6.5 Vectors and Coordinate Systems: Mathematics Communication
So far, you have focused on written mathematical language. In this section, you will practise understanding and taking part in spoken mathematical discussions. You will read and analyse two different dialogues: one between two students, and one between a student and a lecturer.
The goal of this section is not just to understand these dialogues, but to help you feel confident in having similar discussions yourself. You will learn how to have a discussion about mathematics, ask and answer questions, and explain your ideas clearly to someone else. This will help you use English for practical conversations about mathematics, both in class and out of class. To do this, you will need to:
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notice useful phrases and sentence structures in the dialogues,
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memorise them,
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practise saying them,
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and finally, use them in real conversations.
Each line in the dialogues is numbered to help you talk about specific parts easily. For example, if you have a question, you can say: What does the second word from the end of line 8 mean?
By working with these dialogues, you will improve your ability to discuss mathematics in English, ask and answer questions, and communicate your ideas clearly with others.
Dialogue 1: Two students
An example dialogue between two university students, Ruby and Sophie, discussing vectors and scalars. This dialogue occurs in a study space with the two students sitting next to each other.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Ruby: | Hi, Sophie, how are you? I'm in the middle of reviewing the chapter on vectors for our exam, but it’s fairly tough. |
| (2) | Sophie: | I know what you mean. I did a ton of work on that yesterday and I’ve just about got it. What’s bothering you about it? |
| (3) | Ruby: | Well, I'm a bit confused about the difference between a scalar and a vector. |
| (4) | Sophie: | Right, a scalar only has magnitude, which describes the size or amount of something. Temperature, for example, is a scalar. Today it's twenty degrees Celsius, that's all the information we need. A vector however has both magnitude and direction. |
| (5) | Ruby: | OK, can you give me an example of that distinction, please? It’s just words for me at this point, I’m afraid. |
| (6) | Sophie: | I know exactly what you mean. My twin sister’s exactly the same: example, example, example, she’s always saying. Anyway, back to it. Velocity is a vector. It's not enough to say a car is travelling at sixty kilometres per hour. We also need the direction: north, south, east, west? The magnitude is sixty kilometres per hour, and the direction is part of the information too. |
| (7) | Ruby: | That makes sense. What about speed? Or is that the same as velocity? |
| (8) | Sophie: | No, speed is a scalar! Speed is just the magnitude. When we say sixty kilometres per hour, that's just the speed. Velocity includes the direction as well. |
| (9) | Ruby: | Right… So, force would also be a vector? |
| (10) | Sophie: | Exactly, to describe a force, we need to specify both its magnitude, how strong it is, and the direction in which it acts. These two aspects are inseparable, like two sides of the same coin. |
| (11) | Ruby: | Ahhh, now I see why the notes say we can visualise vectors as arrows. The length of the arrow represents the magnitude, and the direction the arrow points represents the direction of the vector… How about adding vectors? Can we just add them like we add numbers? |
| (12) | Sophie: | Not quite! We add vectors by placing one arrow at the end of the other. If we walk four metres east, and then we walk three metres north, the overall displacement from our starting point isn't seven metres. It’s five metres, somewhere northeast. |
| (13) | Ruby: | I see this, and the five metres comes from the standard three-four-five right-angled triangle. What if we multiply a vector by a number? |
| (14) | Sophie: | Multiplying a vector by a positive number changes the magnitude: multiplying by a number larger than one makes the arrow longer; smaller than one makes it shorter. The direction stays the same. If we multiply by minus one, the arrow reverses direction but keeps its length. |
| (15) | Ruby: | I get that, but what happens if we multiply by minus two, for example? |
| (16) | Sophie: | Multiplying by minus two… we can think of this as first multiplying by two, then by minus one. So that would double the magnitude of the vector and reverse its direction. |
| (17) | Ruby: | Thank you, Sophie! That makes perfect sense. How about multiplying a vector by a vector? |
| (18) | Sophie: | I’m sorry, Ruby, but I need to head off to another lecture. Why don't you catch Dr Sterling straight after the next lecture? I'm sure she'll be happy to help. |
| (19) | Ruby: | No worries, Sophie, thank you very much for your help. I’ll do that. |
| (20) | Sophie: | You’re welcome, see you at lunch. |
| (21) | Ruby: | Yeah, see you then. |
Activity 1: Global understanding
From the options below, decide which statements are True, which are False and which are Unknown. Next, select the ‘Answer’ button below the options.
Statement 1: Sophie leaves because she has an appointment with Dr Sterling.
Statement 2: Sophie doesn't know how to multiply a vector by a vector.
Statement 3: For Ruby, working through some examples with Sophie allows her to understand something about the representation of vectors that she had not been able to visualise before.
Statement 4: Ruby understands Pythagoras' theorem [latex]a^2 + b^2 = c^2[/latex].
Answers
Statement 1 is False. Sophie states that she has to leave because she has another lecture. We have no reason to doubt this, so on the basis of what she says, Statement 1 is False. Sophie's comment about Dr Sterling is a suggestion to Ruby that Ruby might try to meet with Dr Sterling.
Statement 2 is Unknown. Sophie knows a lot about this topic and therefore it may be reasonable to infer that she does know how to multiply vectors. However, in the text as it is, we can't claim to know that.
Statement 3 is True. According to Ruby, examples help her understand. Sophie supplies examples and Ruby's realisation in line (11) … now I see why … strongly appears to suggest that the examples they work through allow Ruby to understand why vectors are represented as arrows. Note, in particular, the term Ruby uses is visualise.
Statement 4 is True. We know this because Ruby correctly interprets Sophie's statement (12) to draw on the fact that a right-angled triangle with side length of 3 units, 4 units and 5 units is a well known example of a Pythagorean triangle with integer length sides.
Activity 2: Identifying useful language
Question 1: This is a fairly informal dialogue between two students. This activity focuses on some of the informal language that is used. Each prompt below expresses the same idea as a phrase or sentence in lines (1)-(5) of the dialogue, but in more formal English. Find the informal phrase in the dialogue that matches each prompt. Next, select the ‘Answer’ button below.
- This is somewhat difficult.
- A considerable amount of work/ a load of work/a bunch of work/a great deal of work
- What are you finding troublesome about this?
- Sorry but I can't get the idea from the language alone.
Answers
- it's fairly tough in (1)
- a ton of work in (2)
- What's bothering you about it? in (2)
- It's just words for me at this point, I'm afraid in (5)
Question 2: There are a number of places where Sophie or Ruby says something with a meaning close to I understand you or I understand that (i.e. that some idea or explanation has been grasped). These two categories are subtly different:
- I understand you is more personal: the speaker is acknowledging the other person's feelings or experience.
- I understand that is more cognitive: the speaker is confirming they have grasped an idea or explanation.
The first example of this is in (2) when Sophie says I know what you mean. What exactly is it that Sophie understands? Sophie understands how Ruby is feeling; or perhaps more precisely, she understands her experience of finding the work pretty tough. This phrase is therefore closer to I understand you. It has an additional meaning beyond simple comprehension: it expresses empathy and signals that the speaker shares, or has shared, a similar experience.
Your task: Identify four other utterances in the dialogue with a similar meaning to I understand you or I understand that. Decide whether the meaning is closer to I understand you or I understand that. Next, select the ‘Answer’ button below.
Answers
In (6) I know exactly what you mean. This may be closer to I understand you. Sophie is saying that she understands Ruby's request for more examples because of her own experience with a family member.
In (7) That makes sense. This is closer to I understand that.
In (13) I see this. This is closer to I understand that.
In (15) I get that. This is closer to I understand that.
Question 1: This Language Focus looks at verb-noun pairs and their uses e.g., visualise (verb) and visualisation (noun). Many students find it useful to study academic language in patterns in this way. While understanding the formal relationship between the two forms is important, it is equally important to find examples of how each form is used in context.
In (11), Ruby says …now I see why the notes say we can visualise vectors as arrows. The word visualise is a verb and the noun form is visualisation. These two forms can be used in similar ways, as illustrated by the examples below:
- We can visualise a complex number as a point on the Argand diagram.
- The visualisation of complex numbers on the Argand diagram makes their geometric properties much clearer.
The first of these uses the verbal form visualise; this is a verb because it can be used with a subject: I visualise. The second is the nominal form, visualisation; this is nominal because it can be used with the: the visualisation of ...
Here is another verbal ~ nominal example set with the pair represent and representation:
- A matrix can represent a linear transformation of the plane.
- The representation of a linear transformation as a matrix allows us to perform geometric operations using multiplication.
Your task: Below are ten nouns ending in -(a)tion. What are the verbs from which these nouns are derived? Use a good dictionary to check the form, then find or write one example sentence for each verb form and one for each noun form. Next, select the ‘Answer’ button below.
simplification, evaluation, generalisation, normalisation, expansion, transformation, substitution, factorisation, approximation, assumption
Answers
In addition to the answer in Column 1, some brief examples are provided.
| Verb | Example | Noun | Example |
|---|---|---|---|
| simplify | We simplify the expression by cancelling common factors in the numerator and denominator. | simplification | The simplification of a rational function often requires factorisation as a first step. |
| evaluate | We evaluate the function at [latex]x =3[/latex] by substituting directly into the expression. | evaluation | The evaluation of a definite integral at its limits gives the exact area under the curve. |
| generalise | We can generalise this result to [latex]n[/latex] dimensions. | generalisation | The generalisation of Pythagoras' theorem to [latex]n[/latex] dimensions gives the Euclidean distance formula. |
| normalise | We normalise a vector by dividing it by its magnitude. | normalisation | The normalisation of a vector produces a unit vector pointing in the same direction. |
| expand | We expand the brackets using the distributive law. | expansion | The binomial expansion of gives a polynomial with [latex]n+1[/latex] terms. |
| transform | We transform the coordinates to simplify the equation of the ellipse. | transformation | The transformation of a vector by a rotation matrix preserves its magnitude. |
| substitute | We substitute [latex]u = x^2[/latex] into the integral to simplify the calculation. | substitution | The substitution [latex]u = \sin x[/latex] transforms the integral into a more manageable form. |
| factorise | We factorise the polynomial to find its roots. | factorisation | The factorisation of [latex]x^2 - 5x + 6[/latex] gives [latex](x-2)(x-3)[/latex]. |
| approximate | We can approximate the area under a curve using a Riemann sum. | approximation | The approximation of [latex]\pi[/latex] as [latex]3.14159[/latex] is accurate to five decimal places. |
| assume | We assume that is an even integer and derive a contradiction. | assumption | The assumption that [latex]\sqrt{2}[/latex] is rational leads to a contradiction, proving that it is irrational. |
Question 2: In (12), Sophie says the overall displacement … The term overall is close in meaning to a number of other terms used in mathematics and science to express the idea of a total value, or a value derived from combining a number of other values. Below are some terms with a similar meaning to overall in mathematical and scientific contexts. Note that each is used in slightly different situations, and some have precise technical meanings that differ from their everyday English use.
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total: as in the total area under the curve between [latex]x=0[/latex] and [latex]x=3[/latex].
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cumulative: as in the cumulative distance run by an athlete over several days
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aggregate: as in the aggregate increase in the value of the shares over the quarter
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net: as in the net force acting on an object when two forces are applied simultaneously
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gross: as in the gross income before any deductions are made
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resultant: as in the resultant displacement when we walk four metres east and three metres north is five metres northeast
Your task: Use a good English dictionary to find the meaning of each term. For each one, write a mathematical or scientific example sentence of your own. Then compare your sentences with a partner.
Activity 3: Your turn!
Goal
Practise having a conversation where one student asks another for help. Use the dialogue above as a guide and Focus on communication, not necessarily on solving new math problems. Working together, create a set of confusions that you have or have had and use these to construct the dialogue.
Set-up
In pairs, take the following roles:
Speaker A is the student who has a question or set of questions about vectors and coordinate systems. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don't.
Speaker B is the student that has the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles - one student takes Speaker A and the other student takes Speaker B.
Step 2: Decide on the question or problem that the dialogue will explore. As noted above, this may be a real, actual problem that one of the two students has, or something that both students understand but pretend they do not for the purpose of the dialogue. The problem might be about terminology, concepts, methods, definitions, examples, application or something else.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any useful phrases.
Step 4: Run the dialogue. Speaker A should start, either with a variation on Sophie's opening above, or with something else, for example, Hi [Speaker B]. I'm not sure how to approach [Problem P]. Do you have a moment to help?
Step 5: Switch roles and practise again to experience both asking and answering.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced, how confident you felt, and if there are alternative ways of going through the dialogue.
Dialogue 2: A student and a lecturer
An example dialogue between a university student, Ruby, and her lecturer, Dr Helen Sterling, discussing the dot product and cross product. This dialogue takes place in a lecture room, just after a lecture.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Ruby: | Hi, Dr Sterling, do you have a moment? I had a couple of questions from the lecture. |
| (2) | Dr Sterling: | Of course, Ruby. Let's stay here for a few minutes. What would you like to ask? |
| (3) | Ruby: | Thank you, I've been revising vectors for our exam, but I'm a bit confused about the dot and cross product. I understand their definitions, but I don't get why and where we would want to use them. |
| (4) | Dr Sterling: | Great question. First, let’s step back and note that we have two different kinds of vector products! While there’s only one way to multiply two numbers, there are various ways of multiplying two vectors, such as the dot product and the cross product. Which one shall we start with? |
| (5) | Ruby: | The dot product please. |
| (6) | Dr Sterling: | Okay. So the dot product gives us a number that tells us how much of one vector points in the direction of another vector. Imagine pulling a toy train with a string, on rails along the floor. The force you apply is a vector, and the displacement of the train along the rails is also a vector. |
| (7) | Ruby: | Okay, I'm following... |
| (8) | Dr Sterling: | Their dot product is the work done. If you pull in the same direction as the train moves, then all the force goes into moving the train, and the dot product is large and positive. But if you pull at an angle, then some of the force is wasted trying to pull sideways, and the dot product is smaller. |
| (9) | Ruby: | What if I pull perpendicular to the motion? |
| (10) | Dr Sterling: | Then you do no work on the train, and the dot product is zero. That's the key idea: the dot product measures how aligned two vectors are. |
| (11) | Ruby: | That makes sense. How about the cross product? |
| (12) | Dr Sterling: | Okay, so the cross product is quite different. It gives you a new vector, not just a number. |
| (13) | Ruby: | Is that why the dot product is also called the scalar product, because it gives us a scalar? Whereas the cross product is also called the vector product because it gives us a vector, right? |
| (14) | Dr Sterling: | Wonderful, that's exactly right. |
| (15) | Ruby: | So what kind of vector is this? |
| (16) | Dr Sterling: | This new vector points in a direction that is perpendicular to both of the original vectors. The fact that the new vector is perpendicular is very important because it is useful for problems where you need to find a direction that is at right angles to two given directions. |
| (17) | Ruby: | Can you give me an example of when we'd use that? |
| (18) | Dr Sterling: | Sure, consider a ball on a string, rotating in a horizontal circle above your head. The cross product of the radial position vector and the momentum vector of the ball gives the angular momentum vector. The direction of this new vector indicates the axis of rotation. |
| (19) | Ruby: | Thank you, Dr Sterling. I think I understand them much better now. |
| (20) | Dr Sterling: | You're welcome, Ruby. These concepts are fundamental across STEM, so it's great that you're taking the time to understand them. |
Activity 1: Global understanding
In the previous dialogue, we saw how Ruby asked Sophie for examples because they helped her to visualise the mathematics they were discussing. Examples, as well as counter-examples and non-examples, are a central part of mathematical thinking, and it is no surprise to see them appearing again in the dialogue between Ruby and Dr Sterling. Next, select the ‘Answer’ button below the options.
Your task: Read the dialogue and decide whether each of the following statements is True, False, or Unknown. Next, select the ‘Answers’ button below the options.
Statement 1: The conversation begins with the discussion of a specific problematic example.
Statement 2: Ruby asks for examples of both the dot product and the cross product.
Statement 3: Ruby's question in (9) is a specific case of a principle Dr Sterling describes in (8).
Statement 4: By the end of the dialogue, Ruby feels able to offer additional examples of the dot product and the cross product to a fellow student.
Answers
Statement 1 is False: The conversation does not begin with an example at all. Ruby opens by stating that she is comfortable with the definitions of the dot product and the cross product, but that she does not understand the conceptual difference between them or where each would be used. The problem that Ruby has is partly conceptual and partly one of application.
Statement 2 is False: Ruby does ask for an example of the cross product in line (17). However, it is Dr Sterling who volunteers an example of the dot product in line (6); Ruby does not ask for it. The two examples therefore arise in different ways.
Statement 3 is True: In line (8), Dr Sterling describes what happens when you pull a box at an angle to its direction of motion: part of the effort is wasted, and the dot product is smaller. In line (9), Ruby asks specifically about pulling perpendicular to the direction of motion. This is a particular case of pulling at an angle, specifically, at an angle of 90°. Ruby is therefore asking about one specific instance of the more general principle Dr Sterling has just introduced.
Statement 4 is Unknown: Ruby states clearly that her understanding has developed during the conversation. However, she does not say anything about whether she would feel ready to explain these concepts to another student. We cannot claim to know this on the basis of the dialogue alone.
Activity 2: Identifying useful language
In the dialogue with Ruby, Dr Sterling offers several positive and encouraging comments to Ruby. She also takes the time to point out when an idea is important. The first two examples of these two ideas are:
- First example of a positive and encouraging comment number: (4): Great question.
- First example of pointing out when an idea is important: (10): That's the key idea.
Your task: Locate two further examples of each communicative function in the remainder of the dialogue. Next, select the ‘Answer’ button below the options.
Answer
Positive and encouraging comments:
- (14): Wonderful, that's exactly right.
- (20): ...so it's great that you're taking the time to understand them.
Comments that point out when an idea is important:
- (16): The fact that the new vector is perpendicular is very important because …
- (20): These concepts are fundamental across STEM …
Question for reflection and discussion: To what extent, and in what ways, can using positive language help to create and sustain productive relationships among people engaged in academic work?
You might find the following questions useful to guide your thinking:
- Think of a time when someone used encouraging language with you. How did it affect your confidence or motivation?
- Are there any risks to using too much positive language in an academic context? Could it ever be unhelpful or misleading?
- Is positive language used differently in academic settings compared to everyday conversation?
Activity 3: Your turn!
Goal
Practise having conversations about mathematics problems about vectors and coordinate systems in English. Focus on communication, not necessarily on solving new math problems.
Set-up
In pairs, take the following roles:
Speaker A is the student with a question about the mathematics of vectors and coordinate systems. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don’t.
Speaker B is the lecturer, professor or seminar leader with the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles - one student takes Speaker A and the other student takes Speaker B. Think about how this student-lecturer dialogue might be different to the student-student dialogue in the beginning of the section. The lecturer is highly likely to know the answers to the problems but might also ask helpful questions or provide partial examples in order to guide the student.
Step 2: Decide a problem from the topic of vector and coordinate systems to discuss.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any phrases that you feel may be useful.
Step 4: Run the dialogue. Speaker A should start, either with a variation on Ruby's opening utterance above, or with something else, for example, Excuse me Dr Brown, would you happen have some time to help me out with [Problem P]. It shouldn't take too long.
Step 5: Run the dialogue a second time with the roles reversed.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced; how confident you felt; and if there are alternative ways of going through the dialogue.
In this section, you practised reading and performing dialogues (spoken communication) about mathematics. In the next section, you will focus on written mathematical communication.
