MatH LEARNERS MODULE PDF

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LECTURES AND LESSONS FOR MATHEMATICS FO GRADE 10...

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Mathematics Learner’s Module Unit 2

This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected]. We value your feedback and recommendations.

Department of Education Republic of the Philippines

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

Mathematics – Grade 10 Learner’s Module First Edition 2015

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Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD

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Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems , pictures , photos , brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seeking permission to use these materials from their respective copyright owners. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them. Only institution and companies which have entered an agreement with FILCOLS and only within the agreed framework may copy this Learner’s Module. Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publisher and authors directly. Authors and publishers may email or contact FILCOLS at [email protected] or (02) 439-2204, respectively.

Development Team of the Learner’s Module Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and Rosemarievic Villena-Diaz, PhD Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida Editor: Maxima J. Acelajado, PhD Reviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd, Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A. Tudlong, PhD Illustrator: Cyrell T. Navarro Layout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. Pagulayan Management and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr. Printed in the Philippines by REX Book Store Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600 Telefax: (02) 634-1054, 634-1072 E-mail Address: [email protected]

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

Introduction This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students.

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In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants. There are eight (8) modules in this material.

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Module 1 – Sequences Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position

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With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills.

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

Table of Contents Unit 2 Module 3: Polynomial Functions ............................................................ 99

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Lessons and Coverage ........................................................................ 100 Module Map ......................................................................................... 100 Pre-Assessment .................................................................................. 101 Learning Goals and Targets ................................................................ 105 Activity 1 .................................................................................... 106 Activity 2 .................................................................................... 107 Activity 3 .................................................................................... 108 Activity 4 .................................................................................... 108 Activity 5 .................................................................................... 110 Activity 6 .................................................................................... 111 Activity 7 .................................................................................... 112 Activity 8 .................................................................................... 115 Activity 9 .................................................................................... 115 Activity 10 .................................................................................. 118 Activity 11 .................................................................................. 119 Activity 12 .................................................................................. 121 Activity 13 .................................................................................. 122 Activity 14 .................................................................................. 123 Summary/Synthesis/Generalization ........................................................... 125 Glossary of Terms ...................................................................................... 125 References Used in this Module ................................................................. 126

Module 4: Circles .................................................................................... 127 Lessons and Coverage ........................................................................ 127 Module Map ......................................................................................... 128 Pre-Assessment .................................................................................. 129 Learning Goals and Targets ................................................................ 134

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Lesson 1A: Chords, Arcs, and Central Angles .......................................... 135 Activity 1 .................................................................................... 135 Activity 2 .................................................................................... 137 Activity 3 .................................................................................... 138 Activity 4 .................................................................................... 139 Activity 5 .................................................................................... 150 Activity 6 .................................................................................... 151 Activity 7 .................................................................................... 151 Activity 8 .................................................................................... 152 Activity 9 .................................................................................... 152 Activity 10 .................................................................................. 155 Activity 11 .................................................................................. 155 Activity 12 .................................................................................. 157 Activity 13 .................................................................................. 159

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

Summary/Synthesis/Generalization ........................................................... 160 Lesson 1B: Arcs and Inscribed Angles ....................................................... 161

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Activity 1 .................................................................................... 161 Activity 2 .................................................................................... 162 Activity 3 .................................................................................... 163 Activity 4 .................................................................................... 164 Activity 5 .................................................................................... 167 Activity 6 .................................................................................... 168 Activity 7 .................................................................................... 169 Activity 8 .................................................................................... 170 Activity 9 .................................................................................... 172 Activity 10 ..................................................................................174 Activity 11 ..................................................................................175 Activity 12 ..................................................................................176 Summary/Synthesis/Generalization ........................................................... 177

Lesson 2A: Tangents and Secants of a Circle ............................................ 178

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Activity 1 .................................................................................... 178 Activity 2 .................................................................................... 179 Activity 3 .................................................................................... 180 Activity 4 .................................................................................... 188 Activity 5 .................................................................................... 189 Activity 6 .................................................................................... 192 Activity 7 .................................................................................... 194 Activity 8 .................................................................................... 197 Summary/Synthesis/Generalization ........................................................... 198

Lesson 2B: Tangent and Secant Segments ................................................. 199

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Activity 1 .................................................................................... 199 Activity 2 .................................................................................... 200 Activity 3 .................................................................................... 200 Activity 4 .................................................................................... 201 Activity 5 .................................................................................... 204 Activity 6 .................................................................................... 205 Activity 7 .................................................................................... 206 Activity 8 .................................................................................... 207 Activity 9 .................................................................................... 208 Activity 10 ..................................................................................210 Summary/Synthesis/Generalization ........................................................... 211 Glossary of Terms ....................................................................................... 212 List of Theorems and Postulates on Circles ............................................... 213 References and Website Links Used in this Module .................................. 215

Module 5: Plane Coordinate Geometry ............................................... 221 Lessons and Coverage ........................................................................ 222 Module Map ......................................................................................... 222 Pre-Assessment ................................................................................... 223 Learning Goals and Targets ................................................................. 228

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

Lesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate Proof .......................................................... 229

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Activity 1 .................................................................................... 229 Activity 2 .................................................................................... 230 Activity 3 .................................................................................... 231 Activity 4 .................................................................................... 232 Activity 5 .................................................................................... 241 Activity 6 .................................................................................... 242 Activity 7 .................................................................................... 242 Activity 8 .................................................................................... 243 Activity 9 .................................................................................... 245 Activity 10 .................................................................................. 248 Activity 11 .................................................................................. 250 Summary/Synthesis/Generalization ........................................................... 251

Lesson 2: The Equation of a Circle............................................................ 252

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Activity 1 .................................................................................... 252 Activity 2 .................................................................................... 253 Activity 3 .................................................................................... 254 Activity 4 .................................................................................... 263 Activity 5 .................................................................................... 265 Activity 6 .................................................................................... 265 Activity 7 .................................................................................... 266 Activity 8 .................................................................................... 267 Activity 9 .................................................................................... 267 Activity 10 .................................................................................. 269 Summary/Synthesis/Generalization ........................................................... 270 Glossary of Terms ...................................................................................... 270 References and Website Links Used in this Module ................................. 271

All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

I. INTRODUCTION

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You are now in Grade 10, your last year in junior high school. In this level and in the higher levels of your education, you might ask the question: What are math problems and solutions for? An incoming college student may ask, “How can designers and manufacturers make boxes having the largest volume with the least cost?” And anybody may ask: In what other fields are the mathematical concepts like functions used? How are these concepts applied?

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Look at the mosaic picture below. Can you see some mathematical representations here? Give some.

As you go through this module, you are expected to define and illustrate polynomial functions, draw the graphs of polynomial functions and solve problems involving polynomial functions. The ultimate goal of this module is for you to answer these questions: How are polynomial functions related to other fields of study? How are these used in solving real-life problems and in decision making?

99 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

II. LESSON AND COVERAGE This is a one-lesson module. In this module, you will learn to:

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 illustrate polynomial functions  graph polynomial functions  solve problems involving polynomial functions

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The Polynomial Functions

Illustrations of Polynomial Functions

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Graphs of Polynomial Functions

Solutions of Problems Involving Polynomial Functions

100 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015

III. PRE-ASSESSMENT Part 1 Let us find out first what you already know related to the content of this module. Answer all items. Choose the letter that best answers each question. Please take note of the items/questions that you will not be able to answer correctly and revisit them as you go through this module for self-assessment. What should n be if f(x) = xn defines a polynomial function? A. an integer C. any number B. a nonnegative integer D. any number except 0

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Which of the following is an example of a polynomial function?

4  3x  1 x3 3

B. f ( x )  2 x 2 

3 2 x 2

What is the leading f ( x )  2x  x 3  4 ? A. 1 B. 2

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