Lesson video

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.

Here's the outcome for today's lesson then.

By the end, we want you to be able to say, "I can explain how to balance equations with subtraction expressions." These are the key words that you might expect to hear and understand in today's lesson.

We've got unknown, equation, and expression.

I'm gonna get you to practise saying them.

I'll say "my turn" and say the word, and then I'll say "your turn," and you'll need to repeat it back.

All clear? Okay then.

My turn, unknown.

Your turn.

My turn, equation.

Your turn.

My turn, expression.

Your turn.

Okay.

It's all very well being able to say them, but we need to understand what each of them means as well.

So here are the definitions.

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

An equation is used to show that one number, calculation, or expression is equal to another.

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 an example at the bottom there.

We've got an equation, 4 + 7 is equal to 11.

The equal symbol makes that an equation, but it also shows us an expression example, 4 + 7, isolated by itself, is an expression.

Here's the outline for today's lesson then.

We're gonna start by comparing subtraction expressions, then we're gonna move on to balancing equations with subtraction expressions.

Let's start the first part.

But we also need to say hello to Aisha and Izzy.

Two maths friends who are here with us to help us in our thinking.

They'll be responding to some math prompts.

They'll be giving us some hints and tips.

And they might even be revealing some of the answers during feedback.

Hi, Aisha.

Hi, Izzy.

Ready to go? Okay then.

Aisha and Izzy use a balance scale.

You can see they've got 10 cubes on one side and five cubes on the other.

The scale is weighted down to the side that has 10 cubes, because 10 is greater than 5.

"Let's make these scales balance by subtracting" says Izzy.

Different one there, isn't it? We normally think of trying to make scales balance by adding something to the side that isn't weighted down.

"We can subtract cubes until both sides are equal.

I will write down expressions compared." So you can see the red crosses there that represent taking away those cubes.

We've got 10 takeaway 6, which is less than 5.

But now, Izzy takes away one.

So we've got 10 takeaway 6, is equal to 5 takeaway 1.

What do you notice? The difference is constant.

Both expressions have a value of 4.

The minuend and subtrahend have decreased by the same amount.

So you can see that 5 has been taken away from the minuend, but 5 has also been taken away from the subtrahend.

So there's a constant difference.

That means that both expressions have the same value.

The value of the expressions on each side of an equal symbol must be the same or the equation isn't balanced.

Aisha and Izzy look at a missing symbol problem featuring decimal fractions.

35.

43 subtract 24.

34.

That's the first subtraction expression.

That's being compared to the second subtraction expression, which is 35.

28 subtract 29.

16.

How would you solve this? Hmm.

Well, Aisha says, "I'm gonna use column subtraction to calculate the value of each expression." "Wait," says Izzy, "I think there's a more efficient way that's quicker.

Let's examine the numbers." Always a good idea.

Look closely at those numbers to see if you can do something more efficiently.

"Let's round the numbers in each expression to the nearest one." So 35.

43 is rounded to 35, 24.

34 is rounded to 24, 35.

28 is rounded to 35, and 29.

16 is rounded to 29.

"Now we can use these to reason with.

I know that 35 subtract 24 is greater than 10.

I know that 35 subtract 29 is less than 10.

So that means the missing symbol is greater than." And you can see that because on one side, the first subtraction expression, we know it's greater than 10 without having to actually use columns.

And on the second expression, we know that that's gonna be less than 10 without having to use columns.

Great reasoning, Izzy.

Well done.

Aisha says, "I have a different explanation with the same solution." Ah, interesting.

Always worth comparing explanations.

How else could you explain it using the rounded numbers? "Let's compare the parts from each expression." So to start with, Aisha compares 35 and 35, the first part in each of these expressions.

35 is equal to 35.

So she says, "I know that 35 is equal to 35." And then she compares the second parts in each of these expressions.

"I know that 24 is less than 29.

So we know the missing symbol is more than, because the minuends are approximately equal." "But the subtrahend in the first expression is smaller, leading to a greater difference." That's the really important part to remember.

Sometimes you can fall into the trap of thinking that because the subtrahend in the second expression is smaller, that means that that expression is smaller, but it doesn't, because of course we are subtracting the subtrahend.

Okay, your turn.

Let's check your understanding.

True or false? To work out missing inequality symbols, you have to calculate the precise value of each expression.

What do you think? Take some time and I'll be back in a moment.

Okay, that was false.

Why? Well, you can use your understanding of number and reasoning to work out which expression has a greater value.

You don't need to know the exact value, just whether it is greater or smaller.

It's time for your first practise task.

Let's see how you've got on with all of that learning.

Number one.

Use rounding, estimating, comparing, and reasoning to determine the missing symbol in each of the following inequality statements.

So you've got A, B, C, D, and Andeep says, "What do you notice?" between those.

You've also got E and F here to solve as well.

Pause the video and have a go at all of those.

Good luck.

Remember to think carefully.

Don't forget that if the subtrahend is greater, then the difference will be smaller.

Good luck.

Welcome back.

Time for some feedback then.

Make sure you're ready to mark, and I'll give you time at the end of A to D to catch up with any marking should you need it.

Okay then, here's A.

The answer was less than.

And in B, the answer was greater than.

Rounding an estimation were the most useful strategies there.

For C and D, you had, C was less than, and D was greater than.

Each addend has been made one 10th times the size, so the same reasoning can be applied.

And that is in response to what Andeep asked, which was "What do you notice?" You can see that the digits are similar, but actually, if you compare the first part of A and the first part of C, you've got 36 on A and 3.

6 on C, 3.

6 is one 10th times the size of 36.

Okay, pause the video if you need to catch up with marking.

Here's E then.

It was less than.

Did you reason here that 87,253 and 86,809 are approximately 87,000, and so the first expression had a smaller difference because the subtrahend was greater? That would've been a good way of looking at it.

What about F? Well, the answer was less than.

Did you estimate here? The first expression is approximately 60 subtract 30, and the second is 70 subtract 35, and so the first expression had a smaller difference.

Okay, pause the video here if you need more time to discuss those answers or to catch up with marking.

It's time to move on to the second part of the lesson, balancing equations with subtraction expressions.

So, how can the unknown be found? We've got here an equation that features an unknown, 25 subtract 13 is equal to 32 subtract something.

Both of these expressions are subtraction expressions.

Let's see how Aisha and Izzy get on.

"Let's use a letter to represent the unknown," says Aisha.

"Okay, let's use the letter x to represent our unknown." So they replaced that box, which has a missing number in it, with x as an unknown.

Let's work out the adjustment in the minuend.

So the minuend has had 7 added to it.

As Izzy says, "The minuend has increased by 7 and the difference is the same.

So the unknown subtrahend will also be 7 more.

The unknown is 20." "We've balanced the equation." "What do you mean?" says Izzy.

"The unknown value we found makes the equation balance.

Both expressions are of equal value." How can the unknown be found in this one? 32 subtract x is equal to 25 subtract 13.

"Hang on! I've noticed something" says Aisha.

What has Aisha noticed? "These expressions are the same as last time, but they've swapped." "So the unknown is still 20." "Yes, the expressions can be swapped to either side of the equals symbol." Okay, let's check your understanding so far.

Use constant difference to find the unknown.

Explain your answer.

74 subtract 25 is equal to 82 subtract an unknown.

Pause the video and have a go.

Welcome back.

Let's see how you got on.

So, the minuend had 8 added to it from the first expression to the second expression.

Therefore, the subtrahend needed to have 8 added as well in order to maintain that constant difference and ensure that the equation was balanced.

25 add 8 is 33.

The minuend in the second expression has increased by 8, so the subtrahend needs to be 8 more to ensure the difference is the same and the expressions are equal.

Did you get 33? Did you balance the equation? I hope so.

Okay, let's look at the next bit.

How can the unknown be found? This time we've got larger numbers.

437 subtract 274 is equal to x, an unknown, subtract 385.

"The numbers are larger, so using constant difference and adjustment is trickier.

I'll use column subtraction here.

First, I'll work out the value of the known expression." So she writes out the columns underneath as a jotting.

She gets 163.

Now she can write a new equation.

163 is equal to x subtract 385.

"Now we can use the inverse by adding 385 to 163." She writes out the columns again.

"The value of the unknown is 548." "I think you can use adjustment by comparing the subtrahends and using jottings," Izzy thinks differently.

She compares the second part in each of these subtraction expressions.

274 plus 100 equals 374.

374 plus 10 equals 384.

384 plus one equals 385.

So she knows that 274 has been adjusted by adding 111 to give 385, the second part of the second expression.

She writes this in as a jotting.

"Now I need to adjust the minuend as well." So she's got to take 437 and add 111 to it.

x is equal to 437 plus 111, which is 548.

So here's another example.

How can the unknown be found here? We've got 4.

37 pounds subtract 2.

74 pounds is equal to x subtract 3.

85 pounds.

"Wait! I notice something" says Aisha.

What has Aisha noticed? "The digits are the same here.

We can use the previous solution and convert it to pounds and pence." So the previous solution was x was equal to 548.

Izzy says, "So x is equal to 5.

48 pounds" in the money equation.

Great, thinking you two.

How can the unknown be found in this one then? 12.

5 subtract x is equal to 10 subtract 2.

56.

"The unknown is in the first expression this time, but I think we can still calculate.

First, I'll work out the value of the known expression.

I'm gonna count on to find the difference." 2.

56 added to 0.

04 is equal to 2.

6.

2.

6 plus 0.

4 is equal to 3.

3 plus 7 is equal to 10.

So the difference between 10 and 2.

56 is 7.

44.

So now we have a new equation, 12.

5 subtract x is equal to 7.

44.

"So we could subtract 7.

44 from 12.

5 to find the unknown, but surely adjustment is quicker.

12.

5 is 2.

5 more than 10, so the unknown minuend has to be 2.

5 more than 2.

56." 2.

56 is equal to 2.

5 plus 0.

06.

2.

5 plus 2.

5 is equal to 5.

And 5 plus 0.

06 is equal to 5.

06.

Now we have the complete equation.

It's balanced.

"So the unknown value is 5.

06, and that was much more efficient." I suppose sometimes, when you are faced with decimal numbers, it's very easy to quickly go into a long-winded way of solving it, one that you are more used to, but don't forget how useful adjustment can be.

Time to check your understanding then.

Izzy has used adjustment to complete the inequality.

Is she correct? So look at her jottings.

Has she got the right answer? Pause the video, and see what you think.

Welcome back.

Izzy is incorrect, because she has adjusted by subtracting 6.

If the minuend is 6 more, then the subtrahend must also be 6 more to balance the equation.

This changed from takeaway 6 to add 6.

All right then, it's time for your second practise task.

Number one, balance these equations by finding the unknowns.

Think about whether calculating or adjustment is the more efficient strategy.

X has been used to represent the unknown in each of these equations.

For number two, you've got to solve each of the following problems by representing it as an equation.

A, Izzy starts with 30 pounds and spends 19.

50 pounds of this.

Aisha starts with an unknown amount of money and spends 21.

90 pounds of this.

They now have the same amount of money.

How much did Aisha have to start with? Alex starts with 15 pounds and spends 4.

99 pounds of this.

Jun starts with 12.

70 pounds and spends an unknown amount of money.

They now have the same amount of money.

How much did Jun spend? Pause the video here, have a go at those, and I'll be back in a little while with some feedback.

Welcome back.

Let's start with number one, A and B.

For A, did you use adjustment here and notice the relationship between the two minuends? So, the minuend had 10,200 subtracted from it going from the first part of the first expression to the first part of the second expression.

That meant that you also needed to subtract 10,200 from the subtrahend to give you the value of the unknown.

78,500 subtract 17,250 is equal to 68,300 subtract 7,050.

Let's look at B then.

Did you use adjustment here and notice the relationship between the two subtrahends? This time, if you look at the subtrahends, you could see that 800 was added from the subtrahend in the second expression to the subtrahend in the first expression.

That meant that 800 also needed to be added from the minuend in the second expression to the minuend in the first expression, which was our unknown.

55,451 plus 800, that would have given you 56,251.

The equation is balanced.

Here's C.

This time you may have calculated the first expression and then added.

So, the value of the first expression was 8.

32.

If you use the inverse, that led you to 8.

32 added to 32.

86 being equal to x.

So 41.

18 was the value of the unknown, and the equation is balanced.

There it is.

Now let's look at D.

For this one, you may have calculated the second expression and then added.

The second expression featured the known parts.

The second expression's value was 141.

7.

You could then have used this to give you the unknown by using the inverse.

You'd end up with x being equal to 141.

7 plus 4.

36.

That's 146.

06, which means that the equation is now balanced.

Okay, pause the video here if you need some extra time to mark any of those.

Here's number two then.

A, you might have started by writing out an equation that looked like this with an unknown.

You could have chosen any letter for the unknown if you wanted to.

We've used x just to be consistent with what's gone before.

30 pounds subtract 19.

50 pounds is equal to x subtract 21.

90 pounds.

If you compare the minuends and the subtrahends, you'd see that the subtrahend, both of those parts were known.

19.

50 pounds had 2.

40 pounds added to it to give 21.

90 pounds.

And that meant that 30 pounds, the minuend, also needed to have 2.

40 pounds added to it to give the value of the unknown and ensure that there was constant difference so this was an equation.

Aisha spent 2.

40 pounds more than Izzy, so her starting amount needed to be 2.

40 pounds more, which is 32.

40 pounds.

Okay, here's B.

Same sort of thinking, but this time we can see that Jun started with 2.

30 pounds less than Alex.

So he had to spend 2.

30 pounds less, which is 2.

69 pounds.

On this occasion, if you look at the jottings, you can see that the known parts were actually the minuend.

And the minuend was 2.

30 pounds less in the second expression.

So the subtrahend also needed to be 2.

30 pounds less.

Pause the video here so you can have some time to mark carefully, accurately, and also discuss anything you are not sure about.

It's time to summarise our learning today.

Subtraction expressions can be compared.

Comparing minuends and subtrahends can be more efficient than calculating the value of subtraction expressions.

Equations with subtraction expressions and unknowns can be balanced.

If the minuend and subtrahend are changed by the same amount, the difference stays the same.

If you change the minuend, the difference changes by the same amount.

My name's Mr. Tazzyman, and I've enjoyed that lesson today.

I hope you did too.

Maybe I'll see you again soon.

Bye for now.