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Hi, my name is Mrs. Wheelhouse.

Welcome to today's lesson on "Securing understanding of percentages" from the unit, "Understanding Multiplicative Relationships: Percentages and Proportionality." Let's get started on today's lesson, which I hope you're gonna really enjoy.

In today's lesson, you're going to learn how to manipulate percentages and amount with a representation.

Here are some keywords that we're gonna be using in today's lesson.

Feel free to pause the video if you want to review them.

Our lesson is broken into three parts today.

We're gonna start with part one on finding percentages of amounts.

When finding percentages of amounts, it's always important to have a strategy of how you're going to find that percentage.

For example, how many different ways can you find 75%? Here's a spider diagram, and I've shown one way of calculating 75%, but how many other ways can you think of? Pause video and write these down now.

Welcome back.

Now I'm gonna share some of the ways that I came up with to find 75% of an amount.

You may have these, you may have others, see if we match.

But remember, even if we don't, it doesn't mean yours are wrong.

I couldn't write down all the ways of finding 75%.

So these are just examples.

So you could calculate 25% and then multiply by 3.

You could, of course, find 25% and subtract it from the 100%.

You could calculate 1% and multiply by 75, you could calculate 80% and take away 5%.

As I said, these are just some examples.

Let's consider a bar model showing 240, how 10% be represented in this bar model? That's right.

I would break it into 10 equal amounts.

So you can see here, I've shown the 10 percents all the way along and therefore, each part represents 24 because if 100% is our whole, therefore, 240, 10% will mean 240 divided by 10, which gives us 24.

Well, how would we find 5%? That's right, I halve it.

Well, if I'm halving each of those bars, that means that half of a bar is worth 12 because the whole bar was worth 24.

24 divided by 2 is of course, 12.

What about finding 1%? Oh, okay, I can break the 10% into 10 equal parts.

Hmm, let's see if we can zoom in and see what's going on here.

So what I've done is I've now shown 10%, which was 24, and I've broken it into 10 equal parts, so I can find 1%.

Well, 24 divided by 10 gives us 2.

4, and that means each of the bars here represents 2.

4.

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

And if we know these, we can think of them as like the building blocks for any percentage.

So for example, how could we calculate 32%? Well, 32% remember, would be three lots of 10% and then little bit more, in fact, 2% more.

Well, remember we knew 10% was just 240 divided by 10, which was 24, and we know that 1% was 2.

4, so 2% must be 4.

8.

If I sum these together, I'm going to get my 32%.

So three lots of 24 add 4.

8, giving us a total of 76.

8, and that's the 32%.

So 32% of 240 is 76.

8.

Do you see how we calculated that? Because we knew 10% and 1%, we were able to find the 30% and the 2% we needed to get 32% in total.

Let's find 23% of 500.

Now you can use a bar model if it helps, but if you don't need to, that's absolutely fine too.

First, we're going to plan how to find 23%.

In this case, I think it's easiest to find 10% and then I can double to get 20% and to find 1%, and then I can multiply by 3 to get the 3% I need.

I'm just showing on my bar model to make it clear to myself what it is I'm doing.

So let's start by finding the 10%.

Oh, 10% will be 50 and 1% remember, is dividing the 10% by 10.

So 50 divided by 10 is 5.

Well, 3% must be three lots of 5, which is 15.

So in order to find my 23%, I'm going to have 10% plus 10% plus 3%, or I can write that as 50 add 50 add 15, which gives me a total of 115.

In other words, 23% of 500 is 115.

It's now your turn.

I'd like you please to find 42% of 340.

You can use the bar model if it helps, but if you don't want to, that's fine too.

Pause the video while you do this now.

Welcome back.

Let's see how you got on.

Well, first you had to plan how to find 42%.

Now remember, you may have a different plan.

It's absolutely fine if you do, remember, there's lots of different ways to find the percentage of an amount.

I'm going to start by saying I want 10% and 1% and then I'm going to use those to find 40% and 2%.

So let's start with the 10% first.

To calculate 10%, I do 340 divided by 10, which is 34, and to calculate my 1%, I do 34 divided by 10, this gives me 3.

4.

Now my 2% just means doubling the 3.

4, which gives me 6.

8 and the 40%, I'm gonna do by doing four lots of the 10%.

In other words, 34 multiply by 4, and then add my 6.

8, giving me a total of 142.

8, and that's my 42% of 340.

Well done if you got that right.

Here, we have two of our Oak pupils and they've each worked out 48% of 600.

Can you explain each of their methods? Pause the video and do this now.

Right, let's start with Alex.

Alex found 48% by summing 40% and 8% together.

What about Sam? Well, Sam found 48% by subtracting 2% from 50%.

So two different plans here, but both of them worked and they each found correctly, 48% of 600.

Well done to both Alex and Sam.

It's now time for your first task.

In question one, I'd like you to work out the percentage of the amount and use the bar models to help you.

Pause the video and do this now.

Welcome back.

In question two, you have some percentage bars to show the percentage of days that experience different weather conditions.

I'd like you please to work out how many days were sunny, how many were cloudy, and how many were rainy.

Pause the video and do this now.

Welcome back.

In question three, I'd like you to work out 29% of 120 using the two different bar models.

So what we've got here are two bar models and each is showing different ways to calculate 29%.

Doing each method, I'd like you to find that 29% of 120, please.

Pause the video and do this now.

Welcome back.

In question four, I'd like you please to work out 99% of 200 and show two different methods for doing this.

I'd like you to demonstrate why your method works by using the two bar models.

So a bit like what we just did in question three, only this time, you need to show me on the bar model rather than me giving you the bar model with a method shown.

Pause the video and do this now.

Right, time to go through some answers.

For question one, I asked you to work out the percentage of the amount using the bar models.

So first of all, we had to work out 42% of 800 and we can see that represented here on our bar model at the top.

So we had four blocks and then a little bit of the next block.

Well, this tells us that I'm going to calculate my 10%, my 1% and I'm going to want four lots of 10% and two lots of the 1%.

So that's 10% being 80 and then my 2% is only 16.

Summing these gives me a total of 336.

Similar method here for working out 61% of 440, I filled in my 10%, which was 44 and my 1%, which was 4.

4.

Summing all of these gives me 268.

4.

In question two, I want you to work at how many days was sunny, cloudy or rainy.

Well, we can break this up and we know that the 20 days can be broken into 10 equal size blocks where each block represents two.

50% is five of these blocks, and this means the total of sunny days was 10, cloudy days was 30%, so that would've been three of the blocks, so that's six days and rainy was the remaining, which must be four days.

In question 3A, we wanted you to work out 29% of 120 in this particular way, which meant finding 10% and finding 9%.

Well, 10% of 120 is 12 and 1% is 1.

2.

So I filled in my two 10% blocks and for the 9% block, I need to do nine lots of 1.

2, which is 10.

8.

This means that 20% of 120 is 34.

8.

In part B, however, I want you to work out what 30% was and then subtract 1%.

Well, this means doing three lots of 12 and then subtracting 1.

2, which also gives 34.

8.

It's quite nice that you knew you must have the right answer here because the two answers had to be the same.

In question 4A, I asked you to work out in two different ways.

Now, I've just given an example of how I did this with my bar model, but you may have done something different.

What I did in the first one is I calculated 10% and 1%, and then I did nine lots of my 10%, so that would be nine lots of 20, and then I used my 1%, which was 2, multiplied it by 9 to get to my 9%, so that's 18.

And then I summed this to give 198.

In part B, we did something slightly different.

We said, well, I know 100% is 200, 1% is just 2.

Well, that means I just need to do 200 total, take away the 1%.

So that's just 200 take away 2 giving me 198.

Which one of those did you think was quicker? I know my money is on B.

I think that was a lot faster and a lot less work.

It's now time for the second part of today's lesson and here, we're going to be considering double number lines.

How might we be using these? Let's find out.

Fractions, percentages and ratio all show proportion.

Double number lines are excellent visual tools as they show that proportional relationship.

However, it is important to remember when drawing double number lines that the lines are scaled number lines, both lines must start from zero and the numbers illustrate the proportionality.

So let's use our bar model of 240 to illustrate this.

Remember, both of our lines had to start from zero, which they do, and we have is scaled lines.

So where 100% is on the top line is equivalent to where 240 is on the bottom line.

And that's because my top line is showing the percentage and the bottom line is showing the actual amount.

Remember, our 240 is representing our whole or our 100%.

Now here, I broke my bar model into 10 equal pieces and I knew that each piece represented 10% and that was 24.

This means on my double number line that the first segment is representing 10% and this is equivalent to 24 on the amount line.

So how can we use our double number line to calculate 40%? Well, we can work out the next few multiples of 10%.

So for example, 20%, well, that's just doubling.

So remember to keep it proportional because 10% double to 20%, 24 is double to become 48, and we can keep doing this.

You can see here added did on another 10%, which was equivalent to adding on 24 on the amount line, and this worked all the way up to 40%.

So 40% of 240 is 96.

Of course, we could have used the multiplicative relationship, which I suspect a lot of you were saying to the screen.

That's right, I could just multiply by four.

So 10% multiplied by 4 is 40% and therefore, 24 multiplied by 4 is 96.

So 40% of 240 is 96.

Working out 10%, 5% and 1% still remains useful even with double number lines.

So what I'd like you to do is just show 100% 50% and 10% of 800 on a double number line.

Pause the video and have a go at this now.

If you've done that, how could you use your double number line to work out what 65% is? If you'd like, you can pause and have a go at it and then come back and check with me.

Are you ready? Let's check it.

So here's our double number line.

Remember, I'm starting at zero and that's both the amount and the percentage.

I've got 100% at the other end and that's equivalent to 800 because that's the amount or my total that I'm starting with, and I have my equally spaced segments.

I've put on 50% halfway and I know that's 400.

I've put on my 10%, which is 80 and I put on my 5%, which was 40.

Now, I said to you to put on 100%, 50% and 10%.

So you may not have put on the 5% straight away, unless of course, you'd already moved on to working out 65%.

In which case, I think that 5% is very useful.

So using the 5%, I could just straight up multiply by 13 to get to 65% because five multiplied by 13 is 65.

And if I do that, I can do the same to the 40 because of that lovely multiplicative relationship.

And this tells me that 65% is 520.

The double number line made it really quick and easy to see how I could move from the 5% to the 65%.

Of course, I could have started at the 50% added on 10% to get 60% and added on 5% to get to 65 as well.

And you can see here that still gets us to 520.

We could also have used the multiplier between this percentage and the amount.

But what do I mean by that? That's right, I can move from the percentage line to the amount line because there is a multiplicative relationship there as well and that's multiplying by 8.

So if I want to calculate 65%, I could have just done, 65 multiplied by 8 and that gives me 520.

Wow! Double number lines are really useful.

So depending on the numbers involved, some approaches might be more efficient than others and that's the great bit about maths.

There are so many ways to do things, you can pick the way that you think is the best and that might mean the easiest for you.

It might mean the quickest, it might mean the less amount of working, the great bit is, you get to decide.

So let's see which way you'll pick here.

Using a double number line, work out 35% of 460.

Pause and do this now.

Welcome back.

Which method did you go for? I mean, there are lots of different ways, but I'm gonna start by maybe considering or identifying the 50%, 10%, 5%, 1%, they're normally pretty helpful.

I'm gonna start, of course, by putting on the zero or 0% and 100% of 460, 'cause I know I need those.

I calculated 10%, so that was 46 and I calculated my 5%, which was 23.

Now from here, there were lots of different things I could do.

I've opted for multiplying the 5% value by 7 because that will get me to 35% and therefore, multiplying 23 by 7.

Well, that takes me to 35% is 161, but remember, other methods would've worked too.

Let's consider if I multiply the 10% by 3 and then add it on the 5%.

Again, that also takes me to 161.

Oh Jacob, that's an interesting question.

Jacob wants to know if this still works when using 1%? "Of course," says Sophia.

Double number lines show proportion.

All right, says Jacob, "Let's work out 49% of 600." And he wants to know how Sophia would've worked this out.

And she says, "Well, I'm gonna draw the double number line, put on my 0%, my 100%, 10%, 50% and 1%." Now Sophia may not need all of these, but she's going to put them on so she can then work out what her approach is going to be.

So she's going to do four lots of 10% to get 40% and then she's going to find 9% by multiplying 1% by 9.

To get 49%, she's going to add on.

So you can see that 240 added on the 9% that was 54, gets her to 294.

All right, says Sophia, "How would you do this, Jacob? Have you got an alternative?" And Jacob says, "Well, I'm gonna work out 50% by halving 100%." Great approach so far, Jacob.

Can you spot what you think Jacob is going to do here? So right, he's gonna subtract 1%.

So 300, which was our 50%, subtract 6, which is our 1%, gives us 294.

So Sophia and Jacob have used two different methods here.

Which method do you think is more efficient and why? Well, personally, I think Jacobs is more efficient as it was a lot quicker and had a lot less working out.

And if there are less steps, there's less chance of me making an error.

However, both methods work.

So really, it's a matter of personal preference.

It's now time for our second task.

I'd like you to use double number lines to work out 45% of 700 and 31% of 1,200.

Pause the video and do this now.

Welcome back.

In question two, I'd like you to draw double number lines to work out 52% of 900, 98% of 3,400.

Pause and do this now.

Welcome back.

Let's look at some answers.

So to calculate 45% of 700, I first of all, calculated what 50% was and then I calculated 5% and took it away.

Well, 350 takeaway 35 gives me 315.

Now remember you may have done it a different way, but you should still have got to the same answer.

With 31% of 1,200, I calculated 10% and I calculated 1%.

I then multiplied the 10% value by 3, so 120 multiplied by 3 gives me 360 and then I added on 1% or 360 add 12 is 372.

In question two, I asked you to draw double number lines.

Well, you should have started off by writing down the zero and the 0% on the left hand side and 100% and 900 at the other end of your diagram.

50% is half of this, of 450 and I suspect you would've put the 1% on as well, which is nine.

Personally, I'm now gonna do 50%, add two lots of the 1%, so that will be 450 add two lots of 9, which is 18, giving me a total of 468.

With part B, of course, I'm wanting to calculate 98% and I suspect a different method is going to be better here.

So in fact, what I've done is I've calculated 1%, which is 34, and then I'm gonna subtract 2% from the 100%.

So in other words, 3,400 subtract two lots of 34, giving me a total of 3,332.

It's now time for the final part of day's lesson and we're gonna be considering ratio tables.

Remember, fractions, percentages and ratios all show proportion.

So another representation would be to consider a ratio table.

Now that 10%, 5% and 1% are still very important regardless of how we're considering proportion.

So let's consider our double number line and think about how this can be written as a ratio table.

So here's our double number line, and in this case, I'm going to use 900 being my 100%.

I'm gonna put this in a ratio table.

So here, I've got amount being my first column and percentage being my second column.

So 900 member is 100%, my 90 is 10% and I'm gonna put nine as my 1%.

How do we work out these amounts if we don't have a double number line? Given that ratio tables show proportion, we can identify a multiplicative relationship here.

We can see that the relationship between 990 is simply divided by 10, and that's the same for the percentage.

We can see it down here as well, how to move from 10% to 1%.

And again, we could also move from 900 straight to 9 dividing by 100, and therefore, 100% divided by 100 is 1%.

So we can use ratio tables to work out the percentage of any amount and it's any percentage of that any amount.

So let's consider 72% of 400.

Well, what is the 100% here? That's right, it's the 400.

That's my total.

I'm going to identify what 10% is, so divide by 10, I'm then gonna identify 1%, so I divide it by 10 again, what would I do to find 72%? Now remember, there are lots of ways to do this, this is just one example.

I'm going to multiply the row that talked about 10% by 7.

I'm going to multiply the row that talked about 1% by 2, and now I can sum the two rows that show 70% and 2%.

In other words, 280 add 8 is 288.

Now Andeep has worked out 47% of 120 using a ratio table, and his correct answer is below.

Can you use the ratio table in a different way to find 47% of 120? Pause the video and do this now.

Welcome back.

Let's see what way you came up with.

Now there are lots of different ways.

This is just an example of a different way that you could have done.

So I started off with the same ratio table as Andeep.

Only this time, I found 50% using my table and found 3% and then I subtract it.

Is that the way you went for or did you come up with a different way? Using the ratio table you see here, work out 40% of 350, 21% of 350 and 99% of 350.

I wonder if you use the same method for all three questions.

Pause the video and have a go at these now.

Welcome back.

Let's go through some answers.

Well, to calculate 40%, I just multiplied the row that talked about 10% by 4, so my answer is 140.

To calculate 21%, I took the 10% row and I multiplied by 2.

So in other words, 20% is 70.

I then added that row to the 1% row, so 70 add 3.

5 gives me 73.

5.

To calculate 99%, I used what was there.

I didn't have to add anything extra in or calculate anything new.

I simply took the 350 and subtracted 3.

5 because that's 100%, subtract 1%, to give me 99% is 346.

5.

It's now time for our final task.

Using the ratio table, work out 61% of 840, 59% of 840 and 98% of 840.

Pause and do this now.

Time for question two.

Using any representation you'd like, work out 23% of 720, 11% of 940 and 49% of 56.

Pause and do this now.

Welcome back.

Question three.

Write what you think are the advantages and disadvantages of using a bar model, a double number line and a ratio table when trying to find a percentage of an amount.

Now remember, these are your views, so they may be different to other people's.

So fill in the table now, please.

Welcome back.

Let's go through some answers.

So for question one, A, you should have 512.

4.

For B, you should have 495.

6.

And then for C, you should have 823.

2.

In question two, remember, we said you could use any representation you prefer, but you should have got the same answers as me, regardless of what you went for.

So in A, we should have 165.

6.

In B, 103.

4 and in C, 27.

44.

Remember, in question three, it was what you think the advantages and disadvantages are.

So you might've said a bar model is a good visual representation.

You might have also said that for double number line and for a ratio table, you may have said it's concise or easy to draw.

For disadvantages, you may have said a bar model takes time to draw.

A double number line takes time to draw and it's got to be scaled.

And for the ratio table you might say, well, it's not very visual and I find that less useful.

Let's summarise what we've learned today.

We know fractions, percentages of ratio all show proportion.

And proportion can be represented a number of ways.

And in this lesson we focused on bar models, double number lines and ratio tables.

Well done, you've worked really hard today and I hope that you enjoyed the lesson.

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