Sequences and Series



MYP ASSESSMENT COVER

|Name | |Date |13/10/2014 |

|Class |MYP5 Extended |

|Unit Title |Sequences and series, and their applications |

|Key Concept |Relationships |

|Global Context |Scientific and technical innovation |

|Assessed Criteria |Criteria C: Communicating |

| |Criteria D: Applying mathematics in real-life contexts |

| | |

|Approaches to Learning |Self-Management Social Communication |

| | |

| |Research Thinking |

|Criteria |Level of Achievement (0-8) |

|Criteria C: Communicating | |

|Criteria D: Applying mathematics in real-life contexts | |

|Deadlines |

|Your DRAFT is due |6/10/2014 |

|Your FINAL is due |13/10/2014 |

CRITERION C: COMMUNICATION

|LEVEL OF |DESCRIPTOR |

|ACHIEVEMENT | |

|7-8 |The student is able to: |

| |i. consistently use appropriate mathematical language |

| |ii. use appropriate forms of mathematical representation to consistently present information correctly |

| |iii. move effectively between different forms of mathematical representation |

| |iv. communicate through lines of reasoning that are complete, coherent and concise |

| |v. present work that is consistently organized using a logical structure. |

|5-6 |The student is able to: |

| | |

| |usually use appropriate mathematical language |

| |usually use appropriate forms of mathematical representation to present information correctly |

| |usually move between different forms of mathematical representation |

| |communicate through lines of reasoning that are complete and coherent |

| |present work that is usually organized using a logical structure. |

|3-4 |The student is able to: |

| | |

| |use some appropriate mathematical language |

| |use appropriate forms of mathematical representation to present information adequately |

| |communicate through lines of reasoning that are complete |

| |adequately organize information using a logical structure. |

|1-2 |The student is able to: |

| |use limited mathematical language |

| |use limited forms of mathematical representation to present information |

| |communicate through lines of reasoning that are difficult to interpret. |

|0 |The student does not reach a standard described by any of the descriptors below. |

CRITERION D: Applying mathematics in real-life contexts

|LEVEL OF |DESCRIPTOR |

|ACHIEVEMENT | |

|7-8 |The student is able to: |

| | |

| |identify the relevant elements of the authentic real-life situation |

| |select appropriate mathematical strategies to model the authentic real-life situation |

| |apply the selected mathematical strategies to reach a correct solution to the authentic real-life situation |

| |justify the degree of accuracy of the solution |

| |justify whether the solution makes sense in the context of the authentic real-life situation. |

|5-6 |The student is able to: |

| | |

| |identify the relevant elements of the authentic real-life situation |

| |select adequate mathematical strategies to model the authentic real- life situation |

| |apply the selected mathematical strategies to reach a valid solution to the authentic real-life situation |

| |explain the degree of accuracy of the solution |

| |explain whether the solution makes sense in the context of the authentic real-life situation. |

|3-4 |The student is able to: |

| | |

| |identify the relevant elements of the authentic real-life situation |

| |select, with some success, adequate mathematical strategies to model the authentic real-life situation |

| |apply mathematical strategies to reach a solution to the authentic real- life situation |

| |discuss whether the solution makes sense in the context of the authentic real-life situation. |

|1-2 |The student is able to: |

| | |

| |identify some of the elements of the authentic real-life situation |

| |apply mathematical strategies to find a solution to the authentic real- life situation, with limited success. |

|0 |The student does not reach a standard described by any of the descriptors below. |

Would you like to create your own reality series? What would you do?

What kinds of people or things might be involved? Maybe you could

focus on seats in a theater or the distance a ball travels when it bounces.

What’s that? You thought this project was going to be about a television

reality series? Sorry, this project focuses on arithmetic and geometric series

(and sequences). In some cases, this might be more interesting than what

you find on television!

Arithmetic and geometric sequences and series arise in many situations.

• Depreciation of a car can be an arithmetic sequence if the car

depreciates by a certain dollar amount every year. The sequence

becomes geometric if the car depreciates by a percentage each

year.

• Investments can be represented by arithmetic sequences or series if

a set dollar amount is added at given intervals.

• If salary increases are given at a particular percentage per year, a

geometric sequence can be used.

• It is also interesting to calculate the distance traveled by a ball as it

bounces. If a ball bounces to 80% of its previous height, you can use

this as the common ratio to evaluate the geometric series. Be sure

to take into account the distance up and down between each

bounce.

• An athlete in training might add a set distance to each workout. An

arithmetic series can be used to calculate the total distance after

one month of training.

In our studies of Sequences and Series, we discussed many topics about arithmetic and geometric sequences and series. You have seen how to find the nth term, write explicit and recursive formulas and find the sum of each type of series. You learned that a geometric series can be finite or infinite and depending on the common ratio, the series might have a partial sum or it might not. Now you can use these concepts in real-life scenarios.

Your task is to choose several situations that can be represented by arithmetic or geometric sequences and series. Choose at least one situation that is arithmetic and one that is geometric. If you wish, search the Internet for real-life scenarios to help you get started. If you choose to use something you find during your research, be sure to change the numbers to make your application unique. Find something fun that you find interesting. Whether it is investment related, physics related, or something else entirely, choose what interests you most to make your own reality series!

The creation of a well-developed Reality Series Project should have:

· Real-life series (Choose real-life situations which use arithmetic or

geometric sequences and series. You must have at least one of

each type for this project.)

· Research (Include any research you did to discover the real-life

applications of sequences and series. If you created the real-life

applications yourself, explain your thinking. If you used ideas from

other sources, show how you changed the terms, common

difference, or common ratio to make your application unique.)

· Diagrams or pictures (Include a diagram or picture of the situations

you have chosen. Either write out the 1st several terms, or use

pictures to represent what is taking place. For example, if a ball is

bouncing you might want to show the distance traveled in the 1st

several bounces.)

· Formulas (Write the recursive or/and explicit formulas for each

sequence in the series. Then write the series using summation

notation.)

· Show what you know! (Use as many of the concepts about

arithmetic and geometric sequences and series as you can to

describe your real-life situations. For example, show how to find the

nth term of your arithmetic sequence or describe how to evaluate

the 1st n terms. Discuss whether your geometric series is finite or

infinite and how you know. Pretend you have only 2 terms in the

sequence. Describe how you could write a rule for the nth term.

Make up your own questions about your sequences and series and

then answer them yourself.)

References:

Identify any external resources you used to complete the project, such as web sites, articles, books, etc.

Key: “Sequences and series, and their applications” project

The assessment task was open-ended, allowing students to follow their interest by picking real-life situations (situations which use arithmetic or geometric sequences and series) that they wished to explore. Students must have at least one of each type of sequence described and analysed to achieve the highest level in both criteria. It was assessed against Criteria C and D.

The assessed part of the task was a report that should have included the following:

- Coversheet (Name, Subject name, Date, Year group, Title)

- Introduction

- Background information of the topic of interest is given. Student should explain why the topic of his/her interest was chosen; indicates where the data was taken from (using the correct referencing) and also mentions the reliability of the data.

- The student upon her/his investigation generates question(s) to lead her/his exploration.

- The main Body of the Report – Analysis

- Students should include any research they did to discover the real-life applications of sequences and series. If students created the real-life applications themselves, they should explain their thinking. If students used ideas from other sources, they should show how they changed the terms, common difference, or common ratio to make their application unique.

- Students should identify the relevant elements of the authentic real-life situation

Both Sequences and Series should

- include appropriate mathematical strategies to model the authentic real-life situation

- include appropriate forms of mathematical representation. Students should move effectively between different forms of mathematical representation.

- include mathematical strategies to reach a correct solution to the authentic real-life situation:

o include either written out the 1st several terms, or students should use appropriate forms to represent what is taking place.

o include recursive or/and explicit formulas for each sequence in the series.

o include the series using summation notation.

o describe how a rule for the nth term could be written.

- Include justification of the degree of accuracy of the solution.

- Include justification of whether the solution makes sense in the context of the authentic real-life situation.

- Student suggests improvements to the method when necessary.

- Conclusion:

- Students describe any startling conclusions she/he discovered.

- Student revisits the question(s) generated at the beginning of the investigation and answer them where appropriate.

- Student evaluates the investigation process: describe problems she/he encountered with the project, and suggests improvements.

- Bibliography:

- Using proper and appropriate referencing conventions.[pic]

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