Ilide.info-permutation-g-10-a-detailed-lesson-plan-nacar-jm-prd20e43be139130f1a984c855876a5e24
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University of Antique Sibalom, Antique**We aren't endorsed by this school
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MATH 101
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Computer Science
Date
Jan 13, 2025
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14
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Father Saturnino Urios University Teacher Education program Butuan City A Detailed Lesson Plan in Teaching Mathematics X Submitted By: Submitted to: Jonnah Mariz G. Nacar Mr. Al C. Betantos Student –Teacher Cooperating Teacher Dr. Arlyn M. Floreta Supervising Instructor
A Detailed Lesson Plan In Mathematics X I. Learning Content Permutation Reference: Esparrago, Mirla S. et. al. Next Century Mathematics. Grade 10. Permutation. Phoenix Publishing House Inc. Quezon City. pp 389 –401. II. Learning Objectives At the end of one hour, the students will be able to: a. Illustrate the permutations of objects b. Derives the formula for finding the number of permutations of nobjects take rat a time c. Affirm one’s choice for a simple lifed. Solves problem involving permutation III. Learning Materials Visual Aid IV. Procedure Teacher’s ActivityStudent’s ActivityA. Preliminary Activities Before we start our class, everybody please stand for a prayer. Frank please lead the prayer. Good morning class. Before anything else please arrange your chairs and make your surrounding clean. How are you today? Is anyone absent? That’s good to hear. Now please settle down for we are about to start our new lesson. Remember to always listen, participate and enjoy. Understood? (prayer) Good morning ma’am(arranging chairs and making surrounding clean) We’re fine, Ma’am!No one Ma’am!Yes Ma’am!
B. Lesson Proper a. Introduction Motivation Before we proceed to our lesson proper, I have here an activity for you to work on. I am going to divide the class into 4 groups. Are you ready? In this activity, I will give each group a 4-digit number. What I want you to do is to create a 4 –digit password and list it all in a whole sheet of paper. After creating a 4 –digit password, I have here in the board a cellphone that needs to be unlocked. What you are going to do is to guess or input the correct password for it to open. Say for example, Liam forgot his cellphone unlock code. Help him remember the code with the following hints: Clues: Password is made up of 4 characters All characters are letters Letters of the code are all found in his name. The first letter as he remembers, starts with letter M. As soon as each group receive the numbers, start listing all the possible codes you can create. Clues are given as your guide in decoding the password. From the created passwords choose one which you think is the correct password and input it to unlock the cellphone. Choose someone from your group who’ll write the code. Did you understand? Okay good. I will give you 5 tries to unlock the cellphone. You may now start writing down the possible passwords. Thank you everyone. Later we will check if the numbers you input are correct. So, group 1, from the given 4 –digit number how many possible passwords you created? How about you here in the other group did you created the same number of passwords as the group 1 did? Yes Ma’am!Yes ma’am!(after every group has written their answer) Okay Ma’am!CLUES My 2nddigit is twice the difference of a number and 1 is 4, I am that number. My 3rddigit is twice my last digit. My last digit has the smallest value in our counting numbers. If you add the 2ndand 3rddigit I am your 1stdigit. WHAT IS MY CODE?
Very good everyone! There are exactly 24 4 –digit password that can be created from the numbers I gave you. So how did you come up with all those passwords? Yes Ella? Yes, very good, thank you Ella. In order for you to create a 4 –digit password you arranged the given number, right? So, let’s try to check if your answers are correct. Very good! All 4 group got the correct password. So, the activity that we had has something to do with our topic for today. Our lesson for today is all about the arrangements or orders of certain object in which we call permutation. b. Interaction Discussion From the activity that we had what do you think is permutation? Yes Mae? Very good! Thank you, Mae. Any other idea? Yes Jay? Yes, thank you Jay. Good answer. From what you have said, permutations refer to an arrangement of objects in a definitive order. It is also called an “arrangement number or order”.Say for example, what are the possible arrangement of 3 objects when placed in a row? Let’s say those three objects are represented by the letters A, B and C, so how many possible ways can it be arranged? Let me give you one possible arrangement. Arrangement ACB. Now who can give me another one? Yes Liam? Yes, very good Liam. Thank you. Another? Yes Dion? Okay very good. What else? Jerome? Yes, very good. Is there more? Yes matt? We created 24 passwords Ma’am!Yes Ma’am! We also created 24 4 –digit password. We arranged the numbers Ma’am!Yes Ma’am!Ma’am permutation involves arrangement of objects. Permutation is all about orders or combination Ma’am.BCA Ma’am!CBA Ma’am!
Very good. Is there more arrangement? So, there are exactly 6 possible ways to rearranged the 3 letters. To list it all again here are the arrangements mentioned: ABC BCA CBA ACB BAC CAB So, what if there are more objects to be rearranged or reorder? Such as rearranging numbers 0 –9? Do you think it is easy to list all the possible ways? Yes, that’s right it would take too long for us to list all possible arrangements. So instead of listing it all, permutations have an easier way of finding solution to it. Are you ready to listen how it is done? Okay now there are several rules in permutations. Let us discuss the 1strule in permutations. 1stRule: can everybody read rule number 1 please. Thank you, class. Now factorial of ndenoted by n! is calculated by the products of integer numbers from 1 to n, where n > 0 or just simply n! = (n) (n –1) (n –2) (n –3) … (3)(2)(1). We can apply n! for any distinct objects, meaning objects differ from each other. So, from my example a while ago in which 3 objects are to be rearranged, can you remember how many ways we rearranged it? Yes Miel? Very good! Thank you Miel. Yes 6 arrangements it is. Okay using the formula of Rule 1 we can derived an answer. Let us identify what is our n first.Gian what is our n? Yes, that’s correct. after identifying the nwhat’s the next step? So, from the formula, we have? Yes Carol? Okay nice answer. Thank you. BAC Ma’am!CAB Ma’am!No more Ma’am!No Ma’am!Yes Ma’am!Rule 1: Linear Permutation The n –factorial (n!) The number of permutations of nobjects arranged at the same time. 6 Ma’am!3 Ma’am!n = 3 n! = 3 2 1n! = 6
Another example. How about 8 distinct objects arrange in a row? Frank? Please indicate what is our nand solve in the board. Thank you, Frank, you see it is easier to know how many possible ways we can arrange distinct objects using the factorial formula. Did you understand about the 1strule? So, let’s proceed to the 2ndrule of permutations. Kindly read, everyone? Thank you! This permutation rule is denoted in many ways. P (n, r) nPr Pnrn refers to the number of objects while our rrefers to how many ways can we fill or arrange the objects. Always remember that r < n. Say for example how many ways can 6 students be seated from a groufie if only 4 seats are available? Anyone willing to answer the problem? Yes Joshua. Indicate what is our n and r and solve in the board. Do you think Joshua’s answer is correct?Very good! Thank you, Joshua, you may now take your seat. From Joshua’s solution in how many ways can we arrange 6 students in a groufie with only 4 seats available? n = 8 8! = 8 7 6 5 4 3 2 1 8 = 40320 Yes Ma’am!Rule 2: Permutation without repetition The number of permutations of nobjects taken rat a time without repetition where r < n. P (n, r) = !()!nnrn = 6 and r = 4 P (6,4) = 6!(64)!P (6,4) = 6!2!P (6,4) = 7202P (6,4) = 360 Yes Ma’am!360 ways Ma’am!
Yes, very good! It is read as 6 students taken 4 at a time has 360 possible arrangements. Understood? Now how about if numbers or letters are repeated? Say for example the word BOOK. Okay Josh try to list the number of possible ways we can rearrange the word BOOK. Thank you, Josh. Now class have you notice something from what Josh has listed? That’s correct. As what you have notice some arrangements are repeated therefore when there is already repetition of letters we will use another formula. Unlike in the 2ndrule wherein objects are distinct here on the 3rdrule, objects can be repeated. Thus, I introduce to you the 3rdrule of permutation. Everybody please read. Thank you, class. This rule as what I have told you earlier involves object that are repeated. So, the p, q, ror any letters represented in the denominator are the objects of any given example. So, from the example given what object is repeated? That’s right! Thank you Mae. So, in the 3rdrule we must identify first the objects in the given example the word book has how many objects? That’s right the next thing we do is to list each individual object. How many are B? Yes, what other objects are there? Yes Ma’am!BOOK BOKO BKOO OOKB OKOB KOOB OKBO KOBO OOBK KBOO OBOK OBKO We expected to have 24 results but some are repeated and resulting to only 12 arrangements. Rule 3: Permutation with Repetition Permutation of n things not all different or the number of permutations of nobjects of which pare alike, qis alike rare alike and so on given by: !! ! !...np q rThe letter O Ma’am!4 Ma’am!There is only one B ma’am!2 O’s and only 1 K Ma’am!
Very good. since we already identified the objects let us try to find the number of ways we can rearrange the letter of the word by using the 3rdrule of permutations formula. Shall I call on Jessa to answer please. Indicate the objects okay. Thank you, Jessa. Again, we cannot use the 2ndrule of permutation when there is object repetition instead we will use? Very good! you seem to understand about the 3rdrule. I will give you another example. In how many ways can 5 blue balls, 3 red balls and 2 yellow ball be arranged in a row? Again, you have to identify our nin the problem. So, what is our n? Matt? Yes, we only have to count the total number of objects since they are all the same. Group the balls that has the same color and count them. Now we want to know how many ways can we arrange the balls. So, we are going to use the formula of the 3rdrule. Anyone want to try? Yes Frank. Identify the objects first before solving. Others please also solve while on your seats. Do you have the same answer as Frank’s?Okay very good Frank’s answer is correct.How about the word PAPAYA? Yes Pia? Identify each object first. The rest please answer as well. Very good! you seem to already get the idea of the 3rdrule of permutations. Now if you notice, we have been arranging and ordering objects in a linear manner. How about if we arrange n = 4; B = 1; O = 2; K = 1 !! ! !np q r= 4!1!2!1!= 12 The 3rdrule Ma’am!10 Ma’am!n = 10; Blue = 5; Red = 3; Yellow = 2 10!5!3!2!= 2520 Yes Ma’am!n = 6; P = 2; A = 3; Y = 1 6!2!3!1!= 60
objects in a circular manner? Do you think the manner of solving id still the same with linear? Alright now say for example 3 people are sitting around a circular table. List the possible ways on how we can rearrange these 3 people. You can try by representing them as A, B and C and by drawing it. Then compare the result to linear permutation. Got it? So, what are the result? Mae can you write it on the board? Alright thank you Mae. How about the linear permutation? Who can list it? Yes Jay. Okay thank you Jay. So, there are how many arrangements are there in circular permutation? And how about in linear permutation? Alright now, what you answered about linear permutation is correct however in circular permutation there are only 2 arrangements. Do you have any idea why? Alright this follows the introduction of our 4thand last rule of permutations. Everyone please read. Okay thank you everyone. In circular permutation we are dealing with circular arrangements. Objects are then arranged in circular manner not like the other rule which we had just discussed that are linearly arranged. Say for example, we would like to find out how many ways can a family of 8 be seated or arranged around the circular table? So, let us indicate our nfirst, what is our n? Jessa? No Ma’am!Yes Ma’am!ABC BAC CBA ACB BCA CAB 6 ways Ma’am!Also 6 ways Ma’am!No Ma’am!Rule 4: Circular Permutation The number of permutations of ndistinct objects arranged in a circle, given by: (n –1)! 8 Ma’am!
Yes, that’s right. So, following the formula who can solve in the board? Yes Joshua. Good job. Now here’s another example. In how many ways can we arrange 10 trees around a circular garden? Yes, Liam answer on the board and indicate our n. Very good Liam. Now that you already understand about circular permutation, how is it different from linear permutation? anyone? Matt? Okay thank you Matt. Any other explanation? Jessa? Alright thank you Jessa. Yes, what you said is somewhat true about the two permutations. So, the main difference of circular and linear permutation is that in circular permutation there is no fix starting and end point unlike in linear permutation we know which point or object is first and last. So, going back to the example I gave you about 3 people sitting around the table, arrangements ABC, CAB, and BCA is just the same. Any idea why? Great! That’s correct. so, you already know what are the concepts and the rules of permutations. Any more questions or clarifications about our topic? That’s good to hear.Generalization Now that you already know the rules of permutation and what it is all about, what do you think are the uses of permutation in our life? Yes, that’s good. What else?(n –1)! = (8 -1)! = 7! = 5040 n = 10 (n -1)! = (10 –1)! = 9! =362880 They differ in arrangement Ma’am!In linear Ma’am they are arranged in a row while in circular they are arranged in a circular manner. Because they were just rotated Ma’am not stating which one is first and last in the order. No more Ma’am!We can use permutations in creating passwords or pin codes Ma’am! we can use permutation in creating passwords pin number for our ATM cards, cellphones or maybe to our houses lock that avoids us from burglars
Yes, that’s right another?Yes, that’s good to hear that somehow you do understand the importance and application of permutation in our daily life. Good job class. c. Integration Since everybody already understand our topic which is permutation I am going to give a you group activity. I will group you into four and in each group, I will give a problem involving permutation. You are going to choose one from your group who will present and explain the group’s answer. And after it we will check all your answers. Am I clear? Okay now in explaining your answers do not forget to state what rule of permutation you used, okay? G1. In how many ways can the letters/ word EMAIL be arranged? How many starts with letter A? explain. G2. In how many ways can we arranged in a row the letters/ word CONDENSED? Show your correct solution. G3. How many 3-digit numbers can be written from digits 5,6,7,8 and what rule of permutation did you used? List all the possible outcome. G4. In how many ways can you arrange the 4 chairs in a circular table? Give an illustration to your answer. It can also help us think through that there are many possible options or things that could happen in our life. So, we should not give up and to always try harder Ma’amYes Ma’am!Okay Ma’am!G1. Using rule 1 –linear permutation, we have 5 distinct objects therefore the word EMAIL can be arranged in 5! = 120 ways. So how many starts with letter A, we can solve it by assigning the first letter with A. A_ _ _ _, since we only have four letters blanks available to be fill we use rule 1 again thus 4! = 24. There are 24 possible arrangement that starts with letter A. G2. Since CONDENSED is a 9 –letter word with repeated letters we used rule 3. C = 1 N = 2 E = 2 O = 1 D = 2 S = 1 Thus, we solve it by 9!1!1!2!2!2!1!= 45360
Good job everyone. I guess you understand fully our discussion about permutations. Evaluation Therefore, there are 45360 ways in arranging the letters of the word CONDENSED. G3. We used the rule 2 to answer the question because we take 4 numbers 3 at a time, P (4,3) = 4!(43)!= 4!1!= 24 There are 24 3 –digit numbers that can be formed using numbers 5,6,7 and 8 567 678 785 856 568 675 786 857 576 685 756 867 578 687 758 865 586 657 768 875 587 658 765 876 G4.
Now everybody please return to your proper seats and get ½ sheet of paper for a short quiz. Direction: list down all the permutation of the following objects. 1. Four letter M, A, T, H taken 2 at a time. 2. Four numbers 1, 0, 8, 9 taken 3 at a time. 3. Show how many ways can the word MATHEMATICS be arranged? If you’re done you can pass your paper.Assignment Direction: in a ½ sheet of paper solve the problem. 1. How many different signals, each consisting 10 flags hung in a vertical line can be formed from 4 red flags, 3 blue flags, and 3 green flags. Pass it next meeting. Goodbye class! Goodbye Ma’am!