Lesson video

Hi everyone.

Welcome to our lesson today, we'll be looking at multiplicative relationships.

It's one of my favourites because it appears so much in everyday life.

Really excited that you're learning with me today.

So let's make a start.

Hi everyone and welcome to this lesson on finding a percentage with a multiplier and it's under the unit Understanding of Multiplicative Relationships, Percentages and Proportionality.

By the end of lesson, you'll be able to find a percentage of a quantity using a multiplier, and today's lesson will be looking at some key words.

The first key words are equivalent fractions.

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

For example, one half is exactly the same as two over four.

Four fifths is exactly the same as 40 over 50.

A non example would be one fifth is not the same as three sevenths.

We'll also be looking at the associative law today and the associative law states that a repeated application of the operation produces the same result regardless of how pairs of values are grouped.

And we can group using brackets.

For example, three add four in brackets, add our 10, is exactly the same as three add bracket, four add 10 as they still both give 17.

Now, a non example would be the use of division.

48 divided by two in our brackets, divided by six is four and that is not the same as 48 divided by our six and two in our brackets, as this gives us an answer of 16.

In other words, addition follows the associative law.

Multiplication follows the associative law, but division and subtraction do not.

Today's lesson will be broken into two parts.

We'll be reviewing percentages of amounts and then we'll be using a decimal multiplier.

So let's make a start.

Well, first of all, we know percent comes from the Latin phrase per centum, meaning by the hundred.

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

This means there are lots of different ways to work out a percentage of an amount and you'll see lots of different ways, but it is essential you understand the reasoning behind the working out.

So let's have a look at a bar model showing 240.

How would 10% be represented in this bar model? Time to have a little think.

Well hopefully you could spot, if you divide by 10, that means you have your 10% because we know 10% is 24.

And you can see the a hundred percent is made up of 10 lots of 24.

Now how do you think we can find 5% if we know 10%? Well, if we divide the 10% by two, we've got our 5%, so that means 5% would be 12, 24 divided by two.

You could have also done 240 divided by 20.

That would also give you 5%.

But a bar model is a really nice visual way to show you what those percentages represent.

How do you think we can find 1%? Well, there's lots of different ways how we can find 1%.

For me, I'm going to focus on this 10%.

I'm going to divide it into 10 equal parts.

It's a bit small here, so I'm going to zoom in so we can see what 1% represents.

Well, if we split our 10% into 10 equal parts, that means each part, each 1% should be 2.

4.

So using a bar model can really help work out 10%, 5% and 1%.

And from here it'll enable us to work out any percentage knowing these values.

Now it's time for your check.

I'm going to do the first question and I'd like you to try the second question.

I'm going to use bar models to help me find 23% of 500.

Now, to do this, first things first, I'm gonna have a strategy in mind.

How do I find 23%? Well, for me, I'm going to break 23% into 10%, 10% and 3%.

So that will really help me work out 23% in total.

On a bar model that would look around about here.

So let's work out our 10%.

Well, to find 10%, we simply divide 500 by 10, thus giving me 50.

So that represents my 10%.

Find 1%, remember I'm going to divide that 10% by 10, thus giving me 1% is five.

Now if I know 1% is five, 3% would be made by five multiplied by three, which is 15.

So now I know my 10%, my 10% and 3%, which makes 23%.

So I'm going to add them all together to give me 23% of 500 is 50, add 50, add my 15, which is 115.

And I would like you to try this question.

Using a bar model, see if you can work out 42% of 340.

See if you can give it a go and press pause if you need more time.

Great work.

Let's see how you did it.

For me, I'm going to work out 10% added to another 10%, another 10%, another 10% and 2%.

That would give me my 42%.

So let's work out 10% first.

Well 10% would be found by 340 divided by 10, which is 34.

1% is found by dividing that 10% by 10, so 34 divided by 10 is 3.

4.

Now from here I can work out 2% because it's two lots of my 1%.

3.

4 times two is 6.

8.

Now I have everything I need to work up my 42%.

42% is basically four lots of 10% and a 2%.

Working this out, four lots of 34, add 6.

8 is 142.

8, which is my 42% of 340.

Really well done.

If you've got this one right.

Now, we can also find any percentage of an amount using a double number line.

So let's look at 42% of 340.

First draw a number line indicating a hundred is 340.

Now I want you to think, how do you think we can find 10%? Well, just like before, we're going to divide by 10.

A hundred divided by 10 is 10%.

So that means 340 divided by 10 is 34.

Now let's think about a strategy defined 42%.

I'm going to find 40% and 2% and then solve.

Well, if we know 10%, how do you think we can find 40%? Well, to find 40%, we simply multiply by four.

So I have my 40% here, which is 136.

But now, how do you think we can find 2%? There's lots of different ways, really.

I'm going to divide my 10% by five.

This will give me my 2%.

Because of that multiplicative relationship, I'm going to divide 34 by five as well, giving me 6.

8.

So now I know 40% and I know 2%.

So what is 42% of 340? Well, it's summing the 136 and the 6.

8, is going to be 142.

8.

Same answer, but I've used a double number line.

It's so important having a strategy to find a percentage of an amount and finding 10%, 5%, and 1% can generally help us do this.

Now I'd like you to do is do a check question.

I want you to use a double number line and work out 36% to 460.

See if you can give it a go and press pause if you need more time.

Well, there are lots of different ways you can use a double number line, but identifying 10%, 5% or 1% generally does help.

Let's work out 10%.

Well, 10% has to be 46, so that means 5% would have to be 23.

And from here I can even work out 1%.

1% would have to be 4.

6.

Remember, 1% can be found by dividing 10% by 10.

So 46 divided by 10 is 4.

6.

From here, how's this gonna find 36%? Well, I could find 35% by simply multiplying 5% by seven, giving me 35%, which is 161.

And then I can add on that 1%, which I know is 4.

6.

36% is then 23 times seven, which is 161.

Add our 4.

6, which is 165.

6.

There are so many different ways you could have worked out 36% of 460.

Did you get the same answer but in a different way? If you did.

Well done.

So ratio tables can show the same information as a bar model and double number line.

So let's look at 42% to 340 again.

We know as a bar model it looks like this and we know as a double number line, it looks like this.

But let's use a ratio table now.

We still know a hundred percent to 340.

So from here we can apply the same process to find 10%.

In other words, we divide by 10.

So 34 is 10%.

But how do you think we find 1%? Well, we divide by 10 again to give us 1%.

Then what strategy would you use to find 42%? For me, I'll times the 10% by four.

Then at times the 1% by two.

Then I'm simply summing the 40% and 2% to give me exactly the same answer, 142.

8.

So it doesn't matter which method you prefer, but some take a little longer than others.

Fundamentally, finding 10%, 5%, and 1% generally helps.

So now what I want you to do is use a ratio table.

I want you to use a ratio table to work out 23% of 840.

See if you can give it a go, and press pause if you need more time.

Great work.

Let's see how you got on.

Well, we know 840 represent a hundred percent.

That means I'm gonna find 10%.

It's always a good starting point.

So that's 84.

Well, I'm gonna find 1%.

And again, it's always helpful.

So that's 8.

4.

Now I'm gonna think of a strategy.

How am I going to find 23%? Well, if I multiply 10% by two, I've got my 20%, which is 168.

If I multiply my 1% by three I have, using an area model, I can work out 3% to be 25.

2.

That means I know 23% is the sum of 20% and 3%, which is 193.

2.

Great work.

If you've got this one right.

Now, it's time for your task.

Without a calculator, using the most efficient method for you, work out the following.

See if you can give it a go and press pause for more time.

Great work.

Regardless of the method you chose, as long as it's efficient and it's correct, you should have got these answers.

Massive well done.

Great work everybody.

So let's move on to the second part of our lesson where we're using a decimal multiplier.

Now without a calculator, let's work out 68% of 430 pounds.

Now here's one way you might work this out using a ratio table.

So here's my a hundred percent, 50%, 10%, 1%, 2%, and then from here we can work out 68% by adding the 50%, 10%, 10% and subtract 2%.

So working this out, summing them and the subtraction of 2% gives me £292.

40.

Now using a calculator, can you work out the answer of 0.

68 times 430? You should get 292.

4.

What do you notice? Why does this calculation give you the same answer? Well, let's look at why that works.

Well, if we're looking at 68% of 430 pounds, we know that finding percentages of a given amount is like finding a fraction of amount.

For example, 68% of 430, this is the same as 68 over a hundred of 430.

Now remember when finding fractions of amounts, we use multiplication in the place of the word of.

So 68% to 430 is the same as 68 over a hundred multiply by 430.

This must be equivalent to 0.

68 times 430 since 0.

68 is the decimal equivalent to 68 over a hundred.

So knowing these decimal equivalents allows us to use a multiplier in order to find a percentage of an amount.

What I want you to do is I want you to work out the decimal equivalent of the following percentages.

See if you can give it a go.

Press pause for more time.

Great work.

Hopefully you spotted 26% as a decimal is equivalent to 0.

26.

89% is equivalent to 0.

89 54% is equivalent to 0.

54.

3% is equivalent to 0.

03.

2% is equivalent to 0.

02.

and 5.

8% is equivalent to 0.

058.

Well done.

Knowing those decimal and percentage equivalents, we're able to calculate a percentage of an amount using a multiplier.

So see if you can pair up the question with the correct decimal multiplication.

See if you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

24% of 600 is the same as 0.

24 times 600.

Remember that word of? It means times.

24% as a decimal, 0.

24.

Next 42% of 600 is 0.

42 times 600.

4.

2% of 600 is 0.

042 times 600.

And 2.

4% of 600 is 0.

024 times 600.

So recognising that we can calculate a percentage of an amount using a decimal multiplier simplifies the questions and working out, especially if we have a calculator.

For example, work out 24% or 182, what do you think the decimal multiply would be? Well, it'd be 0.

24.

That word of becomes times 182.

And if we had a calculator, we can just put it into our calculator and work it out, but we can still work it out using a written method using that associative law.

So remember 0.

24 times 182 is equal to 24 times 182, times 0.

01.

Using our area method, I'm going to work this out to be 4,368.

So that means I know 24 multiplied by 182 is 4,368, but remember, I'm still multiplying by my 0.

01.

Then, working out this answer, I get 43.

68.

So without a calculator, I can work out 24% of 182 by using a decimal multiplier and it gives me the answer 43.

68.

Now I want you to have a look at this check question.

Jun has shown the working out for 3% of 812.

Where do you think he's made his mistake and what did he calculate? Well, hopefully you spotted the decimal calculation should be 0.

03 times 812, as 0.

03 represents 3%.

Jun has worked out 30% of 812, not 3%.

Well done if you spotted this.

Now it's time for your task.

What I want you to do is write the calculation showing the decimal multiplier and you do not need to calculate the answer, only show the calculation.

See if you can give it a go and press pause for more time.

Well done.

Let's move on to question two.

Now I want you to work out 27% of 560, 18% of 244 and 6% of 94.

Be sure to show all you're working out.

Great work.

So let's have a look at question three.

Can you explain the strategy behind each student's working out for finding 48% of 64? See if you can give it a go.

Press pause for more time.

Great work.

Let's move on to question four.

Question four says, a hard cheddar cheese contains 37% water, 33% fat, 23% protein, 3.

5% carbohydrates and 3.

5% minerals and other compounds.

From a 250 gramme block of cheese, can you work out how many grammes of water, fat and protein there are? So you can give it a go.

Press pause for more time.

Great work.

Let's move on to our last question, which is question five.

In 2016/2017, the UK distributed approximately 720 billion pounds to a number of sectors.

How much money did the following sectors receive for welfare, health, pensions, and culture? See if you can give it a go.

Press pause for more time.

Great work everybody.

Let's go through these answers.

Well, for question 1a, remember we didn't want the answer.

I just wanted the calculation showing that decimal multiplier.

Well done if you got this one right.

For question two, using that decimal multiplier and an area method, if that's your choice, you should have got these answers.

Massive well done.

For question three, can you explain the strategy behind each student's working out? Well, looking at Alex's, hopefully you can spot he found 10% and then he multiplied it by four to give 40%.

Then Alex found 1% and multiplied it by eight to give 8%.

Then he summed them together to give 48%, which is 30.

72.

Then Sam has actually found 1% and then by two to give 2%.

And he's found 50% and from here subtracted the 2% from the 50%, thus giving 48%.

Great work if you spotted those two strategies.

For question four, we have a bit of context here, so you had to work out what is 37% of 250.

Well, it's 92.

5 grammes.

For fat, it was 33% of 250, which is 82.

5 grammes.

And for protein it was 23% of 250, which is 57.

7 grammes.

For question five, the calculation would be 0.

24 times 720, which gives 172.

8 billion pounds dedicated to welfare.

For health, they receive 0.

20 times 720, which is 144 billion pounds.

And for pensions, they receive 0.

13 times 720, which is 93.

6 billion pounds.

And lastly, for culture, they would've got 0.

02 times 720, which is 14.

4 billion pounds.

Great work everybody.

I do hope you enjoyed this lesson like me, especially with those real life context questions.

We've looked at finding a percentage of an amount using double number lines, bar models, ratio tables, and using a decimal multiplier.

You can also use an equivalent decimal multiplication to find percentages of amounts.

For example, 68% of 430 can be found by calculating 0.

68 times 430.

This is particularly efficient when using a calculator to work out percentages of amounts.

Great work, everybody.

It was wonderful learning with you.