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(2010CI 403 Lesson PlanEsther Song )

([Sequences, Arithmetic Sequences, and Arithmetic Series] )

Overview

The lesson plans presented cover sequences and series, particularly arithmetic sequences and finite arithmetic series. This includes exploring different types of sequences (aside from arithmetic) and patterns. Students are expected to learn how to find the nth term in a arithmetic sequences and the finite arithmetic series using the general formulas.

I chose sequences and series, because it allows students to think about patterns and relationships between terms. The arithmetic sequence provides. I chose to study arithmetic series rather than geometric sequences; I thought that it was a more natural progression. It seems intuitive to look at a finite sequence {1 4 7 10 14} and ask, well what if we add that whole lot? It seems less intuitive to present a geometric sequence and then back to arithmetic sequences (but it’s an arithmetic series this time) then to geometric series.

I recognize that these three topics could potentially be covered in 1 or 2 lessons (but perhaps not 3). I spend extra time on these topics for two main reasons. I spend an entire lesson on just sequences (without equations or strict notation). Rather, I allow students to explore sequences and patterns. Students can deduce and reason the rules that regulate sequences. The second reason that I may take longer on series and sequences is because instead of lecturing and providing the equations or solutions to examples, I decided that students could learn and teach themselves the same information by working through problems. This practice usually takes longer than having the information provided, but students will likely do more thinking and remember more of the material.

My instructional strategies include stronger emphasis on group collaboration, problem solving, and student discussion. According to Bloom's taxonomy, the higher levels involve application, analysis, evaluation, and creation. In a classroom where instruction consists only of lecturing can only realistically involve the lower levels of Bloom's taxonomy: understanding and remembering. In my classroom, I want to spend as much time as possible working in higher levels so that students experience more growth and increasing reasoning. I ask students to solve challenging but approachable problems like the warm-up on Day 2 where students are asked to sum integers 1 through 100.

Students are expected to work in groups, ask each other questions, and grow together. The emphasis is placed on cooperation rather than competition. This is especially important in a mathematics classroom because many students feel that they are “just not good at math.” Students will be less likely to participate if they feel they are constantly being evaluated by performance. Group collaboration also provides more opportunities for individual student contribution and holds them accountable to contributing. My lesson plan is also structured so that I will be more like the “guide on the side” and not the “sage on the stage.” I expect that I will be doing much less of the talking and students will then be able to argue and reason for themselves in arguments without someone telling them that an idea is right or wrong. This usually ends any sort of reasoning and the focus returns on the right answer.

With respect to technology use, I expect that I will have use of a SMART board. In this lesson, I do not use any mathematics-specific software or websites. Calculators are allowed but not highlighted. They are only used for simple arithmetic. They may be helpful for students who have not had similar levels of access to quality education and resources.

I recognize my 3 day unit plan is perhaps less conventional in its tabular form. I wrote it the way I would find most helpful if I were to quickly glance at it during the lesson to easily maneuver through the dense lesson plan. General titles with times should be easily visible. This form of organization makes the lesson plan more readable and separates sections clearly.

Objective/Purpose

There are a variety of things I hope my students gain from this lesson sequence. First, I want them to be see mathematics a little bit differently. I spend a lot of time on sequences so that students have the opportunity to explore patterns, deduce patterns, and create their own. Part of their final assessment piece is that they do exactly that. They search whatever resources to find an interesting sequence, which really means, an interesting pattern. I also hope that students will increase their reasoning skills. They are given challenging problems that should spark discussion, arguments, and multiple ways approaching a problem. Mathematics should be sensible, not magical. I felt that with arithmetic sequences and series, this would encourage such beliefs. By the end of the unit, they should appreciate sequences. They should NOT be mystified by where the equations for arithmetic sequences and arithmetic series come from. They should be able to approach problems with a little more confidence than before.

Specific mathematical and pedagogical objectives are listed in each individual lesson. In general, the mathematical objectives were centered on modeling problems with equations. Pedagogical objectives included student collaboration and student centered discussion. I tried to meet these objectives by constantly having group activities. No work is expected to be done alone. Even the final assessment is done in pairs. I try to have questions that will give students the space to think about things, and I try not to give the “ah-ha moment” away. I lecture once in three days and mostly to introduce notation.

Meeting the Needs of Students

[Background: There are 4 ELL students of varying abilities (1 fairly English proficient English speaker, 2 somewhat proficient, and 1 who only speaks minimal English). There are 2 students with IEPs for extended testing time due to reading proficiency. There are also various levels of academic motivation in the class; this is expected.]

Before continuing further, I claim that treating students equitably is distinct than treating students equally. If I have two children at the park and one of them falls and his knee start starts to bleed, I take my bandage and I go to the bleeding child. After he is bandaged, he is free to play again (perhaps a little more timidly than before). It makes absolutely no sense for me to bandage both children, and it would make no sense for the child who did not fall to complain that he too is not receiving a bandage. The idea of treating students equitably is to adjust our instruction so that all students have a chance to succeed equally.

Given my beliefs on equity, I have created a lesson that attempts to address the needs of my struggling students without sacrificing rich mathematical content. As I addressed in the overview, I try to engage the varying levels of academic motivation by assigning problems that most people can attempt. Piquing interest is important to motivation. Students may not be able to obtain the final solution and they may even give up after struggling for a short amount of time. My guess is that they will minimally be interested in the solution when it is presented, re-engaging them in a class that they otherwise would have check out in the first five minutes.

I also group students according to need. My first lesson provides an idea of the somewhat intentional groupings. The ELL students are paired together. This is intended to give students a way to lean on each other. Perhaps one student understands one part and the other student understand another part. They can create knowledge by sharing what they know. But these students will be paired with patient students who understand the mathematics. I also provide the homework from lesson 2 and the activity for lesson 1 with Spanish translations.

Since the ELL students are in pairs, I know which tables I need to spend a little extra time with so that directions and questions are clear. I also look out for my students with IEPs whenever there is a written direction or problem. I also spend time describing the assessment project on Day 3 so that students with IEPs will have some understanding of the expectations before having to read a long document about what is required.

Lastly, and I feel that this may sound rash, I allow the ELL students to choose to write in their home language (given that I have some method of translating). I reason that ELL students have several classes that have a high focus on reading and writing, which is mentally exhausting if it is in a language that one does not feel comfortable. I want them to show their mathematical knowledge and to allow them to use the tools they feel can best show their knowledge. I reflect on my own background. I have taken several Spanish classes and in few instances have been required to write an essay in Spanish about this or that topic. What I want to say and what I write are two different things. I noticed that I want to say a phrase that expresses better my meaning but am limited to a small vocabulary. In the end, my essay reads like an elementary piece with little insight. There will be plenty of other opportunities for ELL students to practice their English skills. In this lesson, I’d prefer that the ELL students feel free to express themselves the best way they know how.

Connection to Standards

The 3 day unit plan meets the following standards from the Common Core Math Standards and the NCTM Process Standards.

Common Core Math Standards:

A. High School Algebra: Seeing Structure in Expressions Standard 1

Interpret expressions that represent a quantity in terms of its context.

· Interpret parts of an expression, such as terms, factors, and coefficients.

· Interpret complicated expressions by viewing one or more of their parts as a single entity.

The second lesson plan transitions from match sticks forming into triangles to an expression that models the number of matchsticks for each figure. The same is done when considering the many doors problem in lesson 2.

B. High School Algebra: Creating Equations Standard 1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Lesson 2 and Lesson 3 both are focused on developing equations to find the nth term in a sequence of the sum of n terms in a series. Students then use the arithmetic sequence equation to solve two problems in their homework.

C. High School Functions: Interpreting Functions Standard 3

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

The first day students are exposed the Fibonacci sequence, though only briefly. They will recognize that the sequence is dependent on the prior terms. In lesson 3, there is explicit language that links sequences as functions dependent on several variables.

D. High School Functions: Interpret expressions for functions in terms of the situation they model Standard 5

Interpret the parameters in a linear or exponential function in terms of a context.

Part 2 of the homework assigned in lesson 2 requires students to use the formula for arithmetic sequences. Then after finding the answer, they must apply it to back to the problem so that the answer makes sense.

NCTM Process Standards

A. Problem Solving 

Instructional programs from prekindergarten through grade 12should enable all students to— 

· Build new mathematical knowledge through problem solving

· Solve problems that arise in mathematics and in other contexts

· Apply and adapt a variety of appropriate strategies to solve problems

· Monitor and reflect on the process of mathematical problem solving

Students build new mathematical knowledge, like developing formulas for arithmetic series and sequences, through warm up problems from lesson 2 (match sticks) and lesson 3 (summing integers 1 through 100). Summing integers has several solutions but some are more awkward or cumbersome than others.

B. Communication 

Instructional programs from prekindergarten through grade 12 should enableall students to— 

· Organize and consolidate their mathematical thinking through communication 

· Communicate their mathematical thinking coherently and clearly to peers, teachers, and others

· Analyze and evaluate the mathematical thinking and strategies of others;

· Use the language of mathematics to express mathematical ideas precisely.   

Students are placed in groups during warm ups and activities that require they discuss with each other to find a well reasoned way of solving a problem. They bounce ideas off each other. They are provided opportunities to come up to the board (particularly after time has been spent on a warm-up or an activity) to explain their ideas to the rest of the class.

Assessment

Several forms of assessment are found within the lesson plans. Whenever discussing assessment, I lean toward discussing the philosophy that underlies education. Important questions must be asked before making and evaluating any assessment piece. I ask these questions in order here.

1. What information should be gleaned from assessment for the student and for the teacher? (What is the purpose of assessment?)

2. What is assessed?

3. What are the values that are communicated by what is assessed and what should they be?

4. When evaluating assessments, what is the criterion used to decide what a successful student should be? (e.g. percentile, percentage correct, and individual improvement).

The way that I decided to assess is in response to these questions. My summative assessment is part of a portfolio that will be collected at the end of the summer. Half of the assignment is simply to find a sequence that is interesting. I am trying to promote mathematical curiosity, searching for patterns, and improving disposition toward mathematics. These are the values that I am holding highly. The other half of the points are based on understanding and applying the formulas found for arithmetic sequence and series. That is to say, reasoning and content are important as well. Student’s grades will have nothing to do with how they compare to other students. It will be based on the content of their papers.

Formative assessment happens all the time. This is from listening to groups talk to each other during problems, warm-ups, ticket out the door, homework, etc. These will be graded leniently and I will use them to better understand where my students level of understand is and how I should tweak my teaching so that student learning improves. For nearly every objective, there is some form of formative assessment that will tell me whether that objective is met or not. For example, on day 2, my objective is that students will be able to define an arithmetic sequence. They do this as a whole class activity after the warm up. They will be less likely to forget a definition if they helped to create it.

Day 1: Sequences

Time Period: 50 min

Number of Students: 26

Objectives:

· Mathematical Objectives:

· Students can define a sequence.

· Students can create their own sequences.

· Students can determine the pattern or rule that govern simple sequences.

· Pedagogical Objectives:

· Problematize concepts (sequences)

· Students will work together as a group to understand patterns

· Students will be thinking in higher levels of Bloom’s Taxonomy (i.e. Application, Evaluation, and Creation)

· All students will be participating and engaged in activities during class by reasoning and deducing.

· Students will think of strategies to attack problems

Previous Knowledge:

· Irrational numbers like e and pi

· addition, subtraction, multiplication, division, exponents

Materials:

· SMART board

· Sequence worksheet/handout

· Ticket out the door sheets

***General text and directions in black regular font

***Teacher dialogue in purple

***Anticipated student responses/questions/difficulties in italics

4

Lesson Procedure

Approx Time

Activity

Activity Description

Comments

8:00-8:05

Warm Up:

Yes/No Game

Show pre-yes/no activity on the SMART board. [See Artifact 1.1].

Directions: Deduce the pattern by guessing whether each square follows or does not follow the pattern. Guess by saying yes or no. Before we get started on the warm-up, here is an example.

Underlined yes and no will be revealed to begin process. The idea is for students to figure out that the yes’s are for even numbers. This provides an easy introduction to the yes/no game.

8:05-8:15

Show yes/no activity for sequences on the SMART board. [See Artifact 1.2]. Choose the first three for students to guess in the following order: 3a, 2a, 1d. This should give students a feel for the game without much complication. Students should recognize that the yes’s include some kind of order or pattern.

Throughout the activity, it will be important to ask questions like:

· How many people think __ is yes? How many think it’s a no? Why?

· Who thinks they know what the rule for yes and no is?

· If you think you know, come up with something that would be a “yes”.

Students may have trouble seeing a pattern for 1b, the Fibonacci sequence, if they haven’t seen it before.

Response: 1b is actually a yes. This one is tricky. Can you see why it is a yes?

This sequence is called the Fibonacci sequence. Its terms depend on adding the two terms before it. Any given term is a function of the two terms before it.

8:15-8:25

Define Sequence

Invite students to form definition of sequences: All the yes’s in the warm up activity are categorized as sequences. Based on what you’ve gathered, what do you think a sequence is? Provide wait time for students to come up with answers. Write students answers on board. Allow students to argue their points based on the yes/no game.

Write definition provided by student consensus along the lines of: a sequence is an ordered list of terms

Ask: How many terms can a sequence have? (both finite and infinite)

Keep the yes/no game on SMART board. Write students responses on a side board.

Possible student responses:

· A group of numbers that have some kind of pattern

· Numbers with a common difference/multiple/exponent

8:25-40

Find that sequence game

Have students work in groups of 4. (Two groups of 5)

Directions: We’ll be working in groups today to create and find patterns for sequences. One person in your group will come up with a rule for a sequence. Keep it simple. That person will provide a three term sequence based on the rule. Each person in the group presents a sequence that the first person will either say yes, it follows the rule or no, it does not follow the rule. After each person has presented, the goal is to guess the rule. Switch the rule maker and play again.

Provide an easy example for the game.

Okay I’m thinking of a rule, and here’s a sequence: 2, 4, 6.

(Choose three people to guess sequences. The rule is consecutive even numbers).

Possible guesses:

1. 10, 20, 30 (No, not consecutive)

2. 5, 7, 9 (No, odd)

3. 8, 10, 12 (Yes)

Have students guess what the rule is.

Hand out “Find that sequence” worksheet. [See Artifact 1.3].

Activity adapted from Thinking Mathematically by John Mason (1944).

Pair up the ELL student who speaks minimal English, 1 somewhat proficient English speaker and the Spanish speaking aid with two students who usually understand the mathematics and are more patient.

Pair up the fairly proficient English speaker and the somewhat proficient speaker with two students who understand the mathematics and are usually willing to explain material. [Keep both groups at 4 instead of 5].

Do not pair ELLs with students with IEPs.

Potential issues w example:

Students may get the rule right away and only guess sequences that will be “yes.” This will not be helpful for guessing future sequences during the game. If two students choose a consecutive even sequence right away, pause and explain.

If you think you might know the sequence, that’s good. But you want to also be able to check that it’s not another sequence. In other words, getting a “no” to a sequence is just as helpful (if not more) to figuring out what the pattern or rule is.

Choose two other students to give sequences they think might also be the rule but are not consecutive even numbers.

Walk around to see how students are doing. Make sure questions on how to play the game is answered first. Spend extra time with the group with ELL students and groups with students with IEPs.

Walk around a second round to see what kinds of interesting sequences are being made. Encourage reasoning and arguments based on reasons. Pick one or two student sequences to share with class.

8:40-8:45

Go over Find that Sequence Game

***May need to skip this step if time is short.

On board, write strategies (after asking question 3). Write down student strategies. Ask students to add this to the back of their worksheet.

Lead discussion:

1. What are some of the sequences you created?

2. What was difficult about this game?

3. What were some strategies you made to find the pattern?

This wrap up discussion draws out aspects of problem solving that could be applied in the future. It should also make students think about what strategies are useful when thinking of sequences. Anticipated difficulties are that the rule was too complicated or that there were not enough guesses allowed. But strategies will likely include finding sequences that did not fit the rule to rule out possibilities.

8:45-8:48

Wrap Up

Today, we defined what a sequence was. Who can tell us our definition or a sequence? (Pause). We were also able to create our sequences and figure out their patterns. We now have a more intuitive understanding of sequences. Tomorrow, we will be looking at a specific sequence with certain properties.

8:48-8:50

Ticket out the door

Give students ticket out the door during wrap up conversation. Give them 2 minutes to complete. Collect as they leave class. See Artifact 1.4.

Use this information to see if students have a general idea of forming sequences

Artifact 1.1: Pre-Yes/No Game- Even Numbers (with answers)

1

2

3

a

2, 4, 6, 8

Yes

1, 3, 5, 7

No

101

No

b

-2

Yes

-503

No

Pi

No

c

34

Yes

79

No

200000

Yes

Artifact 1.2: Yes/No Game- Sequences (with Answers)

1

2

3

4

a

…-1, 1, -1, 1, -1…

Yes (alternating)

8, 0, 3, -9, 3, 3, 2

No

5, 2, -1, -4, -7

Yes (decreasing by 3)

1, 2, 3, 4, 5, 0, 6, 7

No

b

0, 1, 1, 2, 3, 5, 8, 13…

Yes (Fibonacci)

0, pi, 2pi, e, e2…

No

…-103, -102, -101, -104, -101…

No

Yes (increasing exponential)

c

4, 40, 400, 4000, 400000

No

1/3, 1/9, 1/12, 1/15…

Yes (divide by 3)

1, 3, 6, 10, 15, 21

Yes (increases by one more each time)

18, 28, 38, 49, 58, 68

No

d

2, 4, 8, 16, 32, 64

Yes (2n)

9, 16, 25, 36, 49

Yes (squared terms)

2222, 4444, 6666, 7777, 8888

No

2222, 4444, 6666, 8888

Yes (increasing by 2222)

Artifact 1.3: Find the Sequence Worksheet

Directions: One person in your group will come up with a rule for a sequence. Keep it simple. That person will provide a three term sequence based on the rule. Each person in the group presents a sequence that the first person will either say yes, it follows the rule or no, it does not follow the rule. After each person has presented, the goal is to guess the rule. Switch the rule maker and play again.

Round 1

3 term sequence

Yes or No?

Rule Maker

Guesser 1

Guesser 2

Guesser 3

Guesser 4

What was the rule? _____________________________________________

Round 2

3 term sequence

Yes or No?

Rule Maker

Guesser 1

Guesser 2

Guesser 3

Guesser 4

What was the rule? _____________________________________________

Round 3

3 term sequence

Yes or No?

Rule Maker

Guesser 1

Guesser 2

Guesser 3

Guesser 4

What was the rule? _____________________________________________

Artifact 1.3 cont. (back side includes directions in Spanish)

Direcciones: Una persona en su grupo va a crear una regla para una secuencia. Debe ser sencillo/simple. Esa persona va a proporcionar una secuencia de tres términos siguiendo la regla. Cada persona en el grupo presenta una secuencia que la primera persona determinará: sí, se sigue la regla o no, no sigue la regla. Después de cada persona que ha presentado, el objetivo es adivinar la regla. Jugar de nuevo con nuevas papeles.

Artifact 1.4: Ticket out the Door

Name_________________________________

What is your favorite sequence? Provide 5 elements of the sequence and describe the rule for their order:

Name_________________________________

What is your favorite sequence? Provide 5 elements of the sequence and describe the rule for their order:

Name_________________________________

What is your favorite sequence? Provide 5 elements of the sequence and describe the rule for their order:

Day 2: Arithmetic Sequences Lesson Plan

Time Period: 50 min

Number of Students: 26

Objectives:

· Mathematical Objectives:

· Students can identify arithmetic sequences and describe properties specific to arithmetic sequences

· Students can derive the formula:

· Students can apply the formula to word problems

· Pedagogical Objectives:

· Problematize concepts (series)

· Students will work together as a group to solve problems and argue their points.

· Students will be thinking in higher levels of Bloom’s Taxonomy (i.e. Application and Analysis)

Previous Knowledge:

· Students understand what a sequence is.

· Students understand how to use variables and what they are.

· Students know how to create and apply single variable equations.

Materials:

· SMART board

· Warm up sheet

Lesson Procedure

Approx Time

Activity

Activity Description

Comments

8:00-8:07

Warm Up:

Matches

On SMART board, have match stick problem up. Each student will receive a copy of the warm up. See Artifact 2.1. Have students work in groups on warm up.

Walk around the classroom to see who was able to figure out 2d and 2e. Look for different approaches to solving this problem. (If a way of solving is interesting but not all the way there, ask the student if (s)he would be willing to come up to the board and show his/her process beforehand.)

If student asks what a sequence is, direct their question to their peers. The problem should be easy enough to start. It may get a little more challenging for finding the 30th and 99th figure.

Possible approaches by students:

· Draw figures with increasing match sticks

· Continually add 2 to the sequence until they have 30 numbers in sequence

· Create an equation like 3+2(x-1)=y. This will make a function.

8:07-8:15

Go over warm-up

Make sure everyone has a sequence like: {3, 5, 7, 9…}. This step should make the following questions easier because the pattern is readily visible.

Ask student with lower interest in math class to solve for 2a and 2c, because they should have high likelihood of success. Ask why those numbers make sense.

Ask someone to come to the board to explain 2d. Hopefully there are multiple ways of doing this.

Do NOT go over 3e. We will go over 3e later in the lesson.

8:15-8:20

Define arithmetic sequence

This sequence is an example of arithmetic sequence. Based on this sequence, what do you think are the properties of an arithmetic sequence? On board, write student responses. Ask students to also take notes.

Write down on board formal definition:

An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same. This difference, d, is also known as the common difference.

· Write on board and ask students to write in notes. Underline arithmetic sequence.

Some students may not know what all the words mean either mathematically or the actual English word. It may be important to go over words like “consecutive” and “difference.”

8:20-8:23

Main Problem w/o sufficient info

On SMART board:

There is a hallway with many doors, labeled 1, 2, 3... and so forth. A woman opens a door and then opens every 4th door afterwards. What number door is the last door she opens?

· Have students copy problem in notes.

· Give about 3 minutes for students to work on problem

· Wait for some students to find that problem is unsolvable.

Students should recognize that there is not enough information. They are missing two important pieces: the door she opens first and how many total doors she opens.

Allow students to struggle with this and recognize what info is missing.

8:23-8:33

Main problem w/ new info

Add on SMART board

Problem revised: She opens the 5th door first. She opens 27 total doors.

· Allow students 7 minutes to work in groups (2-4).

· Move around groups and see how they are doing.

· If reluctant learner understands process, ask him/her to come up to board and explain it.

Move to different groups to see ways they are approaching this problem.

Final solution on board:

5+26*4 = 109 (door 109 is the last door she opens). Have students explain the meaning of the answer.

Some students may start with 5 and add four, 27 times. Some students may be paralyzed by the large problem. To these students, I would recommend finding a solution if she only opened 8 doors. This way, they can find a general rule and see if they can apply it the larger value of 27.

Some students may decide to multiply 4 and 27 and add 5. This makes sense, except that it over-counts a door. I can ask this student to come to the board and explain his/her answer. I will see if the class catches this mistake before addressing it myself.

***Possible STOP here.

This is a good stopping place for us. How does this problem relate to arithmetic sequences? Write a paragraph for homework to respond to these questions. Don’t worry about the homework on the back of the warm up.

If there is not enough time, we can stop the class here and continue the rest for the next day.

8:33-8:43

Problem 1 Generalized

Have the equation on board: 5+(27-1)*4 = 109

1. [As a class] What if the first door the woman opens is the 6th door? Which values change?

What if the first door she opens is the 9th, 15th, 20th door? (Label 5 as First Door)

2. What if the number of total doors she opens changes to 35, 10, or 501? Which values change? (Label 27 as total doors opened)

3. What if he opens every 8th, 9th, or 12th door? Which values change? (Label as difference)

The final door value is dependent on which factors? Pause. Though the

Please write this in your notebooks: (write on board)

Note that the values might change, but the relationship between the variables do NOT change.

4. Generalized equation: First door+(total doors-1)*(difference between doors)=Final door

1. The value 5 changes to 6, 9, 15, or 20 depending. This also changes the final door value.

2. The value 27 changes to 35, 10, or 501. This also changes the final door value.

3. The value 4 changes to 8, 9 or 12. This also changes the final door value.

***Possible STOP here.

This is a good stopping place for us. Write on board: Consider for homework what this problem might have to do with arithmetic sequences. What is an arithmetic sequence? How does this problem relate to arithmetic sequences? Bring a paragraph on what you think. Don’t worry about the homework on the back of the warm up.

8:43-8:50

Further General-ization

Wrap up

Wrap Up

1. Can someone remind us of what an arithmetic sequence is?

2. How does this problem relate to arithmetic sequences?

3. [on board] We can generalize further. For the nth term in a sequence, what will the final term be? For instance, I have an arithmetic sequence {a1, a2, a3, a4,…, an}. What information do I need?

a1 + (n-1)*d = an

4. Do you see how it looks very similar to our earlier equation 5+(27-1)*4 = 109?

Homework is on the back of your warm up. Please complete it and bring it in tomorrow. [See Artifact 2.1 cont.]

Students should be able to gather you need to know the first term of the sequence and the difference between terms.

Artifact 2.1 Matchsticks warm up

Warm it up!

Shown below are triangles formed by match sticks (cerillos). One triangle is added to the previous figure each time.

1. Represent the progression of the number of match sticks as a sequence.

2. How many match sticks are needed for the

a. 5th figure?

b. 6th figure?

c. 10th figure?

d. 30th figure?

e. 99th figure?

Answers:

1. {3, 5, 7, 9,…}

2. Match sticks needed

a. 11

b. 13

c. 21

d. 61

e. 199

Artifact 2.1 cont

On BACKSIDE:

Homework:

1. Re-do number 2c, d, and e using the formula for arithmetic sequences. Show your work.

2. Let's say there's a long hallway with 500 doors. A man comes along and opens the 16th door. Then he opens every third door afterward (19, 22, 25,.... so on) until he reaches the end of the hallway. What number door will be the last door he opens?

Tareas:

1. Volver a hacer número 2c, d y e utilizando la fórmula de las secuencias aritméticas. Muestra tu trabajo.

2. Digamos que hay un largo pasillo con 500 puertas. Un hombre llega y se abre la puerta 16. Entonces él abre todas las puertas tercero (por ejemplo 19, 22, 25 ,... etc) hasta que llega al final del pasillo. ¿Qué puerta número será la última puerta que se abre?

Day 3: Arithmetic Series

Time Period: 50 min

Number of Students: 26

Objectives:

· Mathematical Objectives:

· Students can define a series and arithmetic series.

· Students can use sigma notation for arithmetic series.

· Students can find the arithmetic series of an arithmetic sequence

· Students can derive formula for arithmetic series: Sn=(a1+an)n/2.

· Pedagogical Objectives:

· Problematize concepts (series)

· Students will work together as a group to

· Students will be thinking in higher levels of Bloom’s Taxonomy (i.e. Application and Analysis)

· All students will be participating and engaged in activities during class

Previous Knowledge:

· Add and subtract negative numbers

· Add and subtract fractions

Materials:

· SMART board

· Optional calculators

· Assessment assignment hand-outs

Lesson Procedure

Approx Time

Activity

Activity Description

Comments

8:00-8:05

Warm Up:

Problem

On SMART board: A teacher, annoyed with his student, asks him to find the sum of all integers 1 through 100. To the teacher’s surprise, the student produced the answer in less than one minute. What is the solution and what process did you use to find the solution?

Some students will ask if they can use a calculator. Tell them they might not need it, but they can use a calculator. (This will allow for multiple ways of solving).

Have students work in groups on warm up. Walk around to check on group progress. Allow students to struggle. Make sure groups with ELLs and students with IEPs understand the prompt, but do not help them further with actually solving the problem.

Some students may have seen this problem before but might not remember the way to solve it. Ask them not to give it away to the other students who have not seen this problem before.

8:05-8:10

Warm up cont.

Provide the following hint: What if you only had to sum integers 1 through 10? How would you find this answer? Can you generalize the rule to find the sum of 1 through 100?

Write the numbers 1 2 3 4 5 6 7 8 9 10 on the board.

Walk around to groups to see if this sparks some new ways of thinking.

8:10-8:15

Go over warm up

Invite students to the board to offer solutions or solution PROCESS. One student may have added on a calculator 1 through 100: 1 + 2 + 3 … + 100

Another student may realize that numbers can be added to make sets of 100. (99+1, 2+98, 3+97, so forth).

Ask how many sets of 100 are made. (50 sets). Then add the 50 (without a pair to make another set of 100). Total sum is 50*10+50=5050

Provide last way to think about problem if no one comes up with it. Another way to think about this problem is to add

1 + 2 + 3 +…+100

100+99 +97+…+ 1.

Then we get (100+1)*100/2

8:15-8:20

Other problems

What if I have an arithmetic sequence with 8 values?

{1, 3, 5, 7, 9, 11, 13, 15} Can we use the same method to find the sum of these values?

Pause/Give students time work out values. Some students may realize that the same solving method can be used. This problem will be revisited.

8:20-8:30

Lecture

This is an example of an arithmetic series. [On board]. An arithmetic series is the sum of a finite arithmetic sequence. For example if I have four terms in an arithmetic sequence {a1, a2, a3, a4}, then the arithmetic series is simply a1+a2+a3+a4.

Students are expected to take notes.

This is basically what we’ve been doing for the warm-up problem. Integers 1 through 100 can be represented as an arithmetic sequence with our first term, a1, and equaling 1. Our nth term is 100. The common difference is 1. The answer we found (5050) is the arithmetic series of the finite arithmetic sequence.

Here I am going to write out the series for 1 through 100. 1+2+3+4… (Do this until students start to get restless). This is pretty tiring and you guys get the idea. But having the gist of things is not good enough in mathematics. We need a formal notation that will let us express things like this without working up a sweat. Welcome to summation notation:

So the E looking sign is the Greek letter Sigma. It tells us that we’re going to add the term next to it, in this case n. The bottom gives us are starting value, n=1. Afterwards, it is assumed that n=2, n=3, etc until n reaches 100.

On board:

Whatever is next to the sigma tells us what is being summed. The Under the sigma tells us where to start.

On top of the sigma tells us when to stop summing.

What if I have the following notation:

What does this mean? [Pause. Then call on a few people to see what their responses are].

Again, we know we’re summing whatever is on the right of the sigma sign. We use “n=” so that we know what which term is the variable. My equation is 3n and my variable is n. We start at n=1, but this time we see that n is not alone. (It is multiplied by 3). Our first term is 1*3 = 3. Our second term is 2*3=6. Can someone tell us what our third term and four term is?

(9 and 12).

What about our 5th term? [Someone may guess 15. Another student will likely recognize that the summation ends at 4].

Here I introduce Sigma notation. This will be important for future mathematics classes and topics.

8:30-8:40

Activity

Hand out activity. Students will work in groups. See Artifact 3.1. This activity is fairly difficult. Students who have a good understanding of what is going on can help struggling students.

If the majority of students seem to struggle, bring class together. Finish the notation part together (if necessary). Explain that our last term is n so it cannot be a variable. This is why we choose x.

Some students will have a hard time with the more formal and abstract notation. Refer back to the arithmetic sequence we used in class activity. This will make the problem more tangible.

Some students may be confused that for sigma notation, I used x= instead of n=. If this causes significant problems, I will address it as whole class.

8:40-8:45

Wrap-up

We will go over the activity in class tomorrow. Please have as much of it filled out as possible.

8:45-8:50

Go over assessment

Pass out assignment for portfolio. See Artifact 3.2

This assignment will be added to your portfolio. You will need partners for this assignment. You will get to choose your own partners. Please let me know who they are by tomorrow. This assignment is due in 1 week. Part A asks you to do a little research. Search online or in books for some interesting sequences. I’ve provided a few links that you can search through, but do not feel limited to those. Part B might be a little confusing. Take three consecutive values from the sequence you found. Then create an arithmetic sequence that approximates the sequence from Part A. Part C just asks you to find the arithmetic series of the arithmetic sum. This assignment is worth 50 points. Questions?

After class, call over ELL students. Tell them they will be allowed to write their assignment in their native language if they prefer (assuming I have some access to some sort of translating resources/help). They will still have to use the same notation. Hand them the assignment provided also in Spanish. See Artifact 3.3 [Meet with Spanish aid to see if collaboration on grading can be done. Ask to meet with students with IEPs individually. See if they have any issues with this assessment and change as necessary to their needs.

Artifact 3.1

1. You have an arithmetic sequence with n terms: {a1, a2, a3, …, an} with a difference, d.

Write the equation for an as a function of a1, d, and n.

an=(n-1)d+a1

2. You sum the arithmetic sequence and you use sigma notation: (fill in the blanks)

3. Using what you know from the warm-up problem, derive the equation for the arithmetic series (S) of the arithmetic sequence described in #1.

Try using this step:

a1 + a2 + a3 + …+ an

+ an + an-1 + an-2 +…+ a1

(a1+an)n/2=S

Artifact 3.2

Sequences and Series: Portfolio Piece

As part of the collection of your portfolio, you will write a brief paper (in pairs) to show your knowledge of sequences, arithmetic sequences, and arithmetic series. The following parts must be included in your paper for full credit. This should be typed.

A. The Interesting Sequence [approx 300 words]

Find an interesting sequence (e.g. Fibonacci, triangular, rectangular, hexagonal...) online or in a book, or create your own (non-arithmetic) sequence. Describe the sequence and the rule that it follows. Give some background on the sequence (when it was created, who made it, why). Include figures if they will help visualize the sequence. Include why you think it is interesting. Make this personal.

Resources in finding sequences:

http://blog.idiotworld.com/story/323/Why_are_integer_sequences_interesting_anyhow

http://www.mathpages.com/home/kmath025.htm

http://www.mathsisfun.com/your-own-number-patterns.html

B. Approximate with an Arithmetic Sequences [approx 200 words]

Take three of consecutive values from the sequence you found in part A e.g. {a3, a4, a5}. Determine an arithmetic sequence that approximates the three values for each term. Explain how you chose your arithmetic sequence and why. Provide at least 6 values of your arithmetic sequence and the equation of the nth term in the following form: an=a1+(n-1)d as a function of n.

For example,

If you chose the sequence {1, 4, 16, 25, 36…}, choose three values like {4, 16, 25}. You can graph these values if this helps. Create another sequence that approximates these values perhaps {6, 16, 26}. Expand the arithmetic sequence to 6 terms: {6, 16, 26, 36, 46, 56}.

C. Arithmetic Series

Describe and/or define an arithmetic series. Use the arithmetic sequence you found for part B. Determine the equation for the arithmetic series. Use your equation to find the 12th, 200th, and 5004th term in your arithmetic series. Show your work.

If you have any questions, please see me or email me at [email protected]

Artifact 3.3

Secuencias y Series- Parte de la Carpeta de Trabajos

Como parte de la colección de su carpeta, va a escribir un breve documento (en pares) para mostrar su conocimiento de las secuencias, las secuencias de la aritmética, y la serie aritmética. Las siguientes piezas deben ser incluidas en su papel para el crédito completo. Esto debe ser escrito.

A. La Secuencia Interesante [aproximadamente 300 palabras]Buscar una interesante secuencia (por ejemplo, de Fibonacci, triangular, rectangular, hexagonal, ...) en Internet o en un libro, o crear tu propia (no aritmética) secuencia. Describe la secuencia y la regla que le sigue. Dar algunos antecedentes sobre la secuencia (cuando se creó, que lo hizo, por qué). Incluye figuras si van a ayudar a visualizar la secuencia. Incluya por qué usted cree que es interesante.

Recursos para encontrar las secuencias

http://blog.idiotworld.com/story/323/Why_are_integer_sequences_interesting_anyhow

http://www.mathpages.com/home/kmath025.htm

http://www.mathsisfun.com/your-own-number-patterns.html

B. Aproxima con secuencias aritméticas [200 palabras]Tomar tres de los valores consecutivos de la secuencia que se encuentran en la parte A, por ejemplo {a3, a4, a5}. Determinar una secuencia aritmética que se aproxima a los tres valores para cada plazo. Explicar cómo se eligió a su secuencia aritmética y por qué. Proporcione por lo menos 6 valores de la secuencia de la aritmética y la ecuación del enésimo término de la siguiente forma: an = a1 + (n-1) d como una función de n.

Por ejemplo,Si elige la secuencia {1, 4, 16, 25, 36, ...}, elegir tres valores como {4, 16, 25}. Usted puede graficar estos valores si esto ayuda. Crear otra secuencia que se aproxima a estos valores tal vez {6, 16, 26}. Expanda la secuencia aritmética de 6 términos: {6, 16, 26, 36, 46, 56}.

C. Serie AritméticaDescribir y / o definir una serie aritmética. Usar la secuencia aritmética que se encuentran la parte B. Determinar la ecuación de la serie aritmética. Usa tu ecuación para encontrar el término 12, 200, y 5004a en su serie aritmética. Demuestra tu trabajo.

Si usted tiene alguna pregunta, por favor comuníquense conmigo o enviarme un correo electrónico a [email protected]

Artifact 3.4

Rubric: Sequences and Series Portfolio Piece

Part A

Criteria

Total points

Exceeds Expecations

Student provides details and background of sequence. Student presents the rule used for the sequence in a clear way. Student gives detailed explanation of why the sequence is interesting to him/her. Student provides a figure if applicable.

22-25

Meets Expectations

Student provides a brief background of the sequence. Student explains why the sequence is interesting and generally links it to his/her interests. Student provides a rule but explanation is confusing.

18-21

Below Expectations

Student provides little background or history of the sequence or none at all. Student provides a sequence but the rule is missing. Students give an explanation of why the sequence is interesting in a general way that does not reflect his/her interest in the sequence.

Under 18

Part B

Criteria

Total points

Exceeds Expecations

Student has provided an arithmetic sequence that approximates three of the values of the sequence. The explanation is clear and well thought. Student has provided at least 6 terms in the arithmetic sequence. Student has provided the correct equation.

18-20

Meets Expectations

Student has provided an arithmetic sequence and it approximates three of the values of the sequence. The explanation is not entirely clear and the reason for why created it is not fully established. Student provides at least 6 terms. Student has provided the correct equation for the sequence or has made a small arithmetic error.

15-17

Below Expectations

Student provides an arithmetic sequence that does not approximate three values of the sequence in part A. Student have provided less than 6 terms of the arithmetic sequence. Student has either no equation or something very wrong for their equation.

Under 18

Part C

Criteria

Total points

Exceeds Expecations

Correct equation is used and values for 12, 200, and 5004 are correct. Work is clearly shown.

5

Meets Expectations

Correct equation form Sn=(a1+an)n/2 but there may be small algrebra errors. Values are found from the equation.

3-4

Below Expectations

Unable to determine how equation was obtained or no equation is provided. Values for nth terms are not listed, have no work shown and/or do not match the equation that was obtained.

Under 3