Lesson video

Hiya.

My name's Miss Lambell.

Really pleased that you've decided to join me today to do some maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is, "Ordering Positive and Negative Fractions and Decimals" and this is within our unit, "Comparing Fractions and Decimals, Including Positives and Negatives".

By the end of this lesson, you will be able to order a variety of positive and negative fractions and decimals using an appropriate method of conversion and recognising when conversion to a common format is not required.

Now, we've looked at this previously and I think I've referred to it as number sense.

So we'll be taking another look at that during today's lesson.

A key term that we're going to be using in today's lesson, so I thought it might be worth a quick recap, and that is absolute value.

Remember, this is the distance from zero.

So negative five and five are both five away from zero, so they both have an absolute value of five.

We are going to split today's lesson into three separate learning cycles.

The first of which we're going to look at just concentrating on ordering positive and negative decimals.

Now, you know how to order positive and negative integers.

You know how to order decimals, so we're just gonna put those two together.

We're then gonna move on to positive and negative fractions.

And again, you know how to do to positive and negative integers, you know how to order fractions, so we're gonna put those two together.

And then the final one, we're just gonna bring absolutely everything together and I'm gonna throw it all at you and you are going to be looking at positive and negatives decimals and fractions.

Let's get started on that first one.

Here we have Sam, Jun, Andeep, Izzy and Alex.

All of them have a negative balance on their canteen accounts.

They've spent a little bit more money than they actually had on there and they're each gonna tell us what their balance is.

Sam says, "My balance is minus two pounds 13." Jun, "My balance is minus one pound 23." Andeep, "My balance is minus two pounds 31." Izzy, "My balance is minus one pound 32." And Alex, "My balance is minus one pound 35." We're going to list those in order from who owes the most to who owes the least.

Here are the balances.

So remember, as we move to the left from zero, absolute values increase and as we move right from zero, absolute values also increase.

So here we have all of our numbers are below zero.

So we're looking, the smaller of them will mean the larger absolute value.

So let's rewrite these using their absolute values.

So notice, we just dropped that negative, because we're now not worried about the fact that they're negative, we're just looking at how far away they are from zero.

Now we can put those in order.

Remember, we want the absolute value that is the largest.

That will give us the smallest number, which seems a bit odd, doesn't it? So two pounds 31.

That means minus two pounds 31 is the person who owes the most.

Let's look for the next one.

Two pounds 13 is the next one.

And then one pound 35, one pound 32 and one pound 23.

You may find it useful to do as I did there, which is to write the absolute values underneath.

This may help you to order them.

We're going to now write the following in order from smallest to largest.

There are our numbers.

Will we need to do anything differently, because these are not amounts of money? No, the same applies.

Here's my numbers.

I'm going to write them firstly as their absolute values and how far are they away from zero? And then remember, the one with the largest absolute value is the smallest.

So we're gonna go through each of them.

And to check this, what I can do, is remember, if I start from the right-hand side of the list, negative 0.

057 and I move to the left, the absolute value should get bigger, and they do.

I now know that I've got those numbers in the correct order.

Your turn now.

Write the following in order from smallest to largest.

Pause the video and then when you've got your five numbers in order, come back and check.

Good luck.

Great work.

Here is our order.

Negative 37.

757, negative 37.

75, negative 35.

75, negative 35.

57, and negative 35.

075.

Now we've got a situation where we have a mixture of positive and negative decimals.

Sam says, "If I have a mixture of positive and negative numbers, I like to deal with them separately first." She's going to deal with the negative ones and then the positive ones.

So here's the negatives and here's the positives.

And then remember, we're looking for, the smallest one will have the largest absolute value, so that is negative 0.

84, followed by negative 0.

804 and negative 0.

084.

And then 0.

084, 0.

804 and 0.

84.

Maybe you notice something about those numbers? What do you notice? That absolute values decrease as you get closer to zero and increase when you go past zero.

That's really, really important and remember that image of the arrow on the number line can help you to visualise what's happening.

We now have a check for understanding.

Order these numbers.

Alex has ordered the above numbers.

I'd like you to decide if he's done it correctly.

If you think he hasn't, I'd like you to identify what his mistake is.

So here is Alex's order.

Pause the video, decide if you agree with Alex that they are in the correct order and if you don't agree with him, what mistake has he made? Good luck, you can pause the video now.

Let's take a look then.

It was no, he's not right.

Did you come up with no? You did, fantastic, well done.

This is how I've written it and remember, you may have something a little bit different, as long as it got the same idea.

Going towards zero, he has listed from the smallest to the largest absolute value and away from zero, he has listed them from the largest to the smallest absolute value.

Just remember that image.

Okay, as I move to the left from zero, the absolute values increase, as I move right from the zero, the absolute values increase.

That image can be really powerful in helping us here.

You're now ready to have a go at a task independently.

I'd like you to write the following order from smallest to largest.

You're going to pause the video and then when you've got your answers, you're gonna come back.

Good luck with this.

Great work, now let's check our answers.

I'm not gonna read all of those out.

What I'm going to do instead, is ask you to pause the video, check your answers and then come back when you're ready.

So you can pause the video now to check your answers.

Great work.

Let's now then move on to our next learning cycle.

So ordering positive and negative fractions.

So like I said, we know how to order positive and negative integers, we know how to order fractions, so now we're just gonna put the two together.

We're gonna write these fractions in order from smallest to largest.

It is necessary to convert these fractions to be able to order them? So is it necessary to convert them all into fractions that have the same denominator or all into decimals? What do you think? I'm hoping you said no.

It's like I said in the introduction, it's about that number sense, it's about looking at what do we know about these numbers and we should know enough about these fractions, these are very common, familiar fractions, not to have to convert them to be able to order them.

Hopefully, you decided on that too.

Just that image again to reinforce what's happening.

So let's write these now then in order from smallest to largest.

So firstly, we need to decide which of those is smaller.

So remember, anything below zero, we're looking, the smallest number will have the greatest absolute value.

So from negative one and a quarter, negative half and negative three quarters, which of those has the greatest absolute value? And that's negative one and a quarter.

Now we need to compare negative a half and negative three quarters.

Negative a half has an absolute value of 0.

5 and negative three quarters has an absolute value of 0.

75.

0.

75 is greater than 0.

5, so that the next one in our list.

And then a half.

And then these, it should be obvious which is the greatest one and then which is smaller, one fifth or one quarter? So if I've got my cake split into five pieces and I've got exactly the same cake split into four pieces, which pieces are smaller? It would be the five piece one.

So one fifth then one quarter, then two and a half.

So I mentioned in the introduction about being able to spot whether you have to make the conversion into a common format, and here we didn't.

What about if we wanted to write these ones in order from smallest to largest? What might we do here? So it's not quite so obvious, is it, with those fractions, not quite so obvious.

Five eighths, ooh, yeah, maybe I'm okay and then I get to 11 20ths and then I'm not quite so sure.

We can convert them all into decimals.

Remember, if their denominator is a factor of 10, 100 or 1000, we can easily convert them into decimals.

This is something that you've done in a previous lesson, so it should be very familiar to you.

Negative five eighths, eight is a factor of 1000, so that's gonna be our new denominator and then remember, we're looking for that multiplier.

We need to make sure we're multiplying by one so we don't change the value of the negative five eighths.

So we need to make sure that the numerator is also 125.

And then we get our new numerator.

So I'm just gonna go through one again and then I'm going to just put the answers to the other ones up slowly so that you can follow along with me, but I won't explain all of them.

So eight, we needed to convert that so it had a denominator of 1000.

Eighth multiplied by 125 is 1000.

In order not to change negative five eighths into something different, we need to make sure we're only multiplying by one, which means if the denominator is 125, the numerator must also be 125, because then we've only multiplied by one and then five multiplied by 125 is 625.

So we've converted it to negative 625 over 1000.

Now we can convert that easily to a decimal, negative 0.

625.

Like I said, I'm going to show you in steps the others, but I'm not gonna explain each step, 'cause this is something that you're familiar with from a previous lesson.

So negative three quarters is negative 0.

75.

Now, that's one of the ones that you may know and you may not have need to do the method to get you there.

Let's now take a look at minus 11 over 20.

This time, I'm gonna use 100, because 20 is a factor of 100, and then here are my steps.

And then the final one, three fifths, again, you may remember that three fifths is equivalent to 0.

6.

At any point, remember, you can stop the video, rewind it and go back through that slowly to make sure you understand each step.

Now I'm able to order these.

So negative three quarters was the smallest, that have the greatest absolute value, and we're only looking at negative numbers.

Then we had negative five eighths, then negative three fifths and then negative 11 20ths.

What about writing these ones in order from smallest to largest? What do you we might do here? Will changing them into decimals be helpful here? If we're to change them into decimals, we'd end up with some recurring decimals so it's not such a good method, and the reason for that, if we look, the denominators are not factors of 10, 100 or 1000, and those are the ones where we're going to change them into decimals.

So this is a situation where we need to find a common denominator.

So we go for the LCM.

Remember, that's the lowest common multiple, because that's the most efficient method.

You may choose something different.

It just may take you a little bit longer to get to the end and remember, we wanna try and be as efficient as we possibly can.

The lowest common multiple of three, nine and 18 and six, sorry, is 18.

We can convert them.

We know how to do that, so I'm just going to put those up.

I didn't need to convert negative 11 over 18, because it was already in 18ths.

So I'm just gonna pause a moment and let you have a look through each of those to make sure you understand where those new equivalent fractions have come from.

Now we're able to put those in order.

So remember, we want the one with the largest absolute value first, which is negative 16 over 18, which started off life as negative eight over nine.

Then the next one will be negative 15 over 18, which started life as negative five over six.

Then negative 12 over 18 and then negative 11 over 18.

Now, if you were to write the answers all in 18ths, that will be fine also.

Here now we've got a check for understanding and it's a true or false check.

These fractions are in the correct order from smallest to largest.

So you're gonna pause the video and you're gonna decide if it's true or false, but don't forget, I am gonna expect you in a moment to justify your answer.

So I'm gonna strongly suggest that you actually order those yourself to decide whether it's true or false, because then that hopefully will make the justification easier.

Pause the video, good luck, come back when you're ready.

Well done.

What did you decide on? False, brilliant.

It was false.

Okay, now I'm gonna know whether you just guessed, 'cause I need you please to justify your answer, so is it A or B? A, one fifth is smaller than a half, so negative one fifth is smaller than negative a half or B, the absolute values of negative numbers increase as the numbers get smaller? Which one do you think? And the correct answer is B, absolute values of negative numbers increase as the numbers get smaller.

Think back to that image of the number line with the arrows.

Well done.

Now you're ready to put that into practise.

So we've just got three questions in task B and I'll like you please to use the most appropriate method for converting these so that you can put them in order.

And remember here, we are not using calculators.

Really important not to use your calculator.

So I'm trusting you there.

Pause the video, come back when you're ready.

Good luck.

We're now ready to check through those answers.

A, negative three quarters, negative seven 10ths, negative three fifths, negative 11 20ths.

And there I hope you were able to use your number sense.

B, negative five sixths, negative three fifths, negative a half, negative seven 15ths.

And there hopefully you chose to use a common denominator method.

C, negative three 10ths, negative one fifth, negative four 25ths, negative seven 50ths, and for that one, I'm hoping you chose to change them all into decimals.

That was the most appropriate method there.

Let's move on now to our final learning cycle.

So we can order positive and negative decimals, we can order positive and negative fractions, now let's put all of that together and make sure that if we've got a mixture of any of those, that we can do that.

And we know we can, because we've been successful at the previous two learning cycles.

So like I said, now that we can order negative fractions and decimals, we can now order a mixture of them, but we must make sure that we consider the most suitable method.

So in each situation, I want you to think about, which is the most suitable method, which is the most efficient way of working out the answers to the questions? The methods we have been using are these.

Number sense, so using what we already know about the numbers, converting to decimals, so when all denominators are factors of 10, 100 or 1000, converting fractions with the same denominator or a combination of the above.

Which method? So here, I just want you to think about which method you would use to order these.

Let's have a look at the first one.

I'm just gonna pause for a moment and give you a chance to check which method you think we'll use.

Here I think the most appropriate method is to convert them all to decimals.

We can see that all of the denominators are factors of 10, 100 or 1000 and we already have two decimals.

What about this one? Here I would use number sense.

I know enough about those numbers to be able to compare them.

What about this one? This is a situation where I'll convert them all into fractions.

Write the following in order from smallest to largest.

Now remember, this is the one I just said that we are going to use our number sense for.

So pause the video, use your number sense to put them in order and then come back when you're ready.

Now we're ready to check your order.

Remember, when we're looking at the negatives, the smallest one is the one with the largest absolute value, because it's further away from zero.

So we've here our largest absolute of the negatives is two and a quarter.

So we start with negative two and a quarter.

Our next is negative 1.

9, followed by negative one and three fifths.

And then which is smaller out of 2.

78 and three and two fifths? Well, the 2.

78 only has two ones whereas three and two fifths have three ones.

So 2.

78 and then three and two fifths.

We're now gonna have a look at this one.

Now we've said we're going to convert them all to decimals.

So when we did that little check, what method, what's the most appropriate method, we said we're gonna change them to decimals.

So the first number is a decimal, the second one, we're gonna convert to a decimal.

Notice I've changed the denominator to 100 to give me 0.

35.

This one is easy, because it's already in 100ths, so it's negative 0.

15.

This one is gonna be 0.

28.

This one's already a decimal.

This one, I know I've got three 10ths and it's negative, so negative 0.

3.

And for negative eight over 25, I'm going to convert my denominator to 100 and that gives me negative 0.

32.

Now I can write these in order, remembering that image of our number line and what's happening.

Here we go.

So negative 0.

33.

Notice I've then crossed it off in the list.

I know I've put it in.

And I'm just going to repeat that process until all of my numbers are in order.

Now I've not explained that as we've gone through, because the explanation happened in the first two learning cycles.

So if you need to, go back and watch either of those learning cycles again if you didn't quite understand what I've done on this slide.

Now ordering these from the smallest to the largest and this was the one where we said we were gonna use a common denominator.

Again, I'm just going to go through this quickly, because we did this in the second learning cycle.

So if you don't understand it once I get to the end, then just rewind the video and re-watch that second learning cycle.

So I'm changing them all into 64ths and then remembering that image and then I'm looking for the negative with the greatest absolute value.

Now I'm going to put these in order.

Your turn now.

Write the following in order from smallest to largest.

Pause the video, come back when you're done.

You can do this.

Now we'll check your order.

You should have negative 0.

525, negative 13 over 25, negative a half, negative nine 20ths and negative 0.

405.

I'm sure you got that right.

Well done.

Final task for today's lesson then.

You're going to do exactly what we've just done in the checks for understanding and like I said, if you need to go back and re-watch learning cycle one or learning cycle two, then obviously you can do that.

Pause the video now, have a go at these and remember, no calculators.

You've not seen me use a calculator today, so I want you to this using all of the methods that we've been learning over the last sequence of lessons.

Good luck, you can pause the video now.

And here are our answers.

Pause the video, check your answers and then when you're done, come back.

Now we'll summarise the learning that we've done during today's lesson.

And those are that there are three main ways of ordering fractions and decimals.

Remember, that includes positives and negatives.

The first one of those is number sense.

This is the one that I want you to check every single question before you get started, because sometimes we jump in feet first and go, "Ah, we've gotta change them all to decimals," when actually if we just paused for a moment, we will be able to see that we could use our number sense.

Our second method is converting them all into decimals.

So remember, we're going to use this method if all of our denominators are factors of 10, 100 or 1000.

And the final one is where we have to convert them all into fractions that have a common denominator.

So we don't want to do that where we don't need to, because I don't know about you, but I think that's probably the most challenging one.

Number sense, convert to decimals and then converting to fractions with the same denominator.

Moving forward then, you're always going to check number sense first.

Thank you so much for joining me for today's lesson.

It's been great.

Thank you, bye.