Lesson video

Hi, I'm Ms Bridgett.

I'm going to be taking you through your next unit on simultaneous equations.

And the first lesson we're going to be looking at expressions involving two variables.

You're going to need a pen.

You're going to need some paper and also take a moment to make sure you clear away any distractions.

Okay.

Let's begin.

Okay.

In a stationary shop, pens cost 40p and rulers cost 20p.

Binh is going into this stationary shop and she's got exactly £2 to spend on pens and rulers.

Exactly £2.

So she could buy four pens and two rulers.

So four pens would cost £1.

60, two rulers would cost 40p.

What other combinations could she buy? Remember, she's got to spend exactly £2 and she's got to buy both pens and rulers.

If you finish that, have a think about what if she doesn't have to spend exactly £2? What if she could spend less than £2? What combinations are there? Pause the video and have a try.

Okay.

Here are some of the answers that I got for exactly £2.

So we talked about four pens and two rulers.

I tried to work systematically.

So then I found three pens and four rulers, two pens and six rulers, one pen and eight rulers.

If we have to spend exactly £2 and we have to buy both pens and rulers there are the only combinations that we can buy.

If we are allowed to spend less than £2, there are a lot more answers that we could have.

Okay.

Now in that scenario, we have two variables.

The number of pens that Binh bought and the number of rulers that Binh bought.

And this could change.

We're going to call p the number of pens that were bought and r the number of rulers that were bought.

Just as a reminder, a pen cost 40p and a ruler cost 20p.

What would the total cost be? If p is equal to one and r is equal to one? What is the total cost if p is equal to two and r is equal to two.

Okay.

Let's think about this first one.

So p is equal to one and r is equal to one.

That means we spending one lot of 40p and one lot of 20p.

Which is 60p altogether.

If p is equal to two.

We've got two lots of 40p, and two lots of 20p, which is 120p altogether.

Okay.

You'll see this next task.

I've got a table, a two way table labelled with p and r.

Across the top I've got different values for p.

Remember p was the number of pens that Binh bought.

P is equal to one, two, three and four.

Down the side, I've got r, r represented the number of rulers that Binh bought and here in the table, I've got r as one, two, three, four, and five.

Now in the previous example, we looked at what happened when p is equal to one and r is equal to one.

And we got a value 60.

We also looked at what happened when p is equal to two and r is equal to two.

And we got a value of 120p.

You'll notice I filled them both in, in pence.

It doesn't really matter whether I use pence or pounds, so long as I've consistence and I've chosen to use pence.

What I would like you to do, is to try to fill in the table for all the different values of p and the different values of r.

While you're doing it, keep your maths eyes open.

What patterns are there in the table? What can you spot? What do you notice? And then try and think about why are these patterns happening? So if you spot a pattern, can you explain why that pattern is happening? Pause the video and have a try.

Okay.

These are the values that I got.

Hopefully, you got the same values.

Now let's have a look at some of the patterns that I spotted.

If you look at that top row, 60, 100 140, 180.

I noticed that every time I moved to the right, my value increased by 40.

And it didn't just happen on that row.

It happened at all the other rows.

So the bottom row, the middle row, every time I moved to the right, my value increases by 40.

If I were to look at the columns.

So if I look at that first column, 60, 80, 100, 120, 140.

The values are increasing by 20.

And then the final pattern I spotted was here in the diagonals.

So 60 to 120 to 180 to 240.

Here, as I move down to the right, my values are increasing by 60.

And that happens whatever diagonal I look at.

So why are these patterns happening? As we increase pay, what's actually happening is we're buying one extra pen.

As we move to the right in the table, we're increasing the number of pens that we buy by one.

Now the extra cost of a pen is 40p.

So every time we move to the right, we're increasing the number of pens that we buy by one and therefore, we're increasing the total cost by 40p.

And that is why that is happening.

As we move down through the rows what's actually happening, is we're increasing the number of rulers that we're buying.

Now, the cost of a ruler, the cost of one extra ruler is 20p.

So every time we move down through those rows, down through that column, we are increasing the total by 20.

When we move diagonally, we're moving one down and one to the right.

What we're doing is we're increasing our shopping basket by one extra pen and one extra ruler.

That's an extra 40 pens and an extra 20 pens.

And that's why our diagonals are increasing by 60.

Now these aren't the only patterns, but these are the patterns that I spotted.

So you might well have spotted something completely different.

Okay.

So in the previous example, p was equal to the number of pens, that Binh had bought and r was equal to the number of rulers that she had bought.

We're going to change this scenario slightly for this very final task.

So this time, I'm going to tell you that Binh bought exactly 30 pens and exactly 20 rulers.

But what I'm not going to tell you is the cost of the pens and the cost of the rulers.

So this time, I'm going to define p as the cost of a pen, and r as the cost of a ruler.

So in the previous example, p was equal to the number of pens, now p is equal to the cost of a pen.

In the previous example, r was equal to the number of rulers, now r is equal to the cost of a ruler.

I'm also going to tell you, that she spends less than a pound.

So she's getting a very good deal.

She's getting an awful lot of stationary for not very much money at all.

I want you to try and think about, what could p have been? And what could r have been? So what was the cost of a pen? What was the cost of the ruler? What could the cost of a pen have been? What could the cost of a ruler have been? How many different solutions are there? Pause the video and have a try.

Okay.

I think we could have had p equal to one and r equal to one.

So 1p for the pen, 1p for the ruler.

You remember I said it's very reasonably priced.

So that would have been £30 on the pens and £20 on the rulers.

Another solution would have been p equals two and r equals one.

So 60p on the pens and 20p on the rulers.

A third solution p is equal to one and r is equal to two.

30p on the pens, 40p on the rulers.

And my final solution is p is equal to one, and r is equal to three.

So 30p on the pens and 60p on the rulers.

That's the end of the lesson today.

So what we've been doing today, is looking at situations where we have more than one variable.

So in this case, our variables were p and r.

And p and r either represented the number of pens and the number of rulers or the cost of pens and the cost of rulers.

So thank you for all of your time and attention today.

And I will see you next lesson.

When we are looking at systems of equations.