Advanced Mathematics - Rwanda Education Board

1 Advanced Mathematicsfor Rwanda Secondary SchoolsTeacher s guide Senior FiveFOUNTAIN EmmanuelIcyingeneye PacifiqueFountain Publishers Rwanda Box 6567 Kigali, RwandaE-mail: Ngezahayo Emmanuel, Icyingeneye Pacifique 2017 First Published 2017 All rights reserved; no part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means; electronic, mechanical, photocopying, recording or otherwise without prior written permission from the authors and the : 978-9970-19-411-7iiiContentsSection 1: General introduction ..vSection 2: Effective tips for brighter learners.

2 ViiiSection 3: Effective tips for slow learners ..viiiSection 4: Extension knowledge and ideas for teachers ..xSection 5: Content map ..xiiiSection 6: Example of Lesson plan ..xxiiiSection 7: General Methodology ..xxxSection 8: Description of Units ..xxxiUnit 1 Trigonometric Formulae, Equations and Inequalities .. Trigonometric formulae .. Trigonometric equations .. Trigonometric inequalities .. Applications ..22 Unit 2 Sequences .. Generalities on sequences .. Arithmetic sequences and harmonic Geometric sequences .. Applications ..40 Unit 3 Logarithmic and Exponential Equations .. Exponential and logarithmic functions .. Exponential and logarithmic equations.

3 Applications ..49 Unit 4 Solving Equations by Numerical Methods .. Linear interpolation and extrapolation .. Location of roots .. Iteration method ..63 Unit 5 Trigonometric Functions and their Inverses .. Generalities on trigonometric functions and their inverses .. Limits of trigonometric functions .. Differentiation of trigonometric functions and their inverses .. Applications ..91 Unit 6 Vector Space of Real Numbers .. Vector space 3 .. Operations on vectors .. Linear combination .. Euclidian vector space 3 .. Applications ..107 Unit 7 Matrices and Determinant of Order 3 .. Square matrices of order 3 .. Determinant of order three .. Applications.

4 132 Unit 8 Points, Straight Lines, Planes and Sphere in 3D .. Points in three dimensions .. Straight lines in three dimensions .. Planes in three dimensions .. Sphere in three dimensions ..179 Unit 9 Bivariate Statistics .. Covariance .. Regression lines .. Coefficient of correlation .. Applications ..196 Unit 10 Conditional Probability and Bayes Theorem .. Tree diagram .. Independent events .. Conditional probability .. Bayes theorem and applications ..205 References ..208vSection 1: General General introduction to the new curriculum The curriculum for Rwandan schools at primary and secondary levels has been changed from knowledge and content bases to competence based (CBC).

5 CBC is of great importance in aligning Rwanda s Education to the social and economic demands of society. It also presents answers to concerns about the capability and employability of school General guidance to teachers In Advanced Mathematics -Learner s Book Five, there are 10 units. There are many activities to be done by learners before a new lesson. This will help students to understand well the lesson. Teacher must help students to do those activities. Form groups of at least six students and let them do the activities in those the end of the lesson, there is a series of exercises which summarise the lesson taught. As teacher, let the students do those exercises and correct them on , at the end of each unit, there is a series of exercises which summarise the whole List of equipment needed for the subjectStudents will need geometric instruments for sketching curves and scientific calculators for some General guidance on assessment both formative and summativeAssessment is the use of a variety of procedures to collect information about learning and instruction.

6 Formative assessment is commonly referred to as assessment for learning, in which focus is on monitoring student response to and progress Advanced Mathematics Teacher s guide Book Fiveviwith instruction. Formative assessment provides immediate feedback to both the teacher and student regarding the learning process . Formative and summative assessments contribute in different ways to the larger goals of the assessment must provide oral or written feedback to students discussion or work. For example, a teacher responds orally to a question asked in class; provides a written comment in a response or reflective journal; or provides feedback on student the end of each lesson, teacher must give to the students a small evaluation to see if they understood the lesson.

7 Also at the end of the units, teacher must give the general test which summarises the whole unit. When assessments reflect the stated learning objectives, a well-designed end of unit test provides teachers with information about individual students (identifying any student who failed to meet objectives), as well as provides an overall indication of classroom formative and summative assessments serve different purposes, they should be used ultimately within an integrated system of assessment, curriculum, and instruction. To be effective in informing the learning process , assessments must be directly integrated with theories about the content, instruction, and the learning process and must be valid and reliable for the purposes for which they are Guidance on grading and reportingAcademic achievement may be measured in a variety of ways, including compositions, presentations, oral discussion, student work samples, observations, tests, and the products of project-based learning activities.

8 Teachers should use the most current summative assessment data when determining achievement marks for the progress report. When determining what marks to use on daily and weekly assignments, remember that these Introductionviimarks should not conflict with the grades on the progress report. Teachers should not use letter grades when marking papers. Some options may include: Raw scores or ratios (11/12 correct) Written feedback Rubric scores (if using 4, 3, 2, or 1 on papers, there should be guidance as to what these marks mean).Keep in mind that work that is sent home provides parents with a general impression of how students are achieving in school but does not provide a complete picture.

9 Other assessment data are collected that encompasses the progress report grade and some of these assessments are not sent home. Communication regarding progress should be can be considered as part of the effort grade, but would not be used to grade academic achievement in elementary school since the function of homework is to provide practice in skill marks will be reported on a 4-point scale and cannot be equated to former guidelines for letter grades. A grade of 4 indicates a high level of achievement; it communicates that a student has a strong understanding of all the concepts and skills taught for that standard during the quarter and can demonstrate understanding independently and with very few errors.

10 When determining grades for students, teachers should consider the most current assessment data as evidence of learning. Earlier assessments may no longer be relevant if students have demonstrated further 2: Effective tips for brighter learnersTeacher must encourage learners to;1. explain to each other as best they can the lesson learned. 2. have an exercise book, homework book and note book of have geometric instruments and scientific work out exercises and homework and check their answers to gain practice with every lesson. 5. do their own research on learned lessons or lessons to be learned next time. Section 3: Effective tips for slow learnersMathematics may be challenging for a slow learner, but not impossible.