Lesson video

Hi there, I'm Mr. Tazzyman.

Today, I'm gonna be teaching you a lesson from a unit that is all about solving problems that feature two unknowns.

You've probably come across these kinds of problems lots of times in the past, but you may not have known that that's what we would categorise them as.

We're gonna be looking at the structures that underlie some of these problems to really help you to understand what's going on mathematically.

So, make sure you're ready to learn.

Let's go for it.

Here's the outcome for today's lesson, then.

By the end, we want you to be able to say, "I can represent problems with two unknowns and solve using an efficient strategy." These are two of the keywords that you are gonna expect to see today.

We've got coefficient and efficient.

I'm gonna say them and I want you to repeat them back to me, so I'll say "My turn," say the word, then I'll say "Your turn," and you can repeat it back.

Ready? My turn.

Coefficient.

Your turn.

My turn.

Efficient.

Your turn.

Here's what each of those words means.

A coefficient is a number used to multiply an unknown value.

To solve something efficiently means to solve a problem quickly whilst also maintaining accuracy.

This is the outline for today's lesson, then.

We're gonna start by representing problems with two unknowns, and then we're gonna look at problems with the same coefficient.

Today, Alex and Sofia are gonna be joining us.

They'll help to explain some of the concepts that you'll be learning about.

They'll give us some hints and tips and they'll discuss the maths on screen.

Hi, Alex.

Hi, Sofia.

You ready? Are you ready? Okay, let's learn.

Alex buys four oranges and five apples for 3 pounds 35 pence.

Sofia buys four oranges and two apples at a cost of 2 pounds 30 pence.

How much does each orange cost? How much does each apple cost? "This seems like quite a complex problem," says Alex.

"Let's represent the problem with a drawing.

This will help us to see the problem.

I drew it like this," says Sofia.

Alex has got four oranges there and five apples, and he's got 3 pounds 35.

Sofia's got four oranges but two apples, and 2 pound 30.

"You bought four oranges and five apples for 3 pound 35 and I bought four oranges and two apples for 2 pounds 30." "That must have taken you quite a while to draw each fruit.

There must be a quicker away." I think Alex is right.

Sofia's drawn the fruit brilliantly there, but it takes quite a while to do.

Is that really that efficient? "We could replace each picture with a bar to represent it." "Yes, that's a good idea," says Alex.

"We could use an o to represent each orange." "And we could use an a to represent each apple." There they go.

"I represented it like this," says Alex.

"I spent 3 pound 35 on four oranges and five apples.

You spent 2 pounds 30 on four oranges and two apples." "That's an even more efficient way of representing it, I think," says Sofia.

"We could have represented this problem as two equations as well.

The four, multiplied by o, represents four lots of oranges." So, let's see what Alex has written here.

Alex's equation is 4 multiplied by o, plus five multiplied by a, is equal to 3 pounds 35.

And Sofia's is 4 multiplied by o, plus 2 multiplied by a, is equal to 2 pounds 30.

"The 5 multiplied by a represents five lots of apples.

In Sofia's equation, the 2 multiplied by a represents the number of apples.

When working algebraically, these equations can be shortened." Hmm.

What do you notice? You can see those two new equations.

What's actually happened though? "A-ha," says Sofia, "The multiplication symbols have been removed." Let's read the new equations now.

We've got 4o plus 5a is equal to 3 pounds 35 and 4o plus 2a is equal to 2 pounds 30.

"4o still means 4 multiplied by o or four lots of oranges.

And 5a still means 5 multiplied by a, which is five lots of apples.

In Sofia's equation, the 2a represents 2 multiplied by a or two times the number of apples.

The number each letter is multiplied by can be known as the coefficient." There's one of our key words.

"In this problem, both equations share a coefficient.

In this case, they have both got four lots of oranges." Okay, let's check your understanding so far.

3 multiplied by b, plus 6 multiplied by a can also be written as a, 3 times 6 plus b times a, b, 3b times 6a, c, 3b plus 6a, or d, b3 plus a6.

Pause the video and decide.

Welcome back.

The answer was C, 3b plus 6a.

3b, well that's a coefficient and an unknown, and that represents 3 multiplied by b, and 6a represents 6 multiplied by a.

Both of those have been added together.

"The coefficient always comes before the letter." It's important to remember that.

"It is a more efficient way of writing a multiplication equation." Okay, let's check your understanding again.

Tick the problem that, when written as a pair of equations, would share the same coefficient.

a, Alex buys three bananas and two kiwis for 2 pounds 50, Sofia buys two bananas and four kiwis for 3 pounds 80.

b, Alex buys five bananas and two kiwis for 3 pounds 10, Sofia buys four bananas and three kiwis for 3 pounds 60.

Or c, Alex buys two bananas and three kiwis for 3 pounds, Sofia buys four bananas and three kiwis for 3 pounds 60.

Okay, pause the video and decide which of those you think would share the same coefficient.

Welcome back.

It was C.

Sofia tells us why.

"2 multiplied by b plus 3 multiplied by k equals 3 pounds.

4 multiplied by b plus 3 multiplied by k equals 3 pounds 60.

They both have three lots of kiwi." So, 3 is the coefficient shared between both of those statements.

Okay, it's time for your first practise task.

Number one, represent each problem by drawing an efficient model.

For number two, you've got to write each problem as a pair of equations using coefficients.

Okay, read each question carefully.

Enjoy.

I'll be back in a little while with some feedback, so pause the video here.

Welcome back.

Here's number one, then.

Alex buys two bananas and three kiwis for 3 pounds.

Sofia buys four bananas and three kiwis for 3 pounds 60.

How much does each banana and kiwi cost? And you can see, we've got a couple of different representations here.

We've got Alex and Sofia represented in bars at the bottom left there.

Alex's bar features three parts which are labelled k and two parts labelled b, and Sofia's bar features three parts labelled k, that's where the coefficient is shared, but there are four parts labelled b, k for kiwi, b for banana.

And then, the totals are written alongside those bars.

On the other side, we've got a slightly different representation in which we've got some circles which feature k and b.

We've got three circles labelled k for the kiwis and four labelled b for bananas.

Sofia's arrow encompasses all of those circles and is labelled with 3 pounds 60, which is what she spent.

Alex's arrow is shorter though, and only encompasses three k and two b, and is 3 pounds.

Let's look at b, then.

Alex and Sofia are throwing ball at two targets.

Alex hits Target A four times and Target B three times.

He scores 50 points.

Sofia hits Target A four times and Target B four times.

She scores 60 points.

How many points did they get for each target? Okay, you were just representing these here.

So again, you've got some representations you can see on screen.

Pause the video here if you want to compare yours with these or if you want to see any other representations that anybody else might have come up with.

Let's look at c, then.

We're gonna do the same thing again here.

We have sweets are sold in big bags and small bags.

In three big bags and four small bags, there are 345 sweets.

In three big bags and six small bags, there are 405 sweets.

How many sweets are there in both bag sizes? Here are two representations shown again.

Pause the video and compare yours to these and with anybody else's that they might have come up with.

Okay, let's look at number two, then.

Now, we had to write each problem as a pair of equations using coefficients.

We had the same problems here.

Alex's equation might have read 2b plus 3k is equal to 3 pounds.

Sofia's was 4b plus 3k is equal to 3 pounds 60.

Okay, let's look at b.

Alex was 4A plus 3B is equal to 50.

Sofia was 4A plus 4B is equal to 60.

And let's look at c.

3b plus 4s is equal to 345.

3b plus 6s is equal to 405.

Okay, let's move on to the second part of the lesson, then.

Problems with the same coefficient.

Have a look at this representation and think, what do you notice about Alex's bar and Sofia's bar? Hmm.

Well, Alex says, "We bought the same number of oranges." "You bought three more apples than I did." "I've spotted something.

The difference in cost is equal to three apples." Can you see that? Alex has three more parts labelled a than Sofia.

"I have three more apples than you and so the extra money I spent must be the value of three apples." 3 pound 35 subtract 2 pound 30 is equal to 1 pound 5 pence.

So, we know that the difference represented by the arrow is 1 pounds 5 pence.

What does each part of the bar model represent? Hmm.

"The o represents the cost of one orange." "The a represents the cost of one apple." "The 3 pounds 35 represents the total money spent by me." "The 2 pounds 30 represents the total money that I spent." "And finally, the 1 pound 5 represents the difference in money spent, which is equal to three apples.

If we know three apples is equal to 1 pounds 5 pence, we can now work out the cost of one apple." 1 pound 5 pence divided by 3 is equal to 35 p, so we know that each apple costs 35 p.

We can amend the model.

The cost of five apples is five lots of 35 p.

5 multiplied by 35 p is equal to 1 pound 75.

"If we subtract the cost of five apples from the amount I spent, we could find out the cost of four oranges." 3 pounds 35 subtract 1 pound 75 is equal to 1 pound 60.

"So, four oranges cost 1 pound 60." "We can divide 1 pound 60 by four to find the cost of each orange." 1 pound 60 divided by four is equal to 40 pence.

"So, each orange costs 40 pence.

I can amend the model.

We could have also used my bar model to help calculate the costs.

We knew that each apple costs 35 pence.

So, two apples is equal to 70 pence." 2 multiplied by 35 p equals 70 pence.

"Now, subtract the cost of the apples from what you spent." 2 pound 30 subtract 70 p is equal to 1 pound 60.

"As we know, that means four oranges cost 1 pound 60, so we can divide 1 pound 60 by four to find the cost of one orange." 1 pound 60 divided by 4 is equal to 40 pence.

"So, each apple costs 35 pence and each orange costs 40 pence." Okay, time for you to have a go.

Solve this problem.

Alex buys three apricots and two bananas for 3 pounds 30.

Sofia buys three apricots and four bananas for 3 pounds 90.

How much does each apricot and banana cost? I'd advise using a bar model here and do seek out the coefficient that's the same across both.

Okay, pause the video and give it a go.

Welcome back.

Here's what you might have done for Alex's bar.

We've got three parts labelled a and two parts labelled b, and the total of 3 pound 30.

Now, let's compare that with Sofia's, who has three parts labelled a and four parts labelled b at a cost of 3 pounds 90.

3 pound 90 subtract 3 pound 30 is equal to 60 p.

That's the difference between what they spent.

That means that 60 p is worth the same as two lots of bananas.

60 p divided by two is equal to 30 p, so you can see the bar model's been labelled 30 for each of the bananas.

3 pound 30 subtract 60 p is equal to 2 pound 70, so we know that three apricots cost 2 pound 70.

2 pound 70 divided by 3 is equal to 90 pence.

An apricot is equal to 90 pence.

Okay, it is time for your second practise task.

You have got to solve each of the problems. a, Alex buys two bananas and three kiwis for 3 pounds.

Sofia buys four bananas and three kiwis for 3 pounds 60.

How much does each banana and kiwi cost? For b, Alex and Sofia are throwing a ball at two targets.

Alex hits Target A four times and Target B three times.

He scores 50 points.

Sofia hits Target A four times and Target B four times.

She scores 60 points.

How many points did they get for each target? For c, sweets are sold in big bags and small bags.

In three big bags and four small bags, there are 345 sweets.

In three big bags and six small bags, there are 405 sweets How many sweets are there in both bag sizes? Now, these questions may seem very familiar and that's because they are the same as the questions you faced in the first practise task.

But this time, you've actually got to finish solving them rather than representing them using a model or an equation.

Pause the video here and have a go.

Good luck.

Welcome back.

Let's look at a, then.

We drawn out this model in the first practise task.

What you might have noticed, though, was that the difference between the amount spent by Alex and Sofia was 60 pence.

60 pence divided by 2 is equal to 30 pence.

That meant every banana was worth 30 pence.

3 pounds subtract 60 pence is equal to 2 pound 40, so three kiwis were worth 2 pounds 40.

That meant that if you use division, you could see that one kiwi was worth 80 pence.

Okay, let's look at b, then.

Here's a bar for Alex and here's a bar for Sofia.

The difference between the two, well, that's equal to 10 because 60 subtract 50 is equal to 10.

That means that b is equal to 10.

50 subtract 30 is equal to 20, so four lots of a are equal to 20.

That means that one lot of a is equal to 5 Let's look at c, then.

Here's the first bar.

In three big bags and four small bags, there are 345 sweets.

Here's the second bar.

In three big bags and six small bags, there are 405 sweets.

There is a difference of two small bags between them.

Also, there's a difference of 60.

So, two small bags are worth 60.

60 divided by 2 is equal to 30, so the small bags are each worth 30.

We know that there are four small bags in that first bar worth 120, and 345 subtract 120 is equal to 225.

So now, we know that three big bags are worth 225 sweets.

225 divided by 3 is equal to 75.

B is equal to 75.

A big bag contains 75 sweets and a small bag contains 30 sweets.

Okay, let's summarise today's learning, then.

You can represent problems with two unknowns using pictures or other models.

For example, a bar model.

It is more efficient to draw a model of the problem than to illustrate the problem through detailed drawings.

You can represent problems with two unknowns using two separate equations.

These equations can use a coefficient to represent the multiplication of an unknown value.

Hope you enjoyed that.

My name's Mr. Tazzyman.

Maybe I'll see you again soon on another maths lesson.

Bye for now.