Teaching/Learning Activities; Arithmetic Sequence, Arithmetic Series
PPT: Power Point.
A.S: Arithmetic Sequence.
Large Group Presentation;
Introduction
1. Asking students ‘what is the patterns?’ then show the picture of Pyramids in Chichén Itzά, Mexico using Power Point. Have the student watch the picture especially a series of steps. Then explain that the patterns exist in the real world and everywhere.
2. Using Power Point show the students a sheet it has step-shaped array of square and colored 2, 5, 8, 11 squares. Pretend this colored square are the steps and ask the students how many steps would be colored the next two steps.
3. Show the next colored steps 14, 17 with Power Point.
Stressed the same number are added to make the next steps
4. Explain the definition of sequence, sequence notation, terms, infinite sequence using power point.
Sequence: an ordered list of numbers
Terms:each of the numbers is called term and notated with variable with a subscript that gives the term’s place in the list. For the sequence 2, 5, 8, 11…, we can denote the first term, 1, by a1, the second term, 3 by a2, so on.Show Power Point , if the students need further explanation use chalkboard. Thean denotes nth term of the sequence.
Infinite sequence; A sequence that continues infinitely, is an infinite sequence.
Objective 1.1
5. Explain the definition of the arithmetic sequence as the series of a number that every next term is made with adding same number to former number using previous step. Show previous step picture with the power point.
Give the examples of the Arithmetic Sequence 5, 3, 1, -1, … on the chalkboard.
Ask the students what number is added from 53 and then 31
Ask “Are those same?” and “ Is it A.S?”
Using same method for non Arithmetic Sequence example -1, 1, -1, 1, -1, 1…
Objective 1.2
6.Explain the first term, common difference using PPT as following,
First term: the first number in the sequence, it is usually notated by a1.
Common difference: In the Arithmetic Sequence the number which is added to make the next number. It is usually notated by d.
Stress that Arithmetic Sequence has always same common difference.
Show the PPT3 and ask the students what is the common difference and the first term of this step sequence. Then check with the students 3 steps is common difference and 2 is a first term.
7.(Exercise to find common difference and first term)
Ask the students how they find 3 as a common difference on #3.Student responses would be 5-2, 8-5, 11-8. l. Lead the students to generalize those answers to (next term – previous term). Have example: -3, 0, 3, 6, 9……on the chalkboard then ask the student about a1, common difference then find with the students as -3, -3.
Stressed the point that if it the sequence has the constant common difference of the then it is Arithmetic Sequence and if it doesn’t it can’t be Arithmetic Sequence.
Objective 1.3
8. Explain the rule for finding successive termsusing example of a1 = 5, an = an-1 + 3 , on the chalkboard. Show find the terms at least first four terms. Step by step placing the number, explainhow to get the term using the rule for finding successive terms on the chalkboard like below. After first 2 terms ask the students how does the next term will be and write down the term on the chalkboard.
a1 = 5, an = an-1 + 3 ,
a1 = a1=5
a2 = a1 + 3 = 5 + 3= 8
a3 = a2 + 3 = 8 + 3 = 11
a4 = a3 + 3 = 11 + 3 = 14
Objective1.4
9. Explain that specific term can be made using first term and common difference on the chalkboard and have example as below. Lead the students notice the coefficient of the d and subscript number of the term. After showing some of the terms ask the students how a7 , a12 can be expressed then check the answer as a1+ 6d and a1 + 11d on the chalkboard.
a1 = 5, an = an-1 + 3 ,
a1 = a1
a2 = a1 + d
a3 = a2 + d = (a1 + d) + d= a1 + 2d
a4 = a3 + d = (a2+ 2d) + d= a1 + 3d
………
a7=?
a12 =?
Find a2 and a4of the following of example 1)on the chalkboard
a1 = 2, d =
a2 = 2 + =
a4= 2+ 3·=
SGlet the student find the second examplea1 = 7, d = -1, find a2 and a4 with a partner (There is an explanation of how will the group established and how will the efforts of the group be recognized on the last page.) After 15min later ask the group who can show the process and answer on the chalkboard. Give the chance for two groups and check the answers.
Make sure they using same method as a2 = a1 + d and a7 = a1 + 6d.
10. Closing the unit reminding
- What is Arithmetic Sequence?
- What is first term and common difference?
- How to find the common difference?
- The meaning of the rule of the successive term.
- How to generate the specific term using first term and common difference
use the PPT
For an individual work and formative evaluation following printout will be distributed to the students at the end of #10.
☺Ask the students write down the answer and the process.
After get back students homework assess their work and diagnose what they’ve learned and what they didn’t.
Homework (for Formative Evaluation)
1. Define the ‘Arithmetic Sequence’
2. Find the first term and common difference of the sequence.
1) 17, 20, 23, 26…
2) , , ….
3. Find at least first four terms of the sequence that match the following rule.
a1 = 5, an = an-1 -2,
4. Find the a3 and a5 for each of the sequence.
1) a1 = , d= -1
2)a1 = -3, d=
Objective 2.1
11. (Remind the students #10.)
Using the PPT7 explain nth term of an Arithmetic Sequence an= a1 + (n -1) d as below.
a1 = a1
a2 = a1 + d
a3 = a2 + d = (a1 + d) + d= a1 + 2d
a4 = a3 + d = (a2+ 2d) + d= a1 + 3d
………
an= an-1 +d= a1 + (n-1)d
Havethe students focus the changes of a1,a2, a3 ….an(subscript number of term) and the changes the number of d. So students know that the subscript number of term is one more than the number of d.Ask the students memorize the formula. Ask the students finding the value of 5thterm without evaluating all of the previous terms with this formula on the chalkboard as below.
a5= a1 + (5-1)d
12. Stress the students nth term an= a1 + (n -1) d is consist of a1 andd so first a1, d are needed for finding an= a1 + (n -1) d, have below examples to find thean on the chalkboard.
1) a1 = , d= -1
an= a1 + (n -1) d = + (n-1)(-1)= -n +
Ask the students for examples a1 = -3, d= student will answer the formula and the step as below.
an= a1 + (n -1) d = -3 + (n-1)()= n -
Objective 2.3
13. Using the PPT7ask the students how the specific terms for example a10 can be expressed. Student will answer ‘a1 + 9d ’.
Make surethat every term can be expressed with a1and d. For better understanding usespecial color for the subscript variable..
a1 = a1
a2 = a1 + d
a3 = a2 + d = (a1 + d) + d= a1 + 2d
a4 = a3 + d = (a2+ 2d) + d= a1 + 3d
………
an= an-1 +d= a1 + (n-1)d
Ask the students again how a7 term can be expressed using chalkboard as below students will answer a1 + 6d.
14. Using #13 examples, explain how to find the an, when the two terms are given. Explain it with the students responses on the chalkboard give an example to work with a partner.
Example) a7= 4 ,a13=8 , an =?
a7 = a1 + 6d = 4……….①
a13 = a1 + 12d = 8……..②
②- ①: 6d = 4
d =
Place d in ①a1 + 6· = 4
a1 = 0
an= a1 + (n -1) d = 0 + (n-1) = n-
SG Example)a5= 4 ,a7=-13 , an =?
Objective 2.4
15. Explain how to generate an when the sequence is given. Explain that because of an formula is consist of a1 and d, it is important to find a1, d. Remind the students find d using #7.
Have the example on the chalkboard.
2, 4, 6, 8, 10, …..an= ?
Ask the students step by step. Following steps would help thinking mathematics for the students.
i) what is an formula?...... writean= a1 + (n -1) d
ii) what are needed for the formula?...... writea1, d
iii)a1 = ? , d= ?...... a1 = 2, d= 2
iv)an= a1 + (n -1) d = 2 + (n-1)2 = 2n
SG Give an example and have the student find anwith the partner.
,, , ……
16. Closing the class by reminding the following using PPT8
- What is the an formula?
- Can anbe found when a1 and d are given? How?
- How specific term for example a3, a6 can be expressed?
Can you find a1 and d with those?
How?
- Can you find a1 and d in a sequence? Can you find an witha1 and d?
Individual☺ Assign the homework. Ask the students write down the answer and the process.
After get back students homework assess their work and diagnose what they’ve learned and what they didn’t.
Formative Evaluation.
1. Name the an formula
2.Find the an=
1) a1 = , d= -1
2) a1 = 0, d= 3
3. An arithmetic sequence hasa4 = -24, a18= 8 find the an.
4. Find the an of the following sequence.
-1, 4, 9, 14, ……
♥After give the feedback of the homework, there will be a period of time about a week before the test that the students can have questions for the test.
♫♪ First test will be given
Introduction and Objective 3.1
17.Begin with the class with the pattern explanations in our world. Using PPT9 show the mathematical history about the pattern relating Pythagorean Society and Harmony of numbers and show them there are some different kinds of patterns exist. Explain them Arithmetic Sequence is a basic of the sequence and the sum of the Arithmetic Sequence is also the pattern. Sum of the terms of Arithmetic Sequence is called an arithmetic series and it could be complex pattern but it would be real world pattern. Explain that the meaning of the partial sum and using the Greek letter (sigma) to write a series in sigma notation. Explain how to read and how to generate the series from the sigma notation.
Partial Sum
S1 = a1
S2= a1 + a2 =
S3 = a1 + a2 + a3 =
………
Sn = a1 + a2 + a3+………+an =
How to generate series from the sigma notation?
= (21 + 3) + (22 + 3) + (23 + 3) = 5 + 7 + 9
=1 + 3 + 5+…….+(2n-1)
Individual Give example to the student later do with the students on the chalkboard.
=
=
Objective 3.2
18. Use PPT10 and for better understanding color the important variables and stress the changing variables for sigma notation. Explain that to write as a sigma notation there should be an and it will be generated by #15.
After understanding the sigma notation expression have an example on the chalkboard and give another example for work with a partner.
S1 = a1
S2 = a1 + a2 =
S3 = a1 + a2 + a3 =
………
Sn = a1 + a2 + a3+………+an =
Sequence: 1, 3, 5, 7, ……
Series: using #15 find an = 2n – 1 (There can be some explanation those who don’t understand the changing variables using examples on the chalkboard ak = 2k – 1, at = 2t – 1)
S3 = 1 + 3+ 5 =
Sn = 1 + 3 + 5 + 7 +……+ (2n-1) =
SG Example with a partner:
-2 + 0 + 2 + 4 + ……….+ an
Objective 3.3
19. Explain the Sn formula on the chalkboard ask the student memorize the formula. If it is necessary explain again.
Sn = a1 + a2 + a3+………an
+ Sn = an + an-1 + an-2 +………a1 ** (a2+an-1)=(a1 +d + an-1) =(a1 +an)
2Sn = (a1 + an) + (a1 +an)………(a1 +an)
Sn = n (a1 +an)/2 = (a1 +(a1 +(n-1)d) =(2a1 +(n-1)d)
Objective 3.4
20. Stress that there should be a1, d and n for the sum formula and remind the student how to find the a1, d(see #15) and n[SC1](using an). With example show how the sum can be generated with the formula. When solve the example on the chalkboard ask the student step by step. Give the example to solve with the partner.
Example1)1 + 3+ 5 +……41
Ask the students.
i) What is Sn formula?...... writeSn= (2a1 +(n-1)d)
ii) What are needed for the formula?...... writea1, d, n
iii)a1 = ? , d= ?...... a1 = 1, d= 2
iv) n=?...... 41= 1 + (n-1)2
n=21
v) Sn = (2a1 +(n-1)d) = = 441
SG Example with partner.
2) 8 + 1 - 6 – 13+…..an. Sn =?
Objective 3.5
21. Explain finding Sum given the sigma notation series. Using #17 and #20.
Example)
i) What is Sn formula?...... writeSn= (2a1 +(n-1)d)
ii) What are needed for the formula?...... writea1, d, n
iii) a1 = ? , d= ? n =?
Put the number 1…..a1 = = 1
How to find the d? (see #7)
Put the number 2…..a2 = = 3
d = 3 – 1 = 2
iv) n=?...... n
v) Sn = (2a1 +(n-1)d) = = n2
SG Example with partner; before start to solve the problem remind the students the steps that they need.
=
22. Closing: Using PPT11 remind the students below content briefly.
- Generate the series of
- What do we need for the sigma notation?
- Sum formula
- What do we need for the sum formula?
- What do you need for ?
Individual ☺ Assign the homework. Ask the students write down the answer and the process.
After get back students homework assess their work and diagnose what they’ve learned and what they didn’t.
Formative Evaluation.
1. Generate the series.
1)
2)
2. Generate the sigma notation.
1) 1 +5 +9 +…401
2) 10 + 12 + 14 +….+an
3. Write down the Arithmetic Sequence sum formula.
4. Find the sum
1) 1 +5 +9 +…401
2) 10 + 12 + 14 +….+ an
5.Find the sum
1) =
2) =
♥After give the feedback of the homework, there will be a period of time about a week before the test that the students can have questions for the test.
♫♪ Second test will be given
Small group presentation;SG
After the explaining the each of the objectives, usually have the students examples with the partner.
A partner should well cooperate and help each other so assign them before the class begin. Specially student C (she doesn’t cooperate well with others) needs someone who is very friendly and work well with her. Give the students a chance to answer the examples and share their ideas. Even if the answer is incorrect, do not give some comment that makes them embarrassing.
When they deal with easy examples and definitions they could work by alone.
Teacher always ask the student raise their hands if they need help for examples and keep an eye on the ones who didn’t understand the prerequisites.
Independent study;Individual
For all of the students there are 2 formative evaluations as homework, teacher gives the feedback and keep track the students with their responses. For this reason students should answer the detailed process.
For student B; she is very fast to grasp the concept of the sequences and series. Recommend some of the extra homework for her and give feedback for her own development.
Instructional Materials
1. Text book
2. Power Point Presentation(Computer, Projection material): PPT1-11
3. Chalkboard, colored and white chalk and eraser.
1
[SC1]Think find n