Lesson video

Hi, my name is Mr. Taziman and I'm looking forward to learning with you today.

This lesson is from the unit all about equivalence and compensation with addition.

It'll give you some extra hints and tips on how to face addition problems and solve them mentally, rather than jumping straight to a written method.

Here's the outcome for today's lesson.

I can use equivalence and compensation strategies to solve problems. These are the keywords that you're going to be hearing during this lesson.

I'm gonna say them and I'd like 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? Okay then.

My turn.

Equivalence.

Your turn.

My turn.

Compensation.

Your turn.

My turn.

Unknown.

Your turn.

My turn.

Expression.

Your turn.

Okay.

Let's make sure that we're clear about what each of those words actually means.

Equivalence is when two or more things have the same value.

Expressions are said to be equivalent if they have equal value.

The equal symbol is used to show equivalence.

Compensation is a mental strategy which involves adjusting parts of an equation while maintaining equivalence.

An unknown is a quantity that has a set value, but it is represented by a symbol or letter.

In this lesson, it's often represented by an empty box.

An expression contains one or more values where each value is separated by an operator.

Two or more expressions with the same value can be separated by an equal sign to create an equation.

You can see at the bottom there, examples of both an unknown and an expression.

Here's the outline, then.

It's use equivalence and compensation strategies to solve problems. To begin with, we're gonna start with some equivalence puzzles.

Then we're gonna move on to compensation problems. In this lesson, you'll meet Lucas and Izzy.

They're gonna discuss some of the maths and give us some help to make sure that we fully understand what we're learning about.

Hi Lucas.

Hi Izzy.

Lucas and Izzy discuss equivalence.

62 plus an unknown is equal to 80.

"How can we use equivalence to solve problems?" Says Lucas.

"We can use it to find unknowns." There's the unknown labelled.

"I know that both expressions have the same value." She puts down 80 as a whole, 62 as one of the parts, and then the unknown.

"I know that the difference between 80 and 62 is 18.

The unknown is 18." Lucas sets Izzy a puzzle.

"Use the number cards to complete the equation.

You can only use each card once." "A great challenge! I'll start by pairing up the greatest and smallest." 1.

1 plus 2.

1.

"I'll do that because I know that we need equivalence, so I can't have all the greatest in one expression and all the smallest in the other expression.

2.

1 is large compared to the rest, so I'll put 1.

2 in next.

Now, I'll fill in the other expression.

I'll combine the parts to get the value of each expression." 4.

4 is equal to 4.

4.

The expressions are of equal value, so it's correct.

"Well done!" Says Lucas.

Okay, your turn to have a go at something similar.

Complete the equation using the number cards.

Pause the video, and I'll be back in a little while to reveal the answer.

Welcome back.

Izzy's gonna talk us through it and she says, "I'll start with the greatest and smallest paired." There they are.

And then she puts the others in their place and she adds them together and realises that both expressions have a value of 4.

9.

Lucas and Izzy look at a magic square.

What do you notice? Look at the numbers.

Is there any relationship between the numbers? What happens when you add some of them together? "The rows, columns and diagonals all sum to 15." "They're all equivalent," says Izzy.

Lucas and Izzy look at a magic square with missing numbers.

"Let's find the missing numbers." "We know that the numbers added together in each row, column and diagonal are equivalent to 15." "How?" Six plus five plus four is equal to 15.

"15 is the magic number!" So that's what you call the magic number in a magic square.

It's the total of the rows or the diagonals or the columns.

"I can calculate bottom left." Something plus three plus four is equal to 15.

"It's eight." "Let's try a vertical one." Middle left.

Six plus an unknown plus eight is equal to 15.

"It's one." They find the rest of the missing numbers.

Okay, your turn.

Find the missing numbers in the magic square.

Pause the video and have a go.

Welcome back.

Let's start with the top row here.

Eight plus an unknown plus six is equal to 15.

It's one.

Then we've got the middle left missing number there, eight plus something, an unknown, plus four equals 15.

It's three.

Did you get those? I hope so.

Lucas and Izzy look at another magic square with missing numbers.

What do you notice? "This time, there are decimal fractions." "The value of each row, column and diagonal is different this time too." They start by finding out the magic number.

3.

3 plus 7.

2 plus 1.

8 is equal to.

"Should we use a written method?" Says Lucas.

"No.

Let's look closely at each part.

I've noticed something." What has Izzy noticed? "7.

2 and 1.

8 combined is equal to nine, and nine combined with 3.

3 is 12.

3.

So 12.

3 is the magic number." They find the rest of the missing numbers.

Okay, it's your turn to have a go at some practise tasks now.

For number one, I want you to complete the equations below using the number cards.

There's a, and there's b.

For number two, below is an equation with four missing numbers in each expression.

Complete the equation using each number card once.

Find three ways to complete this.

And for number three, find the missing numbers in each of the magic squares.

Pause the video here and have a go at those.

Enjoy them and good luck.

Welcome back.

Let's start with number 1a, and Lucas says "3.

57 and 9.

43 combine to equal 13, so I'll try them together.

13 and seven is equal to 20, so I'll try the seven this side too.

Now I'll use the other parts." 20 is equal to 20, so there was the solution.

Here's b.

"I can see that 28 and 72 equal 100, so I'll try 328 and 472 together.

They're equal to 800, so I'll put in 200 as the last part.

That's equal to 1000.

Now I'll try these three parts.

They add up to 1000 as well!" So there is the solution.

It's important to say for both 1a and b, you may have put the parts for each expression in a different order, but of course that's okay because addition is commutative, so the parts can swap round as much as they like.

It doesn't affect their total value.

For number two, below is an equation with four missing numbers in each expression.

Complete the equation using each number card once and find three ways to complete this.

"Here were my three solutions!" Says Izzy.

One plus eight plus two plus seven is equal to three plus six plus four plus five.

One plus eight plus four plus five is equal to three plus six plus two plus seven.

One plus eight plus three plus six is equal to two plus seven plus four plus five.

"Each expression needed to be equivalent to 18." And here are the missing numbers for the magic squares.

The rows, columns and diagonals are equivalent to 27.

75 in the first one.

We had a missing number in the top row of 10.

5.

Second row was 9.

25 and 11.

75, and in the bottom row it was 14.

25.

Now onto b.

The rows, columns and diagonals are equivalent to 26.

I'll read from left to right, going from top to down.

We had 0.

4, 9.

2, two, 2.

8, 5.

6, 2.

4, and eight, 10, and 0.

8.

Welcome back.

Let's go to the second part of the lesson, compensation problems. Izzy and Lucas solve a worded problem.

At the start of the year, there are 240 pupils in Key Stage 2 as shown in the table below.

Year three, 58, year four, 61, year five, 65, year six, 56.

Over the year, five children join year three, two pupils leave year five and one joins year six.

How many pupils are there in Key Stage 2 now? "Let's start by making Jottings on the table." Says Izzy.

"Okay, I'll highlight any increases or decreases." "So that's an addition of five in year three." And she's put a jotting of plus five on the table next to the previous year three figure.

Two pupils leave year five.

"That's a reduction of two in year five." One joins in year six.

"So that's an addition of one in year six." "Okay, let's work out each year group total now." "I don't think we need to." What do you think? The question says "How many pupils are there in Key Stage 2 now?" Do they need to calculate all of the new year group figures? "We can be more efficient." Says Izzy.

"We just need to adjust the total using these changes." Izzy shows what she means on a number line.

"The whole was 240." She adds on five to get to 245, subtracts two to get to 243, and adds on one to get to 244.

"There are now 244 pupils in Key Stage 2." "I see, you just worked out all the changes and then compensated the whole.

I think there's an even quicker way." What do you think? "We can work out the combined adjustment of each of the year groups first." Five takeaway two plus one equals four, so they've taken each of the adjustments that they jotted down on that table and put it into one equation to work out the total adjustment.

Then adjust the whole using this.

240 pupils plus four pupils equals 244 pupils.

Okay, it's your turn now.

What's the combined adjustment from the annotated table below? So we're not looking for the new figure of how many children there are in the whole key stage.

We're just looking for the combined adjustment.

Pause the video and have a go.

Welcome back.

You might have written an equation that looks something like this.

Two subtract one plus four plus one is equal to six.

Okay, Lucas and Izzy look at an inequality.

They have to state if it's correct or incorrect.

What do you think? Look closely at those numbers.

Is there anything in those numbers that will give us a clue as to whether this is correct or not? "These two expressions are of equal value." "I agree.

The parts are the same." Can you see that? That expression is exactly the same on both sides of that inequality.

"Addition is commutative, so the whole is unaffected." "It should be an equals symbol." Okay, we've got another one here.

What do you think? 13.

678 plus 45.

623 is less than 45.

623 plus 13.

677 plus 0.

001.

"These two expressions are of equal value." "But they have a different number of parts." "They do, but compare these parts." So look at the ones that Izzy has highlighted there.

What do you notice? "13.

678 has been partitioned in the second expression.

The expressions are equal, so this is incorrect." Here's another one.

18.

35 plus 18.

45 plus 18.

55 is less than 18.

65 plus 18.

45 plus 18.

45.

What do you think? "Okay, let's start by adding up each expression." "Wait, I don't think we need to." What do you think? "The first expression has three parts.

One of these parts is 1/10 less and one is 1/10 more.

That means the expression is equivalent to three lots of 18.

45.

The second expression has two parts of 18.

45, but the other is 18.

65, so it must be greater.

This is correct." Okay, time for the second task.

To begin with, you've got to solve the worded problem below.

Four children took part in a relay race with four laps.

Their combined time was one minute, 36.

8 seconds.

At the next race, Izzy was 1.

2 seconds slower, Lucas was a second faster, Jun was 0.

8 seconds slower and Sam was five seconds slower.

What was the new combined time? And Izzy puts in a helpful tip that sometimes we forget when we're thinking about racing and time.

"Remember that going faster means less time." You want fewer seconds.

Number two, are these correct or incorrect? Explain your reasoning without calculating.

So you don't need to calculate the value of each expression to work out whether these are correct or not.

Okay, pause the video here, enjoy those, and I'll be back in a little while with some feedback.

Here's number one then.

Ready to mark? Let's go for it.

We've jotted down the adjustments on the table next to each of the children.

So Izzy had a plus 1.

2, Lucas, was a subtract one, Jun was plus 0.

8, and Sam was plus five.

So they had a combined adjustment of six seconds extra, so their old time was faster.

Their new time was one minute, 42.

8 seconds.

Okay, number two, let's start with a.

"This is correct.

The second expression is 1/100 less." For b, "This is incorrect.

8,000 has been redistributed.

These expressions have equal value." C, "The first expression can be replaced with equal parts.

This is correct.

The second expression is four p more." For d, "This is correct.

The second expression is 1000 kilogrammes greater." And for e, it was incorrect.

"The parts have swapped, but their value is still equivalent." Okay, here's a summary of today's lesson then.

Equivalence and compensation are useful tools for simplifying problems and helping us to find unknowns efficiently.

This can be applied to missing number problems including magic squares.

Also, we can analyse the accuracy of inequalities by carefully analysing expressions and checking equivalence and compensation.

My name is Mr. Taziman.

I've enjoyed the learning today and I hope you have as well.

I'll see you again next time in another maths lesson.

Bye for now.