Detailed Lesson Plan In Mathematics 10-for Cot.docx

Detailed Lesson Plan in Mathematics 10 I. Objectives: At the end of the lesson the students are expected to: 1. Illustrate the permutations of the distinct objects, 2. Use the formula for finding the permutation of n objects taken r at a time; 2. Solve problems involving permutations II. Subject Matter Topic: Permutations (distinct object) Reference: Mathematics Learner’s Module Grade 10 Materials: III. Teaching Procedure Teacher’s Activity A. Preparatory Activities 1. Prayer 2. Greetings 3. Review/Drill B. Developmental Activities 1. Motivation Now, you will form a group with five members. So this row will be group 1, this row will be group 2, group 3, group 4, group 5, group 6, group 7 and group 8. Form a circle now. Faster. Are you in your groups now? Very good. Now, I will give you 4 cards with the letters M, A, T and H written on each card. Do you have your cards now? Now I want you to put your card on the table face down. Shuffle it three times. Done? Now, from those 4 cards, choose 3 cards and face that card up. Now, you have three letters. In a 1 whole sheet of paper, I want you to record the possible arrangement of those 3 letters. I will give you five minutes to do that. Understood?

Learners’ Activity

(After 5 minutes) Are you done? Now, using all the cards, I want you to record all the possible arrangements of the four cards. I will give you 5 minutes to do that. (After 5 minutes) Are you done? Now, Using all four cards, how many arrangements do you have? Yes group 1? How about the other groups, do you have the same answer with group 1? Okay. Later we will check if your answers are correct. How about the three letters, how many possible arrangements do you have? Yes group 5? Are your answers the same with their answer? Okay. We will check later if your answers are correct. 2. Presentation of the Lesson Now class, say for example, you were riding on a bus with 2 of your friends and there were 3 vacant seats in a row. In how many ways can you arranged yourself? So I need three volunteers here in front. Yes Mikaela, Jason and Arc. Say for example, Mikaela, Jason and Arc were on the bus, and these are the vacant seats. So one possible arrangement is that Mikaela is beside Jason and Arc, or we can represent it in symbols. So one possible arrangement is MJA. Who can give me another arrangement? Yes, Michelle?

Another possible arrangement is Mikaela, Arc and Jason.

Very Good. In symbols, MAJ. Who can give another possible arrangement? Yes Antonette?

Another possible arrangement is Arc, Jason and Mikaela.

That’s right. Another one? Yes Rommel?

Arc, Mikaela arrangement.

and

Jason

is

a

possible

Very good Rommel. How about you, Andrew? Jason, Mikaela and Arc is also a possible Very Good. Is there any possible arrangement arrangement. that was not mentioned? Yes Sushmita? Very Good. Any possible arrangement that was not mentioned? None? Let’s list down the possible arrangements you mentioned. MJA AMJ MAJ JMA AJM JAM

Another possible arrangement is Jason, Arc and Mikaela.

Okay, Mikaela, Jason and Arc, you can now take your seats. Now, how many possible arrangements are there? Yes Francis? There are six possible arrangements. Very good. So there are six possible arrangements for three people sitting on 3 seats on a bus. How about if there were 8 people in a bus? Do we have to list all the possible Do you think it will take us a lot of time? 3. Discussion Now class, instead of listing all the possible arrangement of an object, mathematics has an easy way of solving problems, which is concerned with arrangements. And that is by Permutation. “Permutation refers to any one of all possible “Permutation refers to any one of all possible arrangements of the elements of the given set.” arrangements of the elements of the given set.” For instance, given a set of distinct objects, we can arrange them in one of several ways. Like what we did with the possible sitting arrangements of Mikaela, Arc and Jason. The listed arrangement are the permutations of the distinct objects.

Now, let’s discuss the rules of permutation. Kindly read, Marie? Thank you Marie.

“Rule no. 1: The number of permutations of n distinct objects arranged at the same time is given by n!”

Class, n! = n(n-1)(n-2)(n-3) … 3.2.1. Say for example, 5•4•3•2•1=5! and we read this as “Five factorial”. So 5! = 120. In our example a while ago, how many distinct objects do we have? Yes We have 3 distinct objects. Marc? They are Mikaela, Arc and Jason. And what are they Marc? Very good Marc. Remember class that the object that we are talking about is the subject that is being permuted. It may be an animal, a person, a letter, or any other things. Going back to our example, we have three objects, so to find the possible permutations, we will have 3!. 3! = 3•2•1 =6 3! Is equal to six. Is it the same to our answer a while ago? So instead of listing all the possible permutations of an object, we can use n! in order to find on how many ways can we Yes maam. arranged n objects. Understood? Let’s have another example. In how many ways can 4 people arrange themselves in a row for picture taking? Try to solve this problem by yourselves.

Yes Maam.

Okay. Are done? Who wants to solve on the 4!= 4•3•2•1 board? =24 Yes Harvey? Very good Harvey! Now let’s proceed to rule no. 2. Yes, Jason kindly “Rule no. 2: The number of possible objects taken r at a time is given by read? 𝑛! Thank you Jason. nPr =(𝑛−𝑟)! Say for example, we have 5 passengers and there were only 3 vacant seats. In how many ways can we arranged the 5 passengers? Rule no.2 can answer this question. This means that we will take 5 passengers 3 at a time or 5P3. Substituting to the formula, we have,

5!

5P3 =(5−3)! =

120 2

= 60

So there are 60 possible arrangements of taking 5 passengers 3 at a time. Understood?

Yes Maam.

Okay let’s have another example for rule number 2 and this time you’ll be the one to solve it. In a school club, there are 5 possible choices for the president, a secretary, a treasurer, and an auditor. Assuming that each of them are qualified for any of these positions, in how many ways can the 4 officers be elected? Are you done? Who wants to solve on the board? Yes Kim?

𝑛!

nPr =(𝑛−𝑟)! 5!

5P4=(5−4)! = Is the work of Kim correct?

120 1

= 120

Yes ma’am.

Okay. Very good Kim. Is there any question class?

No ma’am.

C. Generalization How are you going to calculate the different permutations of distinct objects? We can calculate the different permutations 𝑛! by using the formula nPr =(𝑛−𝑟)! Excellent! What is a permutation? Very good class! D. Application This time let’s have another group activity. So this will the group 1, group 2, 3, 4 and 5. Each group will be given an envelope that contains the problem. All you have to do is to solve it and once you’re done the group secretary will write your solution on the board while the group

“Permutation refers to any one of all possible arrangements of the elements of the given set.”

reporter will explain your work in front. You only have 5 minutes for that. Is there any None ma’am. question class? Yes ma’am. Am I clear? (Problems inside the envelope) Group 1 Suppose you secured your laptop using a password. Later you realized that you forgot the 5-digit code. You only remembered that the code contains the digits 1, 4, 3, j and B. how many possible codes are there?

Group 2 Suppose that in a certain association, there are 12 elected members of the Board of Directors. In how many ways can a president, a vice president, a secretary and a treasurer be selected from the board?

Group 3 A dress shop owner has 8 new dresses that she wants to display in the window. If the window has 5 mannequins, in how many ways can she dress them up?

Group 4 In how many different ways can 5 bicycles be parked if there are 4 available parking spaces?

Group 5 If there are 10 people and only 6 chairs are available, in how many ways can they be seated?

(After 5 minutes)

Yes ma’am. (Group secretary writes their solution on the Are you done class? So now, secretary of each group write your board) solutions on the board. (Group reporters’ explain their work.) This time, group reporter explain your work. Very good class! E. Back Home Application

IV. Evaluation Direction: Choose the letter that you think is the correct answer. 1. What is the term that refers to the possible arrangements of things? a. Combination

c. Presentation

b. Organization

d. Permutation

2. Which of the following formula is appropriate in getting the permutations of n object taken r at a time? 𝑛!

a. 𝑟𝑃𝑛 = (𝑟−𝑛)! 𝑛!

b. . 𝑛𝑃𝑟 = (𝑛−𝑟)!

c. . 𝑟𝑃𝑛 = 𝑟! d. . 𝑛𝑃𝑟 = 𝑛!

3. In how many ways can the letters w, x, y, and z be arranged in a row? a.

c.

b.

d.

4. How many 4-digit numbers can be formed from the numbers 1, 3, 4, 6, 8 and 9 if repetition of digits is not allowed? a.

c.

b.

d.

5. Michael received 7 paintings from his boss as a reward for his good work. He wants to hang it on his wall horizontally. In how many possible ways can he arrange it all? (Show your solution) a. 7

c. 210

b. 49

d. 5040

V. Assignment Think about of a situation that involves arrangements. What you’re going to do is to: 1. Write all possible permutations of n object taken all at a time. 2. Write all possible permutations of n object taken r at a time. Note: The value of your n must be greater than 4 and the value of your r must be greater than 3.

For example, if you are watching athletic games. Say ten runners join a race. In how many possible ways can they be arranged as first, second and thirds placers? Then answer.