Lesson video

Hello, I'm Mrs. Lashley and I'm gonna be working with you as we go through our lesson today.

I really hope you're looking forward to it and you're ready to try your best.

Today's lesson outcome is to be able to express the percentage shaded of shapes within other shapes.

There is one main key word that we'll be using during the lesson, which is about area.

I'm sure you're familiar and confident with what area is, but if you wish to pause the video so that you can remind yourself, then please do so now.

So the lesson's got two learning cycles.

In the first learning cycle, we're gonna establish how to find the percentage shaded and then in the second learning cycle, we'll be doing some more calculations and working with the percentage shaded.

So let's make a start now with establishing how to find the percentage shaded of shapes within shapes.

So here we have a logo which has been designed as a square with edge lengths of six centimetres with a smaller square of three centimetre edges removed.

So what would be the area of the logo? How would you work that out? Well, Aisha has worked out the area of the largest square first, and that's by doing six squared, which is 36 square centimetres.

And then the area of the smaller square, which it would be nine square centimetres and therefore the area of the logo is the difference between the two because the smaller square has been removed, has been subtracted.

So the area of the logo is 27 square centimetres.

The logo is then to be printed onto a uniform.

So what percentage of the outer square is being printed? So Aisha says the shaded part of the logo will be the printed part, and this has an area of 27 square centimetres.

That's what she just worked out.

However, the logo has an area of 36 square centimetres.

So what is 27 as a percentage of 36? Can you remember how you work that out? Aisha's done it by thinking of it as a fraction.

So well as a fraction it would be 27 out of 36, which simplifies to three quarters because there is a common factor of nine in both the numerator and the denominator, which is also equivalent to 75 over 100.

75 and 100 have a common factor of 25.

And when simplified is also equivalent to three quarters.

And 75 hundredths is equivalent to 75%.

We can think about the word percent to mean for every 100.

So the percentage of the outer square which is being printed on this uniform is 75%.

So Aisha's method for finding that percentage was by getting to an equivalent fraction with a denominator of 100.

How else could the percentage be found? So just take a moment to think about is that the way you would have done the question or is there an alternative method? So another way would be to go from the fraction to the decimal before going to the percentage.

So then we don't need to have a denominator of 100.

So we can see that 27 is the area of the shaded region and 36 is the area of the outer square which simplifies to three quarters because of the common factor of nine.

So three quarters is a division, three divided by four.

So we're gonna put four on the outside as our divider and three on the inside.

So now we need to go through the steps of short division, which I'm sure you are familiar with, but I'm gonna go through it just in case we need a refresher.

So how many fours in three? Zero.

We cannot get any groups of four out of three.

However, three is equivalent to 3.

0, which allows us to carry our three ones over into the tenths column.

So three ones is equivalent to 30 tenths.

How many groups of four can you get from 30? Well you can make seven groups of four and there would be a remainder of two.

So there is a remainder of two-tenths, which is equivalent to 20 hundredths.

By carrying the two into the next column in our place value then we have now got an equivalent remainder.

How many fours in 20? How many groups of four can you get from 20? Five.

And there's no remainder, so this is our decimal.

0.

75 is equivalent to 75%.

So that's another method that you could use to get from a fraction to its percentage without going via a denominator of 100.

If the smaller square in this logo had been positioned in the corner as opposed to somewhere within the centre, the percentage shaded becomes more obvious.

The smaller square would fit three times within the shaded region and four times within the larger square.

And that's our fraction of three quarters.

So here's a check for you.

What percentage of the shape is shaded? So the shaded area is given to you as 50 square centimetres.

I would like you to calculate the percentage of the whole shape which is shaded.

It is a rectangle.

So pause the video whilst you work that out and when you're ready to check press Play.

So the total area was necessary because you needed to know what percentage the shaded area was of the whole area.

So it was a rectangle.

So is base times perpendicular height, length times width, gives you 80 square centimetres.

So the percentage shaded was the 50 that was given to you, divided by the 80.

In whichever method you want to do for this, decimal is 0.

625 which is a percentage of 62.

5%.

Here we have another shaded region within a shape.

Alex and Izzy are both calculating the percentage of the shape that is shaded.

So here is Izzy's solution to finding the percentage shaded of this given shape.

It's a parallelogram and a triangle removed.

So just take a moment to have a look at Izzy's solution.

And here is now Alex's.

So they both get to the same answer but whose is easier to follow? So on the screen we've got both of their solutions.

So you've got Izzy's on the left and Alex on the right, they both come to the same answer which is the correct answer, but whose is easier to follow? Well I would argue that Alex's is easier to follow because he has written down each stage of his working and in a multi-stage problem, which as you move through mathematics they become more of you need to be able to communicate what each calculation represents.

So Alex has written area of a parallelogram equals eight times three equals 24 square centimetres.

And if you look, Izzy does have that within her working out as well.

Area of a triangle, Alex has written half times five times 1.

5.

So that is half times base times perpendicular height equals 3.

75.

Again, if you look within Izzy's working there are two calculations, a half times 1.

5 and then the answer of that which is three quarters.

She's then written three quarters times five equals 3.

75.

So those two stages of working were both for her area of a triangle, but it's not clear to know that because she hasn't communicated it.

The area which is shaded 20.

25 from Alex's, we can see that came from a subtraction, a subtraction of the two previous worked out answers, whereas Izzy has got that value of 20.

25 but it's not clear where that came from.

You've got to do the sort of middle work to try and figure out why and how and what it represents.

And then lastly, the percentage shaded, which is what the question was on asking for.

Alex has shown this the calculation.

So area that's shaded divided by the area of the parallelogram, which was the whole shape multiplied by 100 to get from a fraction to a percentage.

Whereas Izzy has written the decimal.

Where's that decimal come from? You've got to do a lot of work to work out where that decimal has come from and that's equivalent to 84.

375%.

So I would encourage you that whenever you have got a multi-stage problem that you do, write down little notes as to what the calculation is actually working out.

So a check, using Sam's work solution, add the dimensions to the shape.

You'll recognise the shape, it's the same shape that Alex and Izzy were working with, but the dimensions are different.

So pause the video whilst you work through Sam's solution to work out what the dimensions of the shape are.

Press Play to check.

So from the working out, we've got the area of a parallelogram.

So we know that the area of a parallelogram is based on perpendicular height.

So we know that five and seven need to be placed onto the diagram in those positions.

If you had them the other way round, that's absolutely fine.

From the way the diagram's been drawn, it suggests that the seven would be the base of the parallelogram and then the area of the triangle is half times four times two.

So that tells us that one of the lengths is four and one of the lengths is two, the perpendicular lengths that is, and we can place that onto the diagram.

So moving on, we've got this wonderful looking shape, compound shape with some removed shapes, basic shapes looks like two triangles.

So given that the area of the shaded region is 24 square centimetres and the area of the unshaded region is two square centimetres, what percentage of the whole area is shaded? Sam says, well it's two out of 24, which is a decimal of 0.

083 recurring.

So that's 8.

33% to three significant figures.

Jun says no, it is 24 out of 26, which is a decimal is 0.

923076 all recurring, which is approximately 92.

3% who is correct? So just have a moment to look at those, consider their answers and do they seem sensible? So Jun is correct.

The percentage shaded was given as 24 and we then needed the whole area.

The whole area is the sum of the shaded and the unshaded region, which is 26.

In terms of a sense check.

Sam's is a very low percentage that is shaded.

We don't know that this diagram is drawn accurately, but there is a suggestion from them the values that the shaded region is much larger than the unshaded region.

So therefore the percentage should be much higher like Jun's.

Here's another check for you.

So a shape has an area of 40 square centimetres and an area of 24 square centimetres is unshaded.

What percentage of the shape is shaded? Pause the video and when when you're ready to check your answer, press Play.

It was A, the values that were given was for the unshaded, so you needed to do a subtraction to get how much area was shaded, which would be 16.

So then 16 out of the 40 was shaded.

What percentage is 16 of 40? Well that's 40%.

If you went for B, you can see what you've done there.

You used the given numbers and that gave you the percentage of the shape that was unshaded and you can see that 40% and 60% makes 100% if you sum those together 'cause they're the compliment of each other, whereas C doesn't make any sense because if you've got to the value of 66.

7 which is rounded value, so two thirds, then you've worked out what percentage of the unshaded is shaded.

Well, it can't be unshaded and shaded.

So that one doesn't make any sense.

So we're onto the first task of the lesson and question one has three parts and you need to calculate the percentage shaded in each shape.

So pause the video while you work through question one.

When you press Play, we'll move on to question two.

So here's question two.

You need to calculate the percentage shaded once again, given that the area of the unshaded region is 12 square centimetres.

So pause the video whilst you work this one out.

When you press Play, we've got one more question in task A.

So question three work out the percentage shaded of the shapes and this time I'd like you to give it to one decimal place.

So do use a calculator or if you're doing it by hand that's great, but round your answers to one decimal place.

So pause the video whilst you work through the three parts.

When you press Play, we're gonna go through our answers to task A.

Here's the answers to question one.

So on all three parts you were given the area of the shade of region and the area of the whole shape and you therefore just needed to find the percentage, you didn't need to do any finding of the area.

So for part A it was eight out of 64, what percentage of 64 is eight? Well it's 12.

5%.

Part B, you asked yourself the same question but different numbers.

So what percentage of 64 is 12? Well it's 18.

75%.

And lastly part C, what percentage of 64 is 20? 31.

25%.

You are asking yourself that question, but you can do it by writing it as a fraction, getting a decimal and converting it to a percentage.

Onto question two.

This one you did need to do a bit of calculation yourself.

So you were given the unshaded region but you needed to get the area of the whole shape.

It was a compound shape, it was a rector-linear compound shape.

And so the way that the method is on the screen here is that I've completed the shape, I've sort of made it into a rectangle and then I've subtracted the two additional areas which were two squares, both four by four.

We know that they are four by four by doing a subtraction between the 10 centimetre and the 14 centimetre edges.

So the area of the whole shape is 108.

The area of the shaded region is therefore 108 minus 12, which gives you 96.

So the percentage of the shaded region is by doing 96 divided by 108 and then multiplied by 100.

That's approximately 88.

9%.

I didn't ask you to for a given accuracy here.

So if you've rounded that to 89 and got it from the correct working out, then that's absolutely fine.

Question three, work out the percentage shaded to one decimal place.

There were three parts, part A and part B are on the screen now.

So here you were told the shaded region and the unshaded region.

So you had to recognise that you didn't have the area of the whole shape explicitly and therefore you needed to sum them together.

So for part A, the total area would be 84, that comes from 72 plus 12 and then the percentage which was shaded was 85.

7% to one decimal place.

Similar on B.

So you needed to sum them to find the whole area, which would be 68 and then the percentage which was shaded was 91.

2% to one decimal place.

part C, then the area of the shaded region was given to you but you weren't given the area of the unshaded like the previous two parts.

You needed to calculate that.

It was a triangle so you needed to use half times base times perpendicular height and the triangle had an area of eight.

That triangle is the unshaded region.

So by summing the shaded region and the triangle, we get the total area and then we can find our percentage 90.

5% to one decimal place.

So we're now up to the second learning cycle where we're going to use what we've just seen and calculate the percentage shaded for a variety of different diagrams. So continuing to find the percentage which is shaded and we're gonna have a lot more area calculation on this learning cycle.

So Aisha is doing some cooking.

She says, "I have a sheet of pastry that is 350 millimetres by 230 millimetres.

I've cut out circular pastry lids.

They are circles with a diameter of 35 millimetres.

What percentage of pastry do I have left to reroll?" So the diagram there shows you how she has cut these circular lids out from her rectangular pastry sheet.

So the area of the pastry sheet can be calculated from the given dimensions of a rectangle 350 by 230, so that's 80,500 square millimetres.

We then need to work out how much she has removed.

So just like in the previous learning cycle when we were thinking about shaded and unshaded regions using the diagram, we are trying to get to the area of the shaded, which is the amount of pastry that is left.

So we need to work out how much has been removed.

By doing a quick count, there are six rows of 10 lids, an array of 60.

So we've got 60 lots of an area of a circle which has a diameter of 35.

So we don't want the diameter for finding the area, we want the radius so we can half that which is 17.

5.

The total area of all the circular pastry lids that she managed to cut out and remove was 18,375 pi squared millimetres.

So how much pastry is left? Well that's gonna come from a subtraction.

I'm leaving it in terms of pi, so that I haven't got any rounding errors and now I want to find the percentage of the pastry that is left.

So we want to know the area of the remaining pastry out of the area of the original amount of pastry.

So this would be our calculation.

If we use our calculator using the programmed pi button, we can calculate that that is 28.

3% pastry that she has left to three significant figures.

So here is a check for you.

She has a second pastry sheet, same size and decides to cut the circular pastry lids of the same size out in a different pattern.

So you can see here that this is not now a six by 10 array, but instead she's done them alternating in the gaps.

So what your question is, what percentage of pastry is left? So using the calculations and the values that have already been done, what percentage of pastry is left? Pause the video and when you're ready to check press Play.

Hopefully you got to 19.

9% to three significant figures and I would imagine you were already expecting a lower percentage than the one that she did previously because she managed to cut more lids.

She managed to cut 67 lids this time because of the way that she's arranged them.

So still thinking about circles, the radius of the inner circle is three centimetres and the radius of the outer circle is six centimetres.

So here we've got concentric circles by the looks of it that that means they've got a common centre.

The inner circle, that white one or the unshaded has a radius of three and the outer one has a radius of six.

So without calculating it at all, without picking up your calculator, doing any maths within your head, I just want you to estimate the percentage of the whole shape that is shaded.

Okay, so I've now moved that inner circle.

They're no longer concentric circles, but the circles have the same dimensions as before.

It's just that the inner circle has been placed in a different location.

Do you think the percentage shaded is the same? If you were to do the estimate again, would you stick with your original estimate? Finally, I've moved it again and this time the circumferences are meeting.

I would like you to calculate the actual percentage shaded and compare it to your last estimate.

So pause the video while she grabbed your calculator.

Do the sums to work out what percentage of that diagram is shaded.

Given that the inner circle has a radius of three and the outer circle has a radius of six.

We should have both done the same working.

So the area of the larger circle is 36 pi because the radius was six, so pi times six squared.

The area of the smaller circle or the inner circle would be 9 pi, which means that there is an area of 27 pi remaining, once you have removed, you've subtracted the 9 pi.

What percentage is therefore shaded? Well, it would be 75% because 27 pi out of 36 pi is equal to three quarters and that is 75%.

Did you overestimate or underestimate on your first estimate, did you make it better with the second diagram or was you really far away? I would imagine that you probably underestimated how much was remaining shaded with this removal of the circle from the very first diagram.

So a check for you with a diagram and circle.

So a circle has two congruent circles removed from it.

What percentage of the circle is shaded? So pause the video whilst you do the working out to get to the percentage that is shaded in that diagram.

Given that those two congruent circles have been removed.

Press Play when you're ready to check.

The answer is 50%.

So the area of the large circle would be 16 pi.

The diagram shows us that the maximum distance across is eight, which is the diameter.

So the radius would be four pi r squared gives you the 16 pi.

The area of the smaller circle, well again, you've got a diameter marked there.

So the radius would be two, pie times two squared is 4 pi.

There were two congruent circles.

So we need to take away two 4 PIs and that leaves you with 8 pi.

What percentage of 16 pi is 8 pi? Well it's 50%.

Now we've left circles alone for a moment, but this similar idea of a removal to make a shaded region.

So this shape is made by two squares.

The larger square is twice the size of the smaller square.

So we can see that the larger square has a length of two centimetres, which means that the smaller square would have a length of one.

What percentage of the shape is shaded? We go through the same process.

The area of the larger square would be four, two squared gives you four.

The area of the smaller square would be one, one times one is one.

So how much area is left when you remove that smaller square? Well, there would be three square centimetres left.

So what percentage is three of four? 75%.

Now, we've continued the pattern and we've now got three squares.

The largest square has a length of three.

The middle square has a length of two and the inner square has a length of one.

The area of the largest square would therefore be nine.

The area of the middle square would be four and the area of the small square would be one.

So how much shaded area is there? Well the shaded region would be if we took the whole large square, then removed that middle square, cut it away, but then we need to put back the area of the small square because that is also shaded.

So that's a plus, which gives us six square centimetres of area.

What percentage is that of the larger square, the shape within the shape, the shaded area within the larger square? Well that would be 66.

7% to one decimal place and that's because it was two thirds, six over nine is two over three, two thirds.

Moving the pattern one more along then four squares.

The largest square has a length of four centimetres.

Then the second largest is three.

The second smallest is two and the smallest is one.

So there's always a difference of one centimetre with their lengths.

What percentage of this shape is shaded? Once again, we're gonna go through the process of working out all the individual areas and then by subtracting and adding, we're gonna find our shaded region.

So the largest square would be 16, second largest, which is the one with the length of three centimetres would be nine, second smallest would be four, and our most inner square would be one.

So what is the area of the shaded region? Have a go yourself before I go through the steps.

So how much area is shaded on that diagram? It would be 10.

And that's again by taking the full large square of 16, removing the second largest square, but then adding back on the second smallest because there is shaded region within there and subtracting that inner square because that bit is unshaded that comes out as 10.

What percentage of the shape is shaded? 62.

5%.

Because the fraction is five eighths, which is equivalent to 62.

5%.

Lastly, we're gonna move to five squares.

The measurements are all in centimetres, and once again each square is one centimetre longer or wider than the one before.

What percentage is shaded? Largest square is 25, second largest is 16, then it's nine, then it's four, and finally it is one.

So again, we need to calculate the area of the shaded region by doing some subtraction and some addition and that gives us 15 square centimetres.

So 15 square centimetres is still shaded despite the fact there has been some removal of area.

What percentage is shaded? 15 as a percentage of 25 gives you 60%.

So 60% of the original largest square is still shaded.

If I now put that into a table and I've included a full square with no removal, so that's that first column.

I've got two rows that show us the two areas.

So we've got the shaded area and we've got the area of the whole shape.

And then at the last row is showing us the percentage which is shaded.

So if you've just got one square, then the area is one, the shaded area is also one, 100% of it is shaded.

If we then move on to the first one we looked at where I'd removed the smaller square, we saw 75% and we saw 75% on the logo.

We saw 75% with the circles.

Then we got to three and we ended up with 66.

7%.

Then we got to the four, we got 62.

5% and that last one with five we got 60%.

So the amount that is shaded is decreasing as we increase the amount of squares.

Interestingly though, if you look at the value of the area of the shape, they are the square numbers one, four, nine, 16, 25.

Which isn't too surprising because our largest area came from the square, the largest square and the largest square had the dimension of sort of our term number, if we think of it as a sequence.

But a check for you, the area that is shaded.

So that top row is also a familiar number sequence.

What are these numbers known as? So pause the video, you might want to discuss that.

It might be that you need to go and do a bit of research, look back at your number sequence work, but they are a known number sequence, So press Play when you're ready to check.

They are the triangular numbers.

So one, three, six, 10, 15.

So our area of the shaded is our triangular numbers and the area of the shape is our square numbers.

We're up to the last task.

And in the question one, I want you to calculate the percentage of the shape which is shaded.

So pause the video whilst you're working out both the area of the whole shape and the area of the shaded shape to therefore be able to work out the percentage.

Press Play when you're ready to move on.

Here's question two.

So I'd like you to calculate the percentage shaded in each of these concentric circle designs where the radius increases by one centimetres each time.

So pause the video, it's gonna be very similar to the process of me doing the squares.

And then when you're ready to move on to the last question, press Play.

And question three, which of the following have a shaded region closest to 50% of the whole shape? You'll note here that there are no dimensions, which gives you some choice, but be sensible in the choice that you make depending on how many circles are crossing the circle, the diameter.

Press Pause whilst you work through question three.

And then when you press Play, we'll go through our answers to task B.

Question one, this is part A only, we'll move on to B and C in a second.

You needed to get the area of the whole shape.

It was an L-shape, it was a rector-linear compound shape.

The way the method here is it's been split into two rectangles, but it's area is 170 square centimetres.

The rectangle that has been removed, the unshaded region would be 44 square centimetres, which means that the area of the shaded region is 126 and therefore the percentage of the shape that is shaded would be 74.

1%.

Part B, again, you needed to do some area calculations.

So area of a trapezium is 78.

2.

In this circumstance, it was a square that had been removed, so that would be 25, which meant the shaded region would be 53.

2 square centimetres.

And that means that the percentage of the shape that is shaded is 68.

0% to one decimal place.

Lastly, we're onto a semicircle with a circle removed and the radius of the semicircle is the diameter of the circle.

So the area of the semicircle would meet to half because a semicircle is a sector that is half the area of the full circle, pi r-squared, re know the radius is two root two.

If you were using account base, it's really important to get those brackets around two root two when you do the squaring, that has an area of 4 pi.

The area of the circle which was removed, as I say, the radius of the semicircle is the diameter of this circle.

So you needed to half it to get your radius, and that came out as 2 pi.

Therefore, the area of the shaded region was 2 pi and that meant the percentage was 50%.

Question two, calculate the percentage shaded in each of the concentric circle designs.

Well, for part A, 100% of it was shaded.

There was no real calculation necessary there.

For part B, the radius would've increased by one each time.

So the inner one would have a radius of one and the out one would've had a radius of two.

Find the area of the larger one, find the area of the smaller one, subtract it to get the area of the shade of region only, and then find the percentage that came out as 75%.

Part C, once again, you've now got a circle of one, a circle of two, and a circle of three.

In terms of their radii.

You need to work out their three areas, do some subtraction and some addition now because you'd removed a circle but then added another circle back and that came out as 66.

7 to one decimal place.

D was 62.

5%.

By following the same process, your four circles and now have a radius of one, two, three, and four.

And then finally on E it was 60%.

Radius of one, radius of two, radius of three, radius of four, and radius of five.

And I'm hoping that you notice that interestingly, despite the fact that this is now a circle, when we did it with a square, we get the exact same values.

So 100%, 75%, 66.

7%, 62.

5%, and 60%.

An interesting extension and challenge for you is how many circles or how many squares if you prefer to do it with squares, do you need to get to 50%? And lastly, question three, which of the following has a shaded region closest to 50% of the whole shape? So shape B, because it had 62.

5% shaded compared to C, which had 64% and 66.

7% in A.

I did say to you you needed to be sensible with the dimensions that you chose.

So on A three diameters of the smaller circle is equal to the larger circle's diameter.

So I would've chosen a value that had a factor of three so that I could divide it by three to get the diameter of my inner circle.

And similar for B, there was four smaller circles that fit across the diameter of the larger circle.

So choose a diameter that would be divisible by four.

And lastly on C, choose one that was divisible by five.

So in summary, if you have two areas, then one can be expressed as a fraction of the other and it can be a percentage as well, because we know fractions and percentages are equivalent.

It may be the remaining area of a shape after some has been removed as a percentage of the original area.

And something that we touched on during the lesson is that laying out your work clearly is very helpful when there are multistages to a solution, labelling each calculation so that you can follow it through and so can somebody else reading it.

Really well done today and I look forward to working with you again in the future.