Lesson video

Hi, I'm Mrs. Warehouse, and welcome to today's lesson, which is from the unit Maths in the Workplace.

In this series of lessons, we're exploring how maths is used in different careers.

Let's get started.

By the end of today's lesson, you're going to have looked at some of the ways that maths is used by programmers.

Now, a computer programmer is a person who writes and tests computer programmes.

Our lesson is broken into two parts today.

We're gonna begin by looking at something called algorithms. An algorithm is a set of instructions which can be carried out to solve a problem.

Programmers write algorithms that a computer can follow in order to solve problems. The instructions are followed in order by the computer.

Here is an algorithm for finding the absolute difference between any two numbers.

Step one, take two numbers, a and b, where a is strictly greater than b.

Step two, calculate a, subtract b.

And then step three, the result is the absolute difference.

Jacob and Alex have a go at using this algorithm.

Jacob says, "I used nine and 17 and got the answer negative eight." Alex says, "I used nine and 17 and got the answer eight." Who is right? And how do you know? Pause the video while you work this out now.

Welcome back.

Did you spot that a had to be strictly greater than b? Well, when you calculate, therefore, a subtract b, because a is strictly greater, you should get a positive value, so we should be doing 17 subtract nine, which means that Alex did it right and got the answer eight.

Well done, Alex.

Now, instructions have to be clear for algorithms. Lucas says, "I need some help with my homework.

Laura, can you please take a look at my work?" Oh, lovely Laura says, "Sure thing, Lucas." "Oh, Laura, you said you would help me and I got all my work wrong." Oh dear, Lucas does not look happy.

Laura points out, "You asked me to take a look and I did.

You didn't ask me to help you correct your work." Oh, Laura.

The thing is she's technically right.

Now, of course in reality, Laura would've helped Lucas correct his work because she can infer what Lucas meant from his question.

However, a computer cannot do this and therefore programmers need to be very careful with the instructions they give.

So for example, with Lucas and Laura, Lucas asked Laura to take a look at his work.

Well, as soon as she looked at it, that was the end of it.

He didn't ask her for anything else, so if Laura had been a computer programme, that's all she would've done that.

As humans, though, we can infer what someone means, so Laura knows that take a look at actually means, could you please help me? Now, it's time for your first task.

We're gonna try carrying out some algorithms. So this is algorithm one.

I'm gonna go through the steps with you.

For step one, you take two positive numbers, we're gonna call them A and B.

Step two, draw a table with two columns, put A at the top of the left column and B at the top of the right column.

For step three, in the next row in the left-hand column, write the number that is half of A and you ignore any remainders.

So for example, if the number was 17, half of 17 is 8.

5, but you ignore the remainder, so you would just write eight.

And in the right-hand column, you write the number that is double whatever B is.

For step four, you're going to repeat that until you have a one in the left-hand column.

So in other words, halve whatever's directly above you in column A and double whatever's directly above you in column B.

Then in step five, delete any rows where the left-hand column has an even number in it.

Step six, find the sum of the numbers left in the right-hand column.

So this is our algorithm and we're going to carry this out with different sets of numbers.

So for question one, please carry out that algorithm with the following pairs.

So for part A, I want A to be eight and B to be 10.

Part B, A is 29 and B is 34.

In part C, A is 13 and B is 17.

And then in part D, A is 66 and B is 56.

Once you've done that, I want you to look at your results and see if you can work out.

What does algorithm one do? Now, if you are not sure, come up with your own pairs of values for A and B and try them out to see if you can spot what algorithm one is doing.

Pause the video and carry this out now.

Welcome back.

It's now time for question three.

Question three, I'd like you to write an algorithm for making a drink of squash or a cup of hot chocolate, which is a bit more challenging.

Give your algorithm to a partner and let them try to break your algorithm.

Very similar to the way that Laura didn't do what Lucas intended for her to do.

Is there a way for your partner to misinterpret your instructions? Pause the video now while you work on this.

Welcome back.

How'd you get on? Well, let's start by going through question one.

Here is our algorithm again.

So we were taking our pairs of numbers and we were halving one and doubling the other until we reached a one.

Then we were deleting any rows where the left-hand column had an even number.

So let's look at where A is eight and B is 10.

So you can see here I put A in the left-hand column and I put B in the right-hand column.

I halved in the left-hand column, so eight became four, became two, became one, and I doubled in the right-hand column, so 10 became 20, became 40, became 80.

I then deleted any rows that had even numbers in them for column A.

So that's those three rows were deleted.

This meant that the sum of the values in the right-hand column was just 80.

Well done if you got that right.

For part B, I started with 29 and 34.

Now, remember, when I halve 29, I get 14.

5, but we have to ignore any remainders, so I just write 14 and I carried the sum.

When I delete the even rows, it's just the row with 14 in that gets deleted.

Remember, it's only even in the left-hand column that I care about.

I now sum the values that are left in the right-hand column and I get the total of 986.

For part C, I started with 13 and 17, and at the end of my algorithm, I've got a total of 221.

Part D, I started with 66 and 56 and there were quite a lot of rows here, but when I delete the rows that had even values in the left-hand column and then sum what's left in the right hand column, I get a total of 3,696.

Well done if you've got those all right.

So, what does algorithm one do? Well, the algorithm calculates the product of the two numbers I started with.

In other words, eight times 10 is 80, 29 multiplied by 34 is 986, 13 multiplied by 17 is 221, and 66 multiplied by 56 is 3,696.

In other words, I've got a way to multiply two values that's different to what I might have seen before.

Well done if you got that right.

Now, question three, I asked you to write your own algorithm for making a drink of squash or a cup of hot chocolate.

Did you challenge yourself and go through the cup of hot chocolate? Well done if you did.

So, is there a way for your partner to misinterpret your instructions? Now, there were huge numbers of possibilities here.

Hopefully you've had a lot of fun breaking some algorithms. Here's some examples of ways you could have done this.

So if they've given you the instruction, fill the kettle with water, this could mean putting water until it reaches the very top of the kettle.

Water's absolutely gonna escape, by the way, when that kettle boils up.

There's normally a max line that you have to fill to.

Did they tell you that, though? And equally, where does this water come from? It just says with water.

From where? From the tap? From the toilet? Oh, definitely breaking that algorithm there.

I do not want to have a cup of hot chocolate if that's where you've got your water from.

I really hope you had fun doing this.

It's now time for the second part of our lesson.

We're going to look at logic.

Now, logic is about using reasoning to determine whether something is true or untrue.

This is usually done through a series of statements.

For example, Andeep is taller than Aisha and Aisha is taller than Sarah, so what can you reason about Andeep and Sarah? Pause the video and have a quick chat now.

Welcome back.

Well, you should have been able to reason that Andeep is taller than Sarah.

If he's taller than Aisha and Aisha's taller than Sarah, Andeep must be taller than Sarah.

Now, you must be careful to not infer from a statement.

If today is Saturday, then I will go to the beach.

Think about what this statement tells us about the day of the week and what I might be doing.

Let's have a look.

If today is Saturday, then I will go to the beach.

What can you conclude if, A, today is Saturday, B, today is not Saturday, C, I'm going to the beach today, and D, I did not go to the beach today.

Pause the video and work out what you know if the following statements are true.

Do this now.

Welcome back.

How did you get on? This can be a little tricky, so let's go through this carefully.

If today is Saturday, you can conclude that you will go to the beach.

'Cause our statement said, "If today is Saturday, I will go to the beach," so we do.

Now, if today is not Saturday, what do we know? Well, actually we don't know anything.

I may or may not go to the beach.

My statement at the top just said, "If today is Saturday, I'll go to the beach," what it didn't say was only on Saturdays I go to the beach.

So for part B, I can't actually conclude anything if today's not Saturday, not based off that statement.

In part C, it says, "I am going to the beach today." So, what can I conclude? Well, you might have said, oh, well, if you're going to the beach, it must be a Saturday, but actually that's not true.

If I'm going to the beach today, it may or may not be a Saturday.

Remember, my statement didn't say, I only go to the beach on Saturdays.

In part D, it says, "I did not go to the beach today." Well, then I can conclude today is not Saturday.

Remember, if I didn't go, then it cannot be a Saturday, because if it was a Saturday, then I go to the beach.

So the logic does make sense, but it can be a little tricky when you are first learning about it.

Let's do some practising then on that.

Logic can help us find contradictions.

At Oak National Academy, it is Logic Day.

On Logic Day, all pupils either always tell the truth or they always lie.

In other words, any individual pupil is either telling the truth all day or they're lying all day.

On your way around Oak National Academy, you bump into Aisha and Andeep.

Aisha says to you, "We are both lying today." Is Aisha lying today? What about Andeep? Pause the video while you think about this now.

So, what did you conclude? Is Aisha lying today? What about Andeep? Let's think about this.

If Aisha is telling the truth, then she is lying today.

Well, that's a contradiction, she can't be telling the truth and lying.

Hmm.

Therefore Aisha must be lying, her statement cannot be true.

Now, since we know Aisha is lying, that gives us quite a bit of information about Andeep.

Because Aisha is not telling the truth, Aisha and Andeep cannot both be lying because her statement wasn't true.

So that means Andeep cannot be lying today, if he were, it would make Aisha's statement true and that's not allowed.

Therefore, Andeep is telling the truth.

Quick check that you've got this.

You now meet Alex and Izzy.

Alex says to you, "Exactly one of us is lying today." Is Izzy, therefore, telling the truth? Do you think that's yes, no, or you don't have enough information? Pause the video now while you work on this.

Welcome back.

What did you put? Izzy is not telling the truth today.

If Alex is telling the truth, then Izzy must be lying, because remember, exactly one of the two of them is lying.

Now, if Alex is lying, then there cannot be only one person lying today, because if that was the case, Alex's statement would be true and he wouldn't be a liar.

So if Alex is lying, we cannot have exactly one person lying, therefore Izzy must also be lying.

So actually it doesn't matter whether Alex tells the truth or not today, Izzy must be lying today.

Well done if you reasoned that out.

It's now time for your final task.

For question one, you have the statement, if it is raining, I will use an umbrella.

So, what can you conclude if, in part A, it is raining, part B, it is not raining, part C, I am using an umbrella, and part D, I am not using an umbrella? Pause the video while you work on this now.

Welcome back.

Time for question two.

At Oak National Academy, it is Logic Day.

On Logic Day, all pupils either always tell the truth or they always lie.

You meet Jacob and Jun.

Jacob says, "Exactly one of us is telling the truth today." Is it possible to state whether Jacob is telling the truth or not? Pause the video while you work this out now.

Welcome back.

Question three.

You meet Laura and Lucas now.

Laura says, "I am lying today and Lucas is telling the truth today." What can you deduce, if anything, about Laura and Lucas? Pause the video while you work this out now.

Question four, you meet Sam and Sofia.

Sam says, "One of us is lying and one is telling the truth." Sofia says, "We are both doing the same thing and I am lying." What can you deduce, if anything, about Sam and Sofia? Pause the video and work this out now.

Welcome back.

Let's go through the answers.

So I gave you the statement, if it is raining, I will use an umbrella.

Therefore, if it is raining today, you will use an umbrella.

If it's not raining, then we don't know.

I might use an umbrella when it's not raining, we can't tell.

For part C, I said I am using an umbrella.

We don't know if it's raining or not, but in part D, if I'm not using that umbrella, then it is not raining.

Well done if you've got those all right.

Now, question two.

It's Logic Day at Oak National Academy and we met Jacob and Jun, and Jacob said, "Exactly one of us is telling the truth today." Can we work out if Jacob is telling the truth or not? We can't.

We can work out that Jun must be lying today, but we actually don't know whether Jacob's telling the truth or not.

If Jacob is telling the truth, then Jun is lying, and if Jacob is lying, then Jun's also lying, but we don't know what Jacob is doing.

Well done if you spotted that.

Question three.

I asked you what you could deduce about Laura and Lucas when Laura gives you the statement, "I am lying today and Lucas is telling the truth today." Well, what you can deduce is that they're both lying today.

For Laura to be telling the truth, every part of her statement must be true.

In other words, it has to be true that she's lying and it has to be true that Lucas is telling the truth.

Well, that's a contradiction, so therefore one part being a lie is enough to know that she's lying, and because she's lying, we know Lucas is not telling the truth today.

Part four.

We now met Sam and Sofia.

Sam says, "one of us is lying and one is telling the truth," and Sofia says, "We are both doing the same thing and I am lying." What can you deduce? You can deduce that Sam is telling the truth and Sofia is lying.

Well done if you were able to reason that out.

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

A computer programmer is a person who writes and tests computer programmes.

An algorithm is a set of instructions which can be carried out to solve a problem.

Programmers write algorithms that a computer can follow to solve problems. The instructions should follow logically.

Well done.

I hope you've had a lot of fun today learning some of the ways that programmers use maths in their jobs.

I look forward to seeing you in the future for some more maths.

Goodbye for now.