Lesson video

Hi there, my name is Mr. Tazzyman, and I'm really looking forward to teaching you the lesson today from the unit that's all about equivalence, compensation, and subtraction.

If you're ready, we can get started.

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

By the end of the lesson, we want you to be able to say, "I can use constant difference to balance equations and find unknowns." These are the keywords that you're gonna see and hear during the lesson, and it's important that you know how to say them and that you understand them.

I'm gonna say them and I want you to repeat them back to me.

I'll say, "My turn," say the word, and then I'll say, "Your turn," and you can repeat it back.

My turn, minuend, your turn.

My turn, subtrahend, your turn.

My turn, difference, your turn.

My turn, constant, your turn.

My turn, unknown, your turn.

Okay, let's see what each of these words means to make sure that we all understand everything going forward.

The minuend is the number being subtracted from.

A subtrahend is a number subtracted from another.

The difference is the result after subtracting one number from another.

And there's an equation at the bottom that's been labelled with all of those terms. Seven subtract three is equal to four.

And in that seven is the minuend, three is the subtrahend, and four is the difference.

We've also got: a constant is a quantity that has a fixed value that does not change or vary, such as a number.

And then we have: an unknown is a quantity that has a set value but is represented by a symbol or letter.

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

You can see the same equation that we looked at before at the bottom there, but this time we've covered up the minuend with a box.

That's our unknown.

You probably might be able to work it out.

Here's the outline then for the lesson on using constant difference to balance equations and find unknowns.

To begin with, we're gonna look at finding unknowns using constant difference.

Then we're gonna look at larger numbers and decimal fractions.

Here's two people we're gonna meet throughout.

Hi, Alex, hi, Sofia.

They're maths friends.

They're gonna help us by discussing some of the prompts, giving us some of their thinking and reasoning, and even revealing some of the answers.

Okay, ready to learn? Let's do this.

Sofia is trying to find an unknown minuend.

So, something subtract 0.

45 is equal to 1.

01 subtract 0.

51.

"This is tricky!" says Sofia.

I agree.

"Let's explore constant difference with unknowns." Can you see that we've got balance scales there, and the balance scales are showing us that 10 is greater than 5.

You can see that where the 10 is on one side that's been weighed down.

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

"We can subtract cubes until both sides are equal.

I will write down expressions compared." So they start by subtracting six from that side.

10 subtract 6 is less than 5.

And we know that, because now the side with five on, is weighed down.

So let's subtract one from that side.

Now we've got a balanced set of scales, and the equation reads, 10 - 6 = 5 - 1.

Notice that because both of those expressions are now equal in value, the equals sign has appeared between them.

What do you notice here? Ten subtract six is equal to five subtract one.

Hmm.

"The difference is constant.

Both expressions have a value of four." "The minuend and subtrahend have decreased by the same amount." Can you see that the minuend has decreased by subtracting five, and the subtrahend has also decreased by subtracting five.

How can the unknown be found in this equation? 25 subtract 13, is equal to 32, subtract an unknown.

"Let's work out the adjustment in the minuend." So seven has been added.

We've gone from 25 as a minuend, add seven, so that gives us 32 as a minuend in the second expression.

"The minuend has increased by seven and the difference is the same.

So the unknown subtrahend will also be seven more.

The unknown is 20." Brilliant reasoning.

What about this time? 32 subtract an unknown, is equal to 25 subtract 13.

"Hang on! I've noticed something." What has Alex 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, it's your turn now.

Use constant difference to find the unknown, and then explain your answer.

Little tip, remember to start by comparing either the minuends or the subtrahends, depending on which ones you already know.

Pause the video here and have a go.

Welcome back.

If you look closely, you can see that the minuend was adjusted by adding eight.

That means that the subtrahend also has to have eight added to it, because the two expressions are of equal value, so the difference has to remain constant.

Our missing number was 23.

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

Okay, look at this one then.

How can the unknown be found here? We've got an unknown subtract 31, is equal to 45 subtract 20.

"Let's work out the adjustment in the subtrahend." 11 has been subtracted.

"The subtrahend decreased by 11." "You could make use of the inverse here by changing the direction and operation." "Good point! The subtrahend has increased by 11, and the difference is the same, so the unknown minuend will also be 11 more.

The unknown is 56." There it goes.

Okay, let's look at this one then.

123 subtract 41, is equal to 101, subtract something, an unknown.

Let's work out the adjustment in the minuend.

22 has been subtracted.

"The minuend has decreased by 22 and the difference is the same, so the unknown subtrahend will also be 22 less.

The unknown is 19." Brilliant.

Okay, your turn then.

Use constant difference to find the unknown here.

We've got 104 subtract 27, is equal to 80 subtract an unknown.

Pause the video and give it a go.

Welcome back.

We saw that the minuend had decreased by 24.

That means that the subtrahend needed to do exactly the same in order for the difference to remain constant and this to be a balanced equation.

27 subtract 24 is equal to three.

Okay, it is your turn then to do some practise tasks now.

For Number One, you've got to look and see which of the equations in the sequences incorrect, and explain your reasoning.

For Number Two, I'd like you to use constant difference to find the unknowns in the equations below.

Remember, it's all about comparing either the minuends or the subtrahends, depending on which ones you know both of.

For Number Three, there's a worded problem.

Alex has the same birthday as his uncle.

Alex was born in 2013 and his uncle was born in 1981.

They share the same birthday.

Alex says, "The difference between us will always be an even number of years." Explain whether Alex is correct or incorrect.

Okay, enjoy those questions, give them a good go.

Remember to keep using that constant difference, and I'll be back in a little while to give you some feedback, so pause the video here.

Welcome back.

Let's start with Number One A.

The bottom equation in this sequence was the one that was incorrect.

The difference in the second expression is 20, but it should be 18.

Let's look at B.

It was the second to last equation that was incorrect here.

The difference in the first expression is 62, but it should be 52.

Let's look at Number Two then, Two A.

We could see that if we went from the subtrahend in the second expression back to the subtrahend in the first expression, we were adding 35, so we need to do the same for the minuend: 65 added to 35 is 100, so the missing number was 100 here.

Here's B.

The subtrahend, we added 28 to get to the second expression, so we needed to add 28 to the minuend.

That gave 103.

Let's look at C.

We added 85 to the minuend to get from the first expression to the second expression, so we needed to do the same with the subtrahends.

That gave 111.

Lastly, larger numbers here.

We took away 1,200 from the subtrahend, so we needed to do the same in the minuend.

That gave 310.

Here's Number Three then.

While Alex was correct, the gap in years between Alex and his uncle is a constant difference of 32 years, which is even.

The dates of each year will alternate between odd and even numbers, but the difference will still be constant." It's time for the second part of the lesson then.

Looking at larger numbers and decimal fractions.

Sofia looks back at the problem with an unknown minuend.

Remember this one? Something subtract 0.

45, is equal to 1.

01 subtract 0.

51.

"We can use constant difference to help here." "The subtrahend is 0.

06 less, so the unknown minuend will also be 0.

06 less.

"That means it is 0.

95." Great stuff, Sofia, you've made brilliant progress in this lesson already.

Time to check your understanding of that then.

Use constant difference to find the unknown, and explain how you know.

We've got 0.

87 subtract 0.

23, is equal to an unknown subtract 0.

06.

Now, remember here, it says explain how you know.

Don't forget that part.

It's really crucial to show that you are developing as a mathematician.

Okay, pause the video and give it a go.

Welcome back.

If you compare the subtrahends in both of these expressions, you can see that there was 0.

17 subtracted when we went from the first subtrahend to the second.

That means that the minuend also needs to be subtracted by the same amount.

So it's 0.

7.

Here's the explanation.

The subtrahend in the second expression has decreased by 0.

17, so the minuend needs to be 0.

17 less to ensure the difference is the same and the expressions are equal.

Okay, ready to move on? Let's go for it.

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

We've got an unknown subtract 291,235, is equal to 334,450 subtract 301,235.

Wow, there's a lot of digits there.

Try not to be put off.

We can reason this out.

"Let's work out the adjustment in the subtrahend." "The subtrahend increased by 10,000." So consequently we know that we need to increase the minuend by 10,000.

"Let's use the inverse again here." "Good point! The subtrahend has decreased by 10,000, and the difference is the same.

So the unknown minuend will also be 10,000 less.

The unknown is 324,450." So by changing the direction of the arrows, it became easier to calculate that minuend, that unknown.

Remember, if you do change the direction of the arrows, you must make sure that you use the inverse operation as well.

Okay, your turn then.

Use constant difference to find the unknown.

We've got 267,924, subtract 245,212, is equal to 282,924, subtract an unknown.

Pause the video and give it a go.

Welcome back.

Did you notice that when you compare the minuends there was a increase of +15,000? That meant that we also needed to increase the subtrahend by 15,000.

That gave 260,212.

Okay, here's a worded problem.

Alex and Sofia go to the cinema.

Alex has 12 pound 80 and Sofia has 6 pound 50.

They each buy a cinema ticket for five pounds.

Alex has 7 pound 80 left to spend.

How much does Sofia have left? Hmm, "Because we spent the same we can write equivalent expressions." "I started with 12 pounds 80 and had 7 pounds 80 left after spending 5 pounds." "I started with 6 pounds 50 and spent 5 pounds with what's left being the unknown." So 6 pounds 50 subtract an unknown.

Let's look at the adjustments in the minuends.

There was a 6 pound 30 subtraction from the first minuend to the second minuend.

Now to adjust the subtrahend to find the unknown.

7 pound 80, subtract 6 pound 30, is 1 pound 50.

Okay, it's your turn then.

This is the second practise task.

For Number One, I'd like you to use constant difference to find the unknowns in the equations below.

Some of them feature decimal fractions, and some of them feature larger numbers.

And for Number Two, we've got another worded problem.

At the gift shop on a residential week, Alex has 11 pounds 85 pence, and Sofia has 7 pounds 35 pence to spend.

Both buy a souvenir fridge magnet to take home.

I love a fridge magnet.

Sofia now has 3 pounds 36 left.

How much does Alex have left? Okay, pause the video here, and I'll be back in a little while after you've had a good go at those questions to give you some of the answers and some feedback.

Good luck.

Welcome back.

Let's start with number one.

We're gonna look at A through to D first, because they were the ones that involved decimal fractions.

So on A, you might have spotted that we had a 0.

52 increase when we were going from the minuend on the right-hand expression to the minuend on the first expression.

That meant there needed to be a 0.

52 increase of the subtrahend as well in the same direction, giving 0.

56.

For B, the adjustment was 0.

39, going from the first subtrahend to the second subtrahend.

They were unknown parts.

So then we needed to add 0.

39 to 0.

75, and that gave 1.

14.

Let's move on to C then.

We had two known subtrahends, and we went from the right-hand expression to the left-hand expression, there was a decrease of 2.

7.

So we also needed to subtract 2.

7 from 5.

04, and that gave 2.

34.

Look at D now.

We knew already the minuends.

There was a decrease of 2.

37 between the known minuends, so we had to do the same to the first subtrahend to get our unknown.

That was 0.

04.

Okay, here are One E and F, where we were working with larger numbers.

There was an increase from the right-hand expression to the left-hand expression of 35,000 in the known subtrahends.

That meant that we had to add 35,000 onto the minuend.

So the missing number was 345,000.

F, then: Here we had to subtract 10, because we knew already that 10 had been subtracted from the minuends, so we subtracted 10 from 200,001, that gave us 199,991.

Here's Number Two then, our worded problem.

We had to start by identifying the equation and the unknown.

We had 7 pounds 35, subtract 3 pounds 36, is equal to 11 pounds 85 subtract an unknown.

By comparing the minuends, we could see that 4 pound 50 was added on, so we needed to also add on 4 pound 50 to the subtrahend, and that gave 7 pounds 86 pence.

We've reached the end of the lesson then.

Here's a summary of all the things that we've learned.

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

Constant difference is a useful strategy when trying to find unknown minuends or subtrahends with equivalent subtraction expressions.

By analysing the minuend and subtrahends, calculating the adjustment and sometimes using the inverse, you can calculate unknown minuends or subtrahends.

My name is Mr. Tazzyman.

I hope you've enjoyed the learning today, I know I have, and I hope I'll see you again soon in another maths lesson.

Bye for now.