how to write a math conclusion

How to Structure and Format Your Math IA

How to Structure Your Math IA Format

Are you getting ready to present your Math IA? In this article, we’re breaking down the best way to format yours in a way that promotes independent thinking and personal engagement. By following this guide, you’ll be able to score highly in your mathematics course.

Understanding What Math IA is All About

Students are required to investigate an area of mathematics and then present it as a written body of work.

Based on this, it is clear that the student needs to know the appropriate mathematical language and relevant key terms to engage in the topic.

While there are different perspectives when it comes to formatting your maths IA, we recommend the steps outlined below.

By following this tried and tested structure for your math IA, each student will be able to present their findings consistently and optimally

Personal connection and engagement with the subject matter are key. Familiarise yourself with the steps below, and then dive into your chosen topic. Utilising technology can be helpful in conducting a thorough exploration of the topic.

The use of mathematics to prove and explain concepts is applicable to various contexts. This assessment is a great way for students to expand their knowledge and learn valuable skills.

Your assessment needs to define the concept and aim of the work. This will help to keep your data analysis focused. These ideas need to be conveyed in your writing. This is an area of maths that is not strictly bound to the number!

The examiner will be looking at the quality of the idea and the body of work in relation to the assessment criteria.

How to Structure and Format Maths IA

According to student reviews, IB Maths is a struggle for many students working towards getting their diplomas. Apart from the dreaded  mathematics exam , you’re also expected to write up the exploration of a topic.

You can think of the IA as a written mathematical presentation that will impact your final grade.

Many IB students find it hard to study mathematical concepts. This process can help you reflect on new relevant logic that you may not have known previously.

So, where do you even begin your IA? Start with the IA format!

Math IA Structure: What is the Internal Assessment?

The internal assessment is an individual evaluation that focuses on subject-related work.

Alongside the criteria, samples of the student’s work (oral performances, portfolios, lab reports, and essays) are also submitted to the IB for the final grade.

It should show personal engagement with the topic at hand. In this case, it is about the investigation of and correlation between mathematical concepts.

Breakdown of the Math IA Assessment : Appropriate Mathematical Language

Let’s take a look at the criteria for your Math IA. Knowing how you’re being graded will make it much easier to make sure you’re ticking all the boxes.

Presentation (4 marks)

The first criterion is about the presentation, with the aim of assessing the general organization and coherence of your IA.

Although students tend to focus on the complexity of math that their exploration demonstrates, a full 4 points are rewarded for the clarity of your explanations and structure.

In order to score in the top range here, make sure your IA is clearly structured. We’ve shared the optimal format in the next section.

Mathematical Communication (4 marks)

The second criterion looks largely at the mathematical language you have used, such as:

• Terminology

Ensure that these three components are accurate and consistent throughout your IA.

Terms like “plug in” or “put in” should be replaced with mathematically sophisticated words like “substitute.”

Personal Engagement (3 marks)

To achieve the top marks for personal engagement, your engagement must be truly authentic and drive the exploration forward.

It needs to be independent and unique. It should display a degree of creativity in that you present mathematical ideas in your own way and explore the topic from various different perspectives.

This involves making predictions about things you may be interested in, and then finding ways to manipulate the problem, formula, or question to encompass those areas.

Reflection (3 marks)

The IB needs us to do more than just show what we’ve done.

During the reflective stage, connect the results with the initial aims. By doing this, you can determine findings throughout the process.

It is about evaluating the research to pick up on all evidence that goes beyond what a typical mathematical test would.

The IB is all about learning, so be sure to show the marker your growth throughout the IA.

Use of mathematics (6 marks)

This section looks at the quality of the maths and how relevant it is to the exploration.

The IB is measuring relevancy by checking that you only included maths that is directly intended to answer the research question.

It is also worth noting that the maths produced should be at a similar level to the math you cover in your syllabus. This doesn’t mean that you’re confined to only looking into topics that are covered on your syllabus, but it should be of the same rigour!

How to Format these Sections

Introduction:

Like almost all of your internal assessments, your Maths IA has to begin with a super clear introduction that sets the context and aim of the whole exploration.

It is a great place to show your ‘personal engagement’ with the topic you’ve chosen for your IA.

Be sure to account for your interest in the top, its relevance in your life, your prior knowledge about it, what you wish to achieve, and how you’ll arrive at an answer.

You may also include any personalised problem statements and explain how you aim to achieve a solid investigation on the topic.

The body of your IA exploration should focus on the particular topic you have chosen to investigate and the relevant mathematical material that will address the intended aim of the work.

A pivotal point to consider is the level and clarity of the mathematics you use – the IB rewards a lot of marks for the use and communication of Mathematics, so keep this in mind when you start writing your Maths IA up!

For more info on how to write the exploration, check out our complete  Math IA Guide .

Math IA investigation:

As with all assessments, you also need to include a solid conclusion that summarises the research and work you’ve done.

What conclusions did you reach, and did you succeed in exploring the aim that you set out at the beginning of the Math IA!

Importantly, make sure to also discuss some of the challenges in your IA and what you would/could explore with more time and more words.

Finally, zoom out and think about the further implications of your study.

Did your learning affect your life in any way, or how might it affect the lives of others? How has your involvement allowed you to reflect on different mathematics topics?

There is no specific word count for your Math IA, but the IB advises that the exploration should be around 12-20 pages long.

Fonts and Spacing for Your Math IA:

There are no specific requirements on which font you should use, but going with Arial or Times New Roman is generally recommended, with double line spacing and font size of 12.

You may present your work in a word processing software (like Microsoft Word or Pages), or it can be handwritten.

Diagrams and Graphs:

You should include relevant graphs, tables, and diagrams.

Do not simply place these simply as appendices at the end of the essay – they should be fully and clearly labelled to ensure that the examiner knows what you’ve included and why.

It forms an essential part of your research and shows that you fully understand the examples you have included in your analysis.

Bibliography and Citations:

Your report should include a full bibliography with all sources at the end of the report.

In addition to a bibliography at the end, you must acknowledge all direct quotes that you use throughout your essay.

Topic Ideas for SL and HL mathematics

There are many ideas you can explore for the assessment, including graph theory, surface area, geometry, calculus equations, statistics, linear regression, modelling statistics, the SIR model, etc.

It helps to understood the marking criteria before deciding on your topic to ensure you use premium content in your work that can help you score highly.

In your development, you may investigate the correlation between different topics or ideas within Mathematics SL and HL or your AI SL.

Whatever you choose, remember that you will create a new exploration of complex ideas. It’s wise to form your own opinions based on substantive evidence.

Mathematical Concepts Conclusion: The IB Structure

So there we have it, a well organized exploration of the idea and IB layout of your Mathematics IA !

We hope this will give you the push you need to realise your potential and understand complex and unfamiliar mathematics. As with all mathematics, personal involvement by means of practising is best if you want to score highly in your HL syllabus.

Lanterna Resources and Opportunities for Math IA

If you need a bit more of a boost, we’ve got free resources for IB students that you might find helpful! For more personal engagement, feel free to reach out to your instructor or tutor.

Lanterna also offers the following to support students:

• Revision Courses : These offer a helping hand when it matters the most to boost your grades! Look out for these during the Winter and Easter breaks before the final exam.
• Summer Courses : When you are about to start your first or final year of the IB, the summer presents the perfect opportunity to get ahead.
• Online Private Tuition : One-on-one support from the comfort of your own home, whenever and wherever you need it.

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Definition to Remember:

• Thesis + Wisdom + Catchy Last Line = Conclusion

Rules to Remember:

• Much as your introduction gives readers a first impression of who you are and what you hope to accomplish, your conclusion is your chance to offer final wisdom. For many readers, your concluding words are what they will remember long after they have finished reading your piece. For that reason, your concluding paragraph is critical.
• Always end your essay in a way that reinforces your thesis and your purpose. A conclusion must provide a sense of closure. Readers should recognize your final paragraph as an ending. If you feel compelled to type the words “The End,” you’re not there yet.
• Remember to look ahead. Is there future research that you intend or would recommend? Is there something specific you hope your readers will do with the ideas you have shared? Is there a new direction to turn? How can you use your conclusion to keep your readers thinking, even after they have set your essay aside?
• Remind your readers of your overall thesis . Do not merely repeat your thesis. If you have added sufficient evidence in your essay to support your claim, your thesis should sound different to your readers than it did in the introduction. As you remind your readers of your purpose, allow your thesis to express the fullness of all of the evidence you have brought to bear.
• a related story
• a provocative question or series of questions
• a hypothetical scenario
• a surprising fact or series of facts
• an engaging direct quotation
• a striking statement
• background information or context
• an opposing argument
• the who, what, where, when, and why of the paper’s focus
• a combination of the types listed above
• Finish with a catchy last line that is both conclusive-sounding and memorable. Much like a catchy first line, an effective last line should be concise, poetic, persuasive, and provocative. “Writing well may offer little respect, but writing poorly certainly loses it.” David Hartmann, Director of Client Success

Common Errors:

• Tacking on a placeholder conclusion. Writers are often fatigued by the time they are ready to write that final paragraph, and, unfortunately, it shows. As with any kind of writing, if you are finding the work tedious, imagine how uninterested your readers will be. Always save time to set your work aside and refresh before writing your conclusion; the added effort will always pay off.
• Repeating what has been said already. While many of us were taught in elementary school to use the conclusion as an opportunity to remind your readers of everything you just said, an effective post-elementary school conclusion should aspire for more than merely repetition.

Exercise 13.1

Consider a writing assignment you will need to undertake in the near future. How might you approach a conclusion using each of the following approaches? Be specific as you answer.

• A related story:
• A provocative question or series of questions:
• A hypothetical scenario:
• A surprising fact or series of facts:
• An engaging direct quotation:
• A striking statement:
• Background information or context:
• An opposing argument:
• The who, what, where, when, and why of the paper’s focus:
• A combination of the types listed above:

Exercise 13.2

Consider at least five paragraphs you have written in the past week, whether for work, school, or personal use. Write the last line of each on the lines below. If you had been a member of your own audience, would you have found the last line conclusive but memorable? Why or why not? If not, what revisions would you make?

Exercise 13.3

Select one of the examples from Exercise 13.1 and write a conclusion. Once you have finished, ask yourself the following questions:

• Which of the suggestions listed in Exercise 13.1 have you used to interest and inspire your readers? Why?
• Have you included a repetition of your thesis statement that is a fuller, more complete version of the statement you included in your introduction?
• Have you included a catchy last line?
• If you were a member of your own audience, would you find the conclusion memorable? Why or why not?
• What further revisions do you need to make?

The Simple Math of Writing Well Copyright © 2017 by Dr. Jennie A. Harrop is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Browse Course Material

Course info, instructors.

• Prof. Haynes Miller
• Dr. Nat Stapleton
• Saul Glasman

Departments

• Mathematics

As Taught In

Learning resource types, project laboratory in mathematics.

Next: Revision and Feedback »

In this section, Prof. Haynes Miller and Susan Ruff describe the criteria for good mathematical writing and the components of the writing workshop .

A central goal of the course is to teach students how to write effective, journal-style mathematics papers. Papers are a key way in which mathematicians share research findings and learn about others’ work. For each research project, each student group writes and revises a paper in the style of a professional mathematics journal paper. These research projects are perfect for helping students to learn to write as mathematicians because the students write about the new mathematics that they discover. They own it, they are committed to it, and they put a lot of effort into writing well.

Criteria for Good Writing

In the course, we help students learn to write papers that communicate clearly, follow the conventions of mathematics papers, and are mathematically engaging.

Communicating clearly is challenging for students because doing so requires writing precisely and correctly as well as anticipating readers’ needs. Although students have read textbooks and watched lectures that are worded precisely, they are often unaware of the care with which each word or piece of notation was chosen. So when students must choose the words and notation themselves, the task can be surprisingly challenging. Writing precisely is even more challenging when students write about insights they’re still developing. Even students who do a good job of writing precisely may have a different difficulty: providing sufficient groundwork for readers. When students are deeply focused on the details of their research, it can be hard for them to imagine what the reading experience may be like for someone new to that research. We can help students to communicate clearly by pointing out places within the draft at which readers may be confused by imprecise wording or by missing context.

For most students, the conventions of mathematics papers are unfamiliar because they have not read—much less written—mathematics journal papers before. The students’ first drafts often build upon their knowledge of more familiar genres: humanities papers and mathematics textbooks and lecture notes. So the text is often more verbose or explanatory than a typical paper in a mathematics journal. To help students learn the conventions of journal papers, including appropriate concision, we provide samples and individualized feedback.

Finally, a common student preconception is that mathematical writing is dry and formal, so we encourage students to write in a way that is mathematically engaging. In Spring 2013, for example, one student had to be persuaded that he did not have to use the passive voice. In reality, effective mathematics writing should be efficient and correct, but it should also provide motivation, communicate intuition, and stimulate interest.

To summarize, instruction and feedback in the course address many different aspects of successful writing:

• Precision and correctness: e.g., mathematical terminology and notation should be used correctly.
• Audience awareness: e.g., ideas should be introduced with appropriate preparation and motivation.
• Genre conventions: e.g., in most mathematics papers, the paper’s conclusion is stated in the introduction rather than in a final section titled “Conclusion.”
• Style: e.g., writing should stimulate interest.
• Other aspects of effective writing, as needed.

To help students learn to write effective mathematics papers, we provide various resources, a writing workshop, and individualized feedback on drafts.

Writing Resources

Various resources are provided to help students learn effective mathematical writing.

The following prize-winning journal article was annotated to point out various conventions and strategies of mathematical writing. (Courtesy of Mathematical Association of America. Courtesy of a Creative Commons BY-NC-SA license.)

An Annotated Journal Article (PDF)

This document introduces the structure of a paper and provides a miscellany of common mistakes to avoid.

Notes on Writing Mathematics (PDF)

LaTeX Resources

The following PDF, TeX, and Beamer samples guide students to present their work using LaTeX, a high-quality typesetting system designed for the production of technical and scientific documentation. The content in the PDF and TeX documents highlights the structure of a generic student paper.

Sample PDF Document created by pdfLaTeX (PDF)

Sample TeX Document (TEX)

Beamer template (TEX)

The following resources are provided to help students learn and use LaTeX.

LaTeX-Project. “ Obtaining LaTeX .” August 28, 2009.

Downes, Michael. “Short Math Guide for LaTeX.” (PDF) American Mathematical Society . Version 1.09. March 22, 2002.

Oetiker, Tobias, Hubert Partl, et al. “The Not So Short Introduction to LaTeX 2ε.” (PDF) Version 5.01. April 06, 2011.

Reckdahl, Keith. “Using Imported Graphics in LaTeX and pdfLaTeX.” (PDF) Version 3.0.1. January 12, 2006.

Writing Workshop

Each semester there is a writing workshop, led by the lead instructor, which features examples to stimulate discussion about how to write well. In Spring 2013, Haynes ran this workshop during the third class session and used the following slide deck, which was developed by Prof. Paul Seidel and modified with the help of Prof. Tom Mrowka and Prof. Richard Stanley.

The 18.821 Project Report (PDF)

This workshop was held before students had begun to think about the writing component of the course, and it seemed as if the students had to be reminded of the lessons of the workshop when they actually wrote their papers. In future semesters, we plan to offer the writing workshop closer to the time that students are drafting their first paper. We may also focus the examples used in the workshop on the few most important points rather than a broad coverage.

• Download video

This video features the writing workshop from Spring 2013 and includes instruction from Haynes as well as excerpts of the class discussion.

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How to Effectively Write a Mathematics Research Paper

Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on logic and a specific type of language, including symbols and regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition .

Structure and Style

Formal Exposition

The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper. The theorem-proof format, definitions, and logic fall under this style.

Informal Exposition

Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper . Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style.

Conventions of Mathematics

Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. Each area incorporates detailed guidelines to assist the authors.

Related: Do you have questions on language, grammar, or manuscript drafting? Get personalized answers on the FREE Q&A Forum!

Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.

Definitions

In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.

Theorems and Equations

Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end.

Math Symbols and Notations

Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers. Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources .

Protocols for mathematics writing get very specific – fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.

LaTeX and Wolfram

Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.

LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.

Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.

The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.

Good Writing and Logical Constructions

Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Authors must apply a strict, logical construct when writing a mathematics research paper.

There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.

• Concept of a math paper
• Title, acknowledgment, and list of authors
• Introduction
• Body of the work
• Conclusion, appendix, and references
• Publication of a math paper
• Preprint archive
• Choice of the journal, submission
• Publication

The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.

Good advice. For me, writing an essay on mathematics was very difficult. I did not have enough time and knowledge to write a quality essay. I worked a lot in the library and read many articles on the Internet. I studied information about essay writing. But I couldn’t finish the essay in full. I had to look for professional writers on the subject of mathematics. He helped me finish a few paragraphs. The work was delivered on time and on an excellent assessment.

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• Conclusion | Definition & Meaning

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Hypothesis and Conclusion

If-then statement, a implies b, conclusion|definition & meaning.

 The term conclusion in maths is used to define us about the problem that we solve and when we produce the final result at the end then that stage of processes is called as conclusion.

Figure 1 – Give the Right Conclusion to the problem 

When you solve a maths question, you have to end the problem by calculating the last answer and pulling a conclus ion by writing the answer.  A conclusion is the last step of the maths problem. The conclusion is the final answer produced in the end . The answer is completed by writing the arguments and statements by telling the answer to the question. The ending statement of a problem is called a conclusion.

Drawing conclusions refers to the act of thinking of interpreting a series of premises or some ideas and, from them, suggesting something that leads to a meaningful finding. It is normally regarded as a conscious way of learning .

As a rule, a mathematical statement comprises two sections : the first section is assumptions or hypotheses , and the other section is the conclusion . Most mathematical statements have the form “If A, then B.” Often, this statement is written as “A implies B” or “A $\Rightarrow $ B.”  The assumptions we make are what makes “A,” and the circumstances that make “B” are called the conclusion .

To prove that a given statement “If A, then B” is said to be true, we will require some assumptions for “A,” and after doing some work on it, we need to conclude that “B” must also hold when “A” holds.

If we are asked to apply the statement “If A, then B,” firstly, we should be sure that the conditions of the statement “A” are met and true before we start to talk about the conclusion “B.”

Suppose you want to apply the statement “x is even $\Rightarrow$ x2 is an integer.” First, you must verify  that x is even  before  you  conclude that x2 is an integer.

In maths, you will, at many times, confront statements in the form “X $\Leftrightarrow$ Y” or “X if and only if Y.”  These statements are actually two “if, then” statements. The following statement, “X if and only if Y,” is logically equivalent to the statements “If X, then Y” and “If Y, then X.” One more method for thinking about this kind of explanation is an equality between the statements X and Y: so, whenever X holds, Y holds, and whenever Y hold, X holds.

Assume the example: “ x is even $\Leftrightarrow$ x 2  is an integer “. Statement A says, “ x is even,” whereas statement B says, “ x 2  is an integer.” If we get a quick revision about what it suggests to be even (simply that x is a multiple of 2), we can see with ease that the following two statements are identical : If x = 2 k is proved to be even, then it implies x 2 = 2 k 2 = k is an integer, and we know that x 2 = k is an integer, then x = 2 k so n is proved to be even.

In day-to-day use, a statement which is in the form “ If A, then B ,” in some cases, means “ A if and only if B. ” For example, when people agree on a deal, they say, “If you agree to sell me your car for 500k, then I’ll buy from you this week” they straightaway mean, “I’ll buy your car if and only if you agree to sell me in 500k.” In other words, if you don’t agree on 500k, they will not be buying your car from you .

In geometry, the validation or proof is stated in the if-then format. The “if” is a condition or hypothesis , and if that condition is met, only then the second part of the statement is true , which is called the conclusion . The working is like any other if-then statement. For illustration, the statement “If a toy shop has toys for two age groups and 45 percent of toys in the shop are for 14 or above years old, then 55 percent of the toys in the shop are for 13 and fewer years old.” The above statement concludes that “55 percent of the toys in the shop are for 13 and fewer years old.”

In maths, the statement “A if and only if B” is very different from “A implies B.” Assume the example: “ x is an integer” is the A statement, and “ x 3 is a rational number” is the B statement  The statement “A implies B” here means “If x is an integer, then x 3 is a rational number.” The statement is proven to be true. On the other hand, the statement, “A, if and only if B,” means “ x is an integer if and only if x 3 is a rational number,” which is not true in this case.

Examples of Drawing Conclusions

Consider the equation below. Comment if this equation is true or false.

Figure 3 – Example Problem

To calculate its true answer, first, consider the hypothesis $x>0$. Whatever we are going to conclude, it will be a consequence of the truth that $x$ is positive.

Next, consider the conclusion $x+1>0$. This equation is right, since $x+1>x>0$.

This implies that the provided inequality is true.

Simplify the below problem by providing a conclusion by calculating the answer of A.

\[ A= \dfrac{35}{3} \]

The expression given in the question is: $A= \dfrac{35}{3}$

Calculating the answer of A to make a conclusion, The arithmetic operation division is found in the question that is to be figured out in the provided problem. After figuring out the answer to expression A, The conclusion will be given.

\[ A= 11. 667 \]

Therefore, we conclude the question by calculating the answer of $A=11.666$

Consider the equation $0>1 \Rightarrow sinx=2$. Is this equation true or false?

To calculate the correct answer, first consider the hypothesis $0>1$. This equation is clearly false.

calculate the below problem by providing a conclusion by estimating the value of X.

\[ 3+8 \times 2\]

The expression given in the problem is $3+8 \times 2 $.

Multiplication and Plus operation is to be carried out to calculate the answer to the given problem. After figuring out the answer to X  the conclusion will be given.

Thus, we conclude the example by calculating the value of $X = 19$.

All images/mathematical drawings were created with GeoGebra.

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How to Write a Conclusion

Last Updated: July 15, 2023

Template and Sample Conclusion

This article was co-authored by Christopher Taylor, PhD and by wikiHow staff writer, Danielle Blinka, MA, MPA . Christopher Taylor is an Adjunct Assistant Professor of English at Austin Community College in Texas. He received his PhD in English Literature and Medieval Studies from the University of Texas at Austin in 2014. This article has been viewed 480,332 times.

Writing the introduction and body of a paper is a big accomplishment. Now you need to write your conclusion. Writing a conclusion can feel difficult, but it's easier if you plan ahead. First, format your conclusion by revisiting your thesis, summarizing your arguments, and making a final statement. Then, re-read and revise your conclusion to make it effective.

• Let’s say your thesis reads, “Allowing students to visit the library during lunch improves campus life and supports academic achievement because it encourages reading, allows students to start assignments early, and provides a refuge for students who eat alone.”
• You might restate it as, “Evidence shows students who have access to their school’s library during lunch check out more books and are more likely to complete their homework; additionally, students aren’t forced to eat alone.”

• You might write, “According to data, students checked out more books when they were allowed to visit their library during lunch, used that time to do research and ask for help with homework, and reported feeling less alone at lunch time. This shows that opening up the library during lunch can improve student life and academic performance."
• If you’re writing an argument essay, address the opposing argument, as well. You might write, “Although administrators worry that students will walk the halls instead of going to the library, schools that allow students into the library during lunch reported less behavioral issues during lunch than schools that don’t allow students in the library. Data show that students were spending that time checking out more books and working on homework assignments.” [3] X Trustworthy Source Purdue Online Writing Lab Trusted resource for writing and citation guidelines Go to source

• Call your reader to action . For example, “By working with school administrators, Greenlawn ISD can increase academic achievement by letting students use the library during lunch.”
• End with a warning . You might write, “If students aren’t allowed to use the library during lunch, they are missing out on a valuable learning opportunity they’ll never get back.”
• Evoke an image . Write, “Next year, students at Greenlawn could be gathered around a table in the library reading or broadening their minds.”
• Compare your topic to something universal to help your reader relate . You might write, “Everyone knows how stressful it is to have a planner full of assignments, so having extra time to work on them during lunch would be a great relief to many students.”
• Show why the issue is significant. Write, "Giving students more time to spend in the library will help them become more comfortable spending time there, which also helps the library's mission."
• Predict what would happen if your ideas are implemented . Say, “Next year, students at Greenlawn could increase their academic achievements, but results will only happen if they can use the library during lunch.”
• End with a compelling quote . For instance, "As author Roald Dahl once said, 'If you are going to get anywhere in life, you have to read a lot of books.'"

• You could also ask your instructor if you can see an example of a well-written conclusion to give you an idea about what they expect you to write.

• If you want to use an introductory phrase, use a stronger one like “based on the evidence” or “ultimately.” You might also begin your first sentence with a word like “although,” “while,” or “since.” [6] X Trustworthy Source University of North Carolina Writing Center UNC's on-campus and online instructional service that provides assistance to students, faculty, and others during the writing process Go to source
• Additionally, avoid “to conclude,” “in summary,” or “in closing.”

• For example, you may have opened your introduction with an anecdote, quote, or image. Bring it back up in your conclusion. Similarly, if you opened with a rhetorical question, you might offer a potential answer in your conclusion.

• For example, you wouldn’t want to end your essay about allowing students to use the library during lunch by stating, “As the evidence shows, using the library at lunch is a great way to improve student performance because they are more likely to do their homework. On a survey, students reported using the library to do research, ask homework questions, and finish their assignments early.” This leaves out your points about students reading more and having a place to spend their lunch period if they don’t like eating in the cafeteria.

• If you have introduced something you think is really important for your paper, go back through the body paragraphs and look for somewhere to add it. It’s better to leave it out of the paper than to include it in the conclusion.

• If something doesn’t make sense or your conclusion seems incomplete, revise your conclusion so that your ideas are clear.
• It’s helpful to read your entire paper as a whole to make sure it all comes together.

Community Q&A

• Don’t put any evidence or statistics in your conclusion. This information belongs in the body of your paper. [11] X Trustworthy Source University of North Carolina Writing Center UNC's on-campus and online instructional service that provides assistance to students, faculty, and others during the writing process Go to source Thanks Helpful 1 Not Helpful 0
• Make sure you aren’t simply repeating what you’ve written earlier. While you want to restate your ideas, present them in a new way for the reader. Thanks Helpful 0 Not Helpful 0
• Don’t write your conclusion until you’ve written the entire paper. It’ll be much easier to come up with your concluding thoughts after the body of the paper is written. Thanks Helpful 0 Not Helpful 0

• Never copy someone else’s words or ideas without giving them credit, as this is plagiarism. If you are caught plagiarizing part of your paper, even just the conclusion, you’ll likely face severe academic penalties. Thanks Helpful 5 Not Helpful 2
• Don’t express any doubts you may have about your ideas or arguments. Whenever you share your ideas, assume the role of expert. [12] X Research source Thanks Helpful 0 Not Helpful 0

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• ↑ http://writing2.richmond.edu/writing/wweb/conclude.html
• ↑ https://writingcenter.unc.edu/tips-and-tools/conclusions/
• ↑ https://owl.purdue.edu/owl/general_writing/common_writing_assignments/argument_papers/conclusions.html
• ↑ https://writingcenter.fas.harvard.edu/pages/ending-essay-conclusions

About This Article

Writing a conclusion can seem difficult, but it’s easier if you think of it as a place to sum up the point of your paper. Begin your conclusion by restating your thesis, but don’t repeat it word-for-word. Then, use 1-2 sentences to summarize your argument, pulling together all of your points to explain how your evidence supports the thesis. End the paper with a statement that makes the reader think, like evoking a strong image or concluding with a call to action. Keep reading for tips on how to avoid cliches in your conclusion! Did this summary help you? Yes No

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Analyzing Data:

Forming a conclusion:, strategies for analyzing results.

In order to draw meaningful conclusions, it is important to have a strategy for analyzing the data. There are a few different strategies that can be used to analyze results, such as organizing the data into categories, looking for patterns or trends in the data, or using mathematical equations or formulas to determine relationships between different pieces of data. Organizing data into categories is a helpful way to analyze results. This method involves taking the data and grouping it into categories that are related to each other. For example, if you were analyzing test scores, you could group them into categories such as “A” grades, “B” grades, and “C” grades.

This type of analysis allows you to quickly identify patterns or trends in the data. Looking for patterns or trends in the data is another strategy for analyzing results. This involves examining the data to identify any repeating patterns or trends. For example, if you were analyzing test scores, you might look for any trends in how students performed on each section of the test. This type of analysis can help you identify areas where students are struggling and areas where they are excelling. Using mathematical equations or formulas is another way to analyze results.

This method involves using mathematical equations or formulas to determine relationships between different pieces of data. For example, if you were analyzing test scores, you could use an equation to calculate the average score of all students who took the test. This type of analysis can help identify relationships between different pieces of data and can help you draw meaningful conclusions. In summary, analyzing results and drawing conclusions are important skills for students of all ages. Understanding the process of collecting data, analyzing it, and forming conclusions based on evidence can help students develop their critical thinking skills.

By following the strategies outlined above, students can ensure they are completing each stage of the process successfully.

Shahid Lakha

Shahid Lakha is a seasoned educational consultant with a rich history in the independent education sector and EdTech. With a solid background in Physics, Shahid has cultivated a career that spans tutoring, consulting, and entrepreneurship. As an Educational Consultant at Spires Online Tutoring since October 2016, he has been instrumental in fostering educational excellence in the online tutoring space. Shahid is also the founder and director of Specialist Science Tutors, a tutoring agency based in West London, where he has successfully managed various facets of the business, including marketing, web design, and client relationships. His dedication to education is further evidenced by his role as a self-employed tutor, where he has been teaching Maths, Physics, and Engineering to students up to university level since September 2011. Shahid holds a Master of Science in Photon Science from the University of Manchester and a Bachelor of Science in Physics from the University of Bath.

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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10.

• Idea behind hypothesis testing
• Examples of null and alternative hypotheses
• Writing null and alternative hypotheses
• P-values and significance tests
• Comparing P-values to different significance levels
• Estimating a P-value from a simulation
• Estimating P-values from simulations

Using P-values to make conclusions

• (Choice A)   Fail to reject H 0 ‍   A Fail to reject H 0 ‍  
• (Choice B)   Reject H 0 ‍   and accept H a ‍   B Reject H 0 ‍   and accept H a ‍  
• (Choice C)   Accept H 0 ‍   C Accept H 0 ‍  
• (Choice A)   The evidence suggests that these subjects can do better than guessing when identifying the bottled water. A The evidence suggests that these subjects can do better than guessing when identifying the bottled water.
• (Choice B)   We don't have enough evidence to say that these subjects can do better than guessing when identifying the bottled water. B We don't have enough evidence to say that these subjects can do better than guessing when identifying the bottled water.
• (Choice C)   The evidence suggests that these subjects were simply guessing when identifying the bottled water. C The evidence suggests that these subjects were simply guessing when identifying the bottled water.
• (Choice A)   She would have rejected H a ‍   . A She would have rejected H a ‍   .
• (Choice B)   She would have accepted H 0 ‍   . B She would have accepted H 0 ‍   .
• (Choice C)   She would have rejected H 0 ‍   and accepted H a ‍   . C She would have rejected H 0 ‍   and accepted H a ‍   .
• (Choice D)   She would have reached the same conclusion using either α = 0.05 ‍   or α = 0.10 ‍   . D She would have reached the same conclusion using either α = 0.05 ‍   or α = 0.10 ‍   .
• (Choice A)   The evidence suggests that these bags are being filled with a mean amount that is different than 7.4  kg ‍   . A The evidence suggests that these bags are being filled with a mean amount that is different than 7.4  kg ‍   .
• (Choice B)   We don't have enough evidence to say that these bags are being filled with a mean amount that is different than 7.4  kg ‍   . B We don't have enough evidence to say that these bags are being filled with a mean amount that is different than 7.4  kg ‍   .
• (Choice C)   The evidence suggests that these bags are being filled with a mean amount of 7.4  kg ‍   . C The evidence suggests that these bags are being filled with a mean amount of 7.4  kg ‍   .
• (Choice A)   They would have rejected H a ‍   . A They would have rejected H a ‍   .
• (Choice B)   They would have accepted H 0 ‍   . B They would have accepted H 0 ‍   .
• (Choice C)   They would have failed to reject H 0 ‍   . C They would have failed to reject H 0 ‍   .
• (Choice D)   They would have reached the same conclusion using either α = 0.05 ‍   or α = 0.01 ‍   . D They would have reached the same conclusion using either α = 0.05 ‍   or α = 0.01 ‍   .

Ethics and the significance level α ‍  

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A conclusion is a statement arrived at by applying a set of logical rules known as syllogisms to a set of premises . The process of drawing conclusions from premises and syllogisms is called deduction .

This entry contributed by David Terr

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Terr, David . "Conclusion." From MathWorld --A Wolfram Web Resource, created by Eric W. Weisstein . https://mathworld.wolfram.com/Conclusion.html

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How to Write a Conclusion for Research Papers (with Examples)

The conclusion of a research paper is a crucial section that plays a significant role in the overall impact and effectiveness of your research paper. However, this is also the section that typically receives less attention compared to the introduction and the body of the paper. The conclusion serves to provide a concise summary of the key findings, their significance, their implications, and a sense of closure to the study. Discussing how can the findings be applied in real-world scenarios or inform policy, practice, or decision-making is especially valuable to practitioners and policymakers. The research paper conclusion also provides researchers with clear insights and valuable information for their own work, which they can then build on and contribute to the advancement of knowledge in the field.

The research paper conclusion should explain the significance of your findings within the broader context of your field. It restates how your results contribute to the existing body of knowledge and whether they confirm or challenge existing theories or hypotheses. Also, by identifying unanswered questions or areas requiring further investigation, your awareness of the broader research landscape can be demonstrated.

Remember to tailor the research paper conclusion to the specific needs and interests of your intended audience, which may include researchers, practitioners, policymakers, or a combination of these.

Table of Contents

What is a conclusion in a research paper, summarizing conclusion, editorial conclusion, externalizing conclusion, importance of a good research paper conclusion, how to write a conclusion for your research paper, research paper conclusion examples.

• How to write a research paper conclusion with Paperpal? 

Frequently Asked Questions

A conclusion in a research paper is the final section where you summarize and wrap up your research, presenting the key findings and insights derived from your study. The research paper conclusion is not the place to introduce new information or data that was not discussed in the main body of the paper. When working on how to conclude a research paper, remember to stick to summarizing and interpreting existing content. The research paper conclusion serves the following purposes: 1

• Warn readers of the possible consequences of not attending to the problem.
• Recommend specific course(s) of action.
• Restate key ideas to drive home the ultimate point of your research paper.
• Provide a “take-home” message that you want the readers to remember about your study.

Types of conclusions for research papers

In research papers, the conclusion provides closure to the reader. The type of research paper conclusion you choose depends on the nature of your study, your goals, and your target audience. I provide you with three common types of conclusions:

A summarizing conclusion is the most common type of conclusion in research papers. It involves summarizing the main points, reiterating the research question, and restating the significance of the findings. This common type of research paper conclusion is used across different disciplines.

An editorial conclusion is less common but can be used in research papers that are focused on proposing or advocating for a particular viewpoint or policy. It involves presenting a strong editorial or opinion based on the research findings and offering recommendations or calls to action.

An externalizing conclusion is a type of conclusion that extends the research beyond the scope of the paper by suggesting potential future research directions or discussing the broader implications of the findings. This type of conclusion is often used in more theoretical or exploratory research papers.

Align your conclusion’s tone with the rest of your research paper. Start Writing with Paperpal Now!  

The conclusion in a research paper serves several important purposes:

• Offers Implications and Recommendations : Your research paper conclusion is an excellent place to discuss the broader implications of your research and suggest potential areas for further study. It’s also an opportunity to offer practical recommendations based on your findings.
• Provides Closure : A good research paper conclusion provides a sense of closure to your paper. It should leave the reader with a feeling that they have reached the end of a well-structured and thought-provoking research project.
• Leaves a Lasting Impression : Writing a well-crafted research paper conclusion leaves a lasting impression on your readers. It’s your final opportunity to leave them with a new idea, a call to action, or a memorable quote.

Writing a strong conclusion for your research paper is essential to leave a lasting impression on your readers. Here’s a step-by-step process to help you create and know what to put in the conclusion of a research paper: 2

• Research Statement : Begin your research paper conclusion by restating your research statement. This reminds the reader of the main point you’ve been trying to prove throughout your paper. Keep it concise and clear.
• Key Points : Summarize the main arguments and key points you’ve made in your paper. Avoid introducing new information in the research paper conclusion. Instead, provide a concise overview of what you’ve discussed in the body of your paper.
• Address the Research Questions : If your research paper is based on specific research questions or hypotheses, briefly address whether you’ve answered them or achieved your research goals. Discuss the significance of your findings in this context.
• Significance : Highlight the importance of your research and its relevance in the broader context. Explain why your findings matter and how they contribute to the existing knowledge in your field.
• Implications : Explore the practical or theoretical implications of your research. How might your findings impact future research, policy, or real-world applications? Consider the “so what?” question.
• Future Research : Offer suggestions for future research in your area. What questions or aspects remain unanswered or warrant further investigation? This shows that your work opens the door for future exploration.
• Closing Thought : Conclude your research paper conclusion with a thought-provoking or memorable statement. This can leave a lasting impression on your readers and wrap up your paper effectively. Avoid introducing new information or arguments here.
• Proofread and Revise : Carefully proofread your conclusion for grammar, spelling, and clarity. Ensure that your ideas flow smoothly and that your conclusion is coherent and well-structured.

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Remember that a well-crafted research paper conclusion is a reflection of the strength of your research and your ability to communicate its significance effectively. It should leave a lasting impression on your readers and tie together all the threads of your paper. Now you know how to start the conclusion of a research paper and what elements to include to make it impactful, let’s look at a research paper conclusion sample.

How to write a research paper conclusion with Paperpal?

A research paper conclusion is not just a summary of your study, but a synthesis of the key findings that ties the research together and places it in a broader context. A research paper conclusion should be concise, typically around one paragraph in length. However, some complex topics may require a longer conclusion to ensure the reader is left with a clear understanding of the study’s significance. Paperpal, an AI writing assistant trusted by over 800,000 academics globally, can help you write a well-structured conclusion for your research paper. 

• Sign Up or Log In: Create a new Paperpal account or login with your details.  
• Navigate to Features : Once logged in, head over to the features’ side navigation pane. Click on Templates and you’ll find a suite of generative AI features to help you write better, faster.  
• Generate an outline: Under Templates, select ‘Outlines’. Choose ‘Research article’ as your document type.  
• Select your section: Since you’re focusing on the conclusion, select this section when prompted.  
• Choose your field of study: Identifying your field of study allows Paperpal to provide more targeted suggestions, ensuring the relevance of your conclusion to your specific area of research. 
• Provide a brief description of your study: Enter details about your research topic and findings. This information helps Paperpal generate a tailored outline that aligns with your paper’s content. 
• Generate the conclusion outline: After entering all necessary details, click on ‘generate’. Paperpal will then create a structured outline for your conclusion, to help you start writing and build upon the outline.  
• Write your conclusion: Use the generated outline to build your conclusion. The outline serves as a guide, ensuring you cover all critical aspects of a strong conclusion, from summarizing key findings to highlighting the research’s implications. 
• Refine and enhance: Paperpal’s ‘Make Academic’ feature can be particularly useful in the final stages. Select any paragraph of your conclusion and use this feature to elevate the academic tone, ensuring your writing is aligned to the academic journal standards. 

By following these steps, Paperpal not only simplifies the process of writing a research paper conclusion but also ensures it is impactful, concise, and aligned with academic standards. Sign up with Paperpal today and write your research paper conclusion 2x faster .  

The research paper conclusion is a crucial part of your paper as it provides the final opportunity to leave a strong impression on your readers. In the research paper conclusion, summarize the main points of your research paper by restating your research statement, highlighting the most important findings, addressing the research questions or objectives, explaining the broader context of the study, discussing the significance of your findings, providing recommendations if applicable, and emphasizing the takeaway message. The main purpose of the conclusion is to remind the reader of the main point or argument of your paper and to provide a clear and concise summary of the key findings and their implications. All these elements should feature on your list of what to put in the conclusion of a research paper to create a strong final statement for your work.

A strong conclusion is a critical component of a research paper, as it provides an opportunity to wrap up your arguments, reiterate your main points, and leave a lasting impression on your readers. Here are the key elements of a strong research paper conclusion: 1. Conciseness : A research paper conclusion should be concise and to the point. It should not introduce new information or ideas that were not discussed in the body of the paper. 2. Summarization : The research paper conclusion should be comprehensive enough to give the reader a clear understanding of the research’s main contributions. 3 . Relevance : Ensure that the information included in the research paper conclusion is directly relevant to the research paper’s main topic and objectives; avoid unnecessary details. 4 . Connection to the Introduction : A well-structured research paper conclusion often revisits the key points made in the introduction and shows how the research has addressed the initial questions or objectives. 5. Emphasis : Highlight the significance and implications of your research. Why is your study important? What are the broader implications or applications of your findings? 6 . Call to Action : Include a call to action or a recommendation for future research or action based on your findings.

The length of a research paper conclusion can vary depending on several factors, including the overall length of the paper, the complexity of the research, and the specific journal requirements. While there is no strict rule for the length of a conclusion, but it’s generally advisable to keep it relatively short. A typical research paper conclusion might be around 5-10% of the paper’s total length. For example, if your paper is 10 pages long, the conclusion might be roughly half a page to one page in length.

In general, you do not need to include citations in the research paper conclusion. Citations are typically reserved for the body of the paper to support your arguments and provide evidence for your claims. However, there may be some exceptions to this rule: 1. If you are drawing a direct quote or paraphrasing a specific source in your research paper conclusion, you should include a citation to give proper credit to the original author. 2. If your conclusion refers to or discusses specific research, data, or sources that are crucial to the overall argument, citations can be included to reinforce your conclusion’s validity.

The conclusion of a research paper serves several important purposes: 1. Summarize the Key Points 2. Reinforce the Main Argument 3. Provide Closure 4. Offer Insights or Implications 5. Engage the Reader. 6. Reflect on Limitations

Remember that the primary purpose of the research paper conclusion is to leave a lasting impression on the reader, reinforcing the key points and providing closure to your research. It’s often the last part of the paper that the reader will see, so it should be strong and well-crafted.

• Makar, G., Foltz, C., Lendner, M., & Vaccaro, A. R. (2018). How to write effective discussion and conclusion sections. Clinical spine surgery, 31(8), 345-346.
• Bunton, D. (2005). The structure of PhD conclusion chapters.  Journal of English for academic purposes ,  4 (3), 207-224.

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We have now looked at a number of different graphs and charts, all of which were potentially misleading. We hope that from now on if you have to work with a graph or a chart, you will always consider the following points:

look carefully at any horizontal or vertical scale that is given;

consider each graph or chart separately, don't compare them unless you are sure that they have the same scales;

if it is not easy to interpret the graph or chart, trying reading off some values.

IB Math IA Guide - Math IA Shenanigans That No One Will Tell Yeh!

Ace Your IB Math IA with our ultimate guide for 2023! Get top marks and ace your IA with ease. Discover proven tips, tricks and strategies to nail your Math IA today!

Table of content

Criterion a - mathematical presentation: (levels- 0, 1, 2, 3, 4), criterion b - mathematical communication: (levels- 0, 1, 2, 3, 4), criterion c - personal engagement: (levels - 0, 1, 2, 3), criterion d - reflection: (levels - 0, 1, 2, 3), criterion e - use of mathematics: (levels - 0, 1, 2, 3, 4, 5, 6), introduction, body of your exploration.

IB Math students will tell you how they’re always on the edge of their seats for some help, but IB Math IA takes that anxiety to an entirely different level. The reality is far from frightening; nonetheless, IB Math IA can be handled well with a unique IB Math IA topic in hand and lots of coffee! 

But does that guarantee a dependable 7?

It takes more than just a perfect IB Math IA topic to ace.

How’s that, you’d ask.

From researching several IB Math IA examples to planning the mathematical working of your exploration, your IB Math IA structure will get you into trouble if you don’t give it the time it demands. With all the varied content available in bulk online, the process is bound to become anything but easy.

But worry not!

You are at the right place - The Ultimate Guide to IB Math IA!

This article covers IB Math IA rubrics, process key pointers, the structure of the investigation, and interesting IB Math IA topics that will stimulate your mind and help you begin your exploration!

You should also know about the updated course structure of IB Mathematics. Students are allowed to opt for any one of the following four courses in Math:

• Analysis and Approaches ( AA HL ) - Higher Level
• Analysis and Approaches  (AA SL)  - Standard Level
• Applications and Interpretation  (AI HL)  - Higher Level
• Applications and Interpretation  (AI SL)  - Standard Level

For more information about choosing your course, check out our latest article:  Confused about IB Math AA & IB Math AI?

Also, here’s a great surprise for all students!

Patrick Jones , the creator of PatrickJMT Math Videos, acknowledged as the best Math teacher globally with over 1.2 million subscribers on YouTube, has gotten on board with our team at Nail IB! How great is that! He is already working on creating an entire Nail IB video course, and it will prove to be a wholesome guide for you as you tread on your IB Math journey! You should check out his excellent, world-renowned content  here !

Before moving any further, we insist you check out our Free IB Resources for  IB Mathematics SL and IB Mathematics HL.  These are specially assembled for your benefit and will surely assist you on your IB Math journey!

For an absolute hold on IB Math, check out our premium notes designed and curated specially for you, be it  IB Mathematics SL  or  IB Mathematics HL . These bundles are not just limited to messages but offer past year papers and How-to Guides for Extended Essays, Internal Assessments, and more; examples included! You’re in for a smooth ride with these by your side;)

You can also stream our webinar on  How to Write an IB Math Internal Assessment in under 30 minutes  and hear directly from a recent IB graduate to understand the fundamental pointers and some fantastic hacks to lay the foundation of your IB Math IA. Getting the proper guidance ensures you a 7 in the subject you have feared for too long. Click  here  to watch it now!

First things first, let’s understand the criteria. Unless we acknowledge the requirements against which our exploration is scored, it’ll be equivalent to a shot in the dark. The conditions, irrespective of whether you opt for SL, HL (AA or AI), are as follows:

Assessment is done on the conciseness, brevity, and clarity/coherence of your investigation. The proper structure must be given to your IA. As per IB guidelines,

a coherent exploration is,

• Logically developed
• Easy to follow and,
• Meets the Aim.

Also, a well-organized exploration,

• Includes Introduction
• Describes the Aim of the investigation and
• Has a Conclusion

Assessment is done on the appropriateness of the mathematical terminology, notation, and symbols used to progress the exploration. Marks and notes should be correctly used as are used in IB textbooks. For example, x2 should not be written as x^2.

If used, different mathematical representation tools such as tables, graphs, and diagrams must be relevant to the working and be commented on/explained well. Avoid inconsistent use of Mathematical terminology. Applying ICT Tools(for example-  GeoGebra and Desmos ) should be made wisely. For Calculations, Graphic Display Calculators can also be used, but that doesn't outdo math formulas' importance.

Assessment is done on the personal involvement shown. The sure shot way to ensure Engagement is, first and foremost, by going ahead with a topic that interests you (something unique or that affects real-life situations). Personal Engagement is seen throughout the exploration by:

• Independent thinking and creativity showed by the student
• Making the Math Idea your own
• Investigating the idea from varied perspectives
• Exploring different possibilities

Avoid portraying superficial interest. Opportunities for demonstration of personal Engagement should be noticed.

Assessment is done on the evaluation and analysis of the investigation. Mentioning the significance of your exploration results, discussing possible limitations, and justifying why you chose the procedure you did can portray a fair reflection of the IA. Merely explaining your results will get you only a score of 1 out of 3. According to IB guidelines, a review should be meaningful and critical.

A meaningful reflection includes

• Considering limitations in the work
• Comparing different Mathematical Approaches
• Linking to the Aim
• Commenting on the Learning

A critical reflection entails,

• Considering  What Next
• Discussing the implications of the results
• Discussing the strengths and weaknesses of the approaches
• Considering different perspectives

Reflection is an analysis of the student's work, seen throughout the exploration, not just the Conclusion.

Assessment is done on the implementation of Mathematics in the IA. It is essential to understand that the Math used should be on par with the course, nothing too simple and nothing you need help understanding. Also, the Mathematics used should be fully understood and engrained by you. Unfamiliar Mathematics, if used, should be explained well by giving personal examples. As per IB guidelines, students are expected to produce work that is,

• Commensurate  with the level of the course(should either be part of the syllabus at a similar level or slightly beyond)
• Relevant  Mathematics used means Math which supports the development of the exploration towards the completion of its Aim

To score higher levels in Criterion E, it is crucial to understand the meaning of the following terms:

• Precise  Mathematics is error-free and uses an appropriate level of accuracy at all times
• Sophistication  means the Math should be commensurate with the HL syllabus
• Rigour  involves clarity of logic and language when making Mathematical arguments and calculations.

Here  is a fully annotated sample IB Math IA, going through which you can gain a lot of insight  (Read the annotations properly).

Another example,  the Breaking the Code  investigation(fully annotated)- is given here for IB Mathematics HL. The IA explores encryption and decryption in the context of Mathematics.

• Going through the report, we see that the document needs more structure in the beginning since the Introduction does not mention the Rationale or Aim. You don't want to be committing such a blunder.
• Moving further, we see that the body encompasses Math, which is well explained, thereby excelling on criterion E when graded on the SL scale. To excel in HL, the math should have been more rigorous than descriptive.
• Toward the end, the report needs to include a Reflection on the results obtained, which doesn't fare well for the IA. Potential implications of the topic need to be included, and the Conclusion seems bland.

Another interesting annotated sample- for IB Mathematics SL- Regularisation of Irregular Verbs: When can I use the words swimming and know correctly? is given for reference  here . Understand which key points have been missed and which have been taken care of. The more you go through sample IAs, the better your chances of preparing an investigation that'll be scored well.

Like any other exploration, your IB Math IA expects you to give it a fair shot of effort and interest. If you see it as a burden, it will undoubtedly become one. Equally important is to draw an analogy between the topic of your choice and the math involved. Taking care of these points in general, let's understand all that goes into making one's IB Math IA (a brief outline):

• It can have a personal story attached; mention it in your Introduction. If not, explain how the topic underhand impacts real-world situations and motivates you to land on it. Your passion for the IA idea shows in your work, and you don't want to be doing your investigation just for the sake of it.
• f you're curious about  Fibonacci numbers, the Golden ratio , and nature alike, try looking for relationships among them on the Internet. This will entail going through many research papers, publications, and journals and finally settling on mathematical findings and proofs that will help you investigate that particular something you wish to explore. For example, if you want to study how  Traffic Jams have math running in the background , research it in detail since Traffic snarls are an imminent pain for us all. Only you will have to pick up parts you think are relevant and understandable to you.
• ​​​​​​​​​​​​​​ The cycle of Inquiry, Action, and Reflection in learning is vital. Learning the implication of  Plagiarism, Collusion, and Duplication of Work  is essential to keep one's IA transparent and impressive.
• ​​​​​​​​​​​​​​ Besides citing references in the bibliography section, ensure you include it in the body as a footnote or in the exploration itself. Citing credible sources shows how transparent your work is and helps examiners cross-check for correctness. Acknowledging the author's work is essential to the IA-making process.

With this, let's discuss the Structure/Layout of the Investigation. There are numerous guidelines available all over the Internet. Regarding the IA length, though you should keep 6 -12 pages as the prescribed length, your focus should be on including all that pertains to your idea and ruling out everything that's not. So don't set out with a mindset to refrain from exceeding six pages; set out to include everything you know needs to be. Similarly, it is advised to make sure the Math used is suitable for SL and HL levels.

Without any further adieu, let's highlight what the layout of the IA should look like:

• Sets the background of your exploration and gives an argument for your topic choice. Your Rationale tells why you chose so and so topic.
• ​​​​​​​​​​​​​​ This is where you define your investigation's objective and tell what you wish to achieve with this idea.
• ​​​​​​​​​​​​​​ Let's say, for instance, you have opted for Math HL; for a simple mathematical investigation that scores six on the Math SL grade scale, you might end up with a mere four on the HL. The difficulty of your opted Math subject should reflect in your Internal Assessment. It would help if you also outlined the areas of mathematics you will cover in your investigation.
• Elaborate on the method you used for the exploration and justify why you chose to proceed with that particular method.
• Use relevant mathematical tools like labelled graphs, charts, etc., for your mathematical work and explain them in the IA context.
• State your results relating them to the Aim of your Internal Assessment. The significance and impact of these results should be highlighted, as well. In addition, briefly tell how the exploration was helpful to you and all you have gained from it. Possibilities of extension should be mentioned. The bibliography is for you to cite the sources used by you in the making of the IA. We suggest you use the  Citation Machine  for additional guidance in the bibliography.

Now that we're comfortable with the IB Math IA structure let's look at some interesting IB Math IA topics that will get your creative juices flowing and help kickstart your Math IA journey today!

• Simulating models to study and forecast weather patterns. (You could come up with a personal account that led you to land on something like this)
• Exploring the different probabilities associated with a game of your choice; for example, Solitaire(if you're a game buff).
• Investigating the  Math associated with the Global Positioning System(GPS)  and the intricacies of the technology involved.
• Exploring  Fermat's little theorem  or  Goldbach's conjecture (one of the most significant unsolved problems in Mathematics).
• Finding the volume and surface area of an egg, apple, mango or any other real-world object using Calculus's power (Simulation could be used). A good IA on  modelling  manages to score an easy 15-16 marks out of 20.
• Investigating the structural designs of bridges that prevent collapse under loading.
• Studying complex roots graphically.
• Exploring how guitar frets are arranged in Pythagoras Ratios.
• Comparing which will prove beneficial: lump-sum payment of a lottery prize or fee done in instalments?
• Understanding how ISBN codes and Credit Card Codes can be cracked.

And that's a wrap!

Just like any other IA, IB Mathematics IA needs to be started early so that you don't end up compiling just anything at the last minute!

Give it the time it needs, and it will surely pay off. It might seem heavy, but once you decide to pursue an idea of your liking, there will be no turning back! Keep in mind the essential pointers and win the battle courageously!

Want some A-quality guidance? Look no further; at Nail IB, we have assembled premium content for you to ace your IBs, and you should check out our resources for a smooth IB experience. Click  here  for top-notch IB resources or to assess how your prep is going! Our exclusive Nail IB course, created by Patrick Jones, will be out soon too, so stay tuned, as there is no way you would want to miss the holy grail every Math IB student wants!!

This article will serve as a solid foundation for your Math IB Internal Assessment.

IB Resources you will love!

55234 + free ib flashcards, 136 + free ia samples, 3962 + ib videos by experts, 20099 + ib sample practice questions, ib resources for 30 + subjects.

Tecniche di programmazione in MATLAB

Scrivi codice efficiente, robusto e ben organizzato utilizzando le feature di MATLAB®. Porta le tue abilità di codifica al livello successivo imparando le competenze che ti trasformeranno da qualcuno che scrive codice MATLAB funzionante in qualcuno che sviluppa applicazioni MATLAB di alta qualità.

Moduli del corso

Introduction.

Overview of the content covered.

• Course Overview

Structuring Data

Explore choices for storing data within a MATLAB application.

• MATLAB Data Types
• Organizing Data in a Table
• Extracting Data from a Table
• Storing Data in a Cell Array
• Extracting Data from a Cell Array
• Organizing Data in a Structure
• Extracting Data from a Structure
• Review - Structuring Data

Manipulating Heterogeneous Data

Manipulate data in tables, cell arrays, and structure arrays.

• Extracting Multiple Elements from Cell and Structure Arrays
• Function Handles
• Applying Scalar Functions to Arrays
• Converting Data Types

Optimizing Your Code

Use common techniques for improving performance when storing, accessing, and processing data.

• Measuring Performance
• Finding Bottlenecks
• Vectorization
• Memory Usage
• Preallocation
• Improving Memory Usage in Functions
• Review - Optimizing Your Code

Creating Flexible Functions

Write functions that can handle different numbers and types of user input values.

• Creating Flexible Function Interfaces
• Creating Multiple Interfaces with Wrapper Functions
• Converting Input Types
• Querying Function Inputs
• Setting Default Input Values
• Allowing Any Number of Inputs
• Matching Text Inputs to an Allowed Set
• Allowing a Variable Number of Outputs
• Changing the Function Interface with Anonymous Functions
• Review - Creating Flexible Functions

Creating Robust Applications

Create robust applications that withstand unexpected input and produce meaningful errors. Use built-in MATLAB functions and programming constructs, and employ techniques for handling error conditions.

• Course Example - Structuring the Satellite Tracking Code
• Restricting Access Using Private Functions
• Writing Local Functions
• Generating Custom Warning and Errors
• Validating Function Inputs
• Catching and Handling Errors
• Review - Creating Robust Applications

Verifying Application Behavior

Create tests to verify the application is behaving as expected.

• Why Use a Testing Framework
• What Is a Test
• Writing and Running a Test Script
• Avoiding Bugs in Comparisons
• Writing Test Functions
• Verifying Behavior
• Passing Commands as Inputs
• Adding Pre- and Post-Test Tasks

Debugging Your Code

Use integrated tools to debug applications.

• Repairing Satellite Tracking Code
• Errors and Debugging
• Code Analyzer
• Debugging Run-Time Errors

Organizing Your Projects

Use folder reports, MATLAB Projects, and version control to organize and manage your projects.

• Organizing Satellite Tracking Code
• Running Folder Reports
• Organizing Code with MATLAB Projects
• Managing a Code Base with Source Control
• Review - Organizing Your Projects

Learn next steps and provide feedback on the course.

• Additional Resources

Formato:Autogestito

Lingua:Italiano

Lingua Esercizi pratici con feedback automatico Accedi a MATLAB tramite il browser web Report sui progressi e certificato del corso condivisibili Fondamenti di MATLAB

Impara le funzionalità di base di MATLAB per l'analisi dati, la modellazione e la programmazione.

MATLAB per l'elaborazione e la visualizzazione dei dati

Crea visualizzazioni personalizzate e automatizza le attività di analisi dati.

Deep Learning con MATLAB

Impara la teoria e la pratica della costruzione di reti neurali profonde con dati di immagini e sequenze del mondo reale.

MATLAB Onramp

Come iniziare rapidamente con le nozioni di base di MATLAB.

Fondamenti di MATLAB