 3.1 z = 4i (4) Solution Marks (4) Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 7 DBE/PAT 2021 NSC 3.2 z  2 5  2i (6) Solution Marks (6) [10] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 8 DBE/PAT 2021 NSC QUESTION 4 APPLY THE COMPLEX NUMBER TO DETERMINE THE RESULTANT FORCE OF TWO VECTORS BY ADDITION The picture below shows a force diagram of the addition of two vectors. Now use knowledge of complex numbers to add two vectors. Given in the diagram below: Force 1: z1  4 2 cis  6 Force 2: z 2  cis  2 The diagram below shows the two vectors. Using a protractor: Step 1: Measure in degrees. Step 2: Measure in degrees. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 9 DBE/PAT 2021 NSC 4.1 Using a protractor, measure angles and (2) Solution Marks   ______    ______  (2) 4.2 Write z1 and z2 in polar form. (2) Solution Marks (2) 4.3 Determine resultant force zR if zR  z1  z2 . (1) Solution Marks (1) 4.4 Express zR in polar form r cis  . (1) Solution Marks (1) Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 10 DBE/PAT 2021 NSC 4.5 Represent the resultant force in an Argand diagram (3) Solution Marks (3) [9] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 11 DBE/PAT 2021 NSC QUESTION 5 APPLICATION OF COMPLEX NUMBERS IN ELECTRONICS (RLC circuits) The diagram below models an AC circuit with two impedances, z1   30  10i   and z 2  10  30i   connected in parallel. z1  z 2 The total impedance is z T  z1  z 2 Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 12 DBE/PAT 2021 NSC z1  z 2 Determine the total impedance, z T  z1  z 2 (6) Solution Marks (6) [6] TOTAL: 40 Copyright reserved

Downloaded from hlayiso.com TECHNICAL MATHEMATICS PRACTICAL ASSESSMENT TASK 2 GRADE 12 2021 SURNAME AND NAME SCHOOL TERM: 2 MARKS: 30 This task consists of 12 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 2 DBE/PAT 2021 NSC TECHNICAL MATHEMATICS TASK 2 TOPIC: EUCLIDEAN GEOMETRY AIM: To verify the midpoint theorem and proportionality theorem and apply the two theorems to solve problems INSTRUCTIONS AND INFORMATION 1. This PAT Task 2 worksheet consists of SIX questions. 2. Answer ALL the questions. 3. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 3 DBE/PAT 2021 NSC ACTIVITY 1 Objective  To investigate the relationship between the line joining the mid-points of two sides of a triangle and the third side  To apply the midpoint theorem to solve problems in a real-life context Materials required  Mathematical instruments (ruler and protractor are essential)  Pen  Pencil  Cotton/String  Scissors  Tracing paper  Cellophane tape  Paper glue Theory Midpoint theorem: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and it is half the length of the third side. Procedure Step 1: Draw any ΔABC in the space provided below. Solution Step 1 Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 4 DBE/PAT 2021 NSC Step 2: Use a ruler to measure the lengths of AB, AC and BC and record them in the space below. Solution Step 2 AB = _________; AC = __________ and BC = _____________ Step 3: Mark the midpoints D and E of the sides AB and AC respectively. Join D and E. Solution Step 3  Length of AD = _________ and AE = __________ Step 4:  Place a piece of cotton/string along the length of BC.  Mark off and cut so that the length of the cotton is the same as BC.  Fold the piece of cotton/string in half and place that against DE.  Use cellophane tape to stick the cotton piece along the length of DE.  Does the folded half fit exactly along the length of DE?  Use a ruler to measure the length of DE and record the value. Solution Step 4  Does the folded half fit exactly along the length of DE? ____________  Use a ruler to measure the length of DE and record the value: DE = _________ Step 5:  Use a tracing paper, trace and cut out a copy of ADE .     Clearly label angles A, D and E at the respective vertices.    Place your traced cut-out of ADE so that D is placed over B of ABC .    Use paper glue to stick the cut-out of ADE so that D is placed over B of ABC .    Does D fit exactly over B ?    Use a protractor to measure the size of ABC and ADE and record them below. Solution Step 5  ____________    ABC  _______  and ADE  ______  Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 5 DBE/PAT 2021 NSC Observations and Conclusion 1. Answer the following questions:   1.1 Compare the sizes of ABC and ADE Solution 1.1 1.2 Fill in the missing word: The line segment DE is ………………………. to BC (corresponding s are = ) Solution 1.2 1.3 Compare the lengths of DE and BC. Solution 1.3 1.4 Explain the relationship between the lengths of DE and BC, i.e. length of DE = … BC Solution 1.4 DE = … BC Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 6 DBE/PAT 2021 NSC 1.5 Explain the relationship that you observe between the line segment joining the midpoints of two sides of a triangle and the third side. Solution 1.5 2. The picture below shows a large Christmas tree and the diagram alongside models the picture. Points P and Q are the midpoints of sides AB and AC respectively of ABC . The slogan 'MERRY CHRISTMAS' is attached to a steel cable surrounding the Christmas tree from P to Q. A P Q MERRY CHRISTMAS B C Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 7 DBE/PAT 2021 NSC Use the midpoint theorem to determine the length of the slogan from P to Q, by first determining the length of BC (the base diameter) if the circumference of the base circle of the tree is 40,84 m. Show your calculations in the space below. Use the following formula: Circumference of circle = 2 r Solution Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 8 DBE/PAT 2021 NSC ACTIVITY 2 Objective  To investigate the basic proportionality theorem Materials required  Mathematical instruments (ruler and protractor are essential)  Pen  Pencil Theory Proportionality theorem: The line drawn parallel to one side of a triangle divides the other two sides proportionally. Equal ratios form proportion. Procedure Step 1: Draw ΔKLM in the space provided below. Solution Step 1 Step 2: Measure the lengths of KL, KM and LM. Record them below. Solution Step 2 KL = _________; KM = __________ and LM = _____________ Step 3: Draw line segment NP with N on KL and P on KM such that NP || LM Solution Step 3 Show in Step 1 above. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 9 DBE/PAT 2021 NSC Step 4: Measure the lengths of KN, NL, KP, PM and NP. Record them below. Solution Step 4 KN = _________ , NL = _________ , KP = _________ , PM = _________ and NP = _________ Calculations, Observations and Conclusion 1. Answer the following questions: 1.1 Calculate the following: KN KP 1.1.1 and NL PM Solution KN KP 1.1.1  _______ and  ______ NL PM KN KP 1.1.2 and KL KM Solution KN KP 1.1.2  _______ and  ______ KL KM NP 1.1.3 LM Solution NP 1.1.3  LM Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 10 DBE/PAT 2021 NSC 1.2 Compare the values of the ratios calculated above. Solution 1.2 1.3 Explain the relationship observed between the line segments that are divided by the line drawn parallel to the 3rd side of a triangle. Solution 1.3 1.4 Write down the relationship between the lengths of NP and LM, i.e. the length of NP = … LM Solution 1.4 NP = … LM 1.5 Explain the relationship that you observe between the line segment joining the midpoints of two sides of a triangle and the third side. Solution 1.5 Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 11 DBE/PAT 2021 NSC 2. A company has been contracted to place metal spikes on a triangular building along the length of side AD to prevent birds from roosting and messing on the building. The cost of the spikes, including labour, is R165 per meter. The diagram below models the side view of the building, as shown. A 25 m B C 8m 6m D E If the length of AE = 25 m, CE = 6 m, BD = 8 m and BC // DE , use the proportionality theorem to determine how much will it cost to fit the spikes by first determining the length of AD. Show calculations in the space below. Solution TOTAL: 30 Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 12 DBE/PAT 2021 NSC Name: ………………………………… School: …………………………………… The following marking criteria will be used for marking. (Each criterion will have 4 levels at most.) No. Criteria Marks 1 Ability to carry out instructions 6 2 Accuracy in measurement and calculations 8 3 Observations and conclusion of Activity 1 8 4 Observations and conclusion of Activity 2 8 TOTAL: 30 RUBRIC FOR MARKING TECHNICAL MATHEMATICS PAT 2021 TASK 2 CRITERIA 1 2 3 4 1. Ability to Unable to Able to carry out Able to follow all Able to follow all carry out carry out 75% 50 % of the the instructions the instructions ___x 2 = __ instructions of the instructions instructions 2. Accuracy in Unable to Able to do some Able to do Able to do measurement measure correct correct accurate and accurately and measurements measurements measurements calculations errors in and had some and correct and accurate ___x 2 = __ calculations errors in calculations with calculations calculations few errors of without errors rounding off 3. Observations Unable to Able to establish Able to establish Able to establish and conclusion observe the one part of the both both of Activity 1 relationships theorem, i.e. that relationships, i.e. relationships, i.e. and the line segment that the line that the line ___x 2 = __ conclusion is parallel to the segment is segment is was not third side or that parallel to the parallel to the related to the the line segment third side and it is third side and it is theorem is half the size of half the size of half the size of the third side the third side the third side with some with perfect explanation explanation. 4. Observations Unable to Able to establish Able to identify Able to clearly and conclusion observe the some ratios that are identify equal of Activity 2 relationship of relationships of equal and hence ratios that form the ratios, the ratios based form proportion proportions based ___x 2 = __ hence the on calculations based on on correct conclusion and conclusion calculations. The calculations. The was not made had some conclusion was conclusion was related to the errors also correct. perfectly theorem explained. TOTAL 30 Copyright reserved

Downloaded from hlayiso.com TECHNICAL MATHEMATICS PRACTICAL ASSESSMENT TASK 3 GRADE 12 2021 SURNAME AND NAME SCHOOL TERM: 3 MARKS: 30 This task consists of 7 pages. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 2 DBE/PAT 2021 NSC TECHNICAL MATHEMATICS TASK 3 TOPIC: CIRCLES, ANGLES AND ANGULAR MOVEMENT AIMS:  To apply and develop mathematical skills, reasoning and demonstrate an understanding of radians and degrees  To convert between degrees and radians  To calculate area, arc length and height of the segment  To apply the knowledge gained in circles, angles and angular movement to solve real-life problems INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions. 2. Resources required are a wall chart, ruler, compass, pencil and protractor. 3. Clearly show ALL calculations, diagrams etc. that you have used in determining your answers. 4. Make sure your sketch is neat and constructed according to the instructions. Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 3 DBE/PAT 2021 NSC ACTIVITY 1 Resources required: ruler, protractor, compass and coloured pencils Step 1: Draw an equilateral triangle PQR with sides of length 6 cm. (1) Solution Marks Step 1 Step 2: Draw a circle through points A and B such that R is the centre, where A is the midpoint of PR and B the midpoint of QR. (2) Solution Marks Step 2 Must be done on the diagram under Step 1. Step 3: Shade the area of the minor sector RAB. (1) Solution Marks Step 3 Must be done on the diagram under Step 1.  Step 4: Write down the length of side AR and the magnitude of R . (2) Solution Marks Step 4 AR = _________  R = _________ Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 4 DBE/PAT 2021 NSC  Step 5: Convert the magnitude of R to radians. (2) Solution Marks Step 5 Step 6: Determine the arc length of the minor sector RAB. (3) Solution Marks Step 6 Step 7: Hence, determine the unshaded area APQB. (5) Solution Marks Step 7 Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 5 DBE/PAT 2021 NSC Step 8: The diagram below shows angles in degrees and radians. (6) Complete the following table by converting between degrees and radians measures if: 180   radians Solution Marks Step 8 ANGLES DEGREES RADIANS 43,5 29  240 191 ..…….  300 4 ………..  25 43 ………. 108,1 ………. x  ........... ……….. TOTAL 360 ……….. [22] Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 6 DBE/PAT 2021 NSC ACTIVITY 2 The picture below shows a traffic circle at one of the busy intersections in a certain municipality. The municipality wants to pave the inner island of the traffic circle and plans to do so in three consecutive days by paving a third of the island each day. The inner island has a circumference of 18 m. The diagram below the picture models the inner island of the traffic circle. O is the centre of the circle. K and L are points on the circle. PICTURE Inner island (circle) O DIAGRAM K L 2.1 Calculate: 2.1.1 The length of the radius OK correct to ONE decimal place. (1) Solution Marks (1)  2.1.2 The magnitude of the obtuse angle KOL (in degrees). (1) Solution Marks (1) Copyright reserved Please turn over

Downloaded from hlayiso.com Technical Mathematics 7 DBE/PAT 2021 NSC 2.2 Determine the height of the minor segment of chord KL. (6) Use 4h2  4dh  x2  0 Solution Marks (6) [8] TOTAL: 30 Copyright reserved