From the options below, decide which statements are True, which are False and which are Unknown. in relation to the text.

(Note: If you believe a statement is mathematically true, but the information is not provided in the dialogue, you must select Unknown).

Next, select the ‘Answer’ button below the options.

Statement 1: Luke is fairly comfortable working with lines in just two dimensions but is finding the generalisation to 3D space more challenging.

Statement 2: There are fewer ways for two lines to relate to each other in 3D space than there are in 2D space.

Statement 3: A point on a plane can be defined using a starting point and a normal vector.

Statement 4: A normal vector that points in the exact opposite direction can be used to calculate a point on the plane in the exact same way.

In mathematical discussions, explanations are often built step-by-step across several sentences. Each sentence has a specific function (a job that it does to help build understanding).

Consider line (4) from the dialogue, repeated below. In this extract, Callum responds to Luke's comment that in three dimensions things get a bit more abstract. The sentences have been numbered in square brackets [1] to [4] for convenience.

The fundamentals are the same. [1] In 2D we can uniquely determine a line using two points passing through it. [2] We can also define the line using a point passing through it and a vector along it. [3] It is the same idea in 3D, the only difference is that we have one more coordinate. [4]

Here is an analysis of the function of each sentence in Callum's explanation:

• [1] A statement that lines in 2D and 3D space are identical at the most basic level.
• [2] A definition of a line in 2D.
• [3] A second, complementary definition of a line in 2D, using a single point and a vector.
• [4] A generalisation of the definitions in [2] and [3] to 3D. This serves as a summary statement and echoes the statement in [1].

Your task: Now consider two other multi-sentence explanations that Callum gives, specifically sections (10) and (14). Re-read each extract below and make a note of the function of each sentence, following the pattern shown above. You may need to look at the surrounding context of each extract. Next, select the 'Answer' button below to compare your notes with the suggested analysis.

(10) Now, here’s where 3D differs from 2D. [1] In 2D, two lines either intersect or are parallel. [2] But in three dimensions, two lines can either intersect, be parallel and not intersect, or not be parallel and not intersect. [3]

(14) Exactly! [1] Imagine the plane as a flat surface, like a sheet of paper. [2] The normal vector is a vector that points straight up at right angles to every direction on the plane. [3] Any point on the plane can be described in terms of our starting point and the normal vector. [4]

Goal

Practise having conversations about lines and planes in 3D space 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 that has a question about a problem in lines and planes in 3D. 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 a problem from the topic of lines and planes in 3D to discuss.

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 Luke's opening above, or with something else, for example, Hi [Speaker B]. Do you have a quick moment to help me out with [Problem P]. It shouldn't take more than a couple of minutes. 

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.

When reading a chain of emails, it is important to be able to step back from the details (like who is doing which problem) and understand the "big picture" or overall outcome of the correspondence.

From the options below, decide which summary you think most accurately describes the overall outcome and tone of the email exchange between the four students. Next, select the 'Answer' button below the options to check your understanding.

Option 1: Although the email exchange begins well with buy-in from all four students, problems arise and a collective decision cannot be reached.

Option 2: The group works well together to select a time and successfully agree on how to divide up the work.

Option 3: Two distinct opinions arise as to how to approach the group task but after some discussion, an overall decision is reached.

Option 4: The group works well together to establish a time, but they experience difficulties around how to divide up the work.

Activity 2.1: Emails are not face-to-face communication, so it is important to choose your language carefully to ensure the tone is appropriate for your audience.

The email chain you just read is interesting because it is between peers (students). Therefore, it uses more informal and relaxed language than an email to a lecturer would. Note: Communicative practices, including email, vary greatly across different cultures. The observations made in this activity are based on typical email practices in UK universities.

Consider three elements of (1), reproduced below for convenience:

1. The address form (how the email starts).

2. The sign-off (how the email ends).

3. The use of positive language (words or phrases that build a friendly relationship or make a positive comment before discussing the task).

Re-read (1). Note down these three aspects and consider how formal or informal they are. Next, select the 'Answer' button below to compare your thoughts.

Hello everyone,

I hope you're all doing well. I'm getting in touch to organise our group work for the lines and planes assignment. We need to complete four problems covering lines, planes, intersections, and distances.

Could we meet sometime next week to divide up the work and go through the assignment together? I'm fairly flexible with timing. What works best for everyone?

Also, do you think we should meet in person or use Teams?

Cheers,
Aisha

Activity 2.2:  In the previous activity, we saw how Aisha opened the email chain using friendly, semi-informal language. However, as an email conversation continues back and forth, the level of formality often changes.

Your task: Re-read the other emails (2) through (6) in the chain. For each email, note down:

1. The address form (how the email starts).

2. The sign-off (how the email ends).

3. Any positive language (words or phrases that build a friendly relationship or show enthusiasm).

In addition to address forms, sign-offs, and positive language, the email chain contains several other useful functional phrases. These are the specific groups of words that students use to negotiate, suggest, and agree on plans.

Your task: Read the paraphrased meanings below. Go back to the email text and identify the exact phrase or sentence that matches each meaning. Once you have found the phrases, select the 'Answer' button below to check your answers and read the language notes.

1. In (1), a phrase that explains why the writer is sending the email.
2. In (1), a sentence that means something like I can meet at almost any time.
3. In (2), an expression of preference for meeting face-to-face.
4. In (3), a phrase which means something like As far as I've understood from the reading …
5. In (4), a phrase which means something like I will need to revise topic X or I am not sure I remember everything about topic X
6. In (5), a three-word phrase used to make a suggestion.
7. In (6), two consecutive short sentences that mean something like That's decided! This arrangement will work. 

You have read and analysed the email correspondence between the students organising a mathematics assignment. Now, it is your turn to practice writing emails of this kind.

Working in a group of 3 or 4, you will create a similar, imaginary email chain to organise a group presentation for a mathematics module.

Step 1: Before writing, discuss the following with your group:

• What is the mathematical topic of your presentation? (e.g., Applications of integration, Solving linear systems using matrices).

• Who will write the first email?

Step 2: The student whom the group chooses to write the first email should draft it and send it to the rest of the group. This email must include the following elements:

1. An appropriate address form and a brief, polite introduction (e.g., checking in with the group).

2. A description of the task you have decided upon (mentioning what needs to be done to prepare for the presentation).

3. A question asking when the group is available to meet.

4. An appropriate sign-off.

(Tip: Look back at Aisha's first email to help you structure this).

Step 3: Once the first email is sent, the other group members should reply to the thread. When writing your replies, try to use the polite and functional phrases you learned in the previous activities. For example:

• Thanks for reaching out...

• I'm fairly flexible with timing...

• How about...

• I'm a bit rusty on...

Continue replying to each other until you have agreed on a time to meet and successfully divided up the presentation tasks.

Step 4: When your email chain is complete, review it together. Did you manage to reach an agreement? How did the formality of your language change from the first email to the last?