Lesson video

Hi, my name is Mr. Peters, and welcome to today's lesson.

In this lesson, we're gonna be thinking about multiplying mixed numbers by whole numbers where we don't bridge one whole each time.

If you're ready to get started, let's get going.

So by end this lesson today, you should be able to say that I can multiply a mix number by a whole number where the fractions do not bridge one whole.

We've got a couple of key words you're gonna be referring to throughout this lesson today.

I'll have a go at saying them and then you can repeat them after me.

Are you ready? The first one is represent.

Your turn.

The second one is mixed number.

Your turn.

And the last one is partition.

Your turn.

So to represent something means to show something in a different way.

A mixed number is an improper fraction written as a whole number plus the fractional part.

Fractional part here must be a proper fraction.

And finally, to partition means to split an object or a value down into smaller parts.

So we've got two learning cycles in this lesson today.

The first cycle we'll be thinking about representing problems and the second cycle we'll be thinking about calculating those problems as well.

Let's get started with the first cycle.

Today we've got Jacob and Sam.

They've got lots of great thinking to share with us as we go through, and they might also have a few questions along the way, which will support us with our learning as well.

Okay, so let's start here with Jacob then.

Jacob is in a band and he goes to band practise three times a week.

Hmm, I wonder what kind of band he's in.

Maybe it's a rock band or a jazz band or a pop band.

Hmm.

Are you in a band? Do you like making music at all? If so, I wonder if you go and practise as much as Jacob does.

Jacob goes to practise three times each week.

And for each practise session they play for two and a quarter hours.

So how long does the band practise each week? Well, let's have a look about how we could represent this.

Hopefully you can see here we've got three wholes.

Each one of these wholes represent one hour, and we know that Jacob's band practise for two and one quarter hours.

So you can see that we've got two circles that are fully shaded and the third circle has been divided into quarters, hasn't it? And one of those quarters has been shaded.

So we can represent this as two and one quarter hours that have been shaded.

So let's have a look at how we've represented this below.

We can see that we've got a number of circles and we can see that each circle represents one hour.

And therefore in the group we've now got two and one quarter hours that have been shaded.

They practise three times a week.

So we can repeat this group three times.

We can go on to represent this as an expression.

We can represent this as two and a quarter multiplied by three, or two and a quarter three times.

Or we can represent it as three lots of two and one quarter hours, three multiplied by two and one quarter.

We know that we can rotate the order of the numbers in these expressions because of the commutative law.

Thinking about what each of the numbers represent in the expressions, the two and a quarter represents the number of hours that they practise for each session.

And the three represents the number of sessions that they practise each week.

Here's a different example.

Sam this time is putting posters up on her bedroom wall.

She wants to put two posters on the wall together side by side.

Each poster is 35 and a third inches wide.

What is the combined width of the posters? Hmm.

Have a look at the image below.

Can you start to see how we've represented this? We can see we've got two posters, haven't we? And each one of these posters has a width of 35 and a third inches, and they need to be next to each other, which is why they're placed that way.

And we're looking for the combined width, aren't we, how long they are when they're put next to each other.

So we can represent this as an expression, can't we? We've got two lots of 35 and one third, or we've got 35 and one third two times.

And we know we can write the numbers in these expressions in either order because of the commutative law.

And we know that if we were to work out the product, that the product would be equal no matter the order of these numbers in the expressions.

So take a moment to have a think again then, what do each of the numbers represent in the expressions? Well, the 35 and a third represent the width of each of the posters, doesn't it? And the two represents the number of posters that there are joined together, isn't it? Okay, time for you to check your understanding now.

Can you tick the expressions that match this problem? Take a moment to have a think.

The problem is what Sam is suggesting below.

Sam is saying that her mum goes for a run four times a week and each day she runs two and a half miles.

Take a moment to have a think.

That's right.

We could represent it as either A or D, couldn't we? A is four multiplied by two and a half.

The four represents the number of times she goes running, and the two and a half represents how far she runs each time.

We know that we can change the order of these numbers in our expressions because of the commutative law, and that's why D also works.

We've got two and a half multiplied by four.

The two and a half represents the distance that she runs each time.

And the four represents how many times she goes running.

Here's another problem.

Can you write an expression this time yourself that matches it? Let's have a look.

It says I have one and a half avocados for my lunch three times a week.

Take a moment to have a think.

That's right.

We could represent it like this.

This is one and a half multiplied by three, or we could represent it as three multiplied by one and a half.

The three represents the number of times in a week that Sam has an avocado for his lunch.

And the one and a half represents how much of an avocado he has for his lunch on those days.

Okay, on for our first tasks for today.

What I like to do is have a look at these problems here and write two expressions to represent each one of the problems. Good luck and I'll see you back here shortly.

Okay, let's see how we got on there.

So for the first one, a smoothie recipe requires two and a half apples.

I make the smoothie each day before school.

How many apples do I need for the week? Well, if you have the smoothie each day before school, there are five school days in the week, so we're going to need it five times.

And each day we use two and a half apples.

That means we're gonna have five multiplied by two and a half or two and a half multiplied by five.

So we look at B.

It takes three and a quarter hours to drive to my uncle's house.

How long does it take to drive there and back? Well, we're gonna do this journey twice.

We're gonna drive there once to get there, and then we're gonna drive all the way back.

So that's two journeys of three and a quarter hours, isn't it? So we can represent that as two multiplied by three and a quarter or three and a quarter multiplied by two.

And the next one, Sam is making a chain with paperclips.

Each paperclip is four and four tenths of a centimetre long.

Sam uses eight paperclips.

How long is her chain? Well, we know that each paperclip is four and four tenths long and she uses eight paperclips.

So we're gonna represent this as eight multiplied by four and four tenths, or four and four tenths multiplied by eight.

And the last one, a day on Pluto is six and four tenths the length of a day on earth.

How long is a day on Pluto? Well, we're gonna have to think here, aren't we? It's six and four tenths the length of a day here on Earth.

So we can use the six and four tenths, but we need to ask ourself the question, well, how long is a day on earth? Well, we know a day on earth is 24 hours.

So we're looking to create an expression then that would be 24 multiplied by six and four tenths, or six and four tenths multiplied by 24.

The 24 represents the length of time on earth, and the six and four tenths represents how much longer it is for a day on Pluto.

Well done if you managed to get that one too.

Right, and onto cycle B now then, calculating problems. Okay, so let's go back to Jacob and his band practise then.

And let's think about how we could actually calculate how many hours Jacob practises with his band each week.

We know that each band session lasts for two and a quarter hours, doesn't it? And they do this three times in a week, don't they? So have a look at our image above now then.

Previously we've been partitioning it into the number of hours that he practises on each day.

I wonder if we could partition it in a slightly different way this time.

Take a moment to have a think.

That's right, Sam.

We could partition it into the number of whole hours that they practise and the number of parts of an hour that they practise.

So each practise has two and a quarter hours, and that two and a quarter hours can be partitioned into two hours and one quarter of an hour.

And you can see that how we've partitioned it in the image below.

We can say that we have three lots of two whole hours and we also have three lots of one quarter of an hour.

What could we do next then? Take a moment to have a think now.

That's right.

We could multiply the number of whole hours that we have altogether, couldn't we? So let's do that, three multiply by two hours is equal to six hours.

And then we can also multiply the number of parts of an hour that we have.

So three multiplied by one quarter of an hour, that would be three quarters of an hour.

Hmm? What do we do now? Hmm? So what do we need to do now then? That's right.

So far we've partitioned it into the whole hours and the parts of an hour, haven't we? And now we need to recombine the total amount of whole hours with the total amount of the parts of the hours that we had.

So we can use addition to do this, can't we, Sam? You're right.

We can say that six plus three quarters is equal to six and three quarters.

So altogether, Jacob practises for six and three quarter hours with his band every week.

That's right Jacob.

You will be a rockstar before you know if you keep practising that amount.

And I love your rockstar name as well, Jacube Infinity.

I love what you've done there.

Really mathy as well.

I love what you've done there.

Really nice to see that math has inspired your rockstar name.

Okay, and let's revisit Sam's poster problem in her room.

We know that each poster had a width of 35 and a third inches, didn't we? And we had two of those posters.

So when they're placed next to each other, we need to find out how long they are.

We knew the expression we could write.

We could write it as three and a third multiplied by two.

So let's think about how we could record this in different ways now then.

We know the calculation is 35 and a third multiplied by two.

And we can partition this again into its whole numbers and its parts of a whole.

So 35 and a third can be partitioned 35 and one third, and we can multiply the 35 by two.

That would leave us with 70.

And we can multiply the one third by two.

That would leave us with two thirds.

And as we did before, just like Jacobs, we now need to recombine the wholes and the parts, don't we? So we can put 70 plus two thirds is equal to 70 and two thirds.

So the total combined width of the two posters is 70 and two thirds of an inch.

We can now also think about recording it in a slightly different way.

Have a look here.

We know again, the calculation is 35 and a third multiplied by two.

And we could record that as 35 multiplied by two plus one third multiplied by two.

Just like before, we've partitioned the 35 and a third into both its parts and its wholes.

And we've multiplied each of those by two.

And we've written that as one equation from left to right all on the same line.

We can now think about calculating each part of these equations.

So 35 multiplied by two we know is equal to 70.

And one third multiplied by two we know is equal to two thirds.

So altogether we've got 70 and two thirds.

We can now join these together using addition to find the total width.

So we've got 70 and two thirds of an inch being the total width of the posters.

So have a look at the two representations of how we calculated this mass.

Which of these did you prefer? Take a moment for you to have a quick think.

That's right.

There's no right or wrong to which one you prefer here.

You can choose either way of doing it as long as the maths make sense to not only yourself when you try to explain it, but to other people if they were to try and look at it without you there to explain it for them.

Okay, time for you to check your understanding now.

Can you calculate this equation here to start us off with? That's right, four and one fifth multiplied by three.

Well, we can partition the four and one fifth into its wholes and its parts and we can multiply both of those by three.

So four multiply by three plus one fifth multiplied by three.

Four multiplied by three is equal to 12.

And one fifth multiplied by three is equal to three fifths.

So altogether we can say that 12 plus three fifths is equal to 12 and three fifths.

Well done if you've got that.

And here's another one.

Can you explain the mistake this time? Take a moment to have a think.

That's right, Sam.

This person here has actually partitioned the mixed number into a whole number and a fractional part.

They've multiplied the whole number by the four and they've also now tried to multiply the fraction by the four.

However, instead of multiplying the numerator by the whole number, they've multiplied the denominator by the whole number.

And we know that when you multiply a whole number by a fraction, you can multiply the whole number by the numerator to get the numerator.

However, the denominator would stay the same each time, wouldn't it? Well done if you spotted that mistake for yourself.

Okay, so onto our final tasks for today then.

What I'd like to do here is fill in the missing boxes for each of these examples here as you go.

And then for task two, what I'd like to do is find as many different solutions as you possibly can.

For task three, I'd like you to use the slightly different strategy to represent the maths and how you could solve these.

And then for task four, have a go at completing these without any scaffolding to support you through that process.

Good luck with those tasks and I'll see you back here shortly.

Okay, let's go through these then.

The first one is five multiplied by two and one sixth.

So that can be represented as five multiplied by two plus five multiplied by one sixth.

Five multiplied by two is ten, five multiplied by one sixth is five sixths.

And altogether that gives us 10 and five sixths.

For the second one, that's three and two ninths multiplied by four.

Well that can be three multiplied by four and two ninths multiplied by four.

Three multiplied by four is 12.

And two ninths multiplied by four is eight ninths.

And altogether that would give us 12 and eight ninths.

The next one is four and three sevenths multiplied by two.

That can be represented as four multiplied by two and three sevenths multiplied by two.

Four multiplied by two is equal to eight.

And three sevenths multiplied by two is equal to six sevenths.

Altogether that would be eight and six sevenths.

For task two then, how many different solutions can you find for this? Here's one example of how you may have done this.

So we're gonna place a two in both of these to start us off with to see if that works.

Two multiplied by something will be equal to six, and two multiplied by something will be equal to five tenths.

Hmm, I think that's gonna work because we know that two multiplied by three is equal to six and then two multiplied by something gives us 10 twelves.

Well, 10 twelves has an even numerator, doesn't it? It has a numerator of 10 and two multiplied by something is equal to 10.

That would be two multiplied by five.

So we know that two multiplied by three plus two multiplied by five tenths would be altogether equal to six and 10 twelves.

So the original starting expression that we had would've been two multiplied by three and five twelves.

Well done if you got that for yourself or something similar to that.

Okay, and for the next two tasks then, we need to multiply the four and one eighth, both of them by six, don't we? So four multiplied by six, and one eighth multiplied by six.

Four multiplied by six equal to 24.

And one eighth multiplied by six is equal to six eighths, which gives a total of 24 and six eighths.

Three and two elevenths multiplied by five.

Well, three multiplied by five is equal to 15.

And two elevenths multiplied by five is equal to 10 elevenths.

So that'd be 15 and 10 elevenths.

And then finally four and three fifteens multiplied by two.

So four multiplied by two is equal to eight and three fifteens multiplied by two is equal to six fifteens.

Altogether gives us eight and six fifteens.

And then finally, to complete the last calculations, one and two ninths multiplied by three is equal to three and six ninths.

Two multiplied by three and five twelfths is equal to six and 10 twelfths.

And then finally five and two twenty fourths multiplied by six would be equal to 30 and 12 twenty fourths.

Well done if you've got all of those.

Okay, that's the end of our learning for today then.

So to summarise what we've learned, multiplying a mixed number by a whole number can be represented either using the area model or as an equation.

Mixed numbers can be partitioned into their whole parts and their fractional parts to help us make the calculation easier.

And when multiplying a mixed number by a whole number, you multiply the whole number of the mixed number by the integer.

And then you can multiply the numerator of the mixed number by the integer as well.

The denominator would stay the same.

Thanks for joining me again today.

Hopefully you've enjoyed that lesson and becoming a bit more secure in thinking about how you can multiply mixed numbers by an integer or a whole number.

Take care and I'll see you again soon.