Lesson video

Hi, thanks for choosing to learn with me today.

In today's lesson, we're gonna be looking at how maths can help us with our everyday lives.

Let's get started.

In today's lesson on voting systems, you're going to be learning about some other voting systems that already exist, and looking at how they work, and whether or not there are any benefits to them.

Now, in today's lesson, we're going to be looking at two, well, more key phrases than keywords, and you can see them on the screen right now.

We have the Single Transferable Vote, or STV, and with that, the opinions in parliament would match the strength of the support in the country, and we're going to look at the Alternative Vote System or AV, and that's where your constituency gets an MP for the majority support.

Now, don't worry if that doesn't make much sense to you right now, we're gonna explore it in our lesson today.

So our lesson has two parts, and we're gonna begin by looking at the Single Transferable Vote.

So, with the Single Transferable Vote, the opinions in parliament match the strength of support in the country.

So rather than electing just one candidate, bigger areas elect a small group of candidates.

Voters would number the candidates on their ballot paper, starting with one for your favourite, and then they can put numbers next to as many or as few candidates as they like.

So, for example, if there are five candidates and they have an opinion about all five of them, then they'd number them one to five.

But if, for example, there were five candidates and they only really cared about three, they might number them one to three, and just leave the other two blank.

To get elected, a candidate needs to receive enough votes to reach a set value, which is known as a quota.

The quota is based on the number of vacancies and the number of votes cast, in other words, it's not the same all the time.

Now, each voter has one vote.

If your first choice candidate reaches the quota of votes, then additional votes move to each voters' second choice candidate, and we'll look at how that works a little bit.

If no candidate reaches the quota, the least popular candidate is removed and their votes are assigned to the voters' second choice.

We repeat this process until the number of vacancies has been filled.

Now that might seem a little complicated, but don't worry 'cause we're gonna go through an example.

We'll begin by looking at how we work out the quota.

So to decide the quota required to be elected, we use the following formula.

We take the total number of votes, and divide it by the total number of seats available, add 1, and then when we found that result, we add 1 to that.

So, let's see how that works.

There were 109,525 votes cast and three seats to be filled.

So what's the quota for the number of votes required to earn a seat? Remember, we take the total number of votes cast, so that's 109,525, divide it by the number of seats to be filled, 3, add 1, in other words, we're dividing by 4.

When we've done that, we add 1 to the result, and we get 27,382.

25.

Now remember, we can't have 0.

25 of a vote, so we round.

In this case, 27,382 votes are needed.

Your turn now, have a go at the one on the screen, pause while you do this.

Welcome back, how did you get on? Well, if you used the formula correctly, you should have discovered that you've got the answer 17,973.

4, or in other words, 17,973 votes are needed.

Let's do a check now, while we work through and see who wins this election.

So, we worked out our quota, and 17,973 votes are needed, there are four seats to fill.

What I've got here for you is a graph showing how many votes each candidate received.

So what I'd like you to do, is tell me, who if anyone, has been elected? The black horizontal line that you can see going through the graphs shows you where the 17,973 votes are.

Pause and work out now, who, if anyone, has been elected? Welcome back, did you say Candidate A? Well done if you did.

Their bar is clearly above that horizontal black line, so they take the first seat.

Now remember what I said happened, any excess votes are redistributed, and you can see that here.

So this graph shows the update, after we knew the A was elected, and their excess votes redistributed.

So let me ask you now, who, if anyone, has been elected? Pause and make your decision now.

Welcome back, you should have selected Candidate B, they've been elected, so they take the second seat.

Now we've redistributed Candidate B's excess votes, and what I'd like to know is who, if anyone, has been elected? Pause, and make your choice.

Welcome back, you should have said no one was elected.

Now, you might have gone further and even told me that Candidate F, the person with the least votes will now be removed, and their votes reassigned, so let's do that.

There are now four candidates who are still in the running, who's eliminated now? Pause and make your choice Welcome back, you should have said, of course, Candidate G, because they have the lowest number of votes, and the other three candidates, haven't yet met the quota.

So let's redistribute G's votes.

What happens now? Pause, and have a discussion about this.

Welcome back.

Now there's more that happens here.

Candidate E gets eliminated, did you also know that the C and D will get elected? Why is that? Remember, there are four seats to fill, so C and D have to get elected because we already have A and B, and C and D are the only two candidates left.

Oh, now Sofia raises an interesting question.

"Who decides, which votes are 'excess' votes when a candidate reaches the quota?" And you might have seen, that I just redistributed them for you, but how did I know which votes to pick? Well, there are two main methods that people have for this.

The first is random selection, and that's where the excess votes are randomly selected from all the votes a candidate received.

So, for example, a candidate who receives 1,000 votes, but only needed 600 to win, is going to have 400 of their votes selected at random and then reassigned.

Now, you might be thinking, "That doesn't seem that fair.

What about if those 400 people that are chosen, all had a particular person for second, but the other 600 didn't?" Hmm, you're right, it doesn't seem that fair, and that's why we have a second method, and that's fractional transfers, and that's where the excess number is calculated and divided by the number of votes a candidate received.

So, for example, that candidate who received 1,000 votes, but only needed 600 to win, had 400 in excess, only instead of just choosing 400 to reassign, we reassign all of their votes, but each vote is only worth a fraction of what a normal vote is worth.

In this case, each of the reassigned votes is worth 0.

4.

Can you see why that is? Well, what's 0.

4 times 1,000? That's right, it's 400, which is exactly the number of votes that need to be reassigned.

Let's check you've got that 'cause it was quite a bit of maths there, and it might be unfamiliar to you.

So, the quota is 18,000 votes, and a candidate receives 21,000 votes.

Under random selection, how many of their votes are reassigned? Pause the video and make your choice now.

Welcome back, you should have selected 3,000.

Remember, under random selection, it's only the excess that are reassigned.

What about this time? So still the quota is 18,000 votes, and a candidate receives 21,000, but this time, we're doing fractional transfers.

So in this case, how much is a reassigned vote worth? Pause the video, and work this out now.

Welcome back, so what did you get? That's right, you should have said it was 0.

14, and it does go on for a bit.

Remember, we had to take here what our excess was and divide by our total.

It's time now for your first task.

Question 1, To decide the quota required to be elected, the following formula is used: and I'd like you to calculate the quota please for the following elections.

Question 2, What do you notice about the number of votes required? Pause the video and work this out now.

Question 3, 20,000 votes are cast in an election.

There are three seats available.

Here is a table showing the breakdown of the votes.

So the first column shows you where the first vote went to, and then for the people, for example, that voted for Candidate A, I've shown you the breakdown of the second vote.

For Part A, I'd like you to calculate the quota required to win a seat.

Pause, and do this now.

Part B, based on the first vote, who's already won a seat? Pause, while you work this out.

Welcome back, Part C.

Assign all excess votes from any successful candidate, or candidates to the candidate, or candidates who were the most popular second vote.

Pause and do this now.

And then Part D, who has won the final seat? Pause, and do this now.

Welcome back, time to go through our answers.

So for 1 Part A, you should have got 3,145 votes, for B, 2,516 votes, and for C, 2,097 votes.

So what did you notice about the number of votes required? Well, you should have noticed that as the number of seats available increases, the number of votes required to reach the quota decreases.

In Part 3, the quota required to win a seat is 5,001.

So based on that, Candidates A and B have both won seats, because their vote total is over 5,001.

In Part C, I said to assign all excess votes from any successful candidates, to the candidates who are the most popular second vote.

In this case, Candidate E is receiving votes from both A and B.

Now remember, what they receive is the excess, so you can see the excess column there, so that's 812 and 2,819, and we add that to E's total, to bring them to 4,723.

For Part D, who's won the final seat? Well, we can see its Candidate E, because C would now be eliminate, and when C gets eliminated, E will gain enough votes to go over the quota, and D will not.

It's time now for Part 2, where we're going to look at the Alternative Vote.

With the Alternative Vote, your constituency gets an MP the majority support.

Voters put a number by each candidate, one for their first choice, two for their second, et cetera.

And they could assign a number to every candidate or just some.

If more than half of the voters have the same favourite candidate, that person gets elected.

If no candidate has more than half of the vote, the candidate with the fewest votes is removed and their votes are reassigned, based on the voters' second choice.

This process is repeated until a candidate is elected.

Let's check you've got that.

73,345 votes are cast.

Under the AV system, how many votes does a candidate need in order to win? Pause the video now while you make your choice.

Welcome back, you should have chosen, of course, 36,673.

When you halved 73,345, you would've got 36,672.

5, and remember, we need over half, so we definitely need to round up here.

So, here we have a graph, and I've shown you the breakdown of the vote, and the horizontal line shows the number of votes required to win.

Who's removed from the election and has their votes reassigned? Pause and make your choice now.

You should, of course, said, G, they're removed, because they have the fewest number of votes.

So, I've reassigned the votes, do we have a winner? And if you think we don't, who do you think will be eliminated next? Pause and make your choices now.

Welcome back, you're right, we don't have a winner yet, no one's gone past that line, so Candidate F is now eliminated, and their votes are reassigned.

Who do you think will be eliminated next? Well, it might be either E or D, depending on where F's reassigned votes go because they're the next lowest.

Well, I've reassigned the votes, and then Candidate who was eliminated? So after I reassigned F's votes, who got eliminated? That's right, Candidate D.

they have no votes, so those must have been reassigned.

So who has won the election? That's right, Candidate A has won the election.

It's now time for your final task.

In Question 1, 20,000 votes are cast in an election, and you can see a table showing the breakdown of the votes.

For Part A, I want to know, how many votes are required to win the seat? Pause, while you work this out.

In Part B, since no one has over half the vote, which candidate is eliminated? Pause, while you work this out.

Part C, assign the votes accordingly, and produce an updated table.

And then in Part D, tell me which candidate or candidates have won the election? Justify your answer, please.

Pause and do this now.

Welcome back, let's go through your answers.

In Part A, I asked you how many votes are required to win the seat? And the answer is over 10,000.

You could, of course, have been precise, and said it's 10,001, or more votes, either is acceptable.

Remember, though, we just needed over half.

In Part B, no one has those 10,001 votes or more, so we need to eliminate the candidate with the lowest number of votes, which would be Candidate E.

In Part C, I asked you to assign the votes accordingly and produce an updated table.

So when we get rid of Candidate E and we reassign, based on voters' second choice, what happens? Well, you get this table here, so I've updated it, and you can see the new totals.

It then says, which candidate or candidates can win the election? And you need to justify your answer.

Well, let's see what will happen if C will be eliminated.

Now, that's 1,560 votes to reassign, now, even if all their votes go to Candidate D, Candidate D is still going to have the lowest number of votes out of A, B, and D, which means they will be eliminated next.

Now, depending on where those votes go, either A or B could still win this election.

Well done if you said something similar.

It's now time to sum up what we've looked at today.

So we've looked at two different voting systems. We've looked at the Single Transferable Vote, and that's where the opinions in parliament will match the strength of support in the country.

And then we've also looked at the Alternative Vote, and that's where your constituency gets an MP the majority support.

Now, both of these systems are not the ones that we currently use in our country.

What do you think about these? It might be interesting for you to compare these to how we vote here in our country, and see if you can see the pros and the cons of the different voting systems. Did you see how maths can be used to help analyse the vote, and how we use maths to reassign votes fairly? Well done, you've done a great job today, and I really hope you enjoyed learning how we use maths within our voting systems. I look forward to seeing you for more lessons in the future.