Lesson video

Hello there.

My name is Mr. Tilstone.

If I've met you before, it's nice to see you again.

And if I haven't met you before, it's nice to meet you.

In today's lesson, we're going to be adding some fractions, and I'm sure that's something you've already got some experiences with.

So if you are ready, I'm ready.

Let's begin.

The outcome of today's lesson is this, I can make efficient choices about the order when solving additional problems within a whole.

And our keywords, my turn, mixed number.

Your turn.

What is a mixed number? Hopefully, you've got lots of experience of using these.

Let's have a reminder though.

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

So for example, 1 1/2.

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

The first will be adding proper fractions to mixed numbers and the second adding mixed numbers.

For now, let's focus on adding proper fractions to mixed numbers.

Could you think of an example of what that might be? What kinds of questions we might be looking at there? 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.

See if you notice anything about these equations.

1/5 + 2/5, 1 1/5 + 2/5, 2 1/5 + 2/5, 3 1/5 + 2/5.

Hmm, what was the same and what was different about each of those equations? Did you notice? Think about the fractional part.

How would you solve these? Sofia says, "I will use a number line to represent these." Good idea, Sofia.

Number lines are brilliant when it comes to fractions.

So here's a number line.

Here is one, 1/5, plus two, 1/5, equals three, 1/5, or we could say 1/5 + 2/5 = 3/5.

And then we've got 1 1/5 plus that 2/5.

That was the same thing that we had before, the same fraction equals 1 3/5, and this time 2 1/5 + 2/5.

Again, we're adding 2/5 on, equals 2 3/5.

And 3 1/5 + 2/5 = 3 3/5.

Sofia also represents the calculations using bar models.

Again, really helpful with fractions.

Helps you to picture and visualise those fractions.

So 1/5, you can see they're shaded in grey, plus 2/5, are shaded in purple, and altogether you can see 3/5 have been shaded.

So 1/5 + 2/5 = 3/5.

This time 1 1/5, that mixed number, 1 1/5, that's in grey, plus 2/5, that's in purple as it was before.

Equals, what have we got this time? What's our total? 1 3/5.

So what changed and what stayed the same? Well, the fractional parts were the same in each question, weren't they? In each part we were adding 1/5 and 2/5 together.

The only difference is we also had an extra whole number in the second example.

Then we've got 2 1/5 + 2/5, and we can represent that as such.

So we've got two whole bars plus that 1/5.

So you can see hopefully 2 1/5 there, + 2/5 = 2 3/5.

The fractional part is the same as before.

And the same with 3 1/5 + 2/5 = 3 3/5.

The fractional part is the same.

We've just got extra whole numbers.

"Did you notice?" Says Sofia.

"Only the number of whole bars changed.

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

What do you notice? Sofia says, "I notice a pattern.

The sum increases by one each time." Did you notice that? Why is that? Sofia's going to explain.

This is because one addend increases by one each time, but the other addend stays the same.

Well, let's have a look at that.

So we've got 1/5 + 2/5 becomes 1 1/5 + 2/5, so increased by one.

And then that becomes 2 1/5 + 2/5.

So that increased by one.

Also, the mixed number sum always has a fractional part of 3/5.

Did you notice that? Jacob says, "I notice that 2/5 are always added to 1/5." This means that the fractional part of the mixed number sum is always 3/5.

I also notice, he says, "That the whole number does not change within a calculation." Did you spot that? So when it was 1 1/5 + 2/5, it was 1 3/5.

When it was 2 1/5 + 2/5, it was 2 3/5.

The whole number wasn't changing.

So what might come next? Can you predict what would be next in this sequence? Did you predict this one? 4 1/5 + 2/5 = 4 3/5.

We could keep going, couldn't we? Sofia says, "I didn't need to calculate to know the sum, I used what we noticed." Let's look at the calculation and represent it on a part-part-whole model.

They're very useful for looking at the composition of fractions.

So here we've got 1 1/5 + 2/5 = 1 3/5.

Look at the part-part-whole model.

Can you see which the parts are and which the whole is? So let's start with the parts.

What are the parts? The parts are 1 1/5.

Can you see that as a part? And separate part is 2/5.

They are the two parts.

What's the whole, or the total? The total or the whole is 1 3/5.

Can you see that on the part-part-whole model? The parts are 1 1/5 and 2/5.

The total or whole is 1 3/5.

Addition is commutative, so we can change the order of the parts.

I'm sure you knew that already.

3 + 2 = 5 could also be expressed as 2 + 3 = 5.

The order can be switched.

So in this case, we could say 2/5 + 1 1/5 = 1 3/5.

It gives us the same answer if we switch those parts over.

And the part-part-whole model would look slightly different, but it would still give the same answer and it's still got the same parts and the same whole.

Jacob says "We would still add 2/5 to 1/5.

The whole number part would stay the same." Jacob says, "I think that I could now solve this without using resources." Hmm, confident.

The whole number part will stay the same.

The fractional parts have the same denominator, which is, can you see that as five? They're both fifths, so they can be added using unitising.

Have you heard that word before? 1 + 2 is 3.

So 1 1/5 and 2 1/5 is 3 1/5.

And there's the answer, 1 3/5.

Let's have a little check for understanding.

Let's see how much of that you've taken in.

So what is 5 1/5 + 2/5? And you've got three choices.

Is it 8/5? Is it 5 3/10? Or is it 5 3/5? And if you get the right answer, see if you can explain why the wrong answers are wrong and why people might think they're the right answers.

Pause the video.

Let's have a look.

What was the right answer? It was 5 3/5.

The fractional parts have the same denominator.

Did you notice that they were both fifths and can be added using unitising? So in this case, 1 + 2 is 3.

So one, 1/5 and two, 1/5 is three, 1/5.

The whole number part remains the same.

That hasn't changed.

We haven't added any extra whole numbers.

Let's look at a different example.

So this time we've got 1/10 + 3 2/10 + 4 + 1/10.

We've got four addends.

Did you notice anything about those numbers? What have we got? We've got a variety there, haven't we? We've got some proper fractions.

Got 1/10 and another 1/10.

We've got a whole number.

We've got four and we've got a mixed number 3 2/10.

And they don't seem to be in any particular order, do they? How would you add these numbers? Now remember, addition is commutative, so we can mix them up, we can rearrange them.

Jacob says, "This is tricky." Yeah, it looks tricky at the minute, doesn't it? In that order, it feels tricky, but I think we can do something.

He says, "I would just add them one at a time, starting with 1/10, add 3 2/10.

Then I would add the 4 to that, and then finally add the 1/10." Is that what you would do? Does that seem efficient? Sofia says, "Wait, I respectfully challenge you." I really like that phrase, by the way.

If you disagree with somebody, why don't you try that? I respectfully challenge you.

"There is a more efficient way." I think Sofia's right.

She says, "Let's start by partitioning the mixed number." Hmm.

So what's the mixed number? Can you remember? That's 3 2/10.

So if we partition that, what would the parts be? So 3 2/10 could be partitioned into 3 and 2/10.

Now we can combine the whole number parts because we can add in any order.

So we can do 3 + 4 + 2/10 + 1/10 + 1/10.

Now, I don't know about you, but to me that looks a lot more achievable now, it looks a lot easier, a lot more straightforward.

So 3 + 4 = 7.

That was nice and easy, + 2/10 + 1/10 + 1/10.

And what do you notice about those fractions? We can combine the fractional parts.

They've all got the same denominator.

We can use unitising just like we did before.

2 + 1 + 1 is 4, easy.

So two, 1/10 + one, 1/10 + one, 1/10 is four, 1/10 or 4/10.

That leaves us with 7 + 4/10.

Well that's not too complicated, is it? What happens when we combine those two parts? What do we get? We get a mixed number and that mixed number is 7 4/10.

That became so much easier when we thought about the whole number of part first and then added the fractional part second, wouldn't you agree? Let's have a little check.

So here, we've got 5 + 1/9 + 2/9 + 1 3/9 = something.

And what we've done here, we've taken the 1 3/9 and partitioned it into 1 and 3/9.

And then we've rearranged it, we've swapped the addend order.

We're doing the whole numbers first, so 5 + 1, and then we're adding the fractional parts, + 1/9 + 2/9 + 3/9.

Or we could think of it in terms of unitising.

So we know 1 + 2 + 3.

So one, 1/9 + two, 1/9 + three, 1/9 = something.

So that gives us 6 + 6/9.

And when we combine those parts, we've got the mixed number 6 6/9.

So over to you.

See if you can do the same with the equation on the right.

Pause the video and good luck.

How did you get on? How did you approach that? Let's have a look.

Well, you could start by partitioning that mixed number 2 2/8 into 2 and 2/8.

And then we could add those whole numbers.

So 2 + 2 + 1/8 + 2/8 + 1/8.

That gives us four when we add the whole numbers together.

plus 4/8 when we add the fractional parts together.

And then we've got a nice simple job at the end to turn that into a mixed number, which is 4 4/8.

Did you get that correct? Very well done if you did.

You're on track.

It's time for some practise.

Calculate these.

Remember, you can swap the order of the addends over.

So have a good thing before you do each one.

What's the most efficient way that I can do this? Number two, a mixed number and a proper fraction.

Have a sum of 2 3/10.

If all the denominators are 10 and we've included those there, look, you can see the denominators are 10.

What could the mixed number and proper fraction be? There may be more than one answer.

Pause the video.

Good luck with that and I'll see you soon for some feedback.

Welcome back.

How did you get on? Are you ready for some answers? Let's have a look.

So number one, 1 2/7 + 4/7 = 1 6/7.

So the whole number part did not change.

We were adding the fractional part together.

The next one, 2/9 + 4 3/9 = 4 5/9, the same as before.

The whole number part didn't change.

We were adding the fractional part together.

Did you spot the difference? So this time the mixed number was the second addend.

And 3 1/8 + 3/8 = 3 4/8.

The whole number part has not changed.

The fractional part has.

And 1 2/7 plus something equals 1 6/7.

That's a slightly different kind of question, isn't it? So what would you add to 1 2/7 to get to 1 6/7? The whole number part is still the same.

So we're not adding any whole numbers.

We're adding an extra lot of sevenths and it's 4/7.

So 1 2/7 + 4/7 = 1 6/7.

And something plus 4/10 = 2 6/10.

So we are going to be adding a mixed number here.

What's the mixed number going to be? Well, I think we already know the whole number part.

That's got to be two and it's gonna be some tenths.

How many tenths, two and how many tenths? 2 2/10.

And then 4 + 2/9 + 5 1/9 + 3/9.

That seems complicated, but if we rearrange it and partition some of those numbers like the mixed number, we'll be okay.

That's 9 6/9, and then the final one, if we partition that mixed number there, add all the whole numbers together, add all the fractional parts together, we get 14 5/8.

And then a mixed number and a proper fraction have a sum of 2 3/10.

If all the denominators are 10, what could the mixed number and proper fraction be? What did you get? Well, it could be this 2 1/10 + 2/10 = 2 3/10, that works.

We know the whole number has got to be two.

So it could be 2 2/10 + 1/10 = 2 3/10.

So very well done if you've got either of those answers and a special well done if you've got both of those answers.

Let's move on to the next cycle, which is adding mixed numbers.

Sofia drinks 2 1/5 litres of water in a day.

That's good, that's healthy.

And Jacob drinks 3 3/5 of litres of water.

Very good.

Can you visualise that? Can you picture that? How would that look? Well, here's Sofia's water.

She's got a mixed number.

She's got 2 1/5, so she's got a whole number part that's two, and she's got a fractional part 1/5, and you can see that on the measuring jug.

Now, here's Jacobs.

He's drunk even more water.

He's also got a mixed number.

He's got 3 3/5.

So can you see the whole number there? That's three, and can you see the extra fractional part? That's 3/5.

What might the question be? See if you can predict what am I going to ask you now.

Look at the cycle title, adding mixed numbers.

The question is this, how much water do they drink in total? Did you guess that? Well done, if you did.

Sofia says, "Let's start by representing this in a bar model." Good idea.

So here we go.

It's a little bit like a part-part-whole model, isn't it? We've got two parts and we've got a whole.

The parts they're known, they are 2 1/5 and 3 3/5, both mixed numbers.

The total or the whole is unknown at the minute.

We're going to calculate it there.

Sofia says, "We can use the bar model to form an equation." What do you think she's going to write in that equation? To find the total, we need to add the known parts.

So 2 1/5 + 3 3/5 = something.

That's the equation.

How would you add together these mixed numbers? Hmm, think about the work you did in the previous cycle.

Have you got any top tips? What could you do? It looks quite complicated, but could we do something to make it simpler? Sofia says, "Let's start by partitioning the numbers." You've done that before.

So instead of 2 1/5, why don't we take them apart and have 2 and 1/5.

And instead of 3 3/5, why don't we take those apart and have 3 and 3/5? So essentially, now we've got four different addends.

What could we do with those addends? We've got 2 + 1/5 + 3 + 3/5.

I think we could rearrange those to make this a little bit simpler.

What about this? Sofia's got a good idea.

We could combine the whole number parts.

So 2 + 3 = 5, nice and simple.

And then combine the fractional parts.

These parts have the same denominator so they can be added using unitising.

So we know that 1 + 3 is 4, easy.

So one, 1/5 + three, 1/5 is four, 1/5.

Nice and straightforward.

And that gives us 5 + 4/5.

Or we've got a nice easy final step.

Combine those together to create a mixed number total.

5 + 4/5 = 5 4/5.

Wasn't that so much more straightforward when we broke it down? In total, the children have drunk 5 4/5 litres of water.

Let's have a little check.

Is this true or is this false? 5 2/9 + 2 6/9 = 7 8/18.

True or false? And why? Have a good think about that.

If you've got a partner to work with, explain it to them.

See if you can come up with an agreement.

Pause the video.

What do you think? True or false? It's false, but why? First, we add the whole number part.

So 5 + 2 = 7.

And then we add the fractional parts using unitising.

So we know that 2 + 6 is 8, so therefore, two, 1/9 and six, 1/9 is eight, 1/9.

So that makes 7 8/9.

So the denominator part was wrong there.

Well done if you spotted that.

It's time for some final practise.

Solve these calculations.

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

So let's read these problems. Jacob has 10 one-quarter metres of rope.

It's a mixed number.

Sofia has three and two-quarter metres of rope.

It's a mixed number.

How much rope do they have altogether? Sofia runs 5 3/10 of a kilometre, mixed number, on Saturday, and then another 3 5/10 of a kilometre, mixed number, on Sunday.

How far has she run in total over the weekend? Remember those skills, remember you can rearrange the parts of the equation.

Remember that you can partition those mixed numbers.

Lots and lots of tips and tricks to make that easier.

C, Jacob is measuring rainfall.

He presents his results in a table.

So there's a table.

So the left-hand column is the day, and the right-hand column is the rainfall.

You might see some mixed numbers in there.

How much rainfall is there in total over the two days that had the most rainfall? Hmm, you've got a couple of extra things to think about there.

Give your answer as a mixed number of centimetres.

All right, best of luck with that.

If you can work with a partner, I'd recommend that so that you can share strategies.

See you soon for some feedback.

Pause the video.

Welcome back.

How did you get on? How did you find that? How confident are you feeling? Let's have a look at some answers.

Number one, 1 3/7 + 3 2/7 = something.

So we could add the whole number parts together, first, 1 + 3, and then we could add the fractional parts.

3/7 + 2/7, which is 5/7.

So that's 4 5/7.

1 3/7 plus something equals 4 5/7.

That's 3 2/7.

6 3/8 + 4 2/8.

Again, whole numbers could be added together first and the fraction parts, that's 10 5/8.

3 2/8 plus something equals 6 4/8, or the other part would be 3 2/8.

And then 1 3/10 + 1 3/10 + 2 5/10.

Well, they've all got the same denominator.

We could add the whole numbers together first.

So 3 + 2 + 1.

And then we could add the fractional parts together.

3 + 1 + 5 = 9.

So 3/10 + 1/10 + 5/10 = 9/10.

So that's 6 9/10.

Well done if you got that.

And then we've got 2 30/100 + 1 15/100 + five 35/100.

We can still use that same strategy, even though we've got this really big denominator.

So we can add the whole numbers together, 1 + 2 + 5, which equals 8.

And then we could add the fractional parts together.

And that would give us 8 80/100.

Well done if you got that.

And then number two, Jacob has 10 and one-quarter metres of ropes.

Sofia has three and two-quarter metres of rope.

How much rope do they have altogether? Now, you might have represented that as a bar model like this.

The parts are 10 and a quarter and three and two quarters.

The denominators are the same there.

So we can use that same strategy as before.

It's just in a word problem.

This time we can add together the whole number parts and then the fractional parts.

The total of the whole is unknown at the minute, but when we combine those together, 10 and one quarter plus three and two quarters, equals 13 and three quarters.

Jacob and Sofia have 13 and three-quarter metres of rope.

And then Sofia runs 5 3/10 of a kilometre on Saturday and then another 3 5/10 of a kilometre on Sunday.

How far has she run in total over the weekend? We could use a bar model again.

And just like before, we could use that strategy of adding the whole number of parts together and then adding the fractional parts together.

The parts of 5 3/10 and 3 5/10.

The total is unknown.

We need to add together the known parts and whatever strategy you chose, it gives us 8 8/10.

Sofia runs 8 8/10 kilometres in total over the weekend.

And then Jacob is measuring rainfall.

He presents his results in a table.

How much rainfall is there in total over the two days that had the most rainfall? Well, the two days that had the most rainfall were Monday and Thursday.

So we need to add together 2 1/5 and 1 3/5.

And when we do that, and I would add the whole number parts together first and the fractional parts, that gives us 3 4/5.

There was 3 4/5 centimetres of rainfall over those two days.

Very well done if you got that, that was extra challenging.

We've come to the end of the lesson.

Today's lesson has been efficiently solving additional problems within a whole.

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

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

And you've seen lots of examples of that today.

You've worked with lots of examples where the denominator is the same.

To add fractions with the same denominator, the language of unitising can be used.

And you've done also lots of other little things like partitioning the mixed number when adding the numbers together.

Very well done on your achievements and your accomplishments today.

I think you've been amazing.

What do you think? Give yourself a pat on the back.

You deserve it.

I hope I get the chance to spend another math lesson with you, so at some point in the near future.

But until then, have a great day.

Whatever you've got in store, I hope you're successful.

I hope you work really hard and try really hard just like you did in this math lesson.

Take care and goodbye.