Lesson video

Hello, and welcome to this lesson on rounding in different bases.

My name is Mr Maseko.

Now before you start this lesson make sure you have a pen or pencil and something to write on.

Now that you have this, let's get on with today's lesson.

The first thing, try this activity.

We are talking about rounding, and we're looking at rounding to nearest five, 50 and 500, and coming up with rules for those.

So pause the video here and give this a go.

Now that you've tried this, let's see what you've come up with.

First, let's just think about the numbers that would round to five, 50 and 500.

So we're going to start with some boundaries, so zero, five and 10, zero, 50 and 100, and then zero, 500, and 1000.

Now the smallest number that rounds to five is at the halfway point, so it's 2.

5.

So that would round to five.

And the largest number is less than the halfway point between five and 10.

So less than, 7.

5.

And a number less than 7.

5 would round to five.

Now, if we look at that, what do you notice? The boundary is half of what? The value we are rounding to.

Think back to rounding to the nearest 10, when rounding to the nearest 10, the smallest number that can round to the nearest 10 was five.

And five is half of 10 and the largest number, that had to be less than 15.

And so you added five to 10.

So rounding is consistent, but we're doing that with our boundaries and our boundaries are always half of the value we're rounding to.

And we take that away and we added on to find our largest value and we take it away to find our smallest value.

Now for 50, this should now become really obvious, that's 25 and 75, these are boundaries.

And then for 500, that's 250 and 750.

Now for the upper boundary, remember it should be less than it.

And then for the smaller boundary, we can be there and anything above.

And those are similarities to rounding to the nearest 10, 100, 1000.

That should say 1000, I apologise.

Next, when rounding to the nearest 500, what's the smallest number that rounds to 10,000? Subtract 500, so that's 9,500 and then 10000s.

So what's the smallest number that rounds, we are rounding to the nearest 500, so you've got to work from 9,500, you've got to add on 250, which would give us 9,750.

Now at this boundary, if we're here, we round up to 10,000.

So the smallest number is 9,750.

So now we're going to be looking at rounding and rounding in different bases.

This first example, we're looking at 34, which is written in base 10, and we want to round that in base seven.

So 34 in base 10 is three tens and four ones.

Now in base seven, we know that, before we have no 49s, but we know that we have what? Four sevens, which gives us 28.

And then six ones, which gives us six.

So 28 plus six that gives us 34.

Now we want to round this number to the nearest one zero in base seven.

Now what is one zero in base seven? Good, that is seven.

So rounding to the nearest seven.

So you've got to think, remember our boundaries for rounding, that we were talking about in the previous exercise.

What's the smallest number that rounds to seven? Remember to think of your boundary between zero and seven, what's the smallest number that rounds to seven? It's 3.

5.

So anything above 3.

5 would round up.

So we have to look, when we are rounding to any number, we're looking at our units value.

So if we're looking at units value, is it bigger than 3.

5? Yes, so that means this rounds up.

So that means that rounds up.

Because we've bigger than the smallest number that rounds up to seven.

So this rounds up, in that we're adding one more seven, so we're going to end up at five, zero in base seven.

So four six in base seven rounds to five zero in base seven, which makes sense because 34, the nearest seven to 34 would be 35, and 35 is five zero in base seven.

So when deciding to round up or down, we have to look at whether our unit value is bigger than the smallest number that rounds up to what we're rounding to.

Now, let's try this again with a different example.

We're looking at numbers in base five now.

So we are through with 34, but I want to write this in base five.

We have one 25, one five, that makes 30, and four ones.

So one, one, four that is 34 in base five.

So we want to round this to the nearest one zero in base five, which is five to the nearest five.

So let's look at our units digits, and we've got to think a number that the smallest value that rounds up to five, that is 2.

5.

So is our unit value bigger than 2.

5? Yes, so this rounds up.

So this rounds up because we've bigger than 2.

5, which means this becomes one two zero in base five.

And that makes sense because 34, rounded to the nearest five is 35 and one, two, zero, this represents 35.

Now this one I want you to try on your own.

Pause the video here and give this a go.

Now that you've tried this, let's see what you come up with.

What do we have? We have the number 13.

Now this rounded to the nearest 10 in base 10 would be 10.

Because if you look at three, three is less than five, which means this rounds down.

So rounds down.

So if in base 10, this would round down, what about base eights? In base eight, what do we have? In base eight we have one eights and what? Five ones.

In base eight, now we're looking at our units value, we're rounding to the nearest eights.

Between zero and eight what's the smallest number? It's four.

Are we bigger than four? Is our units value bigger than four? Yes, so this rounds up.

So this becomes two zero in base eights.

Look what happened.

In base 10, 13 rounded down.

But when we wrote it in base eight, it rounded up, when we rounds to the nearest eight.

So the nearest eights to 13 is 16.

And that makes sense 'cause two zero represents 16.

So when we're deciding to round up and down, you're looking at that unit value and seeing whether it's bigger or smaller than the smallest value that rounds up to the value you're of rounding to.

Now here's an independent task for you to try.

Pause the video here and give this a go.

Now that you've tried this, let's see what you come up with.

Two three rounded to the nearest five.

Three is bigger than half of five, so this rounds up to three zero in base five.

One two in base seven rounded to the nearest seven.

Two is less than half seven, so this becomes one zero in base seven.

So that rounds down.

One three in base six rounded to the nearest six.

Three is half of six, so where there's a halfway point, we round up.

So this becomes two zero in base six.

Two one in base four rounded to the nearest four.

one is less than half of four, so this becomes what? Two zero in base four.

'Cause it rounds down.

One zero three in base eight rounded to the nearest eights.

Three is less than half of eight, so this becomes one zero zero in base eight, 'cause it rounds down.

What do we notice when rounding even and odd bases? With even bases, we have a halfway point that we can be.

So you can have a number like three at the halfway point.

But with odd bases, our halfway points are usually decimals.

So we never have values in our units place that are decimals, so we're either always above or below, so it's not easier to tell whether we round up or down.

And to round up or down, we have to be, if we're bigger than half the place value we're rounding to, then we round up.

If we're less than half the place value we're rounding, there we round down.

For this independent task we're looking at decimals written in base 10 and in base five.

If you don't want to a clue, pause the video here and give this a go.

If you want to a clue, keep watching.

So those of you that want to try this on your own, pause the video in three, two, one.

For your clue, you've got to think about the place value charts for base five.

So we've got to keep dividing down, so you do one divided by five, that is 0.

2.

That is 0.

2s position in base five.

Then divide by five again and you end up at 0.

04.

Then divide down again, you end up at 0.

08.

So these are the decimal place values in base five.

Now, can you use those to figure out decimals that you can write and decimals that you can't write.

And why you can write them? So pause the video here and give this a go.

Now let's see what we've come up with.

Numbers that are possible.

Values that are possible were any numbers we can make, 0.

2s, 0.

04s and 0.

08s and so on and so forth.

The next one is, would be 0.

0016s and then divide that down again and so on and so forth.

So any decimal values that are in the base five place value chart, we can make those in base 10.

So we could make 0.

8 base 10, how? Because 0.

8 base 10 is zero ones and what? Four 0.

2s.

So 0.

8 base 10, that's the same as 0.

4 base five.

Is there another one we can make? We could make 0.

24 'cause it's a combination of a 0.

2 and a 0.

04.

So 0.

24, that should be the same as 0.

11 in base five.

And now can you figure out the reason why we can't write some decimals in base five and in base 10? And also should we be using the word decimal in base five? If you think that you've got an answer to this, ask you parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.