3.5 Workshop 4 (Communication)

In this workshop, you’ll practice verbally sharing your proofs to the problems you worked on in the Construction workshop with those in your group.

NOTE: This forms some of your preparation for the dialogic assessment so it is very important that you engage in this workshop.

Task 3.19 (10 min).

Warm Up

1.

Allocate a number to each of the remaining non-assessed problems from the Construction workshop for this block.
2.

Create a random list (using e.g. https://www.random.org/lists/) to allocate each member of your group a random number from 1 to 6 inclusive. No number should be allocated to more than one person. The number you get indicates the problem whose proof you’ll be explaining.

NOTE: If you have fewer than 6 people in your group, make sure you allocate ALL 6 proofs amongst those in your group. This will mean some people will be talking about more than one proof.
3.
For each person in your group, allocate another person to be their “Encourager”. When a speaker is talking about their proof, their encourager must:
- •
  
  highlight at least one positive aspect of the speaker’s proof/explanation.
- •
  
  ask at least one question.
- •
  
  invite questions from the other listeners.
- •
  
  keep an eye on time. As a group, you should spend approx. 15 mins on each proof before moving onto the next speaker.
4.

Remind yourselves of the “Guidance for the Communication Workshops” on the IMU Learn course.
5.

Use this time to familiarise yourself with the proof you’ll be explaining to the group. What are the key steps? How will you structure your explanation?
6.

If there’s time, practice explaining your proof to the person next to you.

Task 3.20 (90 min).

Communication Time!

1.

Decide amongst yourselves which order each of you will explain your proofs in.
2.

Take it in turns to talk about your proofs.
3.

If you’ve run out of questions to ask the speaker before the 15 mins is up, check through the suggested questions - have you asked at least one from each category? If so, ask your tutor for some questions about this proof.

Suggestions for Questions to ask

•
Questions regarding the mathematical objects that appear in the proof and their definitions
- –
  
  Can you remind me of the definition of…?
- –
  
  What does that theorem state?
- –
  
  Can you explain why that theorem can be used in this context?
•
Questions regarding the logical links between parts of the proof
- –
  
  Can you explain why you can assume that?
- –
  
  Can you give a bit more explanation about why that step follows?
•
Questions regarding the key ideas of the proof
- –
  
  Where in your proof do you use this property/assumption?
- –
  
  What are the key steps in this proof?
•
Questions regarding the proof and its links to the main ideas seen in the course.
- –
  
  Are there other proofs in the course that use a similar proof technique/idea? Can you briefly describe them?
- –
  
  If we changed this [object, assumption, conclusion], how would your proof change? Would the statement still be valid?

Extending Questions

Prove the following statements using mathematical induction.

1.
For every integer $n\geqslant 0$ , $4^{n}-1$ is divisible by $3$ .
1. (a)
  
  What if, in the induction step, we assumed $P(n-1)$ was true instead of $P(n)$ and wanted to prove $P(n+1)$ ? How would our argument change?
2.

For every $n\in\mathbb{N}$ , $\sum_{k=1}^{n}k^{3}=\frac{1}{4}n^{2}(n+1)^{2}$ .
1. (a)
  
  Are there any other expressions for sums that you can deduce from this result?
2. (b)
  
  Can you think of a visual way to express this result?
3.

Every positive power of 13 can be written as a sum of two squares.
1. (a)
  
  Can you think of an argument that uses weak induction AND an argument that uses strong induction?
2. (b)
  
  Can you generalise this argument for numbers other than 13?
3. (c)
  
  How could this statement and argument be adapted to consider a sum of more than two squares?
4.

H02 (Not discussed in Communication Workshops)
5.

Let $S$ be an infinite set of sets such that if $A,B\in S$ , then $A\cap B\in S$ . If $A_{1},\cdots,A_{n}\in S$ , then $\bigcap_{i=1}^{n}A_{i}\in S$ for all $n\in\mathbb{N}$ .
1. (a)
  
  Can this statement and argument be adapted for unions instead of intersections? Where would the proof change?
2. (b)
  
  Can you think of an example of an infinite set $S$ that satisfies the requirement that $A,B\in S\implies A\cap B\in S$ ?
3. (c)
  
  Does the set $S$ have to be infinite to satisfy the requirement that $A,B\in S\implies A\cap B\in S$ ? Why or why not?
6.
Let $n\in\mathbb{N}$ . Take $n$ lines in the plane such that no two lines are parallel and no three lines meet at a single point. Then these lines divide the plane into $\frac{n(n+1)}{2}+1$ regions.
1. (a)
  
  Would this argument work if we removed the requirement that no three lines meet at a single point? Why or why not?
2. (b)
  
  Would this argument work if we removed the requirement that no two lines are parallel? Why or why not?
3. (c)
  
  Is it always possible to colour every region such that no pair of adjacent regions share the same colour? Can you prove your claim?
7.

Every set $S$ with $|S|\geqslant 12$ can be partitioned into two sets $A$ and $B$ such that $4\Big{|}|A|$ and $5\Big{|}|B|$ .
1. (a)
  
  Can this argument work if we replace 4 and/or 5 with any other numbers? What if we adjust the minimum size of $S$ ?