Mathematics Grade 10

MATHEMATICS Teacher's Guide Grade 10



VISIT DEPED TAMBAYANhttp://richardrrr.blogspot.com/1. Center of top breaking headlines and current events related to Department of Education. 102. Offers free K-12 Materials you can use and shareDEPED COPY Mathematics Teacher’s Guide Unit 1 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 10Teacher’s GuideFirst Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any workof the Government of the Philippines. However, prior approval of the government agency oroffice 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 ofroyalties. 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 seekingpermission to use these materials from their respective copyright owners. . All means havebeen exhausted in seeking permission to use these materials. The publisher and authors donot represent nor claim ownership over them. Only institutions and companies which have entered an agreement with FILCOLSand only within the agreed framework may copy this Teacher’s Guide. Those who have notentered in an agreement with FILCOLS must, if they wish to copy, contact the publishers andauthors directly. Authors and publishers may email or contact FILCOLS at [email protected] or(02) 439-2204, respectively.Published by the Department of EducationSecretary: Br. Armin A. Luistro FSCUndersecretary: Dina S. Ocampo, PhDDEPED COPY Development Team of the Teacher’s GuideConsultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, andRosemarievic Villena-Diaz, PhDAuthors: 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. TernidaEditor: Maxima J. Acelajado, PhDReviewers: Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P.Borja, Maylani L. Galicia, Ma. Corazon P. Loja, Jones A. Tudlong, PhD, andReymond Anthony M. QuanIllustrator: Cyrell T. NavarroLayout Artists: Aro R. Rara, Jose Quirovin Mabuti, and Ronwaldo Victor Ma.A. PagulayanManagement 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 StoreDepartment of Education-Instructional Materials Council Secretariat (DepEd-IMCS)Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600Telefax: (02) 634-1054, 634-1072E-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.

DEPED COPY Introduction This Teacher’s Guide has been prepared to provide teachers of Grade 10 Mathematics with guidelines on how to effectively use the Learner’s Material to ensure that learners will attain the expected content and performance standards. This book consists of four units subdivided into modules which are further subdivided into lessons. Each module contains the content and performance standards and the learning competencies that must be attained and developed by the learners which they could manifest through their products and performances. The special features of this Teacher’s Guide are: A. Learning Outcomes. Each module contains the content and performance standards and the products and/ or performances expected from the learners as a manifestation of their understanding. B. Planning for Assessment. The assessment map indicates the type of assessment and categorized the objectives to be assessed into knowledge, process/skills, understanding, and performance C. Planning for Teaching-Learning. Each lesson has Learning Goals and Targets, a Pre-Assessment, Activities with answers, What to Know, What to Reflect on and Understand, What to Transfer, and Summary / Synthesis / Generalization. D. Summative Test. After each module, answers to the summative test are provided to help the teachers evaluate how much the learners have learned. E. Glossary of Terms. Important terms in the module are defined or clearly described. F. References and Other Materials. This provides the teachers with the list of reference materials used, both print and digital. We hope that this Teacher’s Guide will provide the teachers with the necessary guide and information to be able to teach the lessons in a more creative, engaging, interactive, and effective manner. 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.

DEPED COPY Table of ContentsCurriculum Guide: Mathematics Grade 10Unit 1Module 1: Sequences....................................................................................... 1 Learning Outcomes ................................................................................................1 Planning for Assessment........................................................................................2 Planning for Teaching-Learning ...........................................................................12 Pre-Assessment ...................................................................................................14 Learning Goals and Targets .................................................................................14 Lesson 1: Arithmetic Sequences ..........................................................................14 Activity 1 ....................................................................................................15 Activity 2 ....................................................................................................16 Activity 3 ....................................................................................................16 Activity 4 ....................................................................................................16 Activity 5 ....................................................................................................16 Activity 6 ....................................................................................................17 Activity 7 ....................................................................................................17 Activity 8 ....................................................................................................17 Activity 9 ....................................................................................................18 Activity 10 ..................................................................................................18 Activity 11 ..................................................................................................19 Activity 12 ..................................................................................................19 Activity 13 ..................................................................................................20 Summary/Synthesis/Generalization .....................................................................22 Lesson 2: Geometric and Other Sequences ......................................................22 Activity 1 ....................................................................................................23 Activity 2 ....................................................................................................23 Activity 3 ....................................................................................................24 Activity 4 ....................................................................................................24 Activity 5 ....................................................................................................24 Activity 6 ....................................................................................................25 Activity 7 ....................................................................................................26 Activity 8 ....................................................................................................26 Activity 9 ....................................................................................................27 Activity 10 ..................................................................................................27 Activity 11 ..................................................................................................29 Activity 12 ..................................................................................................29 Activity 13 ..................................................................................................31 Summary/Synthesis/Generalization .....................................................................31 Summative Test .........................................................................................................32 Glossary of Terms.....................................................................................................37 References and Other Reading Materials ...........................................................38 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.

DEPED COPYModule 2: Polynomials and Polynomial Equations.............................. 39 Learning Outcomes ..............................................................................................39 Planning for Assessment......................................................................................40 Planning for Teaching-Learning ...........................................................................45 Pre-Assessment ...................................................................................................47 Learning Goals and Targets .................................................................................47 Lesson 1: Division of Polynomials .......................................................................47 Activity 1 ....................................................................................................48 Activity 2 ....................................................................................................48 Activity 3 ....................................................................................................49 Activity 4 ....................................................................................................49 Activity 5 ....................................................................................................50 Activity 6 ....................................................................................................50 Activity 7 ....................................................................................................51 Activity 8 ....................................................................................................51 Activity 9 ....................................................................................................52 Activity 10 ..................................................................................................52 Summary/Synthesis/Generalization .....................................................................53 Lesson 2: The Remainder Theorem and Factor Theorem..............................54 Activity 1 ....................................................................................................54 Activity 2 ....................................................................................................55 Activity 3 ....................................................................................................55 Activity 4 ....................................................................................................56 Activity 5 ....................................................................................................56 Activity 6 ....................................................................................................57 Activity 7 ....................................................................................................57 Activity 8 ....................................................................................................58 Activity 9 ....................................................................................................58 Activity 10 ..................................................................................................59 Summary/Synthesis/Generalization .....................................................................59 Lesson 3: Polynomial Equations ..........................................................................59 Activity 1 ....................................................................................................60 Activity 2 ....................................................................................................61 Activity 3 ....................................................................................................62 Activity 4 ....................................................................................................62 Activity 5 ....................................................................................................64 Activity 6 ....................................................................................................64 Activity 7 ....................................................................................................66 Activity 8 ....................................................................................................66 Activity 9 ....................................................................................................67 Activity 10 ..................................................................................................67 Activity 11 ..................................................................................................68 Activity 12 ..................................................................................................68 Activity 13 ..................................................................................................69 Activity 14 ..................................................................................................69 Activity 15 ..................................................................................................73 Summary/Synthesis/Generalization .....................................................................74 Summative Test .........................................................................................................74 Glossary of Terms and List of Theorems Used in This Module ...................80 References and Website Links Used in This Module ......................................81 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

DEPED COPY 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.

TEACHER’S GUIDEModule 1: SequencesA. Learning OutcomesContent Standard: The learner demonstrates understanding of key concepts ofsequences.DEPED COPYPerformance Standard: The learner is able to formulate and solve problems involvingsequences in different disciplines through appropriate and accuraterepresentations. Unpacking the Standards for UnderstandingSubject: Mathematics 10 Learning CompetenciesQuarter: First Quarter 1. Generate and describe patternsTopic: Sequences 2. Find the next few terms of a sequence 3. Find the nth term of a sequenceLessons: 4. Illustrate an arithmetic sequence 5. Determine the nth term of a given1. Arithmetic Sequence arithmetic sequence• Finding the Next Term 6. Find the arithmetic means of an• Finding the nth Term arithmetic sequence• Finding the Arithmetic 7. Determine the sum of the first n terms of Means a given arithmetic sequence• Finding the Sum of the 8. Solve problems involving arithmetic First n Terms sequence• Solving Real-Life Problems2. Geometric Sequences 9. Illustrate a geometric sequence• Finding the Next Term 10. Differentiate a geometric sequence from• Finding the nth Term an arithmetic sequence• Finding the Geometric 11. Determine the nth term of a givenMeans geometric sequence• Finding the Sum of the 12. Find the geometric means of aFirst n Terms geometric sequence• Solving Real-Life Problems 13. Determine the sum of the first n terms of a geometric sequence 1 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.

14. Determine the sum of the first n terms of an infinite geometric sequence 15. Solve real-life problems involving geometric sequences3. Other Types of Sequences 16. Illustrate other types of sequences like • Finding the Next Term harmonic sequence and FibonacciWriters: sequenceArnaldo I. ChuaMirla S. Esparrago Essential Essential Understanding: Question: Students will How are sequences understand that used to model and there are realistic solve many problem situations mathematical ideas that can be solved and realistic using the concepts situations? of sequences. Transfer Goal:DEPED COPY Students will be able to apply the key concepts of sequences in finding solutions and making decisions for certain real-life problems.B. Planning for Assessment Product/Performance The following are products and performances that students are expected to come up with in this module. 1. Recognize patterns through observations of different illustrative examples 2. Identify objects or situations in real life where patterns are used 3. Define a sequence 4. Determine the next few terms of a sequence given the nth term 5. Share with a partner the steps on how to find the nth term of a given sequence 6. Develop a “Concept Map” on arithmetic sequence 7. Find the nth term of a given arithmetic sequence through games to earn class points/extra points 8. Explain how to find the sum of the first n terms of an arithmetic sequence through a cooperative learning strategy “Think-Pair-Share” 2 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.

DEPED COPY 9. Explain solutions to problems in finding the nth term, the arithmetic means, and the sum of a given arithmetic sequence in their respective groups 10. Role-play and solve real-life situations involving arithmetic sequence 11. Ask each group to create a problem on arithmetic sequence and to be solved by other groups 12. Brainstorm their own real-life experiences where they use the concepts of arithmetic sequences 13. Present their Performance Task on arithmetic sequence, the Reality Series 14. Generalize the concepts learned through the “Power of Four” activity 15. Conduct a mathematical investigation on illustrating and defining geometric sequence through a paper folding activity 16. Compare and contrast arithmetic and geometric means in their respective groups 17. Derive the nth term of a geometric sequence through investigative approach in their respective cooperative learning groups and answer some exercises 18. Find geometric means between two numbers through a group game 19. Derive the formula for the sum of a geometric sequence (finite and infinite) through investigative approach in their respective groups and answer some exercises 20. Explain solutions to problems on finding the nth term, finding the geometric means, and finding the sum (finite and infinite) of a given geometric sequence in their respective cooperative learning groups 21. Determine and define harmonic and Fibonacci sequences through observations in their respective cooperative learning groups 22. Answer exercises or problems on harmonic and Fibonacci sequences. 23. Formulate and solve real-life situations involving sequences through the “Carousel Method” 24. Role-play their own real-life applications of geometric sequences 25. Present their group Performance Task on geometric sequence – May the Best Man Win 26. Self assess their learning through “Exit Slip/Card” 3 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.

Assessment MapTYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCEPre-Assessment/ Pre-Test: Pre-Test: Pre-Test: Pre-Test: Part I Part I Part IIDiagnostic Part I Determining Illustrating a Solving real-life Product or the next term harmonic problems Performance of a given sequence involving Task Part sequence, sequences harmonic, and Finding the sum Analyzing a real-DEPED COPY Fibonacci of a sequence or life scenario on sequences sum to infinity of sequences a Finding the geometric Formulating and common sequence solving difference, nth equations based term, and sum Finding a certain on the given of the given arithmetic or scenario arithmetic or geometric mean geometric sequence Finding the nth term of an arithmetic sequence Finding the common ratio and a certain term of a geometric sequenceFormative Lesson 1 “Vocabulary Group Activity “Post It!” Cooperative Map” Learning In groups of In their In their respective In their three, ask respective groups, give as respective them to make groups, create many examples groups, a vocabulary five examples of of objects or complete the map on exercises on situations in real activity under sequence. pattern life which “What to recognition. demonstrate Transfer.” Then, let other patterns Present in class groups each group’s recognize the Write their Performance pattern. answers on a Task. posted manila paper 4 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.

TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCE Share with a In groups of partner the three, develop a steps in finding concept map on the nth term of a arithmetic given sequence. sequenceDEPED COPY Games to earn Cooperative class Learning points/extra points in finding In their respective the nth term of groups, complete the given the activity under arithmetic “What to Reflect sequence on and Further Understand” “Think-Pair- Share” in finding Group Activity the sum of arithmetic Role-play and sequences solve a real-life “Power of Four” problem on activity in arithmetic generalizing the sequence in concepts class learned in preparation for In their respective the summative groups, pose a test real-life problem on arithmetic sequence and let other groups solve the problem “Power of Four” activity where the students will do some brainstorming on their personal experiences which reflect the use of arithmetic sequence 5 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.

TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCE Quiz: Lesson 1 Determining Identifying an Solving problems the next few arithmetic on arithmetic terms of sequence means sequences given the nth Solving real-life term problems on arithmetic Finding the Finding an, a1, sequencesDEPED COPYnth term of a sequence n, or d of a sequence Finding the next term of a Finding/ sequence Inserting arithmetic means Finding the sum of arithmetic sequences Lesson 2 “Power of Illustrating and Cooperative Four” defining a Learning Activity geometric sequence by Use Activity 6: Compare and doing Activity 2: Want Sum in contrast Fold Me Up with determining the arithmetic and a partner formulas for the geometric Cooperative sum of geometric sequences Learning sequence With a partner, In their Cooperative study Activity respective Learning 7: How Well groups, let them Do You Know formulate the Complete the Me in defining nth term of activity under harmonic and geometric “What to Reflect Fibonacci sequences by on and Further sequences doing Activity 4: Understand” The Rule of a Geometric “Carousel Sequence Method” 6 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.

TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCESummative Group game on In their groups, finding pose a real-life geometric problem on means geometric The winning sequence to be group gets extra solved by other points. groups Cooperative Role-playing on Learning the students’ personal In their experiences respective applying groups, answer geometric problems on sequencesDEPED COPY finding the nth term, geometric means, and sum of the given geometric sequence Quiz: Lesson 2 Determining Finding the nth Solving real-life the types of problems on sequences term, a1, or r in geometric and giving the sequences next term for a geometric each type sequence Finding the Finding missing term geometric in a geometric means of the sequence given sequence Finding the sum of finite and infinite geometric sequences Post-Test: Post-Test: Post-Test: Post-Test: Part I Part I Part I Part II Determining Illustrating a Solving real-life Product or the next term harmonic problems Performance of a sequence, sequence involving Task Part harmonic, and sequences Fibonacci sequences 7 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.

TYPE KNOWLEDGE PROCESS/SKILLS UNDERSTANDING PERFORMANCE Finding the Finding the sum Analyzing the common of a sequence or given scenario difference, nth sum to infinity of on sequences term, and sum a geometric of the given sequence Formulating and arithmetic and solving geometric Finding equations based means arithmetic and on the given geometric scenario means Finding the nth term of an arithmetic sequence DEPED COPY Finding the common ratio and a certain term of the given geometric sequenceSelf-Assessment Exit Slip/Card: Ask them to write the following on their Exit Slip/Card 1. All things that they have learned 2. Two questions that they need to clarify 3. One question that they still have in mindAssessment Matrix (Summative Test) Levels of What will I assess? How will I How will IAssessment assess? score? The learner demonstrates Paper and Knowledge understanding of key Pencil Test 1 point for 15% concepts of sequences. every correct Part I response Determine the next term of Items 1, 2, 7, 9, the given arithmetic, 11, and 15 geometric, harmonic, or Fibonacci sequenceProcess/Skills Find the common difference Part I 1 point for 25% or ratio, nth term, and sum of Items 3, 4, 5, 6, every correct an arithmetic sequence or a response geometric sequence 8, 10, 13, 14, 16, 17, and 19 8 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.

Levels of What will I assess? How will I How will IAssessment assess? score? Cite examples of harmonic sequences Find the finite sum or the Part I 1 point for sum to infinity of a geometric Items 18, 20, 21, every correct sequence 22, 23, 24, and response 25 Solve for arithmetic and geometric means Find the nth term of an arithmetic sequence Find the common difference and term of the given geometric sequenceUnderstandingDEPED COPY 30% Solve real-life problems involving sequences The learner is able to analyze Part II Rubric for the given scenario on Items 1 to 7 equations sequences and formulate and formulated solve equations based on the and solved given scenario Rubric for the Product/ Pose and solve problems problemsPerformance using the concepts involved posed and in the given scenario solved 30% Criteria: Relevant Complex Authentic Creative Clear InsightfulFormative assessments that will be used in this unit1. Concept Map. It is similar to a Vocabulary Map where the students will define the given vocabulary or concept on how they understand it. Then, 9 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.

they will give examples and non-examples of the concept and samplerelated problems. This can be done individually or with a partner or in agroup. The answers may vary for each student or each group. See theexample below. Two bikers started at the same Equations in the form point and traveled in opposite ax2 + bx + c = 0; a ≠ 0 directions. One biker traveled Examples: 3 kph faster than the other. 2x2 + 17x + 8 = 0 After 1 hour, they were 15 km 5x2 – 20 = 0 apart. Find the speed of each. 3x2 = 300 Non-Examples: x(2x2 + 17x + 8) = 0 2x3 – 16 = 0 5x4 = 80 DEPED COPY Quadratic Equations2. Concept Web. It is a visual display of the characteristics or categories of a term or concept. See the example below. Equations Linear Equations Quadratic Equations The highest Equations in the Equations in the form The highestexponent of the form ax2 + bx + c = 0. exponent of the a≠0 variable is 1. ax + b = 0 variable is 2. One Two 5x + 110 5x2 = 80 8x2 = 0 One Two variable: variables: =0 3x2 – 27 = 0 variable: variables:5x = 8 + 3x 3x + 5y = 8 8x2 = 32 x2 + y2 = 29 6x = 120 7x = 0 This can be done with a partner or in their respective cooperativelearning groups. It can also be done following the suggested format: 10 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.

DEPED COPY 3. Carousel Method. The teacher will prepare a number of problems, each written on a manila paper and will post them around the room (the number of problems will depend on the number of groups in a class). Each group of students will move around the room and will wait for the teacher to say stop. Each group will solve the posted problem right in front of them where the group stopped. Another way of doing the “Carousel Method” is to let the groups move around the room on the teacher’s signal and each group will contribute to the solution of each posted problem as the groups move around the room. 4. Power of Four Method. As the name suggests, the students will form groups with four members and ask them to form a circle. These four will brainstorm on certain concept(s) and anybody in the group could take his or her turn(s) in contributing his or her ideas on the given concept(s). To make sure that everybody in the group will contribute in the activity, each member of the group will be asked to produce 10 strips of colored paper where he/she will write his/her contributions, and place the strips of paper in the cup. 5. Group Game with Class Points or Extra Points. The students will be working with their respective groups to answer the questions that will be posted by the teacher one at a time. They will write their final answer on an illustration board (one per group). When the teacher says “boards up,” the groups will show their answers. The group which will get the correct answer will receive points. The group which will garner the most number of points will merit class points or extra points. 11 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.

DEPED COPY6. Think-Pair-Share. This is a cooperative learning strategy where the learners are given time to think about the solution to a problem and then, the learners share with their partners their solution. 7. Two-Column Chart Method. This can be used to compare or differentiate two concepts or to support a particular concept.C. Planning for Teaching-Learning This module covers key concepts of sequences. It is divided into two lessons, namely: Arithmetic Sequences and Geometric Sequences. Lesson 2 on Geometric Sequences also includes the other sequences, namely: Harmonic and Fibonacci Sequences. In Lesson 1 of this module, the students will be given activities to help them define a sequence. Then, the students will find the next few terms and the nth term of a sequence. This lesson discusses Arithmetic Sequences, provides the students with exercises to illustrate or define an arithmetic sequence; determine its nth term; find its arithmetic means; determine the sum of the first n terms; and use it to formulate and solve real-life problems. Lesson 2 has two parts: Geometric and other sequences. In the first part, the students will illustrate and define geometric sequences, and differentiate it from arithmetic sequences. They will also determine the nth term of a geometric sequence, find the geometric means, determine the sum of the first n terms of a geometric sequence, determine the sum to infinity, and formulate and solve problems involving geometric sequences. In the second part, the students will determine and define other sequences such as harmonic and Fibonacci sequences. In all the lessons, the students will use their prior knowledge and skills to learn and understand the basic ideas of sequences and distinguish their types. They are also given enough activities to process the knowledge and skills learned and further deepen and transfer their understanding of the different lessons. To introduce the main lesson, present to the students the Kilos Kabataan scenario in the introduction. Discuss with them the situation along the following questions: 12 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.

DEPED COPY Was there any point in your life when you extended help to other people or performed some services for your family, school, barangay, community, or country? Did you ever recognize that the numbers given in the scenario could be represented mathematically in order to come up with quantities that create positive change? Elicit responses from the students for them to realize that the above numbers which represent a pattern could represent some situations in real life and which can be solved throughout the discussion of this module. Objectives: After the learners have gone through the lessons contained in this module, they are expected to: 1. generate and describe patterns using symbols and mathematical expressions; 2. find the next few terms and the nth term of the given sequence; 3. define and describe an arithmetic sequence; 4. enumerate the next few terms and the nth term of an arithmetic sequence; 5. insert means between two given terms of an arithmetic sequence; 6. find the sum of the first n terms of an arithmetic sequence; 7. pose and solve problems involving arithmetic sequence; 8. define and illustrate a geometric sequence; 9. compare geometric and arithmetic sequence; 10. give the nth term and the next few terms of the given geometric sequence; 11. insert means between two given terms of a geometric sequence; 12. find the sum of the first n terms of the given finite or infinite geometric sequence; 13. describe harmonic and Fibonacci sequences; and 14. formulate and solve problems involving geometric sequences 13 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.

PRE-ASSESSMENT: Check students’ prior knowledge, skills, and understanding of mathematics concepts related to sequences. Assessing these will facilitate your teaching and the students’ understanding of the lessons in this module.Answer KeyPart I DEPED COPY 6. A 11. A 16. D 21. C 1. D 7. D 12. A 17. C 22. A 2. A 8. D 13. B 18. D 23. C 3. D 9. B 14. C 19. D 24. D 4. A 10. C 15. A 20. A 25. A 5. BPart II 1. 81 organizations; 120 organizations 2. Number of Rounds 12 34 27 81 Number of Organizations 3 9 3. y  3n 4. 243 organizationsLEARNING GOALS AND TARGETS: Students are expected to demonstrate understanding of key conceptsof sequences such as arithmetic, geometric, harmonic, and Fibonacci; and toformulate and solve problems about these concepts. They are also expectedto investigate mathematical relationships in various situations involvingsequences.Lesson 1: Arithmetic SequencesWhat to KNOW This section provides activities which will lead the students to betterunderstand the key concepts of sequences, particularly, arithmeticsequences. Completing these activities will help students further understandsequences. 14 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.

Tell them that as they go through this lesson, they have to think of thisimportant question: “How are sequences used to model and solve somemathematical ideas and real-life situations?” Start the lesson by asking the students their ideas about patterns. Letthem realize that recognizing and extending patterns are important skillsneeded to understand the concepts of arithmetic sequences. Let them doActivity 1 as a springboard for arithmetic sequences.Activity 1: What’s next? Answer Key 1. 2. 20; 36 3. 16;  26DEPED COPY 4. 243; 177 14 5. 5; 5 2 Ask the students if they were able to find the pattern for each exerciseand ask them also about the mathematical skills or principles that they usedto recognize the pattern and to get the next number. Tell them that the givenset of shapes and the given sets of numbers which follow a definite patternare called sequences. Using the definition of a sequence on page 10 of theLearner’s Material, let the students make a “Vocabulary Map” on sequences.After this, the students can do the formative assessments on patterns beforethey work on Activity 2Activity 2: Term after TermAnswer Key 4. 3, 9, 27, 81, 243 5. 2, 4,  8, 16,  321. 5, 6, 7, 8, 92. 1, 3, 5, 7, 93. 9, 6, 3, 0,  3 Provide the students with an opportunity to learn more aboutsequences. Ask them to perform Activity 3. In this activity, students will do thereverse of Activity 2. This time, they will be given sets of sequences and theywill give the general term or the nth term. Guide the students in getting the nthterm if necessary. 15 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.

Activity 3: Getting to Know YouAnswer Key 4. an  1n1. an n  2 5. an  12. an 2n  1 n3. an  2n Before proceeding to the next activity, let the students summarize theconcepts learned in the previous activities. Provide them with an opportunityto relate or connect these activities to their new lesson which is arithmeticsequence. Tell them to work with their respective groups to complete Activity4. Note: Activity 4 is a matchstick activity. In case matchsticks are notavailable, other materials like popsicle stick, tooth pick, or straw may be used.DEPED COPYActivity 4: What do we have in common?Answer Key 1 2 3 4 5 6 7 8 9 10 Number of squares 4 7 10 13 16 19 22 25 28 31 Number of matchsticks1. Yes, the number of matchsticks is three more than the preceding number of matchsticks.2. By adding 33. 3 Ask the students to make a concept map on arithmetic sequences intheir respective groups before performing Activity 5.Activity 5: More of the Matchstick ActivityAnswer Keya5 a6 a7 a8 … an4+3+3+3+3 4+3+3+3+3+3 4+3+3+3+3 4+3+3+3+3 … 4  3  3  3  ...  3 +3+3 +3+3+3 n1 termsa5 4  43 a20 4  193 a50 4  493 an 4  n 13After completing Activity 5, tell the students that it will be easy for them tocomplete Activity 6 by following the rule of an arithmetic sequence. 16 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.

Activity 6: What is missing?Answer Key 6. 17, 14, 11, 8, 5 7. 4, 9, 14, 19, 24A. 1. 3, 12, 21, 30, 39, 48 8. 4, 0, 4, 8, 12, 16 2. 8, 3,  2,  7, 12 9. 1, 7, 15, 23, 31, 39 3. 5, 12, 19, 26 , 33 10. 13, 7, 1,  5, 11, 17 4. 2, 11, 20, 29, 38 5. 2, 4, 10, 16, 22B. 10, 18, 26DEPED COPY Use Part B of Activity 6 to let the students understand the meaning ofarithmetic means. Then, let them answer Activity 7 as a springboard forfinding the sum of a given arithmetic sequence.Activity 7: Summing Up 4. 231 Answer Key 5. -57 1. 22 2. 35 3. -15Ask the students to make a generalization from the above activity.Activity 8: The Secret of Karl Let the students investigate the derivation of the formula for the sum ofan arithmetic sequence in Activity 8 through “Think-Pair-Share.” Givestudents enough time to think about the problem, then tell them to look for apartner to discuss or share their quick solution to the problem. After thesharing, tell each pair to answer the given questions in the activity.Note: To avoid waste of time in pairing, you may instruct the students tochoose their seatmate as partner. 17 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.

Answer Key 3. Multiply 101 by 50 4. 101 50  5 050Think-Pair-Share Questions1. 1012. 50 In this section, let the students apply the key concepts of arithmeticsequences. Tell them to use the mathematical ideas and examples presentedin the preceding section to answer the activities provided.DEPED COPYWhat to PROCESSActivity 9: How well do you know me? Answer Key Item numbers 1, 5, 7, and 10 are arithmetic sequences because each has a common difference. Complete Activity 2 using the nth term of an arithmetic sequencean  a1  n 1d.Activity 10: More on Arithmetic SequencesAnswer Key1. 99 6. 152. d  7; a1  31 7. 63. 8, 31, 54, 77, 100 8. 144. 6 9. 35. 12 10. 5Give class points/extra point-questions before proceeding to Activity 11. 18 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.

Activity 11: What can you insert?Answer Key 6. 1, 2, 5A. 1. 17 7. 14, 12, 10 2. 18, 30, 42 3. 55, 42, 29, 16 8. 1, 5 , 7 , 3 3 3 4. 15, 20, 25, 30, 35 5. 1, 3 9. a  b 2 2B. 1. 46 2. 0 3. 55; 25DEPED COPY 10. 2x, 3x  y 4. 8 3 5. 2Tell them to complete Activity 4 with a partner.B. 1. 46 2. 0 3. -25 4. 8 5. -2Activity 12: SUMthing to Do 3Answer Key 6. 840 7. 1 080A. 1. 1275 2. 2500 8. 214.5 3. 2550 9. 378 4. 1600 10. 50 5. 330B. a1  26; d  6C. 5D. 2600E. 294 19 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.

What to REFLECT on and UNDERSTAND Ask the students to take a closer look at some aspects of sequences.Briefly discuss questions that are still in their minds. Provide them withopportunities to think deeply and test further their understanding of the lessonby doing the Activity, Beyond the Basics. They will work in their respectivegroups to complete the activity.Activity 13: Beyond the Basics Answer Key 1. This is one of the many examples. , , , , , , , , , _____ If the sequence of the number of triangles is 1, 2, and 3, then the next figure is . However, if the sequence of the number of triangles is 1, 2, and 4 (multiplication by 2), then the next figure is . 2. Concept Map - The answers may vary for each group.DEPED COPYSample Problem AArriitthmetiicc Definition Non-examples Examples Sequence Sequence3. Answers may vary. a. What is the first term of an arithmetic sequence whose fifth term is 20 and whose common difference is 4? 20  a1  44; a1  4 b. What is the 6th term of an arithmetic sequence whose first term is 3 and whose common difference is 2? a6  3  52  7 c. What is the common difference of an arithmetic sequence where the first term and the 7th term are 6 and 18, respectively? 18  6  6d; d  2 20 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.

4. 3x  2  x  2  7x 12  3x  2; x  3 If x  3, then the sequence is 5, 7, 9 where the common difference is 2. Thus, the sequence is an arithmetic sequence.5. 3x  2  x  2  4x  5  3x  2; x  36. a. Day 1 234 5 No. of Participants 20 30 40 50 60b. Let n = number of days; an  number of participants Then an  10n  10 or an  10n  1.DEPED COPYc. Day 6 7 8 No. of Participants 70 80 90 S8  20  90   8   440 2Try this: Week 1 2 3 4…n No. of Minutes 12 18 24 30 … 60 an  a1  n 1d; 60  12  n 16; n  9 Before the students move to the next section of this lesson, give thethree formative assessments, which are done by group, to find out how wellthey understood the lesson. Refer to the Assessment Map. After these groupassessments, let the students answer a quiz on this lesson. 21 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.

DEPED COPYWhat to TRANSFER Give the students opportunities to demonstrate their understanding ofarithmetic sequences by doing a practical task. Give each group enough timeto present their respective Performance Task. Use the rubrics for the writtenreport and oral presentation in grading or assessing each group’spresentation. Then, use the “Power of Four” activity for the students togeneralize or sum up all the things learned from this lesson in preparation fortheir summative test. Ask the students also if there are concepts or ideas onarithmetic sequence which need to be clarified.Activity 14: Reality Series Give the students opportunities and enough time to demonstrateunderstanding of arithmetic sequence by innovating a Reality Series. Use therubric for the written report about the chosen real-life situation. Ask thestudents if there are concepts on sequences and arithmetic sequence thatneed to be clarified further.Summary/Synthesis/Generalization: This lesson is about sequences, particularly, arithmetic sequences andtheir applications in real life. The lesson provides the students withopportunities to illustrate sequences and arithmetic sequences using practicalsituations. Moreover, they are given the chance to create problems onsequences and arithmetic sequences as illustrated in some real-life situations.Their understanding of this lesson and other previously learned mathematicsconcepts and principles will facilitate their learning of the next lesson, Geometricand Other Sequences.Lesson 2: Geometric and Other SequencesWhat to KNOW This section provides activities which will introduce the students to thekey concepts of geometric and other sequences. Completing these activitieswill help students further understand sequences. Tell them that as they gothrough this lesson, they have to think of this important question: “How aresequences used to model and solve many mathematical ideas and real-lifesituations?” Start the lesson by recalling arithmetic sequences. Tell them also thatin this lesson, they will learn three more types of sequences. Let them answerActivity 1 as a springboard for geometric sequence. 22 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.

Activity 1: Divide and ConquerAnswer Key1. 4 6.  1 72. 3 7. 23. 1 2 8. a4. 2 9. k k 15. 1 3 10. rDEPED COPY Reiterate to the students the importance of the concept of ratio tobetter understand the next sequence. With a partner, let them completeActivity 2.Activity 2: Fold Me UpAnswer Key3. Completed table Square 123 Area 32 16 8Answers to Guide Questions:1. 32 sq units; 16 sq units; 8 sq units2. Yes, there is a pattern. Each area is found by getting half of the previous area.3. 32, 16, 84. No. There is no common difference.5. 6th term = 1 To test the students’ understanding of geometric sequences, let themanswer orally Activity 3. Encourage maximum participation from the studentsat this point. 23 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.

Activity 3: I’ll Tell You What You Are Answer Key Items 1, 3, and 4 are geometric sequences. Before proceeding to the next activity, the students compare andcontrast arithmetic and geometric sequences through “Power of Four” activity.Then, tell them to form groups of three to complete Activity 4.Activity 4: The Rule of a Geometric SequenceDEPED COPYAnswer KeyAnswers to Guide Questions:1. 2 is the first term.2. 3 is the common ratio.3. The exponent of 3 is 1 less than the position of the term.4. n 15. an  2  3n16. an  a1r n1Answer Activity 5. Require maximum participation from the students.Activity 5: Missing You Answer KeyA. 1. 3, 12, 48, 192, 768 6. 2,  4,  8, 16,  32,  642. 8, 16, 32, 64, 128 7. 256, 128, 64,  32, 163. 120, 60, 30, 15, 15 , 15 8. 27, 9, 3, 1, 1 24 3 9. 1 , 1, 4, 16, 64, 2564. 5, 10, 20, 40, 80, 160 45. 4 , 4, 12, 36, 108, 324 3 10. 5x2, 5x4, 5x6, 5x8, 5x10B. Since we will insert 3 terms, then there will be 5 terms in all. Then a5  2r 4  32; r  2. Note that there are two possible sets of missing terms: 4,8,16 and 4,8,16. Therefore, we have two sets of geometric sequences: 2, 4, 8, 16, 32 and 2,  4, 8, 16, 32. 24 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.

After completing Activity 5, lead the students to the definition ofgeometric means. Let them study the example, then give them 2 or 3exercises about it. Then, in their respective groups, let them be guided byActivity 6 to determine the formulas for the sums of geometric and infinitegeometric sequences.Activity 6: Want Sum?Answer KeyPart 1: Given: 81, 27, 9, 3, … Let S4  81 27  9  3. The common ratio is 1, so we multiplyDEPED COPY 3both sides by 1 . Now, we subtract the second equation from the first 3equation: S4  81 27  9  3   1 S4  27  9  3  1  3 ___________________________ 2 S4  81 1 3 Thus, S4  80  2  120 3Part 3: S1 S2 S3 S4 S5 2 6 14 30 62Answers to Guide Questions1. r  22. Sn increases without bound as n increases.3. No S1 S2 S3 S4 S5 5 -20 105 -520 2 605Answers to Guide Questions1. r  52. Sn is positive if n is odd and negative if n is even. If n is odd, the values of Sn decreases without bound. If n is even, the values of Sn increases without bound.3. The sequence has no finite sum. 25 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.

Recall briefly arithmetic and geometric sequences. Then, with apartner, complete Activity 7 which will lead them to the definitions of harmonicand Fibonacci sequences.Activity 7: How well do you know me?Answer Key1. GS; r  3 6. Neither 7. GS; r   12. AS; d  6 6 8. AS; d  0.2 9. Neither 10. AS; d  93. NeitherDEPED COPY4. GS; r1 55. Neither Give the students enough time to recognize harmonic and Fibonaccisequences and let them study the given examples. Recall the details of thefour sequences briefly before moving to Activity 1 of What to Process.What to PROCESS In this section, let the students apply the key concepts of geometricsequence. Tell them to use the mathematical ideas and the examplespresented in the preceding section to answer the activities provided.Activity 8: Do you remember me?Answer Key1. AS; 40 6. HS; 1 82. GS; 1 7. AS; 14 2 2433. GS; 1 8. HS; 6 354. FS; 89 9. GS; 4865. AS; 7 10. GS; 8 4 125 Use the nth term of a geometric sequence to answer Activity 9. Theactivity can be completed with a partner or in groups of three. 26 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.

Activity 9: There’s More on Geometric Sequences Answer Key1. a5  15002. Since a2  a1r, then a1  3. Therefore, a6  a1r 5  96.3. If we have k 1  4k  2 , then k  1 or k  5. k 3 k 1 34. Using an  a1r n1, we have 768  6 2 n1 or 128  2n1 or 128  2n  21.DEPED COPY Therefore, n  8.5. Since a4  a1r 3 and a2  a1r, dividing a4 by a2 we have r  2. Therefore, a1   3 . 8 Recall briefly the concept of geometric mean before proceeding toActivity 3. Then, give the items of Activity 3 as class points or extra points.Activity 10: Finding Geometric Means Answer Key A. 1. Let a1  16 and a5  81. Since a5  a1r 4, then 81 16r 4. Thus, r  3. 2 If r  3 , the geometric means are 24, 36, 54. 2 If r   3 , the geometric means are 24, 36,  54. 2 2. Let a1  256 and a5  1. Since a5  a1r 4, then 1 256r 4. Thus, r   1 . 4 If r  1 , the geometric means are 64, 16, 4. 4 If r   1 , the geometric means are 64, 16,  4. 4 27 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.

3. Let a1  32 and a4  4. Since a4  a1r 3, then 4  32r 3. Thus, r   1. 2 Therefore, the geometric means are 16 and -8. 4. Let a1  1 and a3  64 . Since a3  a1r 2, then 64  1 r 2 . Thus, 3 3 33 r  8. If r  8, then the geometric mean is 8 . 3DEPED COPYIf r  8, then the geometric mean is  8 . 3 5. Let a1  2xy and a4  16xy 4. Since a4  a1r 3, then 16xy 4  2xyr 3. Thus, r  2y and the geometric means are 4xy 2 and 8xy3.B. Let a3  4 and r  4  1. Since a3  a1r 2, then 4  a1r 2. Thus, 32 8 a1  256.C. Let a1  k and a3  1. Since a3  a1r 2, then 1  kr 2. Thus, r   1. k k k If r  1 , then the geometric mean is 1. k If r   1 , then the geometric mean is 1. kD. Let m, 2, 3, n be the geometric sequence. Then, r  3 . 2 Thus, m  2  3  4 and n  3  3  9 . 23 22E. Let a, 6, b, 24, c be the geometric sequence (variables may vary). Let a1  6 and a3  24. Since a3  a1r 2, then 24  6r 2. Thus, r  2. Since the three numbers are positive, we take r  2. Hence, a  6  2  3; b  6 2  12; c  24 2  48.Tell the students to complete Activity 4. 28 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.

Activity 11: Sum of Terms in a Geometric SequenceAnswer Key B. 1. 256 C. 484 3A. 1. 484 2. 63 2. 1 D. a1  2a1 3. 0 2 1 r 4. 3 5. 6 3.  16 1 2 3 1 r 4. 144 r1 5 2DEPED COPY 5. The sum does not exist since r  1.Note: You may opt to use technology (e.g., scientific calculator, Excel, etc.) indiscussing geometric sequences.What to REFLECT on and UNDERSTAND Ask the students to take a closer look at some aspects of geometricsequences. Briefly discuss questions that are still in their minds, if any. Provide them with opportunities to think deeply and test further theirunderstanding of the lesson by doing the Activity: Miscellaneous Problems inGeometric Sequences. They will work in their respective groups to completethis activity.Activity 12: More Problems in Geometric Sequences Answer Key 1. Concept Web – The answers may vary for each group.GeometricSequence 29 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.