Lesson video

Hi, thanks for choosing to learn with me today.

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

Let's get started.

In today's lesson on elections, we're going to learn how members of parliament are elected.

Now on the screen, you can see the keywords that are going to be used in our lesson today.

And in fact, it's more of a phrase this time.

First Past the Post, or FPTP.

It's the name for the electoral system that we use to elect MPs to Westminster.

Our lesson has two parts.

We're going to begin with the first part on Understanding First Past the Post.

Now First Past the Post is the name for the electoral system used to elect MPs, or members of parliament, to Westminster.

The UK is divided into 650 constituencies.

Each constituency votes to elect an MP.

Each voter gets to vote for one candidate.

And the candidate with the most votes wins.

It doesn't matter, though, how many votes they win by.

So, quick check.

How many candidates can a voter vote for? Is it A, none? B, one? Or C, two? Now remember, we're talking about electing MPs here to Westminster.

So, pause the video now while you make your choice.

Welcome back.

Did you go for one? If you did, you're correct.

Well done.

Now, there are some situations where you might be voting for two or even three, four, five candidates.

One such situation could be voting in your local council elections where there's more than one vacancy.

That's not the particular thing we're thinking about right now.

Let's see if we can model this.

The pupils at Oak National Academy are voting for their form captain, and there are three pupils standing for the position.

We have Izzy, Lucas, and Sam.

Now the other seven pupils vote for who they want to be form captain.

Now using the First Past the Post system, they can each cast one vote only.

Here are the results.

Izzy got two votes, Lucas got three votes, and Sam got two votes.

So quick check.

Who won the election? And how do you know? So is it A, Izzy with two votes? B, Lucas with three votes? Or C, Sam with two votes? Pause the video now and make your choice.

Welcome back.

You should have said it was Lucas.

But how did you know? Well, you knew because he had three votes, which is more than any of the other candidates.

Using the system First Past the Post, it is possible to know who has won an election before all the votes have been counted.

Hmm.

Can you think why that might be the case? Let's look at an example.

There are four candidates standing in an election, and there are 100 people voting.

Remember, we're just modelling this so I've chosen a nice number of people voting.

So a nice round 100, and I've gone for four candidates.

Now candidate A wins the election.

How many votes might they have got? So, what do you think? Pause the video now and chat to the person next to you, or have a think.

Welcome back.

Well, how many votes could they have received? Well, they could have received all the votes.

They could have received 97 votes, and the other candidates get one each.

In fact, there are loads of possible answers here.

So well done.

As long as you've said that candidate A gets more votes than anyone else, you're right.

But what is the smallest number of votes that A could have received? Bear in mind, they did win the election.

Hmm.

It's a slightly harder question.

So, the smallest number of votes A could have received? Well, let's think about it like this.

Let's imagine that each candidate got exactly the same number of votes.

So if everyone received 25 votes, it's a four-way tie.

Now remember, in order to win, A only needs one more vote than the next closest candidate.

Well, let's shift those bars a bit then, shall we? Here we go.

26 votes is the smallest possible number of votes that A could have received.

Any less than that, and A would not have won the election.

Let's check you've got that.

There are now 500 people voting in an election and there are three candidates, A, B, and C.

What is the smallest number of votes that C could have received and still won the election? Pause the video while you either discuss this with someone or have a think for yourself.

Welcome back.

So what is the smallest number of votes that C could have received and still win the election? Well, if I start by dividing 500 by three, that would give me three equal amounts of 166 and two-thirds of a vote.

You can see the six is recurring there.

Hmm.

Okay, well, that doesn't work because we need to have a whole number of votes.

And we do want C to have more so let's move C up a little bit and say C receives 167 votes.

Well, what's left for A and B? Well, if A and B split the remaining votes evenly.

Oh, they can't do that because there's an odd number left.

Okay, well, we'll split it so that one of them receives 167 votes and the other receives 166.

Hang on, that means that C has got the same number of votes as one of the other two candidates.

That's not enough for them to win, therefore.

So we'll now move C up one more vote and say that C receives 168 votes.

And that means that both A and B would receive 166 each.

It's now time for your first task.

In question one, we have 200 people are voting in an election and there are four candidates, A, B, C, and D, that are standing in the election.

For part A, given that candidate B won the election, suggest a possible number of votes that each candidate received.

And then in part B, given that candidate B won the election, what is the smallest number of votes that they could have received? Pause now while you work this out.

Pause the video now while you work this out.

For question two, 400 people are now voting in the election and there are four candidates, A, B, C, D, that are standing.

For part A, given that candidate D won the election, suggest a possible number of votes that each candidate received.

And in part B, given that candidate D won the election, what is the smallest number of votes that candidate D could have received? Pause the video now while you work this out.

Question three, n, where n is a multiple of four, people are voting in an election.

There are four candidates, A, B, C, and D, standing in this election.

Part A, given that candidate A won the election, write a general rule for the smallest number of votes candidate A could have received.

And in part B, write an expression for the proportion of the total vote that candidate A received.

Pause the video while you work this out now.

Welcome back.

Let's go through our answers.

So, for question one, remember, there were 200 people voting and four candidates.

In part A, you were told that candidate B won the election, so suggest a possible number of votes that each candidate received.

Now there were many possibilities here.

However, even if you've got something different to the person next to you, or to someone else, your solution should have certain things in common with theirs, and that's that candidate B has to have the highest number of votes, and the number of votes cast should sum to 200.

In part B, remember, candidate B won the election, so what's the smallest number of votes they could have received? Well, they could have received 51 votes.

The other candidates, therefore, would've received 50, 50, and 49, or something like that.

More importantly, they didn't go past that 51.

In question two, again, candidate D has won the election, but you need to suggest a possible number of votes that each candidate received.

And this time, there were 400 people voting.

Again, many possibilities, but candidate D must have the highest number of votes and the number of votes cast should sum to 400.

So, given that candidate D won the election, what's the smallest number of votes they could have received? Well, that's 101.

And I've given a possible breakdown of the other votes here that they could have received 100, 100, and 99, just showing that that's the smallest number candidate D could have received and won.

Now in question three, we began to generalise, because I don't about you, but it would be really handy to have a formula to work out the smallest number of votes needed to win such that I don't have to work it out for every new scenario.

So, we knew that we started with, given we were looking at multiples of four.

So keeping to what we've looked at so far, I know that I can divide by four and then add one.

In other words, split the vote fairly between my four candidates and then add one to make sure that I definitely win.

And that means one of the other candidates would've had their vote total decreased by one.

In part B, I said to write an expression for the proportion of the total vote that candidate A received.

Well, we take the number of votes that candidate A got, so n over four plus one and then divide by n.

Remember that n was the total number of votes cast.

Well done if you got that right.

If you've got something that's equivalent to the expression that you can see here on the screen, then that's perfect.

It's now time for the second part of our lesson, which is Limitations of First Past the Post.

So earlier, we saw the results of a form captain election at Oak National Academy.

Remember, Izzy had two votes and Sam had two votes, but Lucas won with his three votes.

Now what percentage of the total vote did Lucas receive? Well, we can work that out by saying he received three votes out of the seven that were cast.

And then we want to know what that is as a percentage, so we multiply by a hundred.

Approximately 43% of the vote went to Lucas.

"Wait a minute!" says Izzy.

"You didn't get over 50% of the vote, so why have you won?" Well, now that is true that Lucas didn't get over 50% of the vote, but as Lucas points out, he got more votes than either Izzy or Sam, so he won.

In the First Past the Post system, it is possible for a candidate to be elected even though more people have voted against them, rather than for them.

So a quick check that you've got that.

In the First Past the Post system, it is possible for a candidate to be elected even though more people have voted against them, rather than for them.

Is that true or false? And justify your answer with a reason.

Pause the video and work this out now.

Welcome back.

You should have said that it's true.

A candidate wins the election, remember, if they receive more votes than any other candidate.

So what we're going to do now is consider a model of a general election.

There are four parties, A, B, C, and D, and three constituencies that we are considering.

Now this task is going to look at what happens in each constituency and how this affects the overall outcome.

So, for the first question, we're going to consider the first constituency.

And what you can see on the screen is a breakdown of the vote for this constituency.

So Party A received 5,772 votes.

Party B received 6,889 votes.

Party C received 3,424 votes.

And Party D received 789 votes.

For part A, I want to know which party won the election for this constituency? And then in part B, what percentage of the total vote did each party receive? And you should round to the nearest percent, please.

Pause the video and work this out now.

Welcome back.

Let's look at the second constituency.

Again, on the screen, you can see a table with the breakdown of the vote.

Party A received 4,930 votes.

Party B received 5,381 votes.

Party C received 4,729 votes.

And Party D received 1,291 votes.

So, for part A, which party won the election in this constituency? And then in part B, what percentage of the total vote did each party receive? And again, please round to the nearest percent.

Pause the video and work this out now.

And finally, constituency three.

Here's the breakdown of the vote.

Party A received 21,953 votes.

Party B received 7,116 votes.

Party C received 4,581 votes.

And Party D received 2,093 votes.

So, for part A, which party won the election for this constituency? And then in part B, what percentage of the total vote did each party receive? Please round to the nearest percent.

Pause the video and work this out now.

Welcome back.

Final question.

By considering your answers to questions one to three, please answer the following.

So for part A, which party won the election in the most constituencies? And then in part B, by considering the total number of votes cast across the three constituencies, what percentage of the vote did each party win? And then write a paragraph to explain what this might mean.

Pause the video and do this now.

Welcome back.

Let's go through our answers.

So for the first constituency, we can see here that Party B won the election, as they received more votes than any other party.

Part B, I asked you what percentage of the total vote each party received.

Party A received 34% of the vote.

Party B received 41% of the vote.

Party C received 20% of the vote.

And Party D got 5% of the vote.

Remember, I asked you to round to the nearest percent.

So if you didn't do that, check that your answers round to the percentages you can see on the screen now.

Question two.

So for constituency two, which party won the election here? Again, it was Party B because they received more votes than any other party.

For part B, what percentage of the total vote did each party receive? A got 30%.

B was 33%.

C, 29%.

And D, 8%.

Remember, we're rounding to the nearest percent again.

Now, the final constituency.

Which party won the election here? Well, it was Party A.

They received a lot more votes than the next closest party, which was B.

So what percentage of the vote did they receive? Well, Party A got 61% of the vote.

B got 20% of the vote.

C got 13% of the vote.

And D got 6% of the vote.

Now question four, I asked you to consider your answer to questions one to three when you were answering these.

So which party won the election in the most constituencies? Well, that was Party B.

Remember, they won two out of three.

Let's now consider the total number of votes cast across the three constituencies.

What percentage of the vote did each party win? And you had to write a paragraph explaining what this means.

Well, in total, there were 68,948 votes cast.

And you can see the approximate percentages below.

Bear in mind that I've rounded here to the nearest percent.

So A got roughly 47% of the vote.

B got 28%.

C, 18%.

And D, 6%.

Did those figures strike you as being a little odd, given what you wrote for part A? Well, let's see if you explained it well in your paragraph.

So despite Party A receiving the most votes across the three constituencies, Party B still won more individual constituencies.

So Party A may be more popular, but it doesn't have as many elected members as Party B does.

And this can be the case in real life too.

So our model pretty accurately reflected what can happen when we're electing MPs.

Let's sum up what we've done today.

The First Past the Post system is the name for the electoral system used to elect MPs to Westminster.

The number of MPs a party has in parliament rarely matches their popularity.

And we saw that in our model in our second task.

Now, fewer people can vote for a party, but they can still win more seats.

And we saw that, didn't we, with Party B? Well done.

You've done a great job today and I hope you've enjoyed learning about how the First Past the Post system works, and how we can use maths to analyse the vote breakdown, and whether or not a party's popularity matches how many MPs they can get elected.

I look forward to seeing you for another lesson in the future.