Lesson video

Hi, my name is Mrs. Wheelhouse.

Welcome to today's lesson on checking, understanding of percentages from the unit, understanding multiplicative relationships, percentages and proportionality.

I really hope you enjoy today's lesson.

Let's get started.

The outcome of today's lesson is that you'll be able to manipulate percentages of an amount without a representation.

Now remember we are checking in this lesson, so it may be very familiar with this and that's brilliant if you are, but if you want a little bit more practise, then this lesson is for you.

Let's get going.

We're gonna start just by considering a key word we're using in our lesson today, and that's the word equivalent.

Now, two fractions are equivalent if they have the same value.

So for example, a half is equivalent to two quarters or two over four, and what we mean by that is they have the same value.

You can see some other examples on the screen right now, so feel free to pause if you wish to consider them.

Our lesson today is broken into two parts and the first part is on percentages, fractions and decimals.

Let's get going.

Now percent comes from the Latin phrase per centum, and that means by the hundred.

Now a hundred square is a really nice way to show a percentage.

You can see a hundred square on the screen right now.

The total would represent 100%.

So what do you think 1% would look like? That's right.

I'd only need to fill in one square to represent 1%.

Now, did it have to be that particular square I shaded in? What do you think? You're right.

It doesn't matter what square I filled in, as long as I'm only filling in one square because one square is what represents our 1%.

Remember, 1% simply means it is one part per a hundred or one out of a hundred or 100th, percentages, fractions, ratio and decimals all show proportions and as such we can interchange between them.

You can see my a hundred square on the screen and I've shaded in some of the squares.

I'm not gonna tell you how many I think you can probably see.

What do you think I'm representing here? And we're thinking about this in terms of percentages.

So what percentage of the a hundred square you can see is shaded? That's right, 50% because I have shaded 50 of the squares out of the hundred that you can see.

This one should be really straightforward.

Then what do you think this is as a fraction? Can you give a better answer than you might have thought I was giving you as an answer just a second ago.

What might I mean by that? Right? Brilliant, if you said, I think this is a fraction is 50 hundredths, but did you write that in a different way? Remember our keyword today is equivalent.

So did you write an equivalent fraction to this? In other words, it still has the same value.

Well, you could have done by saying I can take out a factor of 50 from both the numerator and denominator.

In other words, 50 can be written as one multiplied by 50 and a hundred can be written as two multiplied by 50.

I'm writing it as a half multiplied by 50 over 50.

Oh, hang on a second.

50 divided by 50, that's just one.

So this is the same as saying a half multiplied by one, which is just a half.

How would I represent this? It's a ratio.

I don't know about you, but I think I'm going to state the number of shaded to unshaded squares.

In this case there are 50 shaded squares for every 50 unshaded squares.

Now just like I did with my fractions when I wrote an equivalent fraction, I can write an equivalent ratio.

That's right, for every one square that is shaded, there is one square that is not.

Now what fraction have I got shaded here? Feel free to pause the video if you want to count.

Welcome back.

What fraction do you think is shaded here? That's right, it's 17 out of 20.

In other words, 17 twentieths.

I have 17 shaded squares out of the 20 squares in total.

What I'm saying therefore is that for every 20 bars or 20 squares, 17 of them are shaded.

Now how do I show this as a percentage even if I don't have a convenient hundred square grid drawn out for myself? The important thing is that I need to keep the proportion for every 20 bars.

17 are shaded.

That has to stay the same, but I need to do an equivalent diagram that shows a hundred bars rather than the 20 that I have got.

Okay, well let's consider adding another row.

If I do that and I keep the proportion for every 20 bars, 17 are shaded, then that means I'll have 40 bars in total, of which 34 is shaded.

Think about it, all I've done is double.

So 17 has doubled to become 34 and 20 has doubled to become 40.

Now I can keep adding rows, so that'll be another 17 that are shaded.

So 34 goes to 51 out of the 20 bars in total.

So 40 becomes 60 and I can keep this pattern going 68 out of 80 and then 85 out of a hundred.

Now I've got that per a hundred that I was looking for for percentage.

So that's 85%, our knowledge and equivalent fractions allows us to convert so that the denominator of the fraction becomes 100.

Now let's try filling in three fifths of this grid.

What I'm actually asking you of course is to give me what three fifths of 25 is.

Because there are 25 squares in this grid, it means that for every five squares I need to shade three of them.

Now I can do this by considering each row and shading as I go.

Well that was 15 square shaded.

There we go.

So three fifths of 25 must be 15.

The thing is, I can do this without the incremental shading that you saw.

So instead of shading each row one at a time, I could have done it a different way.

I know that when I'm asking for three fifths of 25, I know I can find this by multiplying.

Now remember 25 or in fact any whole number can be written as a fraction 'cause we can write it as 25 of the number over one.

So one would become the denominator.

If I do that, I know how to multiply fractions.

I know I multiply the numerator and this is over the product of the denominators.

In other words, three multiplied by 25 and the denominator is five multiplied by one, but we're pretty good at this now.

So I'm just gonna write five.

Well, three lots of 25 is 75 divided by five, which is 15, so I need to take 15 squares, now to convert this into percentage, I can use my knowledge of equivalent fractions again.

Remember I shaded 15 out of the 25 squares.

How am I going to turn 25 into 100? What will I need to multiply by so that I have an equivalent fraction? That's right, both numerator denominator, we multiply by four.

Remember, I need to multiply both the numerator denominator, otherwise I'm not going to have an equivalent fraction.

Effectively what I'm doing here is multiplying by one and one can be expressed as any number divided by itself.

In this case, four divided by four would give me one.

So what I've then done is gone 50 multiplied by four and 25 multiply by four.

This gives me 60 out of a hundred and that means the percentage that is shaded is 60%.

Let's shade in 45% of 20 blocks.

Well, 45% can be expressed as 45 out of a hundred.

Hang on a second, this is just finding a fraction of an amount and we just did this.

I'm really confident.

I know I need to multiply.

Remember that 20 can be expressed as 20 over one.

So I know I'm going to be finding 45 multiplied by 20 over a hundred multiplied by one.

Hang on a second.

That's not what my working shows.

Sure, the numerator's correct.

I've got 45 multiplied by 20, but why is my denominator now five multiplied by 20 when I thought it was going to say a hundred multiplied by one? Well I know that anything multiplied by one is just itself.

So maybe I could see a hundred there except a hundred can be expressed as five multiplied by 20.

Why do you think I wanted to do that? Did you spot the numerator and the denominator? Both now have a common factor of 20, so I can rewrite this fraction as follows, 45 over five multiplied by 20 over 20.

Ah, this is now very simple to work out because 45 divided by five is nine and 20 divided by 20 is one leaving with nine multiplied by one.

So my answer must be nine.

In other words, 45% of 20 is nine.

Or in this case I've got to shade nine blocks.

Now this shading 36% of this 50 square grid going to be the same as shading in nine 20 fifths of this grid.

What do you think? Pause.

Have a go and then come back and resume this video to check in with me.

Welcome back.

Let's see how you got along.

Well, we know that 36% of 50 is equivalent to saying 36 out of 100 multiplied by 50.

Now remember that means 36 multiplied by 50 over 100.

Did you spot we can rewrite the denominator though? We can write the denominator as two multiplied by 50 and this means we're left with 36 over two multiplied by 50 over 50 or in other words 18 multiplied by one.

So I'm shading 18 of these squares, well now I know what 36% of 50 is.

What is nine 25ths of 50 though? Well let's calculate.

That'll be the same as nine lots of 50 over 25, except I know that I can express the 50 as two multiplied by 25.

This leads me with 18 'cause I have nine times two multiplied by 25 over 25.

Or in other words I have 18 multiplied by one.

So 18 squares, in other words they are indeed the same.

Here's another grid.

What percentage of the grid is shaded? Well, we can start by using fractions.

I can see that I have five hole blocks and then I have five half blocks.

Well, five half is two and a half.

Five holes is five.

Therefore I've got a total of seven and a half.

There are 10 blocks in total.

Do I like that fraction? Well, it's okay that there's a decimal in the numerator at the moment because remember I'm trying to find a percentage, so I need the denominator to be 100, so I'll have to convert.

Well I can do that by multiplying the denominator by 10.

So I must do the same to the numerator so I have an equivalent fraction.

This gives me 75 out of 100.

So in other words, 75% of the grid is shaded.

What percentage of each bar is shaded here? Pause while you have a go.

Welcome back.

How did you get on? Let's start with the top bar.

So here we have six and a half bars that are shaded out of 10, multiplying both numerated nominated by 10 gives us 65 out of a hundred or 65%.

In B, we should have seen that there are two and a quarter or 2.

25 bars that are shaded out of five bars in total.

Multiplying both numerator and denominator by 20 will give us 45 out of a hundred.

In other words, 45%.

Explain how you would shade in 42% of this grid.

Pause the video while you do this now.

Welcome back.

Let's see if your reasoning is the same as mine.

I know there are 25 bars there.

Therefore I want 42 over a hundred multiplied by 25.

Remember, I'm turning that percentage into a fraction to make this very easy for myself, I'm then going to rewrite the denominator as four multiplied by 25 so that I'm left with 42 over four multiplied by 25 over 25.

Well, 42 divided by four is 10 and a half or 10.

5 and 25 divided by 25 is just one meaning that I would need to shade 10.

5 bars of this grid if I wanted to show 42%.

Your turn now.

Question one, what proportion of the grid that you can see on the screen is shaded? Give your answer both as a percentage and a fraction.

In 2A, same thing, what proportion of this grid is shaded? And again, give your answer as a percentage and a fraction.

Pause while you do this now.

Welcome back.

Let's look at the next question.

2 Part B here says what percentage is shaded on the a hundred grid to the right and then shade the same percentage on grids A to G on the screen you can see A to D.

So we'll pause while you do these and then we'll look at the remaining grids.

Welcome back.

Let's now look at grids E to G.

Time to go through some answers.

So what percentage of fraction is shaded on the grid here? We should have seen there were eight shaded squares out of 25 squares in total, which we can convert to be 32 out of a hundred or 32%.

So we have both our fraction eight out of 25 and our percentage of 32, similar working will let us calculate Part A.

We have four shaded squares out of 20 in total.

This means that we have 20% shaded.

Now the a hundred grid that you can see on your right, we need to consider how many squares were shaded.

And in this case we shaded 30 out of a hundred, which is 30%.

If you look at A, A is exactly half of the grid that we had in our diagram, which means that in order to shade 30% here, I'm going to want to shade 15 of the squares, because 30% of 50 is 30 over 100, multiply by 50, which comes down to 15.

And now I want 30%.

So 30% of what I can see in B means 30% of 10, which means I need to shade three blocks.

Now for C, again, I want 30%.

This means 30% of 40 because there are 40 squares there, which means I need to shade 12.

For D, remember I want 30%.

Well, 30% of two means shading, three fifths of a block.

Now what's about E to G? Well here remember I want 30% of E, which means I'm gonna be shading six squares because there are 20 squares here.

In F, remember, I want 30% to be shaded.

Well that's 30% of four, which means I want to shade six fifths of a block and I've represented that here.

But yours may look a little different but still be the same.

So do this check.

What about in G? Well that's 30% of five, which means I want to shade three over two or one and a half bars.

So here are the answers for E to G.

Time to move on now.

And the second part of our lesson is on percentages of amounts.

We know percent comes from the Latin phrase per centum meaning by the hundred.

And we know fractions, decimals, percentages and ratio all represent proportion and we can change between them.

And this means there are lots of different ways to work out the percentage of an amount.

Now you'll see lots of different ways, but it's essential you understand the reasoning behind the working.

So for example, let's find 40% of 600.

We'll use a bar model initially to show this.

So 600 represents our whole or our 100%.

We're going to divide it into 10 equal parts and therefore each part represents 10%.

This means that each individual part represents 60.

Remember we wanted to find 40% of 600.

Well 40% would be four of those bars.

That means 60 times four, which is 240.

Now looking at how we found 40% of 600 means we can work out different multiples of 10 and we can do this without the representation.

Let's see how this would work.

We recognise that 100% was equivalent to 600.

We then found 10% because we did 600 divided by 10 to work out each part being 60 and then we found 40% by multiplying.

Now I want you to have a play.

You can draw a bar model if that would help or given what we've just talked about.

Perhaps you can do it without, pause the video now while you have a go.

Welcome back.

I'm gonna do this without drawing a bar model, but it's absolutely fine if you did do one.

I said, well I know 10% of 800 means breaking that 100% into 10 equal amounts.

So divide by 10 means that 10% is 80, I want 30% so I'll need to multiply by three 'cause I want three lots of my 10% or three lots of 80 is 240 and that's my answer.

For B, similar working.

I first found 10% which was nine, and then doubled to find 20% which was 18.

And then to find 40% of 12, I found 10%, which was 1.

2 and then multiplied by four to give my answer 4.

8.

Now three of our pupils are asked to work out 40% of 20 and you can see you and Alex and Sam's working here.

Pause the video check to see if you think they are correct and then check you can explain what it's they've done.

Pause now.

Welcome back.

Were they all correct? Let's start with Jun.

Jun is correct.

He's used his knowledge on fractions and percentages.

He converted the 40% into a fraction and then multiplied by 20, which is what we saw happen in the first part of our lesson today.

He's therefore correctly calculated that 40% of 20 is eight squares.

What about Alex? Alex is also correct.

He found 10% and then multiplied by four to make 40%.

What about Sam? Sam's also correct.

Sam used their knowledge on equivalent fractions, decimals percentages and wrote 40% as a fraction, then converted to a decimal and then multiplied by 20.

And this works too.

Finding 10% can allow us to find 5% and then you are able to find any percentage which is a multiple of 5% or 10% very easily.

If you know how to find 10%, what would you need to do to find 5%? That's right.

We simply need to divide our 10% amount by two.

Let's work this out now.

I'm going to find 15% of 40.

Now I can calculate this by working out what 10% is, what 5% is, and then summing these two amounts, well 10% of 40 was found by doing 40 divided by 10, which is four and 5% remember is simply half of the 10% value.

Well, four divided by two is two.

I then need to sum these because 10% at 5% is 15% and therefore four plus two is six.

So 15% of 40 must be six.

It's now your turn.

Find 35% of 24.

Pause the video now while you do this.

Welcome back.

You should have started out by working out that you want 30% at 5% or you could have said, I just want to find 5% and then multiply by seven, either works and there are in fact lots of different ways you could have done it.

I'm simply gonna go with finding 10% multiplying and then adding on the 5% or 10% of 24 would be 2.

4, 5% is 1.

2.

Remember 30% is simply 10% times by three.

So 2.

4 times three is 7.

2.

And then summing 7.

2, which is my 30% and 1.

2, which is my 5% gives us 35%, which is 8.

4.

Given that there are lots of different ways to work out a percentage of an amount, some methods are more efficient than others, how many different ways could you work out 45%? I've given you an example by saying, well, I could calculate 5% and then multiply by nine.

What other the ways can you think of? Here's just a couple I'll share with you.

You might have said, well, I'm gonna find 10% multiply by four and then add on 5%.

You could have said, I'm gonna find 50% and then take away 5%.

That'll also work.

Three pupils are asked to work out 55% of 240.

Now they're all but can you explain their method? So how did they get the numbers you can see? Pause the video and have a go at this now.

Welcome back.

Let's start by looking at Jun.

Jun has found 50% and 5% and then summed them together to get 55%.

Laura has found 5% and multiplied it by 11 to make her 55% and Sophia has found 10% multiplied it by five to make 50%, then found 5%, and summed these two results together to get 55%.

It's now time for our final task of lesson.

I'd like you to start please by working out the stated percentage of the given amount, pause the video and do this now.

Welcome back in question 2A, I've asked who do you think is correct and why? So you can see three of our oak pupils and some methods for finding percentages.

See what you think of them.

And then in part B, what percentages would any of the incorrect calculations or calculation, depending on how many are wrong, what percentage would they actually find? Pause the video and do this now.

Welcome back, in question three, I've said to use the values you can see below to fill in the gaps, you can only use each value once and that means there will be one leftover and you need that for part B.

Using that leftover number, create some sentences in the same format as the ones you do using in the first part of the question, how can you use it in the first gap? How would you use your leftover number in the second gap? And what about the third? Welcome back, in question four, I've said on the right there are six students, different strategies for finding 75% of 250.

Now one of them is wrong.

Who is wrong? And can you explain the method used by each pupil? Welcome back, time for some answers now.

So first of all, I asked you to calculate the percentage of the given amounts or 20% of 640 is 128, 30% of 520 is 156, 15% of eight is 1.

2 and 35% of 30 is a total of 10.

5.

In 2F, it said, who do you think is correct and why? Well, Aisha is correct.

10% is just dividing by 10.

But Andy thinks that 20% is dividing by 20 and that's not the case.

20% is found by dividing by five.

Izzy thinks 50% is dividing by 50.

That's not the case to find 50% we divide by two.

Now you might have tried to exemplify your reasoning using a particular value showing that why dividing it by 20 would not give 20%.

Or for example, dividing by 50 wouldn't give 50% and that's great if you did that.

Now this means that you need to tell me what Izzy and Andy would actually be finding.

Well, Izzy would be finding 2% and Andy would be finding 5%.

Now using the values to fill in the gaps, these are just examples.

You might have got other solutions that are also valid.

So I've said you could say 5% of 500 is 25, 20% of 250 is 50, 15% of a thousand is 150, and then you'd be left with 200.

You could also have said 500% of five is 25, 250% of 20 is 50.

1000% of 15 is 150, and that would leave you with a leftover value of 200 again.

Now in this case, I was left with 200.

So I've given examples of where 200 can be used in the first gap, in the second gap and in the third gap.

So remember, this is just an example.

You might have a different leftover value and therefore you'll have different statements here.

Then we moved on to question four, and we have our six students.

The strategy that's incorrect, is Lucas, we're trying to find 75% so we don't divide by 75, Lucas, sorry about that.

I'm afraid you fell into the same trap as our earlier students did.

Do you remember Andy and Izzy, dividing by the wrong amounts? Same thing here for Lucas.

So let's look at the students that were correct.

Andy found 1% and then multiplied to get 75%.

Sophia found 10%, then found 70%, found 5% and summed, Sam here found 25% then multiplied by three to make our 75%.

Aisha found 50% and 25% and summed those, Jacob found 25% and subtracted it from 100% to make 75%.

All of these methods are valid.

There are so many ways to find percentages of amounts.

So let's sum up what we've learned today.

We know percent comes from the Latin phrase percentage meaning by the hundred, and we know we can represent proportion using fractions and percentages and so we can change between them.

There are lots of different ways to work out percentage of an amount.

It's essential you understand the reasoning behind it so that you can choose the best method for you.

Well done.

You've done really well today.

I look forward to seeing you for one of our future lessons.