Lesson video

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Welcome to our first lesson in this fractions topic, which is building on the topic from a few weeks ago.

Today we'll be representing multiplication with proper fractions.

All you'll need is a pencil and a piece of paper.

Pause the video now and get your things if you haven't done so already.

So today we're representing multiplication with proper fractions.

We'll start with a quiz to test your knowledge of fractions from our previous lessons, then we'll learn to represent multiplication, reason about multiplication, and then recognise patterns in multiplication before you move on to an independent task.

There won't be a final quiz in today's lesson.

So let's start with your initial knowledge quiz.

Pause the video now and take the quiz.

Click restart once you're finished.

Great work, so let's start with the do now.

Look at the key vocabulary and match it to the arrows pointing to the fraction.

For the denominator is where the purple arrow is pointing to.

This tells us how many equal parts the item is divided into.

The numerator is where the pink arrow is pointing to.

And that tells us the number of parts that we're dealing with.

And then the orange arrow is pointing to the vinculum, and that separates the numerator and the denominator.

Now here is your next part, so your key vocabulary is unit and non-unit.

Read the sentences and insert the correct word.

So 2/3 is an example of a non-unit fraction as its numerator is not one.

And 1/4 is an example of a unit fraction where it's numerator is one.

Okay, great work.

Now let's look at representing multiplication.

So here we have a multiplication problem, six times 3/4.

And I want you to think about how many different ways you can represent the calculation.

Can you write it with words or numbers? Can you draw a picture to represent it, and have a look at the hints up in the top right hand corner of your screen to help you.

Pause the video now and make some notes.

Here are some of the ways that you may have represented this problem.

So, you know that six times 3/4 is the same as six groups of 3/4.

You could have you use your knowledge of repeated addition to add 3/4 six times.

You know that multiplication is commutative.

So six times 3/4 is the same as 3/4 time six.

You may have chosen to draw it pictorially using a bar model.

So six here is split into quarters, and we are interested in three of those quarters.

You may have applied your knowledge of decimals from our previous fractions unit, and done six times 0.

75, because 3/4 is equivalent to 0.

75, or you may have chosen to turn this into an improper fraction.

So six lots of 3/4 is the same as 18/4, or turned it into a mixed number four and 2/4, and then simplified to four and 1/2.

So you've got lots of representations to use here.

So I'm going to leave them up on the screen and I want you to use these representations that we've looked at together, and this time apply them to four times 2/5.

Pause the video and use the examples as a basis for representing that calculation.

So this is what your representations may have looked like.

Four groups of 2/5.

Repeated addition of 2/5.

Thinking about the commutativity of multiplication, and switching those things around, drawing a bar model representation, or then representing as an improper fraction, and then a mixed number, and finally using your knowledge of decimals.

So this is how we can represent multiplication of an integer, a whole number and a fraction.

Now here we need to think about what are we trying to find, and how has this been represented using a bar model? So our question is 1/2 times 1/4.

Here what we're finding is 1/4 of 1/2.

The blue bar represents 1/2.

So we've got the whole is represented by the whole bar model, and the blue bar is representing 1/2 of the whole.

The red bar is representing 1/4, and we're trying to find 1/4 of 1/2, and we can look at the separate parts here, and see that this red bar represents 1/8.

So 1/4 of 1/2 is equal to 1/8.

Now, today we are just representing using pictorial images.

So we're not going on to solve it, but you can see that if I'm looking at 1/2 on my bar model and finding 1/4 of that, that is equal to 1/8 of the whole.

Let's have a look at another one together.

So we're thinking here what multiplication calculation is represented by the bar model.

So here I've got my whole, and the whole has been divided into five equal parts.

So this represents 1/5, and the 1/5 has been divided into two, and we're looking at one of those parts.

So this is looking at 1/2 and this is looking at 1/5.

So what we're trying to find here is 1/2 of 1/5.

And when you can have a think about how, what that answer would be, but remember we're not solving today we're just representing today.

So let's think about reasoning about multiplication.

We're going to represent, and then we're going to solve this multiplication calculation.

So we've got four times 3/5.

So I can solve this by simply using repeated addition.

So 3/5, plus 3/5, plus 3/5, plus 3/5.

And I know that when I'm adding fractions with the same denominator or the same units, I just add the numerators.

I know that four groups of three is equal to 12.

So four groups of 3/5 is equal to 12/5.

And then that's as an improper fraction.

I know that that as a mixed number is two and 2/5.

Now we're going to come on to the link between the integer and the numerator shortly, but you may have already started to notice how we can get to this solution without using repeated addition.

Let's have a look at another one together.

So we've got six times 2/3, so six lots of 2/3.

So we can again use repeated addition.

And I've represented this also using a bar model.

So six lots of 2/3.

I know that six groups of two is 12.

So six groups of 2/3 is 12/3.

And then if I'm converting that to a mixed number, I've got it represented using a bar model here, 12/3 is equal to four.

So then let's start to look at the patterns, and the link between the integer and the numerator.

So we need to pause the video and have a think about whether you can see any patterns in the numbers here, specifically focus on the integer and the numerator.

Pause the video now.

So if we look at the numerator of the answer in this case, six times 3/4 or six groups of 3/4, we can see that the numerator of the answer is always found by the multiplication of the integer, and the numerator of the multiplier.

And the denominator hasn't changed.

So I've multiplied the integer by the numerator, six times three gives me 18, and the unit is still quarters.

So it's 18 quarters.

And you can see this pattern across the different representations.

Four multiplied by three is equal to 12, and the denominator stays the same.

So it's 12/5.

Again six multiplied by two is equal to 12, and the denominator doesn't change.

So you can start to use that, this principle we'll be using and putting into action in tomorrow's lesson.

Today it's all about looking for patterns and representations.

So now I would like you to pause the video, and see if you can see any patterns in the numbers here.

What I'd like you to focus on specifically is the integer and the answer.

What patterns do you notice here? Pause the video now.

So what you may have noticed is that the answer is always less than the original integer.

So six times 3/4 is four and 1/2.

So the answer four and 1/2 is less than the original integer that's because the integer has been scaled down by the value of the multiplier.

So that four, four and 1/2 is 3/4 of six.

Two and 2/5 is 3/5 of four, and four is 2/3 of six.

So your answer in this case will always be less than your original integer.

So that's a good way to sense check whether you've got the answer correct.

Now it's time to put some of the things that we've been looking at into practise.

So you'll be looking at the representation's of fraction multiplication, and then we'll be using these to help us in tomorrow's lesson of solving multiplication calculations.

So pause the video to complete your task, and click restart once you're finished so that we can go through the answers together.

So you're asked here to match the calculation to the representation and explain how you know.

So the first one, the whole is one and it's divided into thirds, okay.

So I already think I know that it may be concerning this one, and I would just check that that's what I'm looking at.

So I'm now looking at the thirds have been divided into five.

So I'm looking for 1/5 of 1/3.

So 1/5 of 1/3 is the same as 1/3 times 1/5.

Remember we know that multiplication is commutative so we can think of it either way round.

In the second one, the whole this time is five.

It's been divided into thirds, and I'm looking at two of those thirds with my red bar.

So I'm looking for 2/3 of five, which is the same as five times 2/3.

In the third one, this is a slightly different representation.

So this shape has been divided into two equal parts this way.

So it's been divided in half and then the half has been divided into one, two, three, four, five, into fifths.

So we're looking here at 1/2 times 1/5, or 1/5 of 1/2.

And that leaves us with only one leftover, but we'll check it.

The whole is one.

It's been divided into five equal parts, so into fifths, and then the fifths have been divided into four.

So we're looking for 1/4 of 1/5, 1/4 times 1/5.

Now you have to create your own pictorial representation of this calculation.

So you may have drawn something that looks like this, where your whole is two, and you've divided it into two equal parts to show that the whole is two.

Then you've divided your two into fifths.

And we're looking at three of those fifths.

Two times 3/5.

So this is a way that you may have represented it.

And the second one was 1/2 times 1/3.

So this time your whole was one.

So your initial bar model would have looked like that.

And you're looking at the half divided into thirds, and finding 1/3 of 1/2.

Now here you have been given a representation on a bar model and you were asked to use this to help you identify the product, help you identify the whole.

And if we add on the rest of the bar model, we can see that this red bar represents 1/15.

So 1/3 times 1/5 is equal to 1/15.

And here again, thinking about ahead to our next lesson, you may be noticing some patterns about how to do the procedure of fraction multiplication.

But as I said, it's really important to understand how the representation is before moving on to the procedure.

So question four, a similar question.

Using this representation to help identify the product of 1/4 at times 1/5.

If I add on the rest of the bar model, you can see that this red part here represents one out of 20, 1/20.

Great work today.

I know that was a lot of drawing, and lots of representations, but it's so important to understand the representation before we move on to the actual procedure of how to multiply fractions.

And that's what we will be looking at tomorrow.

We'll be looking at multiplying pairs of proper fractions.

I will see you then.