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 of the lesson today then.

By the end, we want you to be able to say, "I can represent the structure of a problem with two unknowns by drawing a model." Here are the key words, systematically and efficient.

I'll say them and I want you to repeat them back to me.

I'll say "my turn," say the word, and then I'll say "your turn," and you can say it back.

Ready? My turn, systematically.

Your turn.

My turn, efficient.

Your turn.

Here's what those words mean.

Working systematically means to solve a problem in an ordered and methodical way to ensure all outcomes are recorded.

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

Here's the outline for today's lesson.

First of all, we're gonna look at analysing some strategies for finding solutions.

Then we're gonna look at drawing bar models.

We are lucky enough to have five friends joining us today.

Andeep, Jacob, Sofia, Aisha, and Alex.

They're all gonna be coming up with some different strategies to solve problems that we're gonna analyse and compare.

Hey you five, are you ready to start? Are you ready to start? Let's go for it.

Year 6 have earned 200 stars.

The stars are either gold or silver.

They have 30 more gold stars than silver stars.

How many are gold? So we don't know how many gold stars there are, nor do we know how many silver stars there are.

That means we have two unknowns.

How many solutions do you think there will be? We know how many gold stars there are relative to the silver stars, so I think there's only one solution.

Let's see how everyone else has tried to solve it.

Here's Sofia.

This is her strategy.

What do you think of it? Is she correct? Let's see.

She says, "I know that half of 200 is equal to 100.

Then we can just add the extra 30 gold stars, so I think there are 130 gold stars." Half of 200 is equal to 100.

100 added to 30 is 130.

130 stars of gold.

That's her jottings.

What do you think? Okay, well here's Aisha's strategy.

Is she correct, I wonder.

She says, "We need two numbers that add together to make 200.

G is equal to 130." She's written G to represent gold.

"And S is equal to 70." She's written S to represent silver.

But she can see that that wouldn't work because there should be 30 more gold stars and silver stars, but in this version, there are 60 more.

Hmm.

"What about if they're both 100, would that work?" No.

"What about if gold is 105 and silver is 95?" No.

"What about if gold was 110 and silver was 90?" Still not right.

"What about if gold was 111 and silver was 89?" No, still not right.

"What about if gold was 125 and silver was 75?" Still not right.

Aisha's used trial and error, but she doesn't seem to have come up with any decent solutions yet.

Aisha says, "I can't seem to find two numbers where there are 30 more gold stars than silver though." Okay, let's have a look at the next person's strategy.

We've got Alex.

"There are gold and silver stars," says Alex, and he starts to draw out some bars.

"There are 30 more gold stars than silver." So he draws an extra 30 onto the gold bar.

Altogether they sum to 200 stars.

200 subtract 30 is equal to 170.

170 divided by 2 is equal to 85.

So he says that there are 85 gold stars and 85 silver stars.

Hmm.

There are 85 gold stars altogether.

Okay, well let's see Andeep strategy then.

Is he correct? He's drawn out a table and he's filled in the numbers.

We've got gold, silver, and the difference between the two.

"I started with 150 and 50 as these sum to 200.

I then just kept adjusting the numbers until there was a difference of 30." Ah, good thinking, Andeep.

There are 115 gold stars.

Let's see what Jacob did.

"I know there are 200 stars altogether." So he draws a bar out with 200 in.

"And there are more gold stars than silver stars." Gold and silver bars have been drawn in, but you can see that he's drawn a larger bar for gold than he has for silver, just to make sure he understands that there is a difference in their value.

"I'm not sure I have enough information to solve this here." Hmm, he's given up.

Well, Andeep said, "We used different approaches, and we've got some different answers too.

We can't all be correct." So let's check your understanding.

You need to tick the strategy that gives the correct answer.

Pause the video and decide who you think has come up with the best strategy that gives the correct answer.

Welcome back.

Well, Andeep strategy gave the correct answer.

115 gold stars.

That's 30 more than 85 silver stars.

So a difference of 30, but they still total 200.

Although Sofia's strategy was wrong, we need to have another look at it.

We can decide what is positive about Sofia's strategy.

How could Sofia's strategy be corrected? She said half of 200 is equal to 100, 100 plus 30 is equal to 130, so 130 stars are gold.

She's close, but we need to make some changes.

Andeep says "Sofia recognises that the number of gold stars is 30 more than the number of silver stars.

So she starts by splitting the stars into two equal groups of 100 and then adding on the 30." However, if there are 130 gold stars and 100 silver stars, this would equate to 230 stars altogether, which is too many.

So she should subtract the 30 extra stars from 200, which leaves 170.

These can then be divided equally.

What's positive about Aisha's strategy? How could Aisha's strategy be corrected? Aisha has recognised that the sum of the stars must add to 200.

However, she has not used trial and improvement.

She's trying random numbers with no real plan.

She could start at 100 stars each and then increase the difference systematically to help find the correct answer.

What's positive about Andeep strategy? How could Andeep strategy be corrected? "Well, Andeep has found the correct answer by using trial and improvement.

However, he has been more systematic and kept a tally of the difference throughout.

His strategy would've taken quite some time to work out the answer." I agree, Jacob.

He's written six rows in a nice neat table.

That's quite time consuming.

What's positive about Jacob's strategy? Well, Andeep says "Jacob has drawn a bar model which recognises that the sum of the stars is 200, and there are more gold stars than silver stars.

However, Jacob's model doesn't include the difference of 30 stars between gold and silver.

He would need to correct his bar model to show that the number of gold stars is the same as the stars with an extra 30." Okay, let's check your understanding then.

What does Alex need to do to find the correct answer? Here's his workings.

What do you think? Pause the video and have a go.

Welcome back.

What did you think? What could Alex have done to find the correct answer? Andeep says, "Alex needs to add the additional 30 stars to the 85 stars to find the total amount of gold stars, which is 115." He nearly got there, Alex, he just missed that last vital step by adding on the difference between the two values.

It's time for your first practise task then.

Explain why you agree or disagree with each statement.

Andeep has used the best strategy because they are the only one to get the right answer.

Number two.

Sofia is the only one who has used an incorrect approach.

Number three.

If Aisha hadn't given up, she would've got the right answer.

Number four.

The bar models by Alex and Jacob show the same information.

Number five.

Aisha and Andeep have used the same strategy.

Number six.

Andeep is the only one who has shown the difference between the number of gold stars and the number of silver stars.

Pause the video here and have a go at those.

Good luck.

Welcome back.

Let's go through these then.

There were some tricky ones here, and hopefully it elicited some good discussion between you and anybody else who you might be working with.

Okay, for number one, although Andeep got the correct answer and work systematically, his strategy would've taken quite some time, and mathematicians always look for more efficient strategies where they can.

In other words, he could have picked something that was quicker and still got the correct answer.

That would be more efficient.

For number two, if Sofia had also halved the 30, adding 15 to the 100 gold stars and subtracting 15 from the silver stars, she would've got the correct answer, so it was not a completely incorrect approach.

For number three, this is not absolutely true.

Aisha may have got there eventually, but it may have taken a long time if she continued to not work systematically.

She was plucking numbers from everywhere.

She didn't really have a clear plan.

And yes, she could have stumbled upon the correct answer, but she might have just kept plucking numbers forever.

Number four then, Jacob's model is missing the information which states that there are 30 more gold stars than silver stars, so this is not true.

Number five, their strategy is similar.

However, Andeep is more systematic, choosing numbers carefully to help solve the problem rather than choosing random numbers.

Number six then, this is not true.

Alex's strategy also clearly shows a difference of 30 between the gold and silver stars, although he just did not add these on to the final number of gold stars.

Okay, I'd imagine that some of those statements have caused quite a lot of discussion points.

You might wanna pause the video now to finish your discussions.

Let's move on to the second part of the lesson then, drawing bar models.

Which strategy is the most efficient? Alex gets 115 gold stars using this strategy involving bar models and some calculations.

And Andeep draws his table out, and he gets 115 gold stars.

We know that both of 'em have the correct answer, but which of these two strategies do you consider to be the most efficient? Hmm.

Alex's strategy shows the structure of the problem.

Andeep's strategy uses trial and improvement.

Mathematicians try to use elegant strategies that apply the structure of the problem being solved.

So, Alex's strategy may be seen to be more elegant as the bar models expose the structure.

Okay, it's your turn then.

What's the same and what's different about these bar models? Pause the video and have a go at thinking about that.

Welcome back.

Andeep says "The bar models show that the gold stars are equivalent to the silver stars plus 30 more stars." Jacob says "The whole is shown in different places.

The bar model on the left shows the whole to the side, whereas the bar model on the right shows the whole at the top as a bar." Okay, ready to move on? Let's see how these bar models can be constructed to represent the problem.

Andeep says, "We know that the gold stars are the same as the silver stars except where there are an extra 30 stars." So we've got the gold and silver bar, but the gold has an extra 30 attached to it.

We know that both the gold and silver bars sum together to a total of 200 stars.

So we've got our total 200.

Subtract 30, is equal to 170.

Now we can divide the remaining stars equally between the gold and silver.

170 divided by two is equal to 85.

So 85 can be written as a label for each of those bars because now they're equal having subtracted the 30.

Finally, add the 85 stars with the extra 30 stars, and there are 115 gold stars altogether.

Okay, that's the first type of bar model used.

Let's move on to looking at the second type now.

Andeep says "We know that there are 200 stars altogether." So we put a bar with 200 on.

"And there are more gold stars than silver stars." Gold and silver have been labelled and put as bars underneath the 200.

Altogether, they total that 200.

"We know that the gold bar is equal to the silver bar and an extra 30 stars." So now we can see that we've got two bars labelled silver.

The first bar labelled silver is next to 30 stars, because silver plus 30 is equal to gold, so we can say that 200 stars are equal to two lots of silver stars and an extra 30 stars.

Two multiplied by an unknown, added to 30, is equal to 200.

You can now subtract the 30 from the 200 and divide the remaining stars by two.

So two multiplied by an unknown, is equal to 200 subtract 30.

Two multiplied by an unknown, is equal to 170.

Two multiplied by 85 equals 170.

So now we can relabel those bars at the bottom as 85.

That means there are 85 silver stars.

Now we can work out the number of gold stars by adding together the 85 and the 30.

85 plus 30 is equal to 115.

There are 115 gold stars.

Okay, your turn.

Draw a bar model to represent the following problem.

In a Premier League record, Arsenal went unbeaten for 49 games.

They won 23 games more than they drew.

How many games did they win? Pause the video here and have a go at that problem.

Welcome back.

There's a bar model to show this unbeaten record.

We can see that we've got win and draw.

The win part is equal to the draw part plus an extra 23.

And in total, we have 49 matches.

You could also have drawn it in this way.

With 49 as the total at the top, and bars of win and draw, where the win bar was 23 greater than the draw bar.

Underneath that, you then end up with two draw bars of equal length and 23.

Okay, it's time for your second practise task then.

You've got to draw bar models to represent each problem.

Number one, the combined price of a melon and a bag of carrots is 4 pounds.

The melon costs 1.

40 pounds more than the carrots.

How much does the melon cost? For number two, Jun has 80 stickers.

He has 30 more non-shiny stickers than shiny stickers.

How many non-shiny stickers does he have? Number three, a bakery makes 35 doughnuts.

They have five more jam doughnuts than custard doughnuts.

How many of each doughnut do they make? Number four, a bakery makes doughnuts.

It makes five more jam doughnuts than custard doughnuts.

If they make 15 jam doughnuts, how many custard and how many altogether? Okay, pause the video here and have a go at drawing bar models to represent each of those problems. Good luck.

Welcome back.

Let's do some marking then.

You'll need to compare your bar models to the ones on screen.

You might have chosen different types of bar model to draw each time.

Here's the bar models for the first one.

We've got melon and carrots on the left there.

The melon bar and the carrot bar is equal except the melon bar also has 1.

40 pounds added onto it.

And you can see the total has been labelled as 4 pounds.

On the right-hand side, we had 4 pounds at the top, then the second row had melon and carrots, with melon being the greater bar.

On the third row, there were two lots of carrots and there was a 1.

40 pounds bar as well.

Pause the video here if you need to compare what you've drawn to what's on screen more closely.

Let's look at number two then.

Similar sort of question in terms of structure.

On the left, you can see we had non-shiny and shiny.

The non-shiny was a bar of equal length to the shiny, except it had 30 extra.

The total was labelled 80.

And on the right, you can see that the total bar at the top was 80.

Then we had non-shiny and shiny bars where the non-shiny was a greater proportion of the total.

And then underneath that in the third row, we had two lots of shiny and a 30 in between.

Again, pause the video so you can compare what you've drawn with what's on screen.

Here's number three then.

In this one, again, it was a similar sort of structure.

We had jam and custard doughnuts with equal bar lengths, except the jam had five extra.

In total there were 35.

And on the right-hand side, again, we had a total bar of 35 at the top.

The second row was two bars, one labelled jam, one labelled custard, and the jam was a greater proportion.

Then in the third row, we had two bars with custard on and five as well.

Again, pause the video so you can compare yours to these.

Okay, let's do number four then.

This had a slight difference to it.

For that left hand bar model, you can see that we had jam and custard with equal bars, although jam had the extra five difference on as well.

But you can see that the total was the total number of jam doughnuts rather than the total of jam and custard, and that was 15.

On the right-hand side, we had the total we didn't know yet.

That was the unknown, so there's a question mark in there.

Then we had 15, that was the number of jam doughnuts, we knew that from the question, and we had a bar representing custard.

And at the bottom, we had two lots of custard and we had five in the middle.

Okay, pause the video if you need to compare your bar models with these.

It's time to summarise the lesson then.

It is often more efficient to solve a problem by identifying the structure of a problem.

You can often represent the structure of a problem using a bar model.

Whilst trial and improvement can find the correct solution, it can often be more time consuming and therefore less efficient a strategy.

My name's Mr. Tazzyman, I hope you enjoy doing the bar models.

Maybe I'll see you again soon.

Bye for now.