Date: 26.03.2013 Teacher: Burcu Yagiz Number of Students: 23 Grade Level: 10th grade Time Frame: 40 minutes

Binomial theorem; Pascal Triangle . 1. Goal(s)  The lesson aims to help high school seniors practice fundamental mathematical skills, including mathematical induction, proofs of some properties of Pascal’s triangle, and proofs of the Binomial Theorem and some of its applications, including one to binomial probability. 2A. Specific Objectives (measurable)  Students should have a general understanding of the binomial theorem and combinations.  Understand the relationship between the expansion of (a + b) ^n and combinatorics. 2B. Ministry of National Education (MoNE) Objectives  Pascal özdeşliğini gösterir ve Pascal üçgenini oluşturur.  Binom teoremini açıklar ve açılımdaki katsayıları Pascal üçgeni ile ilişkilendirir. 2C. NCTM-CCSS-IB or IGCSE Standards:  Students will be able to create growing patterns.  Students will be able to describe growing patterns.  Students will be able to analyze how growing patterns are created.  Students will be able to find number patterns in Pascal’s triangle.  Calculate the number of possible combinations ( ) n r C of n items taken r at a time  Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion 3. Rationale  In each lesson during this period, I will prove one or two properties of Pascal’s triangle or the Binomial Theorem. At the end of the lesson, students put together their worksheets on a construction paper as a project. Proofs are an essential part of mathematics; they are the primary vehicles used to convey mathematical thinking and sharpen students’ problem solving skills.

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By completing worksheets in classes, students begin with constructing Pascal’s triangle in numbers and then in combination symbols. Students learn to calculate number sequences with graphic calculators, such as TI-83/84 Plus. Students then identify and derive Pascal’s rule that underlines the construction of Pascal’s triangle. With the knowledge of Pascal’s rule, students proceed to observe how the Binomial Theorem can be proved by mathematical induction. Students also prove some interesting number patterns embedded in Pascal’s triangle with the Binomial Theorem. The lesson supports NCTM standard that expects high school students to recognize reasoning and proofs as fundamental aspects of mathematics.

4. Materials  Pascal’s triangle template  “Do You Feel Lucky?” problem worksheet  TI-83/TI-84 calculator 5. Resources  http://standards.nctm.org/document/chapter7/reas.htm Reasoning and proof standard for grades 9-12  http://mathforum.org/workshops/usi/pascal/index.html  Advanced Mathematics Richard. G. Brown 6. Getting Ready for the Lesson (Preparation Information)  The lesson employs worksheets, and group activities to help students pick up inductive and deductive reasoning skills. The worksheets implement scaffolding by including clear definitions, step-by-step instructions, numerical examples, formula explanations, and hints to help students comprehend abstract symbol operations.  Prepare worksheets, and transparency on the format and an example of the seq () function available on the TI-83/84 graphic calculator. 7. Prior Background Knowledge (Prerequisite Skills)  Students should have experience in problem solving, particularly in finding patterns.  They should be skilled at interpreting and evaluating with formulas. Some background in the basics of combinations, permutations is expected. Students should be familiar with the TI-83/TI-84 calculators, and particularly should be proficient at using the lists.

Lesson Procedures Transition: The teacher should say: “Hello, I am your mathematics teacher during this period. I am trainee teacher from Bilkent University. I hope we will have a very good lesson together. We will be working on binomial theorem and Pascal triangle today.” 8A. Engage (5 minutes)  This lesson will introduce students to the binomial theorem through a variety of activities.  Students will begin by filling in values of Pascal’s triangle on discovering patterns worksheet.

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1. 2. 3. 4. 5. 6. 7. 8.

(a+b)^1= (a+b)^2= (a + b)^3= (a + b)^4= (a + b)^5= (a + b)^6= (a+b)^7= (a + b)^8=

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Study the following array of numbers. What patterns do you see in the arrangement of the numbers? Describe each pattern using words and symbols.

 “ Do You Feel Lucky? “ activity You have a math quiz for which you are completely unprepared. The quiz has three questions. The bad news is that you have no idea how to do any of them. The good news is each question is true or false. You guess on each question. 1. Determine the number of different ways that you could get every question correct (or wrong, it’s the same answer!) 2. Find the number of ways that you get 1 question wrong. What other number of incorrect answers has the same number of possible outcomes? 3. How many different ways can you get 3 questions correct? Transition: Teacher should give students guide worksheet to prove proof of Pascal triangle with together. B. Explore (10 minutes)  Students can write the expansion of some equations, but they will encounter difficulty about time management and little mistakes.  Thereby, the students will be able to use their TI to solve the coefficients of these expansions by following followed directions.  TI-83/84 Plus seq () Function Format seq(expression, variable, begin, end [, increment]) Arguments Expression = 7 nCr X Variable = X

Begin = 0 (Initial value of the variable) End = 7 (Final value of the variable) Increment = 1 (default value, optional) ++++ Example for Row 7 of Pascal Triangle Input: 2ND LIST OPS 5: seq ( Display: seq ( Input: 7 MATH PRB 3: nCr Display: seq (7 nCr Input: [X,T,θ,n] , [X,T,θ,n] , 0 , 7 , 1 ) Display: seq (7 nCr X, X, 0, 7, 1) {1 7 21 35 35 2... Row 7 of Pascal’s triangle: {1 7 21 35 35 21 7 1}  Encourages students to work together without direct instruction from the teacher  Observes and listens to students as they interact  Asks probing questions to redirect students’ investigations when necessary  Provides time for students to puzzle through problems Transition: Teacher should prove the formula on board again by students’ answers C. Explain (10 minutes)  From these examples we can see that the first term in the expansion of (a+b) ^n is always 1a^n b^0. In successive terms the exponents of a decrease by 1 and the exponents of b increase by1, so that the sum of two exponents in a term is always n. The coefficients of the terms also have a pattern.  Give some definitions about that the array is called Pascal triangle named for the French mathematician Blaise Pascal (1623-1662)  Explain that the numbers in the array are the coefficients of the terms in the expansion of (a+b) ^n, they are called binomial coefficients. Notice that each number is the sum of the two numbers just above it. Hence, from the fifth row of the triangle, we can quickly form the sixth row.  So far, it might seem that to get the numbers in the sixth row of Pascal’s triangle you must first know the numbers in the fifth row, but this is not necessary.

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Provide the students to calculate directly by the formula, but before the formula we have an investigation part to generalize the rule. Students will complete “Do You Feel Lucky?” problem. Teacher will lead class discussion and summary upon completion. Teacher will introduce the Binomial Formula. The students will then solve a combination problem, and relate the solution back to Pascal’s triangle. Do you feel lucky?

You have a math quiz for which you are completely unprepared. The quiz has three questions. The bad news is that you have no idea how to do any of them. The good news is each question is true or false. You guess on each question. 1.

Determine the number of different ways that you could get every question correct (or wrong, it’s the same answer!)

2.

Find the number of ways that you get 1 question wrong. What other number of incorrect answers has the same number of possible outcomes?

3.

How many different ways can you get 3 questions correct?

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At the end of the activity, generalize the formula.

Transition: Teacher should extension for students to understand binomial theorem and Pascal triangle better. D. Extend (5 minutes)  Expects students to use formal labels, definitions, and explanations provided previously  Encourages students to apply or extend the concepts and skills in new situations  Reminds students of alternative explanations. Transition: Teacher should assess students’ learning process. E. Evaluate (5 minutes)

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Observes students as they apply new concepts Assesses students’ knowledge and skills May allow students to assess their own learning and reflect on process Prepare various worksheets about the topics, firstly discovering patterns, “do you feel lucky?” worksheet and TI worksheet and homework worksheet.  Discussion and observation are very important for assessment in that lesson. 9. Closure & Relevance for Future Learning  Closure refers to those actions or statements by a teacher that are designed to bring the lesson to an appropriate conclusion. It is used to help students bring things together in their own minds, to make sense out of what has just been taught. "Any questions? No. OK, let's move on" is not closure. Closure is used to:  Cue students to the fact that they have arrived at an important point in the lesson or the end of a lesson;  Help organize student learning;  Help form a coherent picture, to consolidate, eliminate confusion and frustration, etc.;  Reinforce the major points to be learned...to help establish the network of thought relationships that provide a number of possibilities for cues for retrieval.  At the end of the lesson, bring the lesson to a close with a short, sweet summary of what went on in the lesson and restate the objective. In other words, it is time to "wrap the lesson up and put it in a box." 10. Specific Key Questions:  Can you predict the next row of numbers? ( Comprehension)  Is there a pattern in the sums of the numbers in the rows? (Comprehension)  Do any numbers repeat? (Comprehension)  How long could we continue this pattern? (Comprehension)  What was the rule for one of the patterns that you made? Did anyone else make a different pattern with that rule? (Analysis)  What would be the row after that? (Application)  How do you know? (Analysis)  What is the sum of this row? ( Application)  What is the sum of the first four rows? (Application)  Do you notice a pattern in the sum of the rows? ( Analysis)

 Do you notice any patterns in the triangle? (Analysis)  What patterns do you see in the arrangement of the numbers? ( Application ) 11. Modifications  I should prepare many questions as you can before the lesson so if you have enough time you can ask new questions to the students. I should help to the students if they need help while they are trying to solve the questions In this section, I add suggestions for what I will do to re-teach the lesson or additionally modify within the lesson for students who are having difficulty understanding concepts or skills taught in the lesson. I should be aware of students’ needs, skills and interest. For example, I should choose simple questions firstly. I give time and I should decide their solving time of questions. If they solve immediately, I give more complicated questions. I should follow the solving time of students again. If they solve immediately and correctly I give challenge questions. So, from point of this view, I should have all types of questions in my mind and my lesson plan. And I also add that the questions should be different each other. And almost all questions should have more than one answer. So that different levels of learners and students can utilize the lesson. The question in my lesson should be to provide the "appropriate level of challenge" for each student's level.

Reflective Evaluation of the Lesson 

While preparing this lesson plan, I have learned so many things about binomial theorem and Pascal triangle. I have learned the proof and how I teach them to students. I considered what the students should know before proving I have learned exploration about proofs. . I have also scanned lots of books and websites about this topic. I have investigated usage of technology to integrate this lesson. So, I used virtual TI program. This preparation is very rewarding for me.