Hybrid Math 10 Q3 M1 W1 V2 PDF

Third Quarter Module 1 Week 1 Introductory Message For the facilitator: This module was collaboratively designed, developed and evaluated by the Development and Quality Assurance Teams of SDO TAPAT to assist you in helping the learners meet the standards set by the K to 12 Curriculum while...

Third Quarter Module 1 Week 1 Introductory Message For the facilitator: This module was collaboratively designed, developed and evaluated by the Development and Quality Assurance Teams of SDO TAPAT to assist you in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. As a facilitator, you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module. For the learner: This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. The following are some reminders in using this module: 1. Use the module with care. Use a separate sheet of paper in answering the exercises. 2. Don’t forget to answer Let’s Try before moving on to the other activities included in the module. 3. Read the instruction carefully before doing each task. 4. Observe honesty and integrity in doing the tasks and checking your answers. 5. Finish the task at hand before proceeding to the next. 6. Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it! 2 QUARTER 3 WEEK 1 Let’s Learn After going through this module, you are expected to illustrate permutation of objects, list the possible ways a certain task or activity can be done, and apply the formula for finding the number or permutation of n objects taken r at a time. Let’s Try Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. 1. It refers to the different arrangements of a set of objects. A. permutation B. combination C. selection D. differentiation 2. Which situation illustrates permutation? A. assigning rooms to conference participants B. selecting 15 questions to answer out of 20 questions in a test C. choosing 3 science books to buy from a variety of choices D. forming a committee of senators 3. How many ways can 7 potted plants be arranged in a row? A. 5 040 B. 2 520 C. 720 D. 210 4. What is P(10, 5)? A. 20 240 B. 30 240 C. 40 240 D. 50 240 5. How many four-digit numbers can be formed from the numbers 1. 3, 4, 5, 6, 8, and 9 if repetition of digits is not allowed? A. 720 B. 540 C. 360 D. 840 6. Which of the following expressions represents the number of distinguishable permutations of the letters of the word COMBINATIONS? 12! 12! 12! A. 12! B. C. D. 2!2!2!2! 5! 2!2!2! 7. In how many ways can 7 people be seated around a circular table if two of them insist on sitting beside each other? A. 1,440 B. 720 C. 360 D. 240 8. Which of the following is equal to P [n , (n-1)]? A. n B. n! C. n(n-1) D. n(n-1)! 9. If (7,r ) = 840, what is r? A. 2 B. 3 C. 4 D. 5 3 10. If ( n, 3) = 720, then n =____. A. 11 B. 10 C. 9 D. 8 Alternatively, you may answer these questions online! Use this link on your cellphone, laptop, or desktop: https://forms.gle/jfSM5zKQuC29Qm637 Use proper capitalization to activate the link. You will see your score after completing the test. Make sure to screenshot your work as a proof to your teacher then write your score in the box. Make sure you are connected to the internet! Lesson PERMUTATIONS 1 Look at the pictures shown below. 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 in a menu? Did you know that awareness of these can help you form conclusions and make wise decisions? Find out the answers to these questions and discover the wide applications of permutations. Let’s Recall Activity 1: How many ways? A close friend invited Sofia to her birthday party. Sofia has 4 new blouses (stripes, with ruffles, long-sleeved, and sleeveless) and 3 skirts (blue, white, and black) in her closet https://webstockrevie https://webstockrevie w.net/explore/clipart- w.net/explore/clipart- for the occasions. clothes-striped-shirt/ clothes-striped-shirt/ 1. Assuming that any skirt can be paired with any blouse, in how many ways can Sofia select her outfit? ________ List the possibilities. _________________________________ 4  ______________________________________________________ 2. How many blouse-and-skirt pairs are possible? ______ 3. Show another way of finding the answer in item 1. List down all the possible set of outfits using the other way which is the tree diagram. (Continue the tree diagram) blouse skirt red stripes pink black __________________________________________________ THE FUNDAMENTAL COUNTING PRINCIPLE If there are a ways for one activity to occur, and b ways for a second activity to occur, then there are 𝑎 × 𝑏 ways for both to occur. Note: The counting Principle only works when all of the choices are independent of each other. One choice does not depend upon another choice. If one choice affects another choice, then this simple multiplication process will not yield the correct answer. Let’s Explore Activity 2 Tell Me! Three runners join a race. In how many possible ways can they be arranged as first and second place? 1. Use listing or tree diagram _________________________ 2. How about if it involves ten runners, can you list down the possible ways that they can be arranged as first, second and third place? ____________________________ https://pngtree.com/freepng/carto 3. Is there an easiest way to determine the possible on-vector-free-running- competition-sprint_4754834.html arrangements? _____________________________________ 5  Let’s Elaborate Notice that we are determining the different ways of arranging and selecting objects out of a given number of objects, without actually listing them. On the later part of this module we have discussed about FCP or the Fundamental Counting Principle. And as we continue, we will unleash more about determining ways of arranging and selecting objects. Permutation is defined as an arrangement of elements in an ordered list. It is used to determine how many arrangements are possible in the set of variables considering a sequence and/or a condition. Given that we have n number of elements. To determine how many possible arrangements are there with n elements, we are going to use the formula 𝑷 = 𝒏(𝒏 − 𝟏)(𝒏 − 𝟐) ⋯ 𝟑 ∙ 𝟐 ∙ 𝟏 = 𝒏! Suppose we are to arrange the first three letters of the alphabet A, B, C in a manner that the letters shall be put in a location or space not more than once, hence the arrangement shall be ABC | ACB | BAC | BCA | CAB | CBA 1st 2nd 3rd 4th 5th 6th Using the formula, `𝑃 = 𝑛! = 3! = 3 × 2 × 1 = 𝟔 There are six (6) possible arrangements for the first three letters of the alphabet. Another example A. Evaluate the following expressions 11! 1. 4! 4. 6!5! 11! 2. 5!3! 5. (11−3)! 10! 3. 8! Solutions 1. 4! = 4 ∙ 3 ∙ 2 ∙ 1 = 24 2. 5!3! = ( 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 ) ( 3 ∙ 2 ∙ 1 ) = 720 10! 10 ∙ 9 ∙8 ! 3. = = 10(9) = 90 8! 8! 11! 11 ∙ 10 ∙ 9 ∙8 ∙ 7 ∙6! 11 ∙ 10 ∙ 9 ∙8∙7 55,440 4. = = = = 462 6!5! 6!5! 5 ∙ 4 ∙ 3 ∙ 2∙ 1 120 11! 11 ∙ 10 ∙ 9 ∙ 8 ! 5. = = (11)(10)(9) = 990 (11−3)! 8! 6 Factorial Notation n factorial For every positive integer n, n! = (n)(n-1)(n-2)…(3)(2)(1) However, most problems dealing with permutations are not as simple as the previous examples. With the vast number of possibilities and conditions that any probability problems may fall on, we shall be categorizing the applications of permutation according to its nature of solution. (1) Permutation without Repetition Permutation without repetition is easily identified on a problem involving sets with no identical elements. This is the number of possible arrangements when r elements are chosen from n elements of the set. The formula used for this permutation is 𝒏! 𝑷(𝒏, 𝒓) = (𝒏 − 𝒓)! Example 1 How many permutations are to be made with 3 balls from the 15-ball pool? Solution: In this example, we are to pick 3 balls from the 15 balls. Thus, 3 elements are chosen from set of 15. Given: n = 15, r = 3 𝑛! 15! 𝑃(𝑛, 𝑟) = = (𝑛 − 𝑟)! (15 − 3)! Simplifying the denominator and further expanding the numerator gives us: 15 × 14 × 13 × 12! 𝑃(𝑛, 𝑟) = 12! 12! will be cancelled from the equation, giving us: 𝑃(𝑛, 𝑟) = 15 × 14 × 13 = 𝟐, 𝟕𝟑𝟎 𝒂𝒓𝒓𝒂𝒏𝒈𝒆𝒎𝒆𝒏𝒕𝒔 2,730 arrangements can be done by picking 3 balls from the 15-ball pool. (2) Permutation with Repetition Permutation without repetition is identified on a problem involving sets with identical elements. The repetitions are solved by dividing the total permutation by the product of the factorial of the number of identical elements. The formula used for this permutation is 𝐏(𝐧, 𝐫) 𝐏(𝐧, 𝐫)𝐫𝐞𝐩 = ; where x are the identical elements 𝐗𝟏! 𝐗𝟐! ⋯ 𝐗𝐨 ! 7 Example 2. How many different permutations can be made from the letters of the word DIFFERENT? Solution. In this problem, we need to determine the total number of letters for our n and the number of repetitions per letter x. We have 9 total letters and two (2) Fs and Es, one (1) for the remaining letters. 𝑛! 9! 𝑃(𝑛, 𝑟) = = 𝑥1 ! 𝑥2 ! 2! × 2! Solving the equation, 9! 𝑃(𝑛, 𝑟) = = 𝟗𝟎, 𝟕𝟐𝟎 (2!)(2!) There are 90,720 permutations from the letters of the word DIFFERENT. Example 3. Maria has three vases of the same kind and two candle stands of the same kind. In how many ways can she arrange these items in line? Solution. In this problem, we need to determine the total number of items for our n and the number of repetitions per letter x. We have 5 total objects and three (3) vases and two (2) candle stands. 𝑛! 5! 𝑃(𝑛, 𝑟) = = 𝑥1 ! 𝑥2 ! 3! 2! 5! 5 × 4 × 3! 𝑃(𝑛, 𝑟) = = 3! 2! 3! 2! 5×4 𝑃(𝑛, 𝑟) = = 𝟏𝟎 2 There are 10 ways to arrange the items in line. (3) Circular Permutation Circular permutations, also known as cyclic permutation, is a type of permutation that sets the elements in a circular arrangement, or that has no start point nor and end point. However, an element of the set shall be designated as a fixed or reference element, where the fixed element shall be stationary, and the other elements are not. There are two sub- categories for the circular permutation. First is that the clockwise and the counter clockwise arrangement of the elements can be distinguished, then the total number of circular permutations are 𝑷𝒄 = (𝒏 − 𝟏)! Second is that if the clockwise nor the counter clockwise arrangement of the elements are not distinguishable, then the total number of circular permutations are (𝒏 − 𝟏)! 𝑷𝒄 = 𝟐 8 Example 4. How many ways can 10 persons sit around a table? Solution. The given problem is straight forward to answer. We will be using the first formula for the circular permutation since the direction of arrangement is distinguishable. 𝑃𝑐 = (𝑛 − 1)! = (10 − 1)! = 9! = 𝟑𝟔𝟐, 𝟖𝟖𝟎 There are 362,880 possible arrangements for the 10 persons to sit around a table. Example 5. How many ways can 7 diamonds be formed into a necklace? Solution. The problem above falls under the second case of circular permutation, where the direction of the arrangement is indistinguishable. (𝑛−1)! 𝑃𝑐 = 2 (7 − 1)! 𝑃𝑐 = = 𝟑𝟔𝟎 2 There are 360 possible ways to form the 7 diamonds into a necklace. For more information, we can watch the video lecture about permutations, links provided below: https://www.youtube.com/watch?v=uEI32RFKvqE Activity 3: Answer Me! DIRECTION: Provide the answer of the following given situation. Justify the outcome of your solution. 1. In the player’s jersey there are 2-digit numbers. How many different numbers can we get? _________________ 2. In how many ways can you arrange the letters of the word CHAOS? ________ 3. 5 people need to be arranged in a row. How many possible arrangements are there if Carl and Ian wants to be next to each other? _______________ 9  Let’s Dig In Activity 4 Find Me! A. Calculate each of the following. 1. P (5 , 2) _____ 4. P (7 , 6) _____ 2. P (6 , 6) _____ 5. P (n , 3) = 210 _____ 3. P (10 , 6) _____ B. DIRECTIONS: Answer the following questions. Provide your complete solution. 1. Francis has to choose 5 numbers for his password (from digits 0 to 9) and repetition of digits are not possible. How many different arrangements of numbers could Francis choose? ________________ 2. How many unique 3 digit codes can be created from the 5 digits {1, 2, 3, 4, 5} if repetitions are allowed? ________________ 3. 10 students have appeared for a test in which the top three will get unique prizes. How many possible ways are there to get the prize winners? _________ 4. In how many ways can I create 2 letter words from the letters in the word LOVE? Any 2 letters form a 2 letter word. ________________ 5. How many unique 3 letter passwords can be made from 4 letters if repetitions are allowed? ________________ Let’s Remember In this lesson, we have learned that permutation is the ordered arrangement of given elements from the elements of a given set. We learned that permutations can be categorized into three: 1. Permutation without repetition 𝒏! 𝑷(𝒏, 𝒓) = (𝒏 − 𝒓)! 2. Permutation with repetition 𝐏(𝐧, 𝐫) 𝐏(𝐧, 𝐫)𝐫𝐞𝐩 = ; where x are the identical elements 𝐗𝟏! 𝐗𝟐! ⋯ 𝐗𝐨 ! 3. Circular Permutation A. First is that the clockwise and the counter clockwise arrangement of the elements can be distinguished, then the total number of circular permutations are 𝑷𝒄 = (𝒏 − 𝟏)! B. Second is that if the clockwise nor the counter clockwise arrangement of the elements are not distinguishable, then the total number of circular permutations are (𝒏 − 𝟏)! 𝑷𝒄 = 𝟐 10  Let’s Apply Directions: Answer the problem below. 1. It is in international summits that major world decisions happen. Suppose that you were the overall in charge of the seating in an international convention wherein 12 country- representatives were invited. They are the prime ministers/presidents of the countries of Canada, China, France, Germany, India, Japan, Libya, Malaysia, Philippines, South Korea, USA, and United Kingdom. 1. If the seating arrangement is to be circular, how many seating arrangements are possible? 2. Create your own seat plan for these 12 leaders based on your knowledge of their backgrounds. Discuss why you arranged them that way. Let’s Evaluate Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. 1. The digits 5, 6, 7, and 4 are to be arranged to form a 7-digit integer. How many different integers can be formed? A. 5,674 B. 16,384 C. 840 D. 24 2. In how many ways can the letters of the word REFERENCE be arranged? A. 362,880 B. 181,440 C. 7,560 D. 15,120 3. In how many ways can the letters of the word MATHEMATICS be arranged? A. 6,652,800 B. 39,916,800 C. 4,989,600 D. 55,400 4. A mathematician, a physicist, and an engineer walk into a bar. If there are 3 seats available at the bar, how many ways are there for the mathematician, physicist, and engineer to seat themselves? A. 6 B. 4 C. 1 D. 0 5. The mathematician, physicist, engineer, and computer scientist decided to move from the bar to a (circular) table. They file to the table, staying in order as they sit down, and wind up seated as shown below. 11  How many different seating arrangements at the bar will lead to the seating arrangement around the table in the image above? A. 0 B. 1 C. 4 D. 6 6. Six friends Andy, Bandy, Candy, Dandy, Endy and Fandy want to form a club. They decide that there will be 1 president, 1 secretary and 4 ordinary members. How many ways can they organize this club? A. 720 B. 30 C. 24 D. 2 7. Five children--Myra, Esmond, Yolanda, Carlos, Lin--are playing a game of hide-and- go-seek. Myra counts to 10 and the other four children each go to hide in one of the rooms of the house. If there are 7 rooms that the children could hide in, and each hide in a different room, how many ways can the children hide? A. 5,040 B. 2,520 C. 840 D. 210 8. How many ways can the letters of the word BOTTLES be arranged such that both vowels are at the end? A. 60 B. 120 C. 720 D. 1,440 9. Among 5 girls in a group, exactly two of them are wearing red shirts. How many ways are there to seat all 5 girls in a row such that the two girls wearing red shirts are not sitting adjacent to each other? A. 12 B. 120 C. 24 D. 72 10. In an ice cream shop, there are eight toppings to choose from. The eight toppings are arranged in a round revolving tray. How many ways can these toppings be arranged? A. 5040 B. 10080 C. 2520 D. 5020 Alternatively, you may answer these questions online! Use this link on your cellphone, laptop, or desktop: https://forms.gle/Wn4xb24pPty8AYgdA Use proper capitalization to activate the link. You will see your score after completing the test. Make sure to screenshot your work as a proof to your teacher then write your score in the box. Make sure you are connected to the internet! Let’s Extend DIRECTIONS. Solve the following. Read each item carefully. 1. For the whole class of your own, count the total possible arrangements of the roster of Top 10 of the class. 2. For the whole class of your own, count the total possible arrangements of the Class Officers (President, VP, Secretary, Treasurer, PRO, and Peace Officer) 12  Alternatively, you may answer these questions online! Use this link on your cellphone, laptop, or desktop: http://bit.ly/MATH10WEEK1LETEXTEND Use proper capitalization to activate the link. You will see your score after completing the test. Make sure to screenshot your work as a proof to your teacher then write your score in the box. Make sure you are connected to the internet! References DepEd Grade 10 Mathematics Teaching Guide Arnold V. Garces and Crisselle Espanola-Roces (2018) “Simplified Mathematics Grade 10” St. Augustine Publication, Inc. Geraldo DG. Banaag and Reymond Anthony M. Quan (2013) “Global Mathematics 10” The Library Publishing House, Inc. Nivera, Gladys C. (2014) “Grade 7:Patterns and Practicalities”, Don Bosco Press Inc., Oronce, Orlando A., and Marilyn O. Mendoza (2019) “E-Math 7 Worktext in Mathematics”, Rex Book Store http://www.projectmats.le/documents/quizzes/Sets1.htm Nivera, Gladys C. (2012) “Grade 7:Patterns and Practicalities”, Don Bosco Press Inc. 13  Development Team of the Module Writers: VILMA BOMBITA ROEDER V. FABUL Editors: Content Evaluator: CHONA R. PANOPIO Language Evaluator: Reviewers: GINA C. FRANCISCO MARISOL BARBARA M. FERNANDEZ Illustrator: Layout Artist: Management Team: DR. MARGARITO B. MATERUM, SDS DR. GEORGE P. TIZON, SGOD Chief DR. ELLERY G. QUINTIA, CID Chief MRS. MIRASOL I. RONGAVILLA, EPS-Mathematics DR. DAISY L. MATAAC, EPS – LRMS/ALS For inquiries, please write or call: Schools Division of Taguig City and Pateros Upper Bicutan Taguig City Telefax: 8384251 Email Address: [email protected] 14 ANSWER KEY: Quarter 3: Week 1 Let’s Try: 1.A 2.A 3.A 4.B 5.D 6.D 7.C 8.B 9.C 10.B Let’s Recall Activity 1: How Many Ways? A. 1. Blouses - stripes, with ruffles, long-sleeved, sleeveless Skirt - blue, white, black Possible outfits: Blouse Skirt Blouse Skirt stripes blue long-sleeved blue stripes white long-sleeved white stripes black long-sleeved black with ruffles blue sleeveless blue with ruffles white sleeveless white with ruffles black sleeveless black 2. 12 blouse-and-skirt pairs are possible. 3. blue blue long- Stripes white white sleeve d black black blue blue With sleeveless white white ruffles black black 15 Activity 2: Let’s Explore 1. Runner1 Runner1 First Second Runner 2 Runner 2 Place Place Runner 3 Runner 3 2. 720 ways 3. Yes, Using Fundamental Principle of Counting and Permutation Activity 3: Answer Me! 1. 100 2. 120 3. 12 Let’s Dig in: A. 1. 20 2.720 3.151200 4.5040 5.7 B. 1.27216 2.125 3.720 4.12 5.64 Let’s Apply 1. (n-1)! = (12-1)! = 39916800 2. Sample Answer: Arrangement base on the GDP of every country last 2020. USA, China, Japan, Germany, India, United Kingdom, France, Canada, South Korea, Philippines, Malaysia, Libya Let’s Evaluate: 1.B 2.C 3.C 4.A 5.D 6.B 7.C 8.B 9.D 10.A 16