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Hi, I'm Mr. Tazzyman, and today, I'm going to be teaching you a lesson from a unit that is all about understanding equivalence and how you can use it with calculations involving addition and subtraction.
So make sure that you are ready to learn, so we can get started quickly, here we go.
We'll begin with the outcome of the lesson, then.
I can explain how to balance equations with addition or subtraction expressions in different contexts.
You need to be confident in that, by the end of this lesson.
Here are the keywords that you might hear, perimeter, expression, and area.
I'm gonna say them and I want you to repeat them back to me.
So I'll say my turn, say the word, and then I'll say your turn, and you can repeat it back, ready? My turn, perimeter, your turn.
My turn, expression, your turn.
My turn, area, your turn.
Here are the definitions for those keywords, then.
The perimeter is the distance around a 2D shape.
An expression contains one or more values, where each value is separated by an operator.
Area is the measurement of a flat surface.
It measures 2D space.
This is the outline for the lesson, then.
we're starting with addition expressions with perimeter, then we're moving on to looking at subtraction expressions with area.
These are the friends that we're gonna meet today, that will help us.
Aisha, Izzy, and Alex, hello, you three.
They're gonna be discussing some of the maths prompts that you'll see on screen, and giving us some hints and tips along the way.
Okay, let's get started properly, then, time to learn.
A lawn is an area of grass.
Aisha, Alex, and Izzy are looking at plans for lawns for the Oak Academy garden.
They start by thinking about the perimeter of each lawn, so they can put a paved border around each one.
Aisha says, "I think lawns A and B have the same perimeter." Well, you can see the lawns on the left there, A, B, and C.
For A, the dimensions are 10.
6 metres and 4.
2 metres.
For B, it's 12.
2 metres and 2.
6 metres, and for C, it's 11 metres and 3.
7 metres.
Alex says, "I think lawns B and C have the same perimeter," so they are disagreeing.
Who do you think is right, hmm? Alex says, "I can write an addition expression for the perimeter of lawn A," and he's written 10.
6 plus 4.
2, plus 10.
6 plus 4.
2.
You can see, he's actually written down all four sides there, and he's added them all together.
"I can also think of perimeter as the length, doubled, plus the width, doubled," so there's a slightly different expression been written there, 10.
6 times 2 plus 4.
2 times 2.
Aisha says, "I'm going to double each length to simplify the expressions," and that's been written on the diagram of the lawn there, underneath each letter.
We've got 10.
6 times 2 plus 4.
2 times 2, 12.
2 times 2 plus 2.
6 times 2, and 11 times 2 plus 3.
7 times 2.
She's then calculated each of the multiplication parts in those expressions to give 21.
2 plus 8.
4, 24.
4 plus 5.
2, and 22 plus 7.
4.
Alex says, "I can see that C will have a different length immediately.
The decimal part of the sum will be 0.
4 for lawn C, but 0.
6 for lawns A and B." Izzy says, "I think we can be even more efficient and use what we know about equivalent expressions.
I can also think of the perimeter as the sum of the length and width, doubled.
10.
6 plus 4.
2 times 2." So Izzy's decided to add those two sides together first and then double.
She does the same for the others.
"I'm doubling each time, so just need to compare the addition part of each expression." So she knows that each of those expressions has been multiplied by 2, so she doesn't need to worry about that bit, she can just compare the addition.
"Lawn C has a sum with a decimal part of 0.
7, and A and B have a sum with a decimal part of 0.
8.
So lawns A and B have the same perimeter and lawn C has a different perimeter." Okay, it's your turn.
I'd like you to represent the perimeter of this lawn in different ways.
Pause the video here, and have a go.
Welcome back.
Here's one way, 9.
6 plus 10.
4, plus 9.
6 plus 10.
4.
Here's another way, 9.
6 times 2, plus 10.
4 times 2, and that gives you 19.
2 plus 20.
8.
You could also have put 9.
6 plus 10.
4 times 2, 9.
6 plus 10.
4 has been put into brackets because that is what you need to do first.
That gives 20 times 2.
So all of these are ways of representing the perimeter using an expression.
Aisha, Alex, and Izzy look at a different problem.
Both lawns have the same perimeter.
How long is the side marked x? We've got an unknown, so we don't know what the side is, but we know that both of them have the same perimeter.
A has dimensions of 87.
36 metres and 37.
44 metres.
B dimensions of 56.
16 metres and x, we don't know it yet.
Alex begins with lawn A.
"I'm gonna find the perimeter of lawn A using doubling of the known sides.
87.
36 times 2 plus 37.
44 times 2," and Alex says, "Double 87.
36 is tricky mentally for me.
Double 87 is 174 and double 0.
36 is 0.
72." What do you think, hmm? "Wait! I think we can find the missing length more efficiently." Aisha thinks she's got a more efficient way of doing it.
"So we know that doubling the length and width gives us the perimeter.
The perimeter of A and B is equal.
So we can say the sum of the length and width of each lawn must also be equal." Mm, great reasoning, Aisha.
"I can find the sum of the first expression.
87.
36 plus 37.
44 equals 124.
8, and subtract 56.
16 To find the value of x.
124.
8 subtract 56.
16 is equal to x.
That gives a value of 68.
64." Izzy has a different strategy for finding the missing side.
She started with that same equation, 87.
36 plus 37.
44 is equal to 56.
16 plus x, the unknown.
"I can use the relationship between the known addends.
The difference between 87.
36 and 56.
16 is 31.
2." So she is comparing two of the sides of lawn A and lawn B, and finding the difference, which is 31.
2.
"So the missing length must be 31.
2 more than the other known length of 37.
44.
37.
44 plus 31.
2 is equal to 68.
64.
That is the missing length, 68.
64 metres.
I can show this using bars to represent the sum of the two dimensions of each lawn." So she starts with the first bar, which is the sum of the dimensions of lawn A, 87.
36 and 37.
44.
Then she draws out the bar that represents the sum of the dimensions in lawn B.
56.
16 plus x.
"The first bars in each have a difference of 31.
2, so to maintain equivalence, the bar marked x needs to be 31.
2 more than 37.
44." You can see that they're drawn with an arrow on the bars.
So now she knows that x must be 31.
2 plus 37.
44.
Okay, it is time for your practise task.
Find the missing dimension labelled x for each pair of lawns, which have the same perimeter.
Consider which strategy you find more efficient.
So there's Number 1, with two lawns to compare and an unknown in lawn B.
Here's Number 2, this time the unknown is on lawn A, and there's Number 3, this time, the unknown is in lawn B.
Pause the video here, have a go at these, and I'll be back shortly with some feedback.
Welcome back, let's look at Number 1.
Here are some of the jottings that you may have used.
We'll start with the top part, 28.
65 plus 67.
63 times 2 is equal to 49.
95 plus x times 2.
Of course, both of those expressions feature a times 2, so we can actually disregard that, to get closer to the value of our unknown.
We end up with an equation, which just compares the addition of the two dimensions of each of the lawns, 28.
65 plus 67.
63 is equal to 49.
95 plus x.
You may have calculated the sum of the first expression and found the difference between this and the known addend.
You can see below, 96.
28 is equal to 49.
95 plus x, so 96.
28 subtract 49.
95 is equal to x, and that gives a value of 46.
33.
Pause the video here if you need a bit more time to mark carefully.
Here's Number 2, then, using a similar method to calculate the unknown.
You might have started with 43.
4 plus x times 2 is equal to 36.
9 plus 32.
7 times 2.
Again, you can disregard the doubling, and just think about the addition of the two dimensions for each lawn, that would give you 43.
4 plus x is equal to 36.
9 plus 32.
7.
Again, you may have calculated the sum of the second expression and found the difference between this and the known addend.
It was the second expression you needed to calculate here because that was the one that featured known parts.
On this occasion, the unknown was in lawn A, so you would've ended up with an equation that read, 43.
4 plus x is equal to 69.
6, then it's time to use the inverse, 69.
6 subtract 43.
4 is equal to x, so x is equal to 26.
2 metres.
Pause the video here if you need some more time to mark.
Here's Number 3, then, the same sort of method again, and you would've ended up with an equation after disregarding that doubling, and just looking at the addition of the dimensions, that red 90.
65 plus 37.
1 is equal to 70.
45 plus x.
You may have calculated the difference between one side length in A, and the known length in B, and then adjusted the other length, a slightly different method here, so you would end up with 90.
65 take away 73.
45 is equal to 17.
2.
We've taken the length of the top side of lawn A and the length of the side on lawn B, and found the difference.
Side length x is 17.
2 more than 37.
1.
37.
1 plus 17.
2 is equal to 54.
3.
Pause the video here if you need some more time to mark.
It's time for the second part of the lesson, then, this time, we have subtraction expressions with area.
These lawns have ponds dug into them.
You can see the diagram below, the green represents the lawn, and the blue represents the pond.
Aisha, Alex, and Izzy explore the area of the lawn part and the pond part.
"The area of a rectangle can be found by multiplying length and height.
This whole area is 96 metres squared.
The pond has an area of 12 metres squared, so the lawn part area is the difference between 96 and 12.
The area of the lawn is 96, subtract 12, which is equal to 84 metres squared.
' A and B are made up of lawn and pond.
Both lawns have the same area.
What is the area of the pond in B? And it's important to note here, that both lawns have the same area, but they don't have the same dimensions.
That's where we need to use our understanding of equivalence to help us out.
"I can write the area of each lawn as an expression," says Aisha.
So for lawn A, 5 multiplied by 7, that's the total area of that space, but you need to subtract the area of the pond from that, which is 3 multiplied by 4, That gives you the area of lawn A.
For lawn B, similar concepts, but, of course, we have different dimensions.
The total space is 11 multiplied by 4, but you need to subtract 3 multiplied by the unknown, which is X, we need to work that out.
"The lawns have an equal area, so I know these expressions are equal.
I can calculate the area of lawn A and use that to find lawn B." Okay, to check your understanding, you're gonna help Aisha by calculating the area of lawn A.
Pause the video, and have a go.
Welcome back.
So you might have started with the expression that gave you lawn A's area, 5 multiplied by 7, subtract 3 multiplied by 4.
Then you might have, using your times tables got the value of each of those multiplication parts, giving you 35 subtract 12, which is 23, so the lawn is 23 metres squared, that's the green part in this diagram.
Okay, let's return to that problem, then.
We now know that the lawn part is 23 metres squared.
We don't yet know what B is.
"I know the area of the lawns are equal, so I can use that to help me find the solution for B." Could Aisha should be more efficient here, though, hmm? Well, Alex shares his strategy for finding the area of pond B.
"I know the lawns parts are equal so I can write the expressions as an equation." Good thinking, Alex.
"5 multiplied by 7 subtract 3 multiplied by 4 is equal to 11 multiplied by 4, subtract 3 multiplied by x.
I can find the products of the multiplication expressions." So he replaces each of those multiplication expressions with the product.
The equation now reads 35 subtract 12 is equal to 44 subtract 3 multiplied by x.
"And I'm finding the area of the pond, so I'm just going to rewrite 3 multiplied by x as just p." So now he's got 35 minus 12 is equal to 44 subtract p.
"Now I can use what I know about equal subtraction expressions.
44 is 9 more than 35, so p must be 9 more than 12." That's shown with these arrows.
The minuend is 9 more in the second expression, so the subtrahend also needs to be 9 more than the subtrahend from the first expression.
"The area of the pond part of B is 21 metres squared." Okay, time for you to have a go.
If the area of the pond is 21 metres squared, calculate the missing side x.
Pause the video, and have a go.
Welcome back, let's start with this equation.
3 multiplied by x is equal to 21, x is equal to 7 metres.
Okay, it's time for your second practise task, then.
In each example, the lawn part has the same area in A and B.
Calculate the area of the pond part and unknown length x.
so you've got 1A and 1B.
Remember, they're not the same shape, but they do have equal area, the lawns, that is.
Here's Number 2.
The area of the whole garden is 96 metres squared and the area of the lawn is 60 metres squared.
What could the dimensions of the overall garden be and what could the dimensions of the pond be? Draw labelled sketches of the different possibilities and complete the equation for each sketch.
A multiplied by B, take away x multiplied by y is equal to 60.
Okay, pause the video, and have a go, good luck.
Welcome back, it's time for some feedback, then, so be ready to mark, and let's see how you got on.
Here's 1A.
The area of the lawn in A is 16 multiplied by 8, subtract 7 multiplied by 6, which is equal to 128 takeaway 42.
The area of the lawn in B ,is 12 multiplied by 9, subtract x multiplied by 2, which is equal to 108 subtract p, p is the area of the pond.
128 subtract 42 is equal to 108 takeaway p.
We know this because we know that the area of the lawns for both A and B are equal, so we can use those expressions to create an equation.
You can see that the minuend has been adjusted by subtracting 20, which means the subtrahend also needs to be adjusted in the same way by subtracting 20, so that we maintain a constant difference, and we can say that this is an equation.
P is equal to 22 metres squared, so that means x is equal to 11, and that's because the dimensions of the pond, we already know were 2, and we know that the other side must therefore be 11 because 2 multiplied by 11 is equal to 22 metres squared.
Okay, pause the video here, and mark that carefully if you need to.
Let's do B, then.
Same sort of problem, same concept, different numbers.
The area of lawn A is 13 multiplied by 8 subtract 6, multiplied by 4, which is equal to 104 takeaway 24.
For B, it was 12 multiplied by 12 subtract x times 8, x was the unknown, which is equal to 144 subtract p, which is the area of the pond.
104 takeaway 24 is equal to 144 takeaway p.
If we look at the adjustments here, we can see that the minuend has had 40 added to it, which means that the subtrahend needs to as well, 24 plus 40 gives us the value of the area of the pond, that would be 64 metres squared, because we know that one of the dimensions of the pond is 8, we need to work out what we would multiply that by to get to 64, and that's 8, so x is equal to 8.
Again, pause the video if you need some time to catch up with that marking.
Okay, let's do Number 2, then.
We were drawing some labelled sketches of different possibilities.
We need to start by looking at what we knew already.
We knew that a multiplied by b subtract x multiplied by y was equal to 60.
We also knew that a multiplied by b must be equal to 96, and that's because in the question it said, the area of the whole garden is 96 metres squared.
96 subtract p, the area of the pond, was equal to 60, so p equals 36 metres squared, which is also equal to x multiplied by y, p was the area of the pond, remember.
Okay, here's a possible solution, then, a times b minus x times y is equal to 60.
12 times 8 minus 6 times 6.
96 subtract 36 is equal to 60, so the dimensions you could see on that sketch worked with the equation that we had.
Have a look at this one.
Why is this not a possible solution, do you think? 48 times 2, subtract 4 multiplied by 9 is equal to 60.
96 subtract 36 is equal to 60, but the equation is correct, but if b equals 2 metres and y equals 4 metres, the pond would not fit in the garden, look at the sketch, you can see that the pond actually goes over the boundaries of the garden, so it wouldn't work.
96 has factors of 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 40, and 96.
36 has factors of 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Not all pond sizes will fit in the garden in certain orientations.
The sketches are important to work this out.
Okay, pause the video here if you need some extra time to mark those carefully.
It's time to summarise the lesson, then.
Problems in the context of area and perimeter can be expressed as balanced addition or subtraction equations.
Yo balance an equation with addition expressions, find the unknown value by calculating the value of the expression with known parts, and then find the difference between that value and the known part in the addition expression featuring an unknown.
Equations with subtraction expressions and unknown values can be balanced.
If the minuend and subtrahend are changed by the same amount, the difference stays the same.
My name is Mr. Tazzyman, I've really enjoyed learning with you today, and I hope you did as well.
I'll see you again soon, bye, for now.
