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  1. Concordia Publishing House
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Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials i. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Br. Armin A. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D.

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Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida Editor: Maxima J. Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Maylani L. Galicia, Ma. Corazon P. Loja, Jones A. Quan Illustrator: Cyrell T. Navarro Layout Artists: Aro R. Tuguinayo Jr. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr. All rights reserved. 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.

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Learning Outcomes. Planning for Assessment. Planning for Teaching-Learning. Summative Test. After each module, answers to the summative test are provided to help the teachers evaluate how much the learners have learned. Glossary of Terms. Important terms in the module are defined or clearly described. References and Other Materials.

This provides the teachers with the list of reference materials used, both print and digital. Learning Outcomes Content Standard: The learner demonstrates understanding of key concepts of combinatorics. Performance Standard: The learner is able to use precise counting techniques in formulating conclusions and in making wise decisions. Illustrate the permutation of objects 2. Derive the formula for finding the number of permutations of n objects taken r at a time 3. Solve problems involving permutations 4.

Illustrate the combination of objects 5. Differentiate permutation from combination of n objects taken r at a time 6. Derive the formula for finding the number of combinations of n objects taken r at a time 7. Solve problems involving permutations and combinations All rights reserved. Essential Question: How do the concepts of permutations and in combinations help in forming conclusions and in making wise decisions? Transfer Goal: Students will be able to apply the key concepts of permutations and combinations in forming conclusions and in making wise decisions. Enumerate situations in real life that illustrate permutations and combinations 2.

Formulate equations involving permutations and combinations that represent real-life situations 3. Solve equations involving permutations and combinations 4. Formulate and solve problems that involve permutations and combinations 5. Role-play to demonstrate the applications of the concepts of permutations and combinations in formulating conclusions and in making wise decisions. How will I assess? How will I score? Illustrate permutations and combinations. Planning for Teaching-Learning This module covers the key concepts of Combinatorics, namely, Permutations and Combinations. It is divided into two lessons: Lesson 1 — Permutations and Lesson 2 — Combinations.

D EPED C O PY In Lesson 1, students will identify real-life situations that involve permutations, illustrate permutations of objects, and solve problems involving permutations of n objects taken r at a time. In Lesson 2, students will identify situations that involve combinations, differentiate them from those that involve permutations, and solve problems that involve combinations, or both permutations and combinations. In both lessons, students are given appropriate activities to develop their knowledge, skills, and understanding of permutations and combinations, while utilizing the other mathematics concepts they have previously learned.

As an introduction to the lesson, show the students the pictures below, then ask the questions that follow. Look at the pictures. Have you ever wondered why some locks such as the one shown have codes in them? Have you ever realized that there are several possible ways in doing most tasks or activities, like planning a seating arrangement or predicting the possible outcomes of a race?

Have you ever been aware that there are numerous possible choices in selecting from a set, like deciding which combination of dishes to serve in a catering service or deciding which dishes to order from a menu? Do you know that awareness of these can help you form conclusions and make wise decisions? Encourage students to find the answers to these questions and discover the various applications of permutations and combinations in real life through this module.

Answer Key Part I 1. C Part II 2. B Use the Rubric to 3. C Solutions of some of the problems can be found at the end of this section. Remind them that as they go through the lesson, All rights reserved. Since it is a review activity, this strategy is suggested to allow students to think individually about the answers first, make their own list as required, then discuss with a partner to come up with a final answer, which they will share to the class afterwards. This activity aims to make them list their answers first and then lead them to recall the FCP.

Answer Key A.

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Another way of answering item 1 is through a tree diagram. Possible codes containing the four digits 7, 4, 3, 1: The list must be made systematically to ensure completeness. There are 24 possible codes. The list is quite long. Answers to Guide Questions: a. We determined the different possibilities asked for by listing. We also used tree diagram as well as the Fundamental Counting Principle.

Another way of finding the answers is by making a table.

You may make an example on the dice after this particular activity. It is hard making a list when the list is long. Having recalled the Fundamental Counting Principle, let the students do Activity 2. This activity provides them some more opportunities to use the Fundamental Counting Principle. When students present the solutions to the class, ask some questions regarding the parts of the solution, such as why are the factors being used decreasing, or how do they know how many factors will be used in the multiplication process, and other pertinent questions.

Activity 2. Count Me In! Answer Key 1. We found the answer to each question by multiplying the number of ways that each successive subtask can be done in order to finish the whole task. In situations which involve choosing, we multiply the number of choices for the first position by the number of choices for the second position by the number of choices for the third position, and so on.

Solutions: 1. Divide the class into groups of 4. All groups will answer all items but call on a particular group to share to the class their results in a specific item. Activity 3. Does order matter? In situations 1, 3, 6, 7, and 9, order is important. Number 1: example : 1st place — runner number 8 2nd place — runner number 5 3rd place — runner number 4 Number 3: Example: She may arrange the plants according to height, or according to kind, according to appearance, or any basis she wants.

Number 6: Example: 1 possible result is: President — Mrs. Cavinta Vice President — Ms. Ternida Secretary — Mrs. Perez Treasurer — Mr. Cabrera This is different from other possible results, such as: President — Mr. Cabrera Vice President — Mrs. Perez Secretary — Mrs. Cavinta Treasurer — Mrs. All these arrangements are different from one another. Another possible arrangement, which is different from the first is: Valerie — Gillian — Aira — Khryzza — Joyce There are many other different possible arrangements. Each possible arrangement is called a permutation.

Ask students to perform Activity 4 in small groups. This hands-on activity will emphasize why in some cases or situations, order matters. Activity 4. Assume the four number cards are: You may use other numbers. Each arrangement is called a permutation. Let students summarize what they have learned so far about permutations. Allow them also to make a connection between the Fundamental Counting Principle and permutations based on what they know so far. Activity 5 provides them the chance to practice their computational skills as they solve for the value of P, n, or r in equations involving permutations.

Activity 5. Warm That Mind Up! We calculated the different permutations by applying the formula!

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Aside from the concept of permutations, we applied the meaning of the term factorial, which says that the factorial of a number is equal to the product of the number and all the positive integers less than it. There is some difficulty when the figures are large. Let students do Activity 6. This activity requires students to apply their knowledge and skills to solve simple real-life problems that involve permutations.

You may now ask them to work individually. Mission Possible Answer Key 1. This activity, consisting of more problems on permutations, with some restrictions or conditions, expects students to think critically, instead of using the formula right away. Activity 7. Decisions from Permutations Answer Key 1. We multiply by 2!

Consider first the case that the two said persons always sit beside each other. Like in b , it is as if there are only 11 people. You may now ask them to visualize and cite situations where their knowledge of permutations can help them formulate conclusions and make wise decisions. In this activity, they will explain how to determine if a situation involves permutations, differentiate among the different kinds of permutations, and make a decision based on their knowledge of permutations.

Activity 8. Reason Out Answer Key 1. A situation or problem involves permutations if the order of the objects is important. The circular permutations of n objects refers to the different arrangements of the objects when arranged in a circle; it is obtained with the help of the formula! We use the words distinguishable permutations to refer to the different permutations of n objects when some of them are alike. It is calculated by using the formula!!! The two answers are equal. Bridges blends direct instruction, structured investigation, and open exploration.

It taps into the intelligence and strengths of all students by presenting material that is as linguistically, visually, and kinesthetically rich as it is mathematically powerful. As educators, you strive to make well-informed curriculum adoption decisions. Color covers, black and white pages. All guides are also available in electronic form on the Bridges Educator Site, which includes additional support resources. Includes a wide variety of age-appropriate assessments at each grade level, ranging from interviews, observation tips, and short performance tasks for the youngest students to unit pre- and post-assessments, mid-unit checkpoints, and more extensive performance tasks for Grades 2 and up.

A free account is included with the purchase of every Bridges classroom package. While the content of student books is required, the purchase of pre-printed consumable copies is optional. Also available in Spanish. The Number Corner Student Book presents opportunities for independent practice and engagement with the skills and concepts covered in the daily workouts.

Home Connections—family-friendly assignments that include games and activities as well as worksheets for students to complete independently—offer another source of practice and reinforcement. Bridges classroom packages come complete with the materials teachers need to make math come alive. Bridges includes colorful pre-made components including card decks, spinners, and game boards that support over three dozen Work Places at each grade level.

Each card features a word and multiple illustrations on the front and a working definition on the back. Word resource cards are also available as a free app. All grade levels include children's literature recommendations and most include one or more titles. Find out more. An independent study found that students who use Bridges perform better than their peers.

Read full report. Bridges draws upon decades of research into the best methods for teaching and learning math. View research base. This two-day summer workshop brings our workshop leaders to your district. They work face-to-face with your classroom teachers, walking through unpacking of the curriculum components and exploring math practices in the contexts of the models and strategies.

Each workshop is packed with information, pedagogy, and hands-on activities. Teachers explore resources, dive into Number Corner, and do math that resembles a Bridges classroom. Bridges Getting Started workshops are scheduled either by grade-level bands, such as K—2 and 3—5, or are grade specific. A maximum of 30 participants is allowed in each session, but multiple sessions can be scheduled.

Thank you for sharing your expertise, bringing laughter to the room, and cultivating confidence for our teachers. Learn More. This three-day workshop is designed to equip instructional leaders to lead their own Bridges Getting Started Workshops in-district for new hires or new-to-grade-level teachers.

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Participants will experience a condensed Bridges Getting Started Workshop covering both K—2 and 3—5 grade bands. They'll leave with the knowledge and materials necessary to offer these workshops in their own district.


This face-to-face workshop is appropriate for districts after their first year of implementing Bridges in Mathematics. This week online course is offered in both fall and winter. It is designed to support math coaches and instructional leaders working towards increasing student achievement and teacher content knowledge during a Bridges in Mathematics implementation.

Participants read, reflect, and respond to prompts every two weeks. The readings were helpful because I could directly apply them to the work I have been doing. Prior to the workshop, we work with you to determine these needs. Our workshop leaders come to your district and work face-to-face with your classroom teachers. These leaders offer guidance in specific areas where teachers struggle and model methods for successful implementation.

We appreciated having an expert on-site to answer our questions, share resources, and share their experience. The annual Leadership Institute offers lead teachers, coaches, and curriculum specialists the tools for developing and sustaining a successful implementation of Bridges in Mathematics, Number Corner, and Bridges Intervention. This institute is limited to districts after their first year of implementing Bridges in Mathematics 2nd Edition. I left feeling enthusiastic and excited as we head into our second year of using Bridges. Bridges has been so effective.

It creates an exciting and fun learning environment. Roberts and her family have lived in Hanoi for nearly eight years. Her hobbies include family time - going on dates with her husband, spending time with her five children - from playing volleyball with her three younger ones to Skyping with her two oldest who attend university in the USA, travel and playing music, particularly ukulele and piano.

Ngo of her work at Concordia, adding,. Prior to joining Concordia in August , Ms. Katherine Sharpe is entering her 12th year in teaching. Petersburg, Florida in the United States. In , Ms. Sharpe took part in a Teacher Exchange in Nagano, Japan. The best thing about teaching here are the students and families I am able to work with and the impact I have in their lives. I love to watch my students learn and experience new things. I also love the positive and supportive work environment.

Prior to joining Concordia in August, , Ms. Sharpe taught elementary school in St. Her hobbies include walking, reading, relaxing and socializing with family and friends, going to the beach and being outside.

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Brain Gym is a California USA -based non-profit dedicated to promoting the concept that intentional movement facilitates better learning. Concordia provides varied opportunities to equip students, faculty, and staff for excellence. Prior to joining Concordia in September , Ms. It is working with a family that encourages personal and professional growth. Concordia also acknowledges your strengths and weaknesses and helps its teachers and staff use their talents and skills, wherever needed and helpful. I love Concordia! Wood has been with Concordia since July and has been one of the pillars of our Elementary School program.

Herself an ongoing learner, Mrs. Other Professional Development Mrs. Wood has taught in international schools around the world, including China, Thailand and the Philippines. Her husband, to whom she has been married for 20 years, is an executive chef and they enjoy learning about life and other cultures through travel.

A dedicated professional educator, Mrs. The teachers and the principal have been very helpful collaborating and supporting me in my first year at Concordia. I have really enjoyed this experience and look forward to teaching at Concordia in the years to come. I am glad to be working with Mrs. Nelson and Mrs. Shih joined Concordia two years ago after seven years teaching at Lytle Creek Elementary School in the USA, where she also served as an Academic Coach mentoring new teachers in classroom management and curriculum development.

We teach students from all around the world who are learning to be responsible global citizens who choose to make a difference in our world! We work with a staff who not only care about academics but also care about the physical, emotional, and social growth of the child. I love our school and I love my job! What I like best about teaching at Concordia is that it offers unlimited possibilities to influence students, imparting to them the joy and excitement of learning, the passion of discovery and the magic of an inquisitive mind.

There are so many things that I learn from my colleagues and students everyday! Lubaton taught at international schools in Manilla, Philippines and in Hanoi. Lubaton has a eight-year-old son in the Philippines and her hobbies include cooking, baking, painting, reading and travel. I am looking forward to working in an elite school and continuing to grow as an educator.

Beard of her work at Concordia. Beard and her family lived in Central Asia for 10 years before moving to Vietnam in , and she is fluent in Russian. Beard taught at elementary schools in California, Kyrgyzstan and Hanoi. Snider of what she likes best about teaching at Concordia. An experienced international educator, Mrs. Emily McKinney is beginning her fifth year of working as an educator and her second year working in Hanoi. After working as a substitute teacher and coach in Wisconsin and Chicago, she moved to Hanoi to teach English.

Her experience with Concordia International School Hanoi has been nothing but positive. A truly global educator, Mr. Stephen Conroy is entering his eighth year of teaching middle and high school. A dedicated professional educator and administrator, Mr. Conroy was also named Secondary Assistant Principal for Concordia for the school year. Conroy of his work at Concordia. Originally from Queens, New York, Mr. Robinson taught in the School District of Black River Falls, Wisconsin before joining Concordia at the start of the school year.

Robinson has attended more than 70 hours of Reading Workshop and Writing Workshop training as part of her ongoing educational training. Umphenhour is also certified to teach Special Education for grades K Prior to joining Concordia in Hanoi in , Mr Umphehour taught and served as principal and assistant principal in St. Umphenhour of his work. Jennifer Zimbrick has been teaching for 25 years. Prior to returning to Concordia in August - Mrs. Mills is a recent graduate of doctoral studies and is proud to hold her PhD after several years of study.

She vows to never return to school The upper school team at Concordia is made up of brilliant, passionate educators who push students well beyond the limits they think are possible. In addition to Concordia, Dr. Mills teaches at the university level in the College of Education at Ashford University in San Diego, CA, where she continues to contribute to the online program, and has taught at elementary and international schools both at home, in San Jose and Monterey, California, and Miami Beach, Florida, and abroad, in Rodney Bay, St.

Lucia, Nicosia, Cyprus, and Cairo, Egypt. In the moments when she is not working or researching, Dr. Mills enjoys walking her amazing pups, Kalliope and Cairo, drinking coconut coffee from Cong Caphe, or watching super scary movies in the dark! Prior to joining Concordia Shanghai, Mr. He has expanded our high school curriculum to include both AP World History and Big History, a cutting-edge interdisciplinary course covering the Big Bang to the present.

Additionally, he is piloting ON World History, a new web-based World History course that looks at the Big Questions of World History rather than a memorization of dates, names, and places. Sharpe, Married to Debbie since , they have two boys and five grandchildren. Burns loves to bike and has toured Ireland, Thailand, and the Benelux countries in Europe, in addition to extensive biking around Shanghai, Qatar, and now Vietnam. His other hobbies include history, travel, baseball and board games.

Prior to joining Concordia in July , Dr. He helped raise the standard of instruction by using the Cambridge International Diploma for Teachers and Trainers to train new lecturers. After nine months, Dr. Nelson was promoted to be a head of school, supervising the pre-university program, the American degree transfer program and the English for international students program.

In three years at the School of Pre-University Studies, the school had three students accepted into Oxford and Cambridge and several other students admitted into similarly prestigious universities in the UK, US and Australia. He also developed curriculum and partnerships with universities. I love the high school students. On the whole, they are smart, clever, funny, hard-working and welcoming of new students. I enjoy the opportunities to learn along with the students in my classes.

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We get to discuss and debate topics from literature and psychology to politics. I also appreciate that, for the first time in our 34 years, I am working in the same school with my wife. Nelson loves learning, teaching and encouraging people. He enjoys sports and music and food of all kinds, plus sketching, reading, tech gadgets, traveling and good coffee with good conversation.

His wife Jeni teaches Preschool at Concordia. High school Math and Physics teacher Neil Whitehead has been teaching for 26 years. Following his dramatic turnaround of American School Foundation of Monterrey, Mexico in the high school Math department, which went from very low scores to a globally-recognized program in his very first year with the school, Mr. Whitehead was invited to Washington D. This team was comprised of the top four elite Math students, selected from State Department-recognized schools around the world.

Whitehead prepared and coached the students to compete in a competition called Math Counts against the top four Math students from from each US state. We have great students, professional colleagues and a welcoming community. A passionate mathematician, Mr. Whitehead has also led many community service opportunities during the course of his career, and, together with his wife, Rosy, has raised money to build three libraries in the suburbs of Shanghai for underprivileged students.

High school Science teacher Dr. Brent Kilback has been an educator for 34 years. After a distinguished career that has taken him from teaching positions at Concordia University, Edmonton, to international schools in Guatemala and Papua New Guinea, where he served as head of school, Dr.

Kilback joined Concordia in to return to his core passion of instructing in the classroom. A dedicated scientist devoted to ongoing research, Dr. Kilback has a particular interest in microbiology and has done laboratory research in the area of cytokinesis, spindle fiber assembly and microtubule formation in eucaryotic cell. Samuel Cook is starting his 25th year in education and 20th year working overseas.

Sam has taught mathematics and been a school administrator in China, the Philippines, and Indonesia. He is happy to be joining the Concordia family to teach high school mathematics. Cook said. Sam is an avid runner and is currently training to compete in his first triathlon later this year.

Emanuel Santos has been an educator and a researcher for more than 13 years. Before moving to Vietnam in , Dr. Santos founded two start-up companies and was an invited assistant professor and researcher at the University of Lisbon, Portugal, for six years. Santos has published many scientific articles in the subjects of Big Data, Data Quality and Argumentation Theory, and was also a reviewer for Scientific Journals and International Conferences. Coming from a family of educators, Dr. Santos continues to pursue his passion at Concordia by teaching college courses to energized high school students.

As an educator, it has been always my goal to inspire and provide the students with the best learning experience possible according to their needs and expectations. Concordia provides me with all the conditions I need to perform at my best. In , Dr. Santos visited more than 15 countries during his first around the world trip. He loves to run, cycle and play tennis. He enjoys watching sarcastic comedies every day. My husband and two boys have been fully immersed since January and my daughter and I are so happy to be able to be more intimately involved this school year.

A gifted athlete as well as an educator, Mrs. Sutherland has represented her country on the soccer field. Ian Sutherland. Her hobbies include travel and adventures with her family as well as playing soccer. Linda Bloemberg boasts 19 years in the classroom, with 15 of those years as a school counselor. I enjoy working with like-minded colleagues in a positive environment. McNaull regularly presents to staff and parents on English Acquired Learning techniques and methodology.

They treat teachers with respect and never fail to provide encouragement and share their wealth of knowledge to whoever needs it. Pat Copple has been teaching for 39 years. Married for 44 years, Mrs. Copple and her husband Steve have lived in Asia for 34 years - Hanoi for 24 of those years - and have three children who grew up in Jakarta and Hanoi. An experienced international educator, Ms. Noye on joining this August. A respected international educator, Mrs.

The facilities are beautiful. Everything is new. I appreciate the expertise represented among the faculty and of course, the students are who I like best. Parents are supportive and involved in our preschool program. Concordia maintains a growth mindset instead of a fixed mindset. The Vietnamese staff keep us going with their professionalism, their knowledge and their contacts. Virginia Rojas. The course is offered through the Principals Training Center. Prior to joining Concordia in July , Mrs.

Nelson taught at elementary schools in Florida, Texas and South Carolina USA , before shifting her career overseas, where she taught preschool and elementary school in Kuala Lumpur, Malaysia and Qatar, where she also served as Technology Integration Facilitator. I love to help students learn the Vietnamese language and experience various cultural activities, as well as understanding more about the city and country in which they are living. The students make my day whenever they come to my class and proudly share their stories of applying Vietnamese in daily life, from helping their parents to shop, to ordering food in a restaurant.

A citizen of the world, Ms. Nguyen is fluent in English, speaks Mandarin at an intermediate level and Portuguese at a basic level. Nguyen has visited more than 30 countries. We work together to help the students learn the Vietnamese language and organize different learning activities for a diverse group of students.