Lesson video

Hi, my name's Ms. Lambell.

You've made an excellent choice deciding to stop by and do some maths with me today.

I'm really pleased.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is securing understanding of addition and subtraction with fractions, and that's in the unit arithmetic procedures with fractions.

By the end of this lesson, you'll be able to generalise and fluently use addition and subtraction strategies to calculate with fractions and also mixed numbers.

Some keywords that we'll be using in today's lesson, improper fraction.

Remember, this is a fraction where the numerator is greater than or equal to the denominator.

Mixed number, and this is an improper fraction written as an integer part plus it's fractional part.

Remember, the fractional part must be a proper fraction.

An LCM is our abbreviation for lowest common multiple.

We're going to split today's lesson into two separate learning cycles.

In the first one we'll look at addition and subtraction with mixed numbers, and in the second learning cycle we will concentrate on efficient calculation strategies.

So as good mathematicians, we like to be efficient if we possibly can.

Let's get going with that first learning cycle then.

Here we've got Laura and Andeep and they're talking about how much time they spent on their homework last week.

I'm wondering how much time you spent on your homework last week.

Laura says she spent 2 3/4 hours on her homework.

Andeep, "I spent 3 5/12 hours." How much longer did Andeep spend on his homework? Think about what calculation do you need to do to answer that question.

Sometimes it's easier to think of what would happen if it was integers.

If Laura had spent two hours and Andeep had spent three hours, we know to calculate how many extra hours Andeep did, we would subtract Laura's from and Andeep's.

So that's exactly what we're going to do here.

We're going to do 3 5/12 subtract 2 3/4.

Now, you are confident with adding and subtracting with proper fractions.

We are now going to look at what happens if we've got mixed numbers.

So what we're going to do is we are going to create improper fractions because then we can apply the same methods as we did with proper fractions.

So I need to convert 3 5/12 into an improper fraction that's equivalent to it.

You've done this in previous lessons.

If you needed to, of course, you could go back and revisit that in one of the other videos.

3 5/12 is equal to 41/12 and 2 3/4 is equal to 11/4.

We need to do 41/12 subtract 11/4.

We have to create two fractions that are are equivalent to these that have a common denominator.

And remember, try and use the LCM if you can because that's the most efficient.

And the lowest common multiple of 12 and 4 is 12.

So we are going to create equivalent fractions with denominators of 12.

Well, luckily that first fraction doesn't need to be changed.

But 11/4 becomes 33/12.

Again, you are really, really familiar with this, but if you needed to go back and check, you could.

41/12 subtract 33/12 is 8/12.

And I can simplify that to 2/3 of an hour.

And you've done the simplification before.

What number is the arrow pointing at? So we can see two numbers on the number line.

We've got the 2 2/3, and then we can see the distance between 2 2/3 and the arrow is 1 3/5.

In order to work out what number the arrow is pointing at, we need to add on the 1 3/5 to the 2 2/3.

So just as we did in the previous slide, we're going to change each of our mixed numbers into improper fractions.

We end up with 8/3 plus 8/5.

Then we're going to create equivalent fractions with the same denominator using the LCM, in this case 15.

So we end up with 40/15 plus 24/15, which is 64/15.

Now we need to work out, we need to convert that back into a mixed number.

We know that one whole is 15/15, so I could create four wholes and there would be 4/15 left over.

So my final answer is 4 4/15.

If you needed to here, there's quite a lot there isn't there? You could pause the video.

Just check through each of those steps and make sure you understand how we've got 2, 4 and 4/15.

What number is the arrow pointing at now? We've got 2 2/3, but this time we're moving to the left.

So we need to subtract the 1 3/5.

So that method, remember we're going to use, we're going to convert into improper fractions first, then create our equivalent fractions with a common denominator in this case 15.

Then we can perform the calculation 16/15 and then we want to convert that back into a mixed number if we're left with an improper fraction.

So here we end up with 1 1/15.

Your turn now.

What number is the arrow pointing at? So you can pause the video now and then come back when you're ready.

Good luck.

Great work.

Let's have a look and see if you've got the same answer as I did.

Here are your calculations.

If you need to pause the video and look through them, you could.

Because we know that this is 1 2/3.

Well done.

Did you get that right? Super work, well done.

Now, we're gonna have a look at this calculation here.

What digit is missing in this calculation? So this time we need to follow through exactly the same process, but we have the answer and we need to find the missing digit.

We don't know because it's 3 and something quarters.

We don't know what the numerator of the first fraction is going to be, but we do know that 1 5/6 is 11/6.

And we do know that 1 11/12 is 23/12.

We need to create the common denominator.

So the common denominator here is 12.

So I've converted 11/6 into 22/12.

Now, we should be able to find that missing numerator.

And it's 45.

45 subtract 22 is 23.

Yeah, that's right, isn't it? So now we know that the missing denominator is 45, but we've started off with a fraction that was a mixed number.

So we need to convert that back into a mixed number, which is 3 3/4.

So the missing digit was 3.

Let's just do one more of those together and then I know you'll be ready to have a go at one of those independently.

The process is the same, but we are just going to work out what the missing digit is.

So convert to improper fractions step one.

Change so that we have got a common denominator.

So we ended up with 5/4 plus something over 8 is 29/8.

Common denominator here is going to be 8.

So we end up with 10/8.

Add something over 8 equals 29/8.

Yeah, you are right, the missing denominator, sorry, the missing numerator is 19.

And so now we just need to convert 19/8 back into mixed number form, which is 2 3/8.

The missing digit was 3.

Now, I'd like you to have a go at this question.

What is the missing digit in this calculation? Good luck with this.

Pause the video and then when you're ready, come back and check in.

Well done.

How confident are you that you've got this right? Very confident.

Wow, that's fantastic, brilliant.

Let's have a look.

So we converted what we could.

Then we found our common denominator, which was 6.

So we ended up with something over 6, add 11/6 is 25/6.

So what is the missing numerator? It was 14, but then remember we needed to convert that back into a mixed number.

But notice here I've simplified first 14/6 is actually 7/3, and that was 2 1/3.

Well done if you got that one right.

It's a little bit harder, wasn't it, than the previous one? Because we had to do a little bit of simplifying.

You are now ready to have a go at our first task in today's lesson, task A.

By calculating the value of each side, I'd like you to complete the inequality statement by placing either the greater than or less than symbol between them.

Good luck with these.

I know that you are absolutely ready to tackle this type of problem.

Good luck.

You can pause the video now.

And question number two, you need to find the missing value in each of the diagrams and give your answer as a mixed number in its simplest form.

There's quite a few to do there, so you'll be gone for quite a while, but I'll definitely be here when you get back.

You can pause that video now.

Great work.

And moving in on to question number three.

So this is probably the most challenging of those, but remember you did one of those in our check for understanding earlier and you got it right, didn't you? Let's have a go at finding the missing digits.

Pause the video and then come back when you're ready.

Good luck.

Amazing work.

You're absolutely smashing this.

Let's check your answers.

So the first one, two, sorry, 4 and 2 is 6 and 3 and 3 is 6.

So actually here we don't need to do the whole calculation.

We can just compare the fractional sum in each calculation.

So 3/4 add 1/5 is 19/20.

And 1/2 plus 2/5 is 18/20.

So we can see that the left-hand side is greater than the right-hand side.

B, 4 subtract 2 equals 2 and 3 subtract 1 equals 2.

So again, we just need to look at the fractional parts.

So we need to subtract the fractional parts in each calculation.

4/5 subtract 3/4 is 1/20.

1/5 subtract 2/5 is 2/20.

So we now know that the right-hand side is larger.

And C, here we know that 1 add 2 is 3 and 2 add 1 is 3.

So again, we just need to compare those fractional sums. 5/8 add 3/4 is 1 3/8.

5/6 add 1/3 is 1 1/6.

3/8 is equal to 9/24.

And 1/6 is equal to 4/24.

Both of those fractions, the whole number part, the integer part was one.

So we just need to then compare the fractional part and we can see that the left-hand side is greater than the right-hand side.

Here are the symbols and the which way round each of them needs to be so that you can check yours.

Now, we can go through the answers to question two.

A, the question mark was 4 31/40 B, 3/4, C, 4 1/12, D, 1 39/40, E, 5 7/12, and F, -1 11/12.

And finally question number three.

The missing digits were in A, 2, B, 1, and C, 5.

Well done if you've got all of those right.

Now, we can move on to our second learning cycle.

And in this cycle we're going to be concentrating on efficiency, so being the most efficient mathematicians that we possibly can.

So we'll be looking at strategies to do that.

Here we have Laura and Andeep.

Now, we already know about how much homework they're doing.

We used that information in the previous learning cycle.

So just to recap, Laura spent 2 3/4 hours and Andeep, sorry, Andeep spent 3 5/12.

How long did they spend on their homework in total? That's what we're going to look at first.

We know to find a total of something, we sum the two things.

So 2 3/4 add 3 /12.

Are those two things equivalent? What do you think? Yes, they are as they use the commutative law.

We can put the equal symbol between them.

We can find the sum of the integers and the fractions separately, and then find the sum of those.

So here we can find the sum of our integer part, which is 2 and 3, so 2 add 3.

And then we can sum the fractional parts, so 3/4 add 5/12.

Those two things are equivalent.

We know that 2 add 3 is 5.

And then we know that to add any two fractions together, they must have a common denominator.

In this case, that's 12.

3/4 becomes 9/12.

Now, we can add together the 9/12 and the 5/12 to give us 14/12.

Then we can convert that into 7/6.

So we simplify that fraction.

And we know that 7/6 is 1 1/6, and then we can add those together.

So the total time that they spent was 6 1/6 of an hour.

We'll go back to this problem.

So we actually looked at this exact problem in the first learning cycle.

We looked at how much longer Andeep spent on this homework.

So this was the calculation that we did.

Are those two things equivalent? And again, yes they are.

They use the commutative law.

We can find the difference of the integers and the fractions separately, and then find the sum of those.

Here we can see we've separated out the integer parts and the fractional parts.

Remember we are thinking here about efficiency.

This is much more efficient I think you'll agree than converting both of those fractions into improper fractions because 3 subtract 2 is 1, dead easy.

Now, we're going to do 5/12, subtract 3/4.

And we're super good at this because we've been doing it lots recently.

5/12 subtract 3/4, write them so they have a common denominator.

5/12 subtract 9/12.

So we end up with one add.

What is 5/12 subtract 9/12? That is -4/12.

And we know that 1 add -4/12 is 8/12, and that simplifies to 2/3.

Here, I would like you please to choose the most efficient step in each column.

So we're answering the question 3/8 subtract 1 5/6.

And each of those steps there may be a correct step, incorrect steps.

But what I want you to do is to identify which is the most efficient step in each of the columns.

You can pause the video now and when you've got your answer, you can come back and we'll check.

In the first column it was C, in the second column B, and in the third column it was C.

So the third column was definitely the right answer.

So that was the only one you could have.

If we look at the first column, you could have had B, but that's not the most efficient because it's not used the lowest column multiple.

And again, B in the second column was the most efficient because it used that lowest common multiple.

A is not wrong, but it is not the most efficient.

Laura and Andeep are working out the answer to 2/7 add 3/21.

What is the same and what is different about their calculations? Let's have a look at what they did.

Here's Laura's method.

2/7 add 3/21 is 2/7 add 1/7, which is 3/7.

And here's what Andeep's done.

2/7 add 3/21 is 6/21, add 3/21, which is 9/21, which is 3/7.

So I'd like you to just pause a moment and think about what is the same and what is different about their calculations.

Laura noticed that actually 3/21 can be simplified to 1/7.

So she spotted that that was not in its simplest form.

And by simplifying that first we can see that her method was much more efficient than Andeep's.

Andeep's is not wrong, it's just not quite as efficient as Laura's.

I'd like you now to decide as we go through this, we're going to do this check for understanding together rather than you pausing in the video.

I'll pause to give you a chance to work out the answer.

I want you to think about which is the most efficient method.

So look to look at the calculation and decide is there an opportunity to simplify it first, or will we just use the lowest common multiple method? Let's have a look at the first one.

1/7 add 3/8.

Where do you think it goes? That goes in the lowest common multiple method.

1/7 add 3/8 are already in their simplest form.

Let's take a look at the next one.

5/8 subtract 6/24.

That one goes in the simplify first because 6/24 is actually 1/4, and so therefore we could simplify that first.

Maybe make our calculation a little bit easier.

Let's look at the next one.

5/10 add 7/12.

Where is it going? You're right, yeah, simplify first.

We know that 5/10 is 1/2, so let's do that simplification first just to make our life a little bit easier.

And the next, 2/5 subtract 5/12.

Simplify first or LCM? You are right, yeah, LCM.

2/5 and 5/12 are already in their simplest form.

9 over, sorry, 19/3 add 9/2.

What about this one? We've got some improper fractions here.

Simplify first.

And maybe I gave you a clue there as to where that was going because I said, I mentioned the fact that they were improper fractions.

So change them into mixed numbers first because then we can use that more efficient method that we've just looked at.

Now, you are ready to have a go at this task.

Which of these are equal to 9 3/4? So you've got some calculations there, you're going to work them out.

Some of them are equal to 9 3/4, but others are not.

And you'll see here part i says for those that do not have an answer of 9 3/4, what do you need to add or subtract so that they do? So you're gonna work out the answers, identify those that are equal to 9 3/4, and then decide what you would need to do to make the others have an answer of 9 3/4.

Good luck with this.

You can pause the video now.

I look forward to seeing you when you come back.

Great work.

Let's have a look at the answers then.

So A was correct, D was correct, and E was correct.

I shouldn't really say correct.

They were equal to 9 3/4.

Let's have a look at what we needed to add or subtract in order to make the others be equal to 9 3/4.

So that was part i.

B, we would need to add 7/10.

C, we would add 1/12.

F, we would add 1/4.

G, we would subtract 1/5.

And H we would add 1/6.

How did you get on with those? Brilliant, well done.

We are now ready to summarise our learning from today's lesson.

Let's take a look at what we've done.

One method for adding and subtracting mixed numbers is to convert the mixed numbers to improper fractions and find a common denominator.

That's what we looked at in that first learning cycle.

So our example here, 3 1/2 subtract 5/6.

We converted those firstly into improper fractions and then we would find our lowest common multiple, which in this case would've been 6, so that we could complete that question.

Converting to improper fractions is not always the most efficient way of adding and subtracting mixed numbers, and that's what we've looked at in that second learning cycle.

An alternative method is to deal with the integers and the fractions separately.

And we can see that example there, 3 5/12 subtract 2 3/4.

We can separate out the integer parts and we could do 3 subtract 2, and separate out the fractional parts, and we could do 5/12 subtract 3/4, and then we can find the sum of those two parts.

You've done fantastically well today.

I've really enjoyed doing this math with you today.

I look forward to seeing you again.

Bye.