Lesson video

Hi there.

My name is Mr. Tilstone.

I hope you're having a good day.

Let's see if we can make your day even better by having a successful math lesson.

Today's lesson is all about mixed numbers and I'm sure you've got lots of experience with and confidence with mixed numbers.

So let's begin.

The outcome of today's lesson is this.

I can make efficient choices about the order when solving subtraction problems within a whole.

And we've got some key words.

My turn, mixed number.

Your turn.

And I'm sure you know what a mixed number is by now, but let's have a reminder anyway.

A mixed number is a whole number and a fraction combined.

For example, this is 1 1/2.

Could you think of a different mixed number? Our lesson today is split into two cycles, two parts.

The first will be subtracting with mixed numbers and the second subtracting mixed numbers from mixed numbers.

Let's begin by thinking about subtracting with mixed numbers.

And in this lesson you're going to meet Sofia and Jacob.

Have you met them before? They're here today to give us a helping hand with the maths.

Sofia has been given a set of equations to solve.

She's got 7/8 - 2/8 = mm.

She's got 1 7/8 - 2/8 = mm.

She's got 2 7/8 - 2/8 = mm.

And finally 3 7/8 - 2/8 = mm.

What do you notice? Anything? Are those equations linked? How would you solve them? Sofia says, "I will use a number line to represent each equation." That's a good idea.

Here is a number line.

So we're doing 7/8 - 2/8 = something.

Now, here's 7/8 and we're going to take 2/8 or 2 1/8.

You might like to think of it as from that number, from that fraction, and it's equal to 5/8.

So now what about 1 7/8 - 2/8, what's that equal to? Well, here's 1 7/8, when we subtract 2/8, it gives us 1 5/8.

What changed? What stayed the same? What about 2 7/8 - 2/8? What would that give? That will give 2 5/8.

Again, what's changed, and what's stayed the same? And what about 3 7/8 - 2/8, what's that equal to? That is equal to 3 5/8.

What are you noticing here? What can we say? One thing that changes throughout these calculations is the whole number part, and one thing that's staying the same is the fractional part.

Sofia also represents the calculations using bar models.

So here we've got 7/8 - 2/8 = 5/8.

And we can show that with a bar model.

What about 1 7/8 - 2/8 = 1 5/8.

Let's have a look at that.

Here's 1 7/8, and we're going to subtract those 2/8.

And Sofia asks, "What do you notice?" Hmm? What changed? What stayed the same? The only difference between those calculations was that the second one had a whole number in front.

What about this one, 2 7/8 - 2/8 = 2 5/8.

Let's investigate that one.

What can you see here? Here we've got 2 7/8.

What would happen if we subtracted 2/8? That's what would happen.

What are we left with? 2 5/8.

What about 3 7/8 - 2/8 = 3 5/8.

And this is what happens when we subtract those 2/8.

We're left with 3 5/8.

"Did you notice only the number of whole bars changed each time.

The fractional bar stayed the same." Let's look at these equations in more detail.

7/8 - 2/8 = 5/8.

1 7/8 - 2/8 = 1 5/8.

2 7/8 - 2/8 = 2 5/8.

And 3 7/8 - 2/8 = 3 5/8.

What do you notice? Sofia noticed a pattern.

She says, "The difference increases by one each time.

This is because the whole increases by one each time, but the known part stays the same." So that's the 7/8 - 2/8 = 5/8 part.

That's constant through all of those calculations.

"Also the difference always has a fractional part of 5/8." Did you notice that? And this is what Jacob noticed.

"I notice that 2/8 are always subtracted from 7/8." That was true throughout.

So the fractional part of the difference is always 5/8.

"I also notice that the whole number does not change within a calculation." Did you spot that too? So what equation might come next? What do you think? What would you predict? The last one was 3 7/8 - 2/8 = 3 5/8.

"I didn't need to calculate.

I knew the fractional part would be the same." So the next one must be 4 7/8 - 2/8 = 4 5/8.

Let's look at this calculation and represent it in a part-part-whole model.

So this is 4 7/8 - 2/8 is equal to? Look at the part-part-whole model.

We've got a whole that's 4 7/8 and we've got a part that's 2/8 and we are subtracting the 2/8 from 4 7/8.

What part does that leave us with? What do you notice? Well, Sofia notices, "The whole is 4 7/8 and the known part is 2/8.

The difference is 4 5/8." "I think that I could now solve this without using resources," says Jacob.

Do you think you could do that too? He says, "The whole number part will stay the same." So what do you think, 4 7/8 - 2/8, what is that equal to? Jacob says, "The fractional parts have the same denominator." Did you notice that? They're both 8s.

"So they can be subtracted using unitizing." We can think of it as 7 1/8 and 2 1/8.

"7 - 2 = 5, so 7 1/8 - 2/8 = 5 1/8." And that's what we've got here.

So the answer is 4 5/8.

The whole number part hasn't changed.

Shall we do a little check for understanding to see how you are getting on? Let's have a go.

We've got 5 7/8 - 2/8, what is that equal to? Pause the video and have a go.

Welcome back.

Which one did you think it was? The answer is C, 5 5/8.

Let's explore that.

The fractional parts have the same denominator.

Did you notice the 8s? So they can be subtracted using unitizing.

We know 7 - 2 = 5, so 7 1/8 - 2 1/8 = 5 1/8.

And the whole number part remains the same.

We haven't subtracted any whole numbers, just a fractional part.

Okay, let's have a look at a different example.

A different mixed number.

This time we've got 9 1/4 - 4 = something.

What do you notice this time? What's different? Hmm, well, this time we're not subtracting a fraction from the mixed number, we're subtracting the whole number from the mixed number.

How would you calculate it? What would you do? Jacob says, "Let's start by representing it in a part-part-whole model." Great idea.

Here we go.

"And partitioning the mixed number." So that mixed number part 9 1/4 could be partitioned into 9 + 1/4.

And that gives us the equation, 9 + 1/4 - 4 = mm.

That still seems hard.

I wonder if there's something we could do.

What about if we rearrange the equation so that it was 9 - 4 + 1/4.

Let's do that.

9 - 4 + 1/4 = 5 + 1/4, and then we can turn that back into a mixed number.

We can recombine.

So that gives us 5 1/4.

So, 9 1/4 - 4 = 5 1/4.

Let's do another example, and then you can have a go at your own.

So this is 3 5/6 - 2.

Going to use a part-part-whole model.

So this is 3 5/6 and that's the part that we're subtracting the 2.

What will that leave us with? Well, we could partition that mixed number into 3 + 5/6, so that gives us the equation 3 + 5/6 - 2 = something.

But if we rearrange that and think about the whole numbers first, that gives us 3 - 2 + 5/6 is equal to, and that means that's 1 + 5/6.

And when we combine those, we've got 1 5/6.

That's our mixed number.

That's our answer.

See if you can do the same with 5 6/7 - 2, what is that equal to? Pause the video.

Welcome back.

How did you get on? Did you manage to find an answer? Did you use a part-part-whole model? Did you then partition that mixed number? Did you then do 5 - 2 + 6/7? Did that give you 3 + 6/7? Did you then combine that into a mixed number and turn that into 3 6/7? That's the correct answer.

Well done if you got that.

You're on track.

It's time for some practise.

I think you're ready for this.

Calculate these.

Let's have a look.

We've got 8/10 - 3/10 is equal to, and then 1 8/10 - 3/10 is equal to, and then 2 8/10 - 3/10 is equal to, and then 3 8/10 - 3/10 is equal to? So they're linked those calculations.

I'll leave you to look at the rest of them yourselves and see if you can spot a link.

And number two, a mixed number and a proper fraction have a difference of 2 7/10.

If all the denominators are 10, and they've been included there, they're all 10, what could the mixed number and proper fraction be? And there is more than one possibility here.

Right here.

Pause the video and off you go.

Welcome back.

How did you get on? Number one, 8/10 - 3/10 = 5/10.

So, 1 8/10 - 3/10 = 1 5/10.

So, 2 8/10 - 3/10 = 2 5/10.

And 3 8/10 - 3/10 = 3 5/10, 3 5/9 - 2/9 = 3 3/9, 6 4/5 - 2/5 = 6 2/5, and 6 4/5 - 4 = 2 4/5, 3 5/9 - 2 = 1 5/9.

You might have used a part-part-whole model for these or maybe you didn't.

And 3 7/9 - something = 3 1/9.

So the difference between those is 6/9.

And 2 3/7 - something = 1 3/7.

And again, a part-part-whole model would've been helpful here if you needed it, and that is 1.

Well done if you got those.

And number two, lots of answers for this one we could have.

2 9/10 - 2/10 = 2 7/10.

That would work, wouldn't it? Or we could have 2 8/10 - 1/10 = 2 7/10.

I think you're ready for the next cycle.

In fact, I know you are.

And that's subtracting mixed numbers from mixed numbers.

So, so far we've subtracted with mixed numbers.

This time both of the numbers are going to be mixed numbers.

Sofia has 4 5/10 metres of rope.

She gives a piece of rope to Jacob and has 1 2/10 metres of rope left.

"What might the question be here do you think?" Here's the question.

What length of rope does she give to Jacob? Did you predict that one? Well done if you did.

"Let's start by representing this in a bar model." Bar models are so helpful, aren't they? Here we go.

So we've got 4 5/10 is our whole, and 1 2/10 is a part, and that's what we're going to be subtracting.

So you might see both of those are mixed numbers this time.

One part is 1 2/10 and the other is unknown.

But we can work it out.

The whole is 4 5/10.

"We can use a bar model to form an equation." What could we write? To find the unknown part, we need to subtract the known part from the whole.

What would that look like? How about this? 4 5/10 - 1 2/10 = something.

We don't know what that something is? We're going to work it out.

"How would you subtract these mixed numbers?" What strategy would you use? Well, "Let's start by representing the whole." Here we go.

That is 4 5/10 and you might have some base 10 equipment in your classroom that you can use.

"Partitioning the part that we are subtracting." So 1 2/10 is 1 + 2/10.

"Now, we can subtract one whole from the four wholes to leave three wholes." So take away that one whole, and we left with 1 + 2/10.

"And subtract 2/10 from the 5/10." What would that look like? "These parts have the same denominator so they can be subtracted using unitizing.

5 - 2 = 3, so 5 1/10 - 2 1/10 = 3 1/10." Let's see that.

Here we go.

And that is what we are left with.

"I give 3 3/10 metres of rope to Jacob." So, 4 5/10 - 1 2/10 = 3 3/10.

Those resources were really helpful there for visualising that.

Let's have a little check.

True or false? 9 6/9 - 5 2/9 = 4 8/9.

True or false, and why? Pause the video.

Let's see, was that true or false? That was false.

Why? The whole number parts have been subtracted correctly.

So, 9 - 5 = 4 but the fractional parts have been added, not subtracted.

Well done if you spotted that.

6/9 - 2/9 = 4/9.

Time for some final practise.

Calculate these.

Number two, solve these problems, giving your answer as a mixed number.

Shall we read them? Jacob has 10 3/4 metres of rope.

Sofia has 3 1/4 metres of rope.

How much more rope does Jacob have? Be careful with that word.

How much more? Think about what that's asking and what operations you're going to need.

Sofia runs 5 8/10 of a kilometre on Saturday but only manages to run 3 5/10 of a kilometre on Sunday.

How much further did she run on Saturday? And again, be careful with that wording.

Make sure you understand what operation we're using here.

Are we adding or subtracting? Hmm.

C, Jacob is measuring rainfall.

He presents his results in a table.

What is the difference in rainfall between Monday and Friday? Okay.

Pause the video.

Good luck with that and we'll see you soon for some answers.

Welcome back.

How did you get on? Let's give you some answers.

So number one, 5 5/9 - 1 3/9 = something.

Well, you could subtract the whole number parts and then subtract the fractional parts.

That gives us 4 2/9.

And using that same strategy, 5 5/9 - 1 2/9 = 4 3/9.

5 5/9 - 1 1/9 = 4 4/9.

Did you notice the pattern there? And 9 7/9 - 2 4/9 = 7 3/9, 9 7/9 - 3 5/9 = 6 2/9, and 9 7/9 - 1 2/9 = 8 5/9.

Did you notice the link between those calculations.

And 2 3/4 - something = 1/4.

Hmm.

The difference between those two numbers is 2 2/4.

And I think a part-part-whole model is helpful there.

And then 7 6/8 - something = 3 1/8, and the difference is 4 5/8.

And A, Jacob has 10 3/4 metres of rope.

Sofia has 3 1/4 metres of rope.

How much more rope does Jacob have? So this is a different problem.

We need to use subtraction.

Here we go.

10 3/4 as a whole, 3 1/4 is a part.

We're looking for the difference between the part and the whole to give us the other part.

So, 10 3/4 - 3 1/4, we could think about the whole numbers first and then subtract the fractions.

Perhaps that's one strategy.

We could use partitioning here and think of that 3 1/4 as 3 + 1/4.

That would give us the calculation 10 - 3 = 7.

And 3/4 - 1/4 = 2/4.

Jacob has 7 2/4 metres of rope more than Sofia.

B, Sofia runs 5 8/10 of a kilometre on Saturday and 3 5/10 of a kilometre on Sunday.

How much further did she run on Saturday? So again, we are looking at a different problem here.

So we need to use subtraction.

Here is our whole 5 8/10, and here's part 3 5/10.

We need to find the difference.

We could do it this way, 5 8/10 - 3 5/10.

We could partition that 3 5/10 into 3 + 5/10, and then we could calculate 5 - 3 = 2, and then we could subtract the fractional part.

So, 8/10 - 5/10 = 3/10.

So combine those together to make that mixed number.

And we've got 2 3/10.

Sofia ran 2 3/10 kilometres further on Saturday.

And Jacob is measuring rainfall.

He presents his results in a table.

What's the difference in rainfall between Monday and Friday? Well, on Monday is 5 4/5, and on Friday 2 1/5.

And we are looking at the difference between those.

So we need to use subtraction.

5 4/5 - 2 1/5, we could partition that mixed number into 2 + 1/5 and that gives us 5 - 2 = 3.

And then the fractional part 4/5 - 1/5 = 3/5.

And put them together, we've got 3 3/5.

That's our mixed number.

That's our answer.

The difference in rainfall was 3 3/5 centimetres.

We've come to the end of the lesson.

You've been fantastic today.

Today, we've been efficiently solving some traction problems within a whole.

Mixed numbers can be partitioned in the same way as whole numbers.

If fractions have the same denominator, they can be subtracted.

To subtract fractions with the same denominator, the language of unitizing can be used.

And you've done that lots of times today.

Well done in your achievements and your accomplishments today.

I think you've been amazing and I think you need a little bit of a pat on the back and you need to say, "Well done me." I do hope I get the chance to spend another math lesson with you at some point in the near future.

But until then, have a great day.

Be successful in whatever you are doing.

Take care and goodbye.