Sequence and Series.docx
- 1. DETAILED LESSON PLAN IN MATHEMATICS 10 Content Standard: The learner demonstrates understanding of key concepts of sequences. Performance Standard: The learner able to formulate and solve problems involving sequences. Objectives: 1. Defined and illustrate arithmetic sequence; 2. Determine the nth term of arithmetic sequence; 3. Apply the concept of arithmetic sequence in real life situations. Subject Matter: Topic: Sequence and Series Sub Topic: Arithmetic Sequence Learning Resources: References: Simon L. Chua, D.T., Josephine L. Sy Tan, Arvie D. Ubarro, Ma. Remedios R. Cayetano, SPC, Renato R Guerrero Soaring 21st Century Mathematics http://www.mathscore.com http://www.basic-mathematics.com Materials: Printed Materials Powerpoint
- 2. PROCEDURE Teacher’s Activity Student’s Activity A. Preparation “Kindly all stand, let us start our day with a prayer. May I request Nicole to lead our opening prayer.” “Thank you for that powerful prayer Nicole.” “Good morning class.” “How are you today? “Kim, who is absent for today?” “Oh! That’s great!!” “Kindly arrange your chairs and pick up the pieces of dirt under your table .” B. Review and Motivation “Let’s have a review first from our last discussion, so anyone who can give me the topic of our last discussion. Yes, Adrian? “Very good Adrian, anyone who can give me the definition of sequence? Yes Alleona? “That’s great Alleona” “What do you call to the each number in a sequence? Yes Kyle?” “Very Good Kyle, very good class. So now we can proceed to our next topic, are you ready? “To brighten up your day, I prepared a game for you.” “May I request all of you to join the game.” “So the title of our game is ‘Guess the magic word’, do you know it class?” “I will group the class into 2 groups, later I’ll call the group who are going to answer.” (All students will stand and Nicole will lead the prayer) “Good morning Ma’am” “Feeling good Ma’am.” “Nobody is absent today Ma’am” “Yes Ma’am.” (Students will arrange the chairs and pick up the pieces of dirt.) “Sequence Ma’am.” “A sequence or progression is a list of objects, events or numbers in a definite order.” “Term Ma’am” “Yes Ma’am” “No Ma’am.”
- 3. “I will show set of numbers in screen then identify the next terms of given set of numbers. Each item corresponds to the letter that are already shuffled. When your answers are all correct therefore we can guess the magic word.” “Do you have any question about the mechanics?” “Here are the jumbled letters.” T : 1, -1, 1 A : 17, 21, 25 C : 15, 21, 28 M : 61, 73, 85 R : 13, 18, 23 H : 32, -64, 128 I : 60, 75, 90 E : -25, -31, -37 I : 20, 25, 30 T : 40, 50, 61 “Here are the questions you need to answer, you’ll find answers in the jumbled letter box” 1. 1, 5, 9, 13, _, _, _ 6. 13, 25, 37, 49, _, _, _ 2. -7, -2, 3, 8, _, _, _ 7. -1, -7, -13, -19, _, _, _ 3. 0, 15, 30, 45, _, _, _ 8. 10, 16, 23, 31, _, _, _ 4. 1, -1, 1, -1, _, _, _ 9. 0, 5, 10, 15, _, _, _ 5. 2, -4, 8, -16, _, _, _ 10. 1, 3, 6, 10, _, _, _ “None Ma’am”
- 4. “The first 5 questions will be answer by the 2nd group while the numbers 6-10 will be answer by the 1st group. I’ll give you 3 minutes to look for the answers.” “Okay, let’s start, 2nd group what are the answers for numbers 1-5?” “Great job group 2!” Okay group 1 answer numbers 6-10.” “Very good group 1, therefore class what is our magic word? Anyone? Yes Renz” C. Discussion “Anyone you can guess our topic for today, yes Alliah? “You got it right!” “What have you noticed to our given? What did you do to get the next three terms? Yes Joy? “Very good observation Joy!” “Let me introduce to you our topic for today, which is the Arithmetic Sequence. An arithmetic sequence or arithmetic progression is a sequence which each term after the first term differs from the proceeding term by a constant.” “Take a look at number 9, we have 0, 5, 10, 15, and we are looking to the next three terms, if “For number 1 Ma’am, first look for the next terms of the given so we have 1, 5, 9, 13 if you notice they just add 4 to get the next term therefore the answer is 17, 21, 25 which is the letter A. For number 2, the answer is R the next three terms are 13, 18, 23. Number 3, the answer is I with next three terms of 60, 75,90. For number 4, T with next three terms are 1, -1, 1 and for number 5, the answer is H with the terms 32, -64, 128. “Ma’am for number 6, letter M (61, 73, 85), then for 7 it is E (-25, -31, -37), for 8 it is letter T (40, 50, 61), for number 9 I (20, 25, 30) and for 10, it is letter C (15, 21, 28) “ARITHMETIC Ma’am” “Arithmetic Sequence Ma’am” “There is a pattern and for the other given you need to add same number to get the next terms”
- 5. you notice they have something in common. The first term is 0, the 2nd term is 5, 3rd is 10, to get the next terms you need to add 5, therefore the 4th term is 15, the 5th term is 15+5 which is 20 then for the 6th term 20+5 and it is 25 and so on. You just add 5 to get the next terms. Did you get it class? “The difference between terms consecutive terms is called the common difference, denoted by the letter d. For number 9 our common difference is 5.” “Look at number 3 we have 0, 15, 30, 45 and we are looking for next three terms and for their common difference. Anyone from the class? Yes Ivy? “Very good Ivy” “Class, the formula to get the common difference is d = Subtract the first term from the second term “Identify the common difference of number 1. 1, 5, 9, 13, _, _, _ Yes Russel?“ “Very good Russel, kindly give the next three terms of the given.” “I wonder what if we are looking for the 50th term of the given, yes we can solve it manually but it is time consuming, but we don’t need to worry there is a solution for that problem.” “We called that as the General Term of an Arithmetic Sequence” “ Formula for Arithmetic Sequence The 𝑛𝑡ℎ term (the general term) of an arithmetic sequence with first term a, and the common difference d is 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 “Yes Maam” “The next three terms Ma’am are 60, 75 and 90. The reason why I got those terms is I just add 15 to the 4th term to get the 5th term then 5th term + 15 then I get the 6th term and so on. Therefore the common difference is 15” “4 Ma’am” “17, 21, 25 Ma’am”
- 6. “Let’s have our example 1 Find the 15th term in the arithmetic sequence 2, 5, 8, 11, . . . So first, look for its common difference, to get the d, we need to subtract the 1st term from the 2nd term (5-2= 3), our d = 3. Next lets just substitute the given in our formula, the nth term is 15 that’s what we are looking for, then our first term (𝑎1) is 2 then the common difference is 3. 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑎15 = 2 + (15 − 1)3 𝑎15= 2 + (14)3 𝑎15= 2 + 42 𝑎15= 44 Therefore the 15th term is 44. Did you get it class? “So now its time for you to shine, using the same given for example number 1, kindly look for the 25th term. Yes Ann?” “You got it right Ann, good job!” “Can you follow class?” “Now, let’s proceed try another example.” “Yes Ma’am” “I already got the common difference ma’am and it is 3, then the first term is 2 and we are looking for the 25th term, the nth term is 25. now I’ll substitute it to the formula and our formula is 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑎25 = 2 + (25 − 1)3 𝑎25= 2 + (24)3 𝑎25= 2 + 72 𝑎25= 74 “ Yes Ma’am”
- 7. Find the 10th term in the arithmetic sequence 2, 4, 6, 8, . . . Anyone from the class? Yes Shine? “Very Good Shine! “And for our last problem, using the given for example 2 find the 25th term. Anyone from the class? Yes Nikki? “Very good Nikki!” “I’ll give one real life problem about arithmetic sequence, A writer wrote 630 words on the first day, 760 words and 890 words for the 3rd day, look for the number of words in 5th day? Yes James? “Well done, class! Do you have any questions? Any clarifications?” D. Analysis “Great I’m pleased with your participation” “Before we end what is sequence again?” “What is our lesson for today?” “How about the meaning of arithmetic sequence?” “The first term ma’am is 2, the common difference is 2 because 4-2 is equals to 2 and the nth term is 10. Now we can substitute to the formula, 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑎10 = 2 + (10 − 1)2 𝑎10= 2 + (9)2 𝑎10= 2 + 18 𝑎10= 20 Therefore Ma’am, the 10th term is 20. “Ma’am first term is 2, the common difference is 2, the nth term is 25 then we can now use the formula 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑎25 = 2 + (25 − 1)2 𝑎25= 2 + (24)2 𝑎25= 2 + 48 𝑎25= 50 The 25th term Ma’am is 50.” “1150 words Ma’am. You need to add 130 to get the next term, 890 +130 is 1020 and that is for our 4th day, and 1020 + 130 is 1150 and that is for 5th day” “None Ma’am” “A sequence or progression is a list of objects, events or numbers in a definite order.” “Arithmetic Sequence Ma’am” “An arithmetic sequence or arithmetic progression is a sequence which each term
- 8. “How to get the common difference?” “Who can give the formula of General Term of an Arithmetic Sequence “Very good class, you certainly did well today” E. Generalization “Now class, let’s remember the formula in finding general term of an arithmetic sequence. Plus the definition and the concept of arithmetic sequence not only by mind but also by heart. Okay class?” “Fabulous Class!” “And with that, remember that life is a sequence or series of moments all called now. So don’t waste every second of it and live it to the fullest” after the first term differs from the proceeding term by a constant.” “2nd term – 1st term Ma’am” “𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑” “Yes Ma’am!” Application Activity: Question 1: Find the 10th term of the sequence 1, 6, 11, 16, …(5pts) Question 2: Solve for the 5th term of the sequence 100, 250, 400? (5pts) Question 3: Give 1 own problem about arithmetic sequence then solve it. (10pts) Answers: 1. 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑎10 = 1 + (10 − 1)5 𝑎10= 1 + (9)5 𝑎10= 1 + 45 𝑎10= 46 2. 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 𝑎5 = 100 + (5 − 1)150 𝑎5= 100 + (4)150 𝑎5= 100 + 600 𝑎5= 700
- 9. Assignment 1. Define and give the partial sum of arithmetic sequence. 2. Search and study the definition and formula of Geometric Sequence.