3.6 Integration: Exploring Mathematics Through Text
In the previous section, you practised reading and performing dialogues (spoken communication) about mathematics, and explored written communication through email correspondence. In this section, you will focus on longer written mathematical communication. This book uses several different types of text across the chapters. This gives you exposure to a variety of mathematical writing.
The text below is a fictional extract from an online handbook for the Open Day at the fictional Walsch-Macdonald Northern University. It consists of two introductory paragraphs followed by a longer section with a subheading. Imagine that the section below the subheading represents one of several similar sections in the handbook. Each section informs students of the applications of some of the areas of mathematics they have already studied at school, in order to help them prepare for the presentations at the Open Day and to ask questions.
Your task is to read the text and work through the activities. The activities have a logical structure: they begin with more global or whole-text activities, that examine the purpose and audience of the text, and they gradually narrow to a focus on individual language items. In mathematical writing, there is an inherent connection between ‘macro’ text-external elements (such as audience and purpose) and ‘micro’ text-internal aspects (such as individual words and phrases).
By working through the various activities below, you will be able to:
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Work with extended mathematical prose texts to enhance your understanding of both mathematical concepts and the structure of mathematical writing.
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Identify and develop your command of useful academic phrases and sentence structures within text.
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Apply this language in your own writing and mathematical discussions.
Activity 1: The overall audience and purpose of the text
The purpose of this activity is to help you think about three features of a text: its audience (who reads it), its purpose (why it was written), and its structure (how it is organised). Academic writing takes many different forms. Each form has its own conventions, purposes, and ways of organising information. Noticing these differences will help you become a stronger academic reader and writer.
Question 1: Match the following terms, which occur in the title or in the first paragraph of the text, to their explanations. Next, select the ‘Answer’ button below.
| Term | Explanations | |
|---|---|---|
| Infosheet | When people receive a positive welcome to an event. | |
| Open Day | A concise document providing factual information on a topic in summary form. | |
| a Q&A slot | An event where a university or department invites prospective students to visit, explore facilities, and learn about programmes and student life. | |
| to be warmly invited to attend | An abbreviation for a scheduled time period where an audience can ask questions and receive answers. |
Answers
Infosheet – A concise document providing factual information on a topic in summary form. Check out the related terms handout, blurb, fact sheet and flyer.
Open day – An event where a university or department invites prospective students to visit, explore facilities, and learn about programmes and student life.
A Q&A slot – An abbreviation for a scheduled time period where an audience can ask questions and receive answers. (Q&A stands for Questions and Answers.)
To be warmly invited to attend – When people receive a positive welcome to an event.
Question 2: What do these terms suggest to you about the possible purposeor audience of the text? Select the ‘Answer’ button below.
Answer
These terms suggest that the text is likely to be something to do with an event or meeting to which people are invited. It is likely to be a positive event or meeting (warmly invited) with an opportunity for questions.
Question 3: Read the whole text briefly and select which one of the following four prompts best summarises the purpose of the whole text. Next, select the ‘Answer’ button below.
- To advertise Dr van Leiden’s new book entitled the Applications of Mathematics in the World of Business, Commerce and Industry.
- To invite students to an interactive seminar with a Q&A session and to encourage them to prepare for it.
- To provide a comprehensive list of all possible applications of integration in civil, mechanical and electrical engineering.
- To argue that integration is considered to be more important than differentiation at this university.
Answer
The correct answer is 2. There is no mention of any book, so 1 is not correct. 3 is not correct as there is no attempt at the list being comprehensive, even though it is quite long. Moreover, the applications in the text extend beyond the three branches of engineering in the prompt. 4 is not correct since differentiation is not mentioned and there isn’t any comparison in the text. In any case, few mathematicians would make the statement that either part of calculus is ‘more important’ in general.
Walsch-Macdonald Northern University Open Day
Infosheet for Dr van Leiden’s presentation
One of the many attractions of mathematics is the vast scope of exciting career opportunities it offers. As part of the Open Day next Wednesday, Dr van Leiden, Mathematics Outreach Officer for the School of Mathematics at WMNU, will deliver a presentation, including a Q&A slot, on the Applications of Mathematics in the World of Business, Commerce and Industry. The presentation is titled: ‘From Course Notes to Career Success: starting to think about the infinite (!) possibilities for mathematicians beyond your degree’. This presentation has proven to be very popular with students over several years, and you are warmly invited to attend and ask questions you may have.
To support you with your preparation for this, we have produced a series of accessible and informative texts on areas of mathematics that you are already familiar with: integration, differentiation, vectors and matrices, trigonometry, and probability. You may wish to read through a few of these to give you a sense of how the richness of these areas of mathematics plays out in the commercial, industrial, and financial worlds. We hope these will be of interest in themselves; however, they will also help you to engage with Dr van Leiden’s presentation on the Open Day itself, to ask pertinent questions and to begin thinking through your longer-term plans in mathematics. Each section includes copious references to wider literature which you may wish to follow up on.
1. Some Applications of Integration in Modern Society
As any student of mathematics is aware, integration is one of the fundamental operations of calculus. As students of mathematics we seek to become proficient in calculating integrals in the context of a problem set. However, we may be less confident about the value of integration beyond theoretical mathematics and the role it plays as an essential analytical tool across many disciplines. Because it allows us to find accumulated quantities, such as areas under curves or total change, through the summation of infinitely many infinitesimal contributions [7], integration plays a powerful role in transforming abstract mathematical ideas into powerful methods for solving real-world problems.
Have a think, first of all, about the application of integration to physics and engineering. How can integration deepen our understanding of physical phenomena? In mechanical engineering, for example, double integrals allow precise calculation of aerodynamic forces. Engineers determine the lift force on an aircraft wing by integrating pressure values across the wing’s surface, ensuring that even subtle variations in air pressure contribute to the total force [8]. This precision leads to safer and more efficient wing designs.
Let’s take a second case study from engineering: civil and structural engineers rely on integration to manage water flow and structural stresses. For example, when assessing a dam’s capacity, engineers integrate water velocity fields across the spillway to calculate the total flow rate. Similarly, in construction design, integration within computer modelling tools helps analyse how loads are distributed through beams and columns. This analysis enables the optimisation of material use while maintaining both stability and safety [3].
If we turn now to economics and finance, we find that integration provides a foundation for measuring benefits, costs, and risks. Economists can calculate consumer and producer surplus by finding the areas between supply and demand curves, capturing the added value enjoyed by buyers and sellers. Similarly, financial analysts employ integration in models that price complex derivatives and assess investment risk [4], offering quantitative insights into uncertain market behaviour. Staying with modelling applications, in environmental science and biology, integration supports the modelling of natural processes. Environmental scientists sum pollutant concentrations over given regions to estimate total pollutant loads, while biologists use integration to track changes in animal populations or study how medicines move through the human body [6].
Integration also plays a central role in probability theory and statistics. Probability density functions require integration to calculate event likelihoods across continuous ranges. By summing these infinitesimal probabilities, statisticians can determine event frequencies and update predictions as new data appear. This process forms the basis of Bayesian inference and underpins many machine learning algorithms that refine models through new data [1].
Finally, modern computational science continues to expand the uses of integration. It underlies artificial intelligence and machine learning systems, which integrate information to refine learning algorithms. Numerical methods, such as Monte Carlo techniques, remain essential for solving high-dimensional and analytically complex problems in physics, engineering simulations, climate modelling, and materials science [2]. And in the world of computer graphics, integration is fundamental to creating life-like imagery and animations to enhance the effect of movies and animation. Rendering algorithms, which describe how light interacts with surfaces and materials, depend crucially on integration to simulate realistic shading, reflections, and shadows. For example, Monte Carlo integration techniques once again approximate these complex light interactions to produce the photo-realistic visual effects seen in films, games, and virtual reality [5].
Clearly, integration isn’t simply an arcane or esoteric process in mathematical computation. Some might say that it is this – and I won’t necessarily disagree! – but it is also a powerful practical tool that underlies a whole host of real-world activities and phenomena that provide value and benefit to individuals and society.
References:
[1] Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
[2] Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep learning. MIT Press.
[3] Hibbeler, R. C. (2017). Structural analysis (10th ed.). Pearson.
[4] Hull, J. C. (2017). Options, futures, and other derivatives (10th ed.). Pearson.
[5] Pharr, M., Jakob, W., & Humphreys, G. (2016). Physically based rendering: From theory to implementation (3rd ed.). Morgan Kaufmann.
[6] Reynolds, J., Riley, J., & Brown, G. (2014). Environmental modeling: An introduction. Wiley-Blackwell.
[7] Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning.
[8] Young, H. D., & Freedman, R. A. (2014). University physics with modern physics (14th ed.). Pearson.
Activity 2: Positive language and the language of invitation
In Activity 1, you saw that the main purpose of this text is to encourage students to undertake some preparation for this presentation at the Open Day. Throughout this text, there are various individual words and phrases that use positive language to show that this text is warmly inviting students to an event and to do some preparation. Other sections that look at positive language include Language Focus 1 in Section 1.6 and Language Focus 1 in Section 7.6.
Consider for example, paragraph 1, line 1:
One of the many attractions of mathematics is in the vast scope of exciting career opportunities it offers.
This opening sentence contains the following two strings, both of which talk about mathematics in a positive way and thus, contribute to making the text a positive invitation to the Open Day:
- one of the many attractions of: This suggests that mathematics is an attractive choice of degree and for many reasons.
- the vast scope of exciting career opportunities: This phrase reinforces the point of the previous phrase, by stating that one of the benefits of mathematics is that it opens up a number of career routes and job opportunities.
Your task: Re-read the first two paragraphs and note down any phrases that meet any of the following three criteria. Next, select the ‘Answer’ button below.
- Criterion 1: phrases that create a positive picture about:
- mathematics itself and/or
- about the Open Day.
- Criterion 2: phrases that are to do with inviting students. The term inviting here can mean either ‘simply extending an invitation to attend’ and/or to do with inviting students to do something themselves.
- Criterion 3: phrases that indicate what support is offered to students.
You may also wish to note down any useful language. Here is an example.
Example: You are warmly invited to attend. This meets Criterion 2 as it is a part of the text where students are ‘simply invited’ (as opposed to being invited to do something). Language to note: warmly invited to attend. The collocation warmly invited can also be expressed as a noun phrase: a warm invitation.
Answer
Paragraph 1:
- ‘From Course Notes to Career Success: starting to think about the infinite (!) possibilities for mathematicians beyond your degree’. This meets Criterion 1a and 1b. Regarding 1a, mathematics is portrayed as infinite in a positive sense. For 1b, the phrase Career Success creates a positive impact about the Open Day. Language to note: the use of the bracketed exclamation mark ‘(!)‘ which indicates a humorous use of the term infinite.
- This presentation has proven very popular with students over several years. This meets Criterion 1b. Language to note: has proven very popular with. This is the present perfect form of the structure X proves very popular with Y. Another example of the present perfect form is The changes to the module on discrete mathematics have proven very popular with all students.
- … and you are warmly invited to … ask any questions you may have. This meets Criterion 2: an invitation to do something (i.e. ask any questions). Language to note: ask any questions is a neutral expression. Compare this with raise any questions which may suggest a slightly greater degree of formality in the questions, in some contexts.
Paragraph 2:
- To support you with your preparation for this, … This meets Criterion 3 since it indicates that support is offered to help students to maximise their engagement. Language to note: To support you with X, … This is a common phrase and it means to help you with.
- … we have produced a series of accessible and informative texts on areas of mathematics … This meets Criterion 1b since it speaks positively about the resources provided. Language to note: a series of accessible and informative texts. Here, informative refers to the content itself: the texts convey useful and substantive information. By contrast, accessible refers to how the content is presented: the texts are written in clear, straightforward language and structured in a way that makes them easy to understand and navigate. In university teaching and learning, accessibility encompasses a wide range of strategies, including providing alternative formats (such as audio or large print), captioning for videos, and designing materials that work well with assistive technologies. These approaches ensure that all students, regardless of ability or learning preference, can engage with the content. You can read more about how the University of Leeds approaches accessibility.
- … that you are already familiar with: … Meets Criterion 3 since it describes particular accessible features of the resources for the Open Day. Language to note: familiar with. This adjective + preposition phrase also has a noun form: familiarity with. Note the change of the stress on the words: familiar vs familiarity.
- You may wish to read through a few of these … Meets Criterion 2 as it explicitly invites students to do something. Language to note: You may wish to … Here may makes the phrase a suggestion rather than an instruction.
- … how the richness of these areas of mathematics plays out in the commercial, industrial, and financial worlds. Meets Criterion 1a as it states something positive about mathematics. Language to note: plays out in. If X plays out in area Y, then X takes a role in, or can be seen working in area Y.
- We hope these will be of interest in themselves; … This meets Criterion 1b as it describes a positive feature of the resources for the Open Day. Language to note: be of interest in themselves. The phrase in themselves is another way of saying inherently or intrinsically. For example, Mathematics is interesting in itself, independent of any applications it may have.
- … however, they will also help you to engage with Dr van Leiden’s presentation on the Open Day itself, to ask pertinent questions and to begin thinking through your longer-term plans in mathematics. Meets Criterion 3 as it explains how the resources will support students engaging with the Open Day. Language to note: pertinent questions. The meaning of pertinent is relevant and insightful.
- Each section includes copious references to wider literature which you may wish to follow up on. This meets Criterion 1b in that it notes something positive about the resource. Language to note: X includes copious references to Y. Here copious means many.
Language Focus
In sentence 1 of paragraph 1, there are two adjective + noun combinations. They are:
- vast scope: the adjective vast means very large or extensive. Note that this adjective has no adjectival ending of its own (unlike logical and mathematical).
- exciting career opportunities (where the underlined syllable is stressed): here career opportunities is a compound noun and it is preceded by the adjective exciting. Note that the adjective exciting ends in -ing.
Re-read paragraph 2 and identify any further adjective + noun combinations and note the adjective endings that occur. Next, select the ‘Answer’ button below. Finally, consider trying to memorise some of these adjective + noun to use them in your own writing and speaking about mathematics.
Answer
There are seven more adjective + noun strings:
- infinite … possibilities: this phrase is slightly tongue-in-cheek as career opportunities are technically finite in number. The adjective ending is -ive. This is a playful use of mathematical language.
- accessible and informative texts: here we have two adjectives preceding a single noun. Each adjective has a different ending: -ible and -ive.
- the commercial, industrial, and financial worlds: here again we have a series of three adjectives preceding a single, plural noun. All adjectives have the ending -al.
- pertinent questions: this means questions that are relevant, useful, helpful (as noted in the activity above). A tutor might encourage students to participate in tutorials by saying: We would aspire to ask pertinent questions in tutorials, lectures and in one-to-one supervision meetings. The adjective ending here is -ent.
- longer-term plans: these means plans over the long-term; it is the opposite of immediate plans or short-term plans. In your studies, making plans over the shorter- and the longer-term is usually sensible.
- copious references: copious means many or abundant. We might say that certain theorems have a copious number of proofs. The adjective ending here is -ious.
- wider literature: this string means more books, articles and other sources. Lecturers are likely to say things like: Consult the wider literature on this topic. Note that wider is a comparative adjective ending in -er.
Activity 3: Matching disciplines to applications
Most of the text above consists of a paragraph-by-paragraph listing of areas of applications of integration. For example, in paragraph 4 we have In mechanical engineering, for example, double integrals allow precise calculation of aerodynamic forces. This sentence gives an example of the application of integration in mechanical engineering, in this case, to compute aerodynamic forces.
Re-read the relevant paragraphs and match the areas to the paraphrased descriptions below of how integration is used in that field. Next, select the ‘Answer’ button below.
| Area | Application of integration | |
|---|---|---|
| Civil and structural engineers | Monitoring how animal populations change over time and how substances are absorbed by living organisms. | |
| Economists | Assessing water flow and material stress through mathematical analysis of distributed forces. | |
| Financial analysts | Using probability functions with continuous variables to determine the likelihood of events happening. | |
| Environmental scientists | Measuring the added financial value for buyers and sellers in a market. | |
| Biologists | Creating realistic light, shadow, and reflection visuals in computer-generated images. | |
| Statisticians | Creating mathematical models that use integration to measure investment risk and price financial instruments. | |
| Animators | Determining the total concentration of harmful substances released into a specific environment. |
Answer
| Area | Application of integration |
|---|---|
| Civil and structural engineers | Assessing water flow and material stress through mathematical analysis of distributed forces. |
| Economist | Measuring the added financial value for buyers and sellers in a market. |
| Financial analysts | Creating mathematical models that use integration to measure investment risk and price financial instruments. |
| Environmental scientists | Determining the total concentration of harmful substances released into a specific environment. |
| Biologists | Monitoring how animal populations change over time and how substances are absorbed by living organisms. |
| Statisticians | Using probability functions with continuous variables to determine the likelihood of events happening. |
| Animators | Creating realistic light, shadow, and reflection visuals in computer-generated images. |
Activity 4: Links within and between paragraphs
The text above has the following structure:
- An introduction which orients the readers to the purpose of the text.
- A main body with various examples of the application of integration to the real world.
- A concluding paragraph which summarises the main body.
The main body consists of a series of examples of applications of integration to the real world. Effectively, it is a list of applications of integration. Each paragraph begins with a linking word/phrase, connecting it to those before and maintaining clear, logical flow. We will call this text structure ‘listed paragraphs’. In mathematical writing, listed paragraphs are common when giving several reasons for a method, or, as here, listing applications of a concept.
The linking words/phrases from paragraph 2 of the main body onwards, are presented in the table below in the first two columns.
Your task: Match the linking word/phrase with the approximate paraphrase in the third column, and in the fourth column determine whether the approximate paraphrase is more formal or similar formality compared to the original. Next, select the ‘Answer’ button below.
| Paragraph of main body | Linking word or phrase | Approximate paraphrase | Similar formality (SF) or more formal (MF)? | |
|---|---|---|---|---|
| 2 | Have a think, first of all, about … | Turning now to … | ||
| 3a | Let’s take a second case study from … | Developing the modelling application idea, … | ||
| 3b | Similarly, … | As a final example, … | ||
| 4a | If we turn now to … | Moreover, … | ||
| 4b | Staying with modelling applications, … | It is clear from the above that … | ||
| 5 | … also … | Consider, initially, … | ||
| 6a | Finally, … | What’s more, … | ||
| 6b | And, … | Not dissimilarly, … / Analogously, … | ||
| 7 | Clearly, … | A second case study is drawn from … |
Answer
| Paragraph of main body | Linking word or phrase | Approximate paraphrase | Similar formality (SF) or more formal (MF)? |
|---|---|---|---|
| 2 | Have a think, first of all, about … | Consider, initially, … | MF: initially is more formal than first of all. |
| 3a | Let’s take a second case study from … | A second case study is drawn from … | MF: Let’s is quite informal and drawn from is quite formal. |
| 3b | Similarly, … | Not dissimilarly, … / Analogously, … | MF: The double negative Not dissimilarly is more formal. |
| 4a | If we turn now to … | Turning now to … | SF: A similar paraphrase. |
| 4b | Staying with modelling applications, … | Developing the modelling application idea, … | MF: Developing can be considered slightly more formal in this context. |
| 5 | … also … | Moreover, … / What’s more, … | MF: Note also that Moreover and What’s more, … are at the beginning of the sentence. |
| 6a | Finally, … | As a final example, … | SF: A close paraphrase. |
| 6b | And, … | Moreover, … / What’s more, … | MF: Considerably more formal. |
| 7 | Clearly, … | It is clear from the above that … | MF: This longer paraphrase is can be considered slightly more formal by many readers. |
Activity 5: Referencing
The text contains references. References are an important part of academic writing. Different kinds of writing use references in different way. Consider or discuss the following two questions about the usage and form of references in this text. Next, select the ‘Answer’ button below.
- Sometimes we include information about the author and date inside the text. For example we might write, Brown (2019, p.24) notes that the prime number theorem has received considerable attention in recent years. Here, the author is part of the sentence, and the date and page are in brackets. This is called an ‘in-text citation’. We do not see this in the text above. Instead, each reference is simply a number with a hyperlink. Why do you think this referencing format has been chosen? (Hint: consider the purpose of the text and what the writer expects readers to do with the sources).
- The numbering of the referencing is not in the form 1, 2, 3 … Instead, the first reference is 7, then 8, then 3. Why is this? (Hint: look at how the sources are organised in the reference list at the end of the text).
Answer
- In this context, references are optional supplementary reading. The target audience (prospective students preparing for an Open Day presentation) needs the main text to be accessible and engaging without requiring them to stop and read academic sources. The hyperlinked numbers allow interested students to explore further without interrupting those who just want the main points. This contrasts with academic research papers, where in-text citations show exactly where specific claims come from and are essential to the argument.
- The sources in the reference list at the end of the text are ordered alphabetically, A – Z. So the first reference used in the text is the 7th reference in A – Z order in the reference list.
Activity 6: The structure of a particular paragraph
Consider the penultimate paragraph of the text reproduced below for convenience:
Finally, modern computational science continues to expand the uses of integration. It underlies artificial intelligence and machine learning systems, which integrate information to refine learning algorithms. Numerical methods, such as Monte Carlo techniques, remain essential for solving high-dimensional and analytically complex problems in physics, engineering simulations, climate modelling, and materials science [2]. And in the world of computer graphics, integration is fundamental to creating life-like imagery and animations to enhance the effect of movies and animation. Rendering algorithms, which describe how light interacts with surfaces and materials, depend crucially on integration to simulate realistic shading, reflections, and shadows. For example, Monte Carlo integration techniques once again approximate these complex light interactions to produce the photo-realistic visual effects seen in films, games, and virtual reality [5].
This paragraph covers two applications of integration: firstly, in computational science in general, and secondly, in computer graphics. The individual sentences of the first part can be analysed has having the following logical flow.
Sentence 1: Finally, modern computational science continues to expand the uses of integration. Purpose: This sentence introduces the topic, modern computational science, and explicitly connects it to the overall theme of how integration is used in this area.
Sentence 2: It underlies artificial intelligence and machine learning systems, which integrate information to refine learning algorithms. Purpose: This is a general principle about how integration is important in computational science. Structure: The first part of the sentence is a clause; the second part is a relative clause providing more information about machine learning systems. Specifically:
First part of sentence 2: It underlies artificial intelligence and machine learning systems, … The final phrase is machine learning systems.
Second part of sentence 2: … which integrate information to refine learning algorithms. This relative clause provides more information about machine learning systems.
Sentence 3: Numerical methods, such as Monte Carlo techniques, remain essential for solving high-dimensional and analytically complex problems in physics, engineering simulations, climate modelling, and materials science [2]. Purpose: This sentence gives more detailed examples of uses of integration.
Your task: Consider the remaining second section of the paragraph, and decide whether its structure follows the same three-step model above, or whether the second section is different from the first. Next, select the ‘Answer’ button below.
Answer
Part 2 has the same structure as Part 1 as indicated below.
Sentence 1: And in the world of computer graphics, integration is fundamental to creating life-like imagery and animations to enhance the effect of movies and animation. Purpose: Introduction to the topic.
Sentence 2: Rendering algorithms, which describe how light interacts with surfaces and materials, depend crucially on integration to simulate realistic shading, reflections, and shadows. Purpose: This is a general principle about how integration is important in computational science. Structure: This is broadly the same as the related sentence in Part 1. We have an independent clause and a relative clause as follows:
Independent Clause: Rendering algorithms … depend crucially on integration to simulate realistic shading, reflections, and shadows
Relative clause: … which describe how light interacts with surfaces and materials, … This relative clause provides additional information about Rendering algorithms.
Sentence 3: For example, Monte Carlo integration techniques once again approximate these complex light interactions to produce the photo-realistic visual effects seen in films, games, and virtual reality [5]. Purpose: This sentence gives more detailed examples of uses of Monte Carlo integration techniques.
Activity 7: Specific words and phrases of academic writing
Locate specific words and phrases in the text with the following meanings. Next, select the ‘Answer’ buttons below.
- Consider paragraph 2 of the main body. Identify two words that have a similar form which mean something like exact ~ exactness or highly accurate ~ a high degree of accuracy.
Answer
precise ~ precision. (Note: the grammatically negative of these words are, respectively, imprecise and imprecision.)
- Consider paragraph 3 of the main body. Identify two words that have a similar form which mean, respectively, something like examine methodically and in detail ~ a methodical and detailed examination of something.
Answer
analyse ~ analysis. Note: other words and phrases with a similar form include analyst, analytical, unanalytical, re-analyse. There is a field of mathematics called Real Analysis.
- Identify a five-word phrase in the final paragraph of the text which means something like a mysterious procedure understood by only a few people.
Answer
an arcane and esoteric process (where the underlined syllable is stressed).
Language Focus
Consider the following sentence in the second paragraph of the text: You may wish to read through a few of these to give you a sense of how maths plays out in the commercial, industrial, and financial worlds.
The phrase plays out means has an influence on or can be seen in. This phrase can be characterised as a verb (plays) plus a particle (out). These two-word verbs are also called phrasal verbs. English has many of these verb + particle phrasal verbs.
Your task: The table below gives several examples of other verb + particle phrasal verbs with play. In the table below, match the phrasal verbs on the left, to their meanings on the right. Next, select the ‘Answer’ button below.
| Phrasal verb | Meaning | |
|---|---|---|
| To play around with something: [Lecturer to students in the computer cluster] ‘Just have a play around with the code and see how it works. Trial and error is the way! You’ll pick it up as you go along.’ | To minimise or reduce the perceived or actual impact of something. | |
| To play along with something: ‘Let’s just play along with the ideas in this presentation for a moment and see if makes sense by the end of the hour.’ | To unfold, develop, or show the consequences of something in a particular context. | |
| To play out: [Lecturer to students in the lecture theatre] ‘How does this limitation of the theorem play out in the domain of our problem?’ | To have a go at doing something/to experiment with something to see if it works. | |
| To play up: [Student to student in the computer cluster] ‘The modelling software is playing up again. I just can’t get it to capture what I want.’ | To go wrong/to not function well. | |
| To play down something: [Learning mentor to student seeking advice on examination technique] ‘Never play down the importance of learning the formulae, but also of the value of applying them to various examples.’ | To give a new idea or suggestion some time before making a judgement about it. |
Answer
| Phrasal verb | Meaning |
|---|---|
| To play around with something: [Lecturer to students in the computer cluster] ‘Just have a play around with the code and see how it works. Trial and error is the way! You’ll pick it up as you go along.’ | To have a go at doing something/to experiment with something to see if it works. |
| To play along with something: ‘Let’s just play along with the ideas in this presentation for a moment and see if makes sense by the end of the hour.’ | To give a new idea or suggestion some time before making a judgement about it. |
| To play out: [Lecturer to students in the lecture theatre] ‘How does this limitation of the theorem play out in the domain of our problem?’ | To unfold, develop, or show the consequences of something in a particular context. |
| To play up: [Student to student in the computer cluster] ‘The modelling software is playing up again. I just can’t get it to capture what I want.’ | To go wrong/to not function well. |
| To play down something: [Learning mentor to student seeking advice on examination technique] ‘Never play down the importance of learning the formulae, but also of the value of applying them to various examples.’ | To minimise or reduce the perceived or actual impact of something. |
Activity 8: Generalising academic language
In the third paragraph of the main body we see: This analysis enables the optimisation of material use while maintaining both stability and safety. This sentence can be broken down in the following way:
| Text | This analysis | enables | the optimisation | of material use | while maintaining | both stability and safety. |
|---|---|---|---|---|---|---|
| Grammatical Analysis | subject | verb | object of verb | of-phrase complement to optimisation | temporal linking word followed by -ing verb | object of verb in a compound structure |
A paraphrase of this sentence might be By using this analysis, we both optimise the use of materials and at the same time do not sacrifice anything in terms of either stability or safety. More generally, we have something like X enables optimisation of A while losing nothing with respect to B. This sentence structure therefore allows us to compare the benefits or losses of something at the same time.
Below are some further example sentences following the above scheme.
- The approach taken in this textbook facilitates the effective use of communication of mathematics while also following rigorous mathematical reasoning.
- This method emphasises clarity of presentation while retaining both the necessary concepts and using proper mathematical notation.
Your task: Make sure you are familiar with the grammar of this construction. Then try to create one or two more sentences using this structure.
In this section, you have focused on longer written mathematical communication. To conclude the chapter, the final section provides a glossary of important mathematical terms from the chapter.
