Date: March 18, 2015
Teacher: Athena KOLUKISA
Number of Students: 21
Grade Level: Lise3-E
Time Frame: 40 minutes
Summation Notation
1. Goal(s)
 Students will be able to identify summation notation (sigma), express sum of finite series with sigma notation.
2A. Specific Objectives (measurable)




Students will be able to identify summation notation (sigma).
Students will be able to write a finite sum using sigma notation.
Students will be able to read sigma notation and evaluate a sum.
Students will be able to practice the notation properties with exercises.
2B. Ministry of National Education (MoNE) Objectives

Toplam sembolünü ve çarpım sembolünü açıklar, kullanışları ile ilgili özellikleri gösterir, temel toplam formüllerini modelleyerek inşa eder.
3. Rationale
 The topic of summation notation is a part of algebra. Students will be able to use the notation in next topic which will be sequences and series.
 Geometric sequences are also useful in next topic which will be multiplication notation.
 LYS includes questions from summation notation. Several questions were asked in the past in Turkish exams.
4. Materials
 Board markers
 Computer and interactive board
 Worksheets
5. Resources
 Esen Yayınları 11th grade book
 Materials which was prepared ACI math teachers
 http://www.coolmath.com
 http://www.storyofmathematics.com
 http://www.mathcentre.ac.uk
 http://www.coolmath.com
6. Getting Ready for the Lesson (Preparation Information)
 Make sure worksheets are distributed before the lesson.
7. Prior Background Knowledge (Prerequisite Skills)
 Students will be expected to know some rules about a finite sum.
Lesson Procedures
Transition (2 minutes): Good morning! I introduced myself last Friday so you knew me and you knew my rules for lesson procedure. Today, I will teach summation notation.
Let’s get started, shall we?
I knew the last Monday you learnt mathematical induction you proved some rules.
 Who remember the rules?
 Which was the first rule that you proved with mathematical induction in the lesson?
Now I want to start to lesson with very famous mathematician he lived in 19th century.
The teacher prepares interactive board and his/her presentation about the lesson.
8A. Engage (5 minutes)
•
•
Carl Friedrich Gauss is sometimes referred to as the "Prince of Mathematicians" and the "greatest mathematician since antiquity". He has had a remarkable influence
in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.
At the age of 7, he is reported to have amazed his teachers by summing the integers from 1 to 100 almost instantly (having quickly spotted that the sum was actually
50 pairs of numbers, with each pair summing to 101, total 5,050).
In elementary school in the late 1700’s, Gauss was asked to find the sum of the numbers from 1 to 100. The question was assigned as “busy work” by the
teacher, but Gauss found the answer rather quickly by discovering a pattern.
Transition: Now, work individually on the question. Individual study time is 2 minutes.
8B. Explore (5 minutes)


Teacher writes 1+2+3+……+n = ? on the board.
Teacher walks around and controls students work.
ASK students to decide the exploring activity was harder and why? (Evaluation)
Several other questions can be asked:
a) Which method did you use for the proof? (Comprehension),
b) What is the answer if we use the formula? (Comprehension)
Transition: Let’s now move on to summation notation.
C. Explain (15 minutes).
The teacher writes the title on the board.
Sigma:
• The sigma the 18th letter of the Modern Greek alphabet.
• Σ is used as a symbol for the summation operator.
The teacher explains the terms on the diagram and gives examples.
Examples:
Ask:
How do you write these finite sums using sigma notation?
•
1+2+3+…+n=
•
Answer: ∑𝑛𝑘 𝑘
•
Answer: ∑𝑛𝑘 𝑘 2
2+4+6+…+2n=
•
Answer: ∑𝑛𝑘 2𝑘
•
Answer:
13 + 23 +33 +…+ n3 =
Answer: ∑𝑛𝑘 𝑘 3
1+3+5+…+2n-1=
∑𝑛𝑘(2𝑘
12 + 22 + 32 +…+ n2 =
− 1)
•
1
1
+ +
1.2 2.3
…+
1
𝑛.(𝑛+1)
=
1
Answer: ∑𝑛𝑘 𝑘.(𝑘+1)
Transition (3 minutes): The teacher distributes worksheet which includes formulae and some properties about the topic. Look at the formulae and look at examples which are
next to the formulae. I want you to calculate the sum according to the formulae; individually work time is 2 minutes.
The teacher asks students answers and he/she checks the answers and writes correct answers on the board.
D. Extend (10 minutes)
Transition (1 minute): now I will prove some properties which are on your worksheets.
Properties:
1. ∑𝑛𝑘=1 𝑐 = n.c
Ask students:
• How do we prove it?
• Who wants to share idea about the proof?
Proof:
∑𝑛𝑘=1 𝑐 = c+c+……+c = c.n
2. ∑𝑛𝑘=1 𝑐 .ak= c. ∑𝑛𝑘=1 ak
Ask students:
• How do we prove it?
• Who wants to share idea about the proof?
Proof:
∑𝑛𝑘=1 𝑐. ak = c a1 +c a2 +……+c. an = c.( a1 +a2 +…+an )= c. ∑𝑛𝑘=1 ak
3. ∑𝑛𝑘=1 ak ±bk= ∑𝑛𝑘=1 ak±∑𝑛𝑘=1 bk
Proof:
∑𝑛𝑘=1 ak±bk= a1±b1±a2±b2±a3±b3±……±an±bn
= a1± a2±……±an±b1± b2……±bn
=∑𝑛𝑘=1 ak±∑𝑛𝑘=1 bk
𝑝
4. ∑𝑛𝑘=1 ak=∑𝑘=1 ak+∑𝑛𝑘=𝑝+1 ak
Proof:
∑𝑛𝑘=1 ak= a1+a2+a3+……+ap
+ ap+1+ap+2……+ an
𝑝
=∑𝑘=1 ak+∑𝑛𝑘=𝑝+1 ak
𝑛+𝑚
5. ∑𝑛𝑘=𝑝 ak= ∑𝑛−𝑚
𝑘=𝑝−𝑚 ak+m = ∑𝑘=𝑝+𝑚 ak-m
Proof:
∑𝑝𝑘=1 ak= ap+ap+1+ap+2+……+an
∑𝑛−𝑚
𝑘=𝑝−𝑚 ak+m = ap-m+m+ap+1-m+m+ap+2-m+m+……+an
= ap+ap+1+ap+2+……+an
𝑝
= ∑𝑘=1 ak
∑𝑛+𝑚
𝑘=𝑝+𝑚 ak-m= ap+m-m+ap+1+m-m+ap+2+m-m+……+an
= ap+ap+1+ap+2+……+an
𝑝
= ∑𝑘=1 ak
𝑛+𝑚
∑𝑛𝑘=𝑝 ak= ∑𝑛−𝑚
𝑘=𝑝−𝑚 ak+m = ∑𝑘=𝑝+𝑚 ak-m
E. Evaluate (3 minutes)
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In explain part the teacher evaluates students according to their participation and their answers.
First set of exercises in worksheet, evaluation will be done according to answer sheet.
Second set of exercises in worksheet, evaluation will be done according to answer sheet.
9. Closure & Relevance for Future Learning
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The teacher clarifies the key points of the lesson, tries them together into a coherent whole.
At the end of the lesson, the teachers bring the lesson to a close with a short, sweet summary of summative notation.
The teacher makes to take quick feedback from students about the lesson.
10. Specific Key Questions:
•
•
•
•
•
•
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How Gauss proved the finite sum formula?
How do you write these finite sums using sigma notation?
How do we prove the equality?
Who wants to share idea about the proof?
How do you write these finite sums using sigma notation?
Which method did you use for the proof?
What is the answer if we use the formula?
Reflective Evaluation of the Lesson
While I was preparing the lesson plan, finding engagement activity was hard to me. Sigma notation was not in IB math curriculum, so I searched other resources. I
found some webpages which included sigma notation lesson plans. I analyzed the lesson plans and their examples. Then I wanted to find when we use the notation in real life
or did we use it in real life and I realized that 10 Turkish lira back face included sigma notation. Then I decided to the example in my lesson for hooking students.
Interactive boards are used in ACI effectively in every subject lessons. I have never used the type of interactive board in my lessons so I practice using the board. And I
prepared my lesson with using the board and PPT. My mentor had shared her sources with me; she gave me worksheets about the lesson. I added the examples in my lesson
presentation.
My lesson was in the afternoon so I was nervous about students’ attention of my lesson. Before the lesson, I prepared Emel Hanım’s computer for my lesson. I had a
lesson with Lise3/G after the class I continued same topic with Lise3/E. But the lessons were different classrooms so I had to move my materials one class to other.
I was in the class before the bell rang and I prepared my materials. When the bell rang, Dr. Sands and Arman Bey came to my class but the students didn’t come to the
class on time. I waited them one or two minutes and I had to start my lesson with 3-4 students. During the first five minutes, the students came to lesson with small groups so
my beginning of engagement part interrupted two or three times. And then I hooked the students’ attention, I started to lesson with story about Gauss. I mentioned example
about usage of sigma. Then I started to teach sigma notation. During the lesson I wanted to interact with students and I tried to do it, I asked questions for that purpose but I
didn’t hook all of students in all times.
I planned to complete my entire lesson plan but I didn’t complete. This was my choice because I could complete my lesson plan but I preferred to get feedback from
students. I finished my lesson with summarizing the lesson, I mentioned about key points of the topic.
I was my second time for teaching mathematics in English so I had nervous about it but I think my English was fluency in the lesson. If I have to give feedback myself,
I should been more energetic and hook all students. However I am happy with the lesson.