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Hi everyone.

My name is Ms. Cooh, and today we're looking at arithmetic procedures index laws.

I really hope you enjoy the lesson.

It's going to be challenging.

It's got lots of fun tasks to do.

I know I'm going to enjoy teaching it, so let's make a start.

Hi everyone, and welcome to this lesson on checking and securing understanding of calculating with fractions and decimals, under the unit arithmetic procedures, index laws.

And by the end of the lesson, you'll be able to evaluate an expression involving decimals and fractions.

We'll be looking at these key words, equivalent fractions.

And remember two fractions are equivalent if they have the same value.

For example, 1/2 is equivalent to 2/4 as they have the same value.

4/5 is equivalent to 40/50 as they have the same value.

1/5 is not equivalent to 3/7.

This is because they do not have the same value.

We'll also be looking at the keyword reciprocal, and a reciprocal is the multiplicative inverse of any non-zero number, and any non-zero number multiplied by its reciprocal always gives one.

For example, five and 1/5.

These are reciprocals of one another.

This is because five multiplied by 1/5 is equal to one.

2/3 and 3/2.

These are reciprocals of one another because 2/3 multiplied by 3/2 is equal to one.

A non example would be four multiplied by 0.

These are not reciprocals of each other because if you multiply four by 0.

4, it does not give us one.

Today's lesson will be broken into three parts.

We'll be looking at reviewing calculations with decimals.

Then we'll be reviewing calculations with fractions, and then we'll be using a calculator efficiently.

So let's make a start reviewing calculations with decimals.

Now there are a few different methods to add, subtract, multiply, divide decimals.

We'll be looking at the most efficient methods and then expanding them further.

So let's start with addition and subtraction of decimals.

I want to show you these calculations.

Which one has been set out correctly and I want you to explain.

Have a little think, press pause if you're needing more time.

Well, hopefully you can spot b is set out correctly because when adding or subtracting, ensure to line up the decimal points and those corresponding place values.

Now we're jumping straight into a check.

Using your knowledge on adding and subtracting fractions, I want you to work out a, b and c.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, for question a, I've shown this working out, and you should have an answer of 21.

And for b, some people like to put like an imaginary zero there if you will.

Just so we have something in those decimal places.

So now I know each of my numbers share two decimal places, we can add.

Then, this gives me this working out, giving me an answer of 65.

04.

For c, you might notice it's not in column form, so putting it in column form, putting that imaginary zero in there if you wish, and then working it out gives me a final answer of 25.

154 when you subtract 2.

98 from 28.

134.

Well done if you got this one right.

So now let's have a look at multiplying decimals, and there are lots of different ways.

One efficient approach is to use the associative law and rewrite the calculation using integers and powers of 10.

For example, we have 12.

4 multiply by 1.

I'm going to rewrite this calculation so I have integers and powers of 10.

So I have 124 times 0.

1 multiplied by 14 times my 0.

That's still the same as 12.

4 times 1.

Then, using the commutative law, I'm just going to put my integers together and my powers of 10 together.

Then I can work out 124 multiply by 14, for example, using the area model.

And I've worked it out to be 1,736.

Then I've multiply by 0.

1 by 0.

1 to give me 0.

01.

Then I have a nice simple calculation of 1,736 multiply by 0.

01 is 17.

36.

So there are lots of different ways to efficiently multiply decimals.

For me, I like to use the associative law and the commutative law to work it out.

So now what I'd like you to do is a check.

I want you to work out 3.

7 times 6.

1 multiply by 1.

28.

Choose a method which is more efficient for you.

So you can give it a go.

Press pause for more time.

Well done.

Let's see how you got on.

Well, for me, multiplying 3.

7 by 6.

4 is the same as 37 multiply by 0.

1 multiply by 64 multiply by 0.

Then, grouping together my integers and my powers of 10, I've got 37 multiply by 64, then multiply by 0.

01.

I'm going to use an area model to work out that 37 multiply by 64.

So that gives me 2,368, which I then multiply by my 0.

01.

Giving me a final answer of 23.

68.

Well done if you got this one right.

For b, using the associative law, I'm going to read out my calculation as 21 multiply by 128 multiply by 0.

001.

Working this out using an area model, I know 21 multiply by 128 is 2,688.

Then I'm multiplying it by that 0.

001 to give me 2.

688.

Huge well done if you got this.

Now let's move on to dividing.

When dividing decimals using mental or written methods, one approach is to write the division as a fraction.

And given we know a simple fraction is where the numerator and denominator are integers making the divisor and integer can make a written or mental calculation much easier.

For example, 1.

2 divided by 0.

4, write this as a fraction, 1.

2 over that 0.

Now you might notice the denominators are decimal, so it's not really friendly.

So using equivalent fractions we can multiply the numerator and denominator by 10.

So that means I have 12/4.

Much easier, my denominator is integer, and I can work out 12/4, which is 3, nice and easy.

What I want you to do now is have a look at this check.

I want you to fill in the blank so the fraction is equivalent to the calculation and then I want you to work out the answer to 16.

8 divided by 0.

See if you can give it a go.

Press pause for more time.

Well done.

Well, for me, 16.

8 divided by 0.

8 is the same as 16.

8 over 0.

Not looking particularly friendly there.

So what I'm going to do is multiply the numerator and denominator by 10 to give me 168/8.

Now I'm going to simplify to give me 84/4, simplify again, 42/2.

Simplify again to give me 21/1.

Therefore, any of the divisions by an integer really does allow the answer to be clearly seen and calculated.

So I know my answer is 21.

Well done if you got this.

But not all divisions of decimals will give an integer result, but writing the divisors as an integer will enable us to calculate more efficiently.

For example, let's have a look at 1.

8 divide by 1.

Writing as a fraction, that denominator still doesn't look friendly, so I'm going to rewrite it as 18/15.

Now it might be a bit tricky to divide by 15, so I'm going to see if I can simplify this fraction.

I can and it gives me 6/5.

This is much easier to divide.

Five is much easier to divide, so I'm going to use short division.

Here's my divisor of five.

And I'm saying how many fives go into six.

So using my short division method and remember those trailing zeros, I can work out six divided by five is 1.

So I've worked out the answer, 1.

8 divided by 1.

5 is 1.

2, but this is also exactly the same answer to 18 divided by 15 or six divided by five.

All of these divisions will give us exactly the same result because they are equivalent fractions.

Now it's time for your check.

I want you to work out the following, ensure you show all your working out.

See if you can give it a go, press pause for more time.

Great work.

Let's see how you got on.

Well, I'm going to rewrite 96.

3 divided by 1.

5 as a fraction.

I'll just see if I can simplify it a touch to give me 321/5.

Now this equivalent fraction is much easier to divide.

So using my division methods, I've just worked out my answer to be 64.

So that means 96.

3 divided by 1.

5 is 64.

Let's have a look at b, 5.

52 divided by 1.

Well, writing my equivalent fractions, then I know the calculation can be simplified to 27.

6 divided by six.

Using short division, I can work out my answer to be 4.

Really well done if you've got this one right.

Well done, everybody.

Now it's time for your task.

Work out the following, press pause for more time.

Great work.

Let's move on to question two.

Andeep buys 12 pencils and 13 pens.

Each pencil costs 37p and each pen costs 58 pence.

He pays with a 20 pound note.

You need to work out the amount a change he should get.

See if you can give it a go.

Press pause for more time.

Well done.

Let's have a look at question three.

Can you work out those hidden digits.

a shows the addition using the column method.

b shows the subtraction using the column method.

And c shows multiplication.

See if you can give it a go.

Press pause for more time.

Fantastic.

Let's go through these answers.

For question one, you should have all of these answers.

Press pause if you need more time to mark.

Great work.

Question two.

Here is the working out for the cost of pencils.

Well, there's 12 pencils, each costing 37p.

So in total that's four pounds 44.

For the 13 pens, it's 13 multiplied by 0.

58, which is seven pound 54.

The total is the sum of our four pound 44 and seven pound 54 to give us 11 pound 98.

Then, subtracting this from 20 gives us eight pounds and two pence.

Very well done if you got this.

For question three.

This was a great little puzzle.

You had to really use your knowledge on addition, subtraction and multiplication of decimals.

Well done if you got this.

Great work, everybody.

So let's move on to the second part of our lesson, reviewing calculations with fractions.

When multiplying two proper fractions, you multiply the numerators to get the numerator of the product and multiply the denominators to get the denominator.

For example, 3/5 multiply by 3/4 is three times three over five times four.

Answer to 9/20.

So in general, a over b multiply by c over d is exactly the same as a multiply by c over b multiply by d.

This diagram illustrates it beautifully.

a over b multiply by c over d is equal to a multiply by c, which you can see as the green area over b multiply by d, which is seen as the whole area of a rectangle.

Nice little illustration to show the multiplication of fractions.

What I want you to do is let's have a look at a quick check.

Which of the following is the answer to 2/3 multiply by 9/20.

See if you can give it a go.

Press pause for more time.

All of them are equivalent fractions to the product of 2/3 and 9/20.

Well done if you got this.

So the product of two fractions sometimes needs simplifying.

For example, 5/6 multiplied by 3/5 is the same as 15/30, which then can be simplified to 1/2.

And you can simplify before multiplying in order to be more efficient.

For example, you might have noticed some common factors to be simplified first.

If you multiply 5/6 by 3/5, I can rewrite it as five times three over six times five, and from here, I can then identify my three to be three times one and my six to be three times two.

Then from here, knowing three over three is equivalent to one.

Five over five is equivalent to one.

That means here's my one multiplied by 1/2, thus giving me 5/6 multiply by 3/5 is equal to 1/2.

Now what I want you to do is calculate these answers by simplifying first and show the steps you took.

See if you can give it a go.

Press pause for more time.

Let's see how you got on.

Well, for a, identifying these factor pairs, I can simplify to give me 5/9.

And for b, identifying these factor pairs for our numerator and denominator, I can simplify to 1/2.

And for c, finally, identifying these factor pairs, I can simplify to give me 4/15.

Really well done if you've got this.

Now what I want us to do is go back to that original check question I showed you before, and I want you to work out the simplified answer to 2/3 multiply by 9/20.

See if you can give it a go.

Press pause for more time.

Well done.

Well, let's see how you got on.

The only simplified answer is 3/10.

We do know all of these fractions are equivalent.

However, we were asked to identify the simplified answer only.

Well done if you've got this one.

Now let's have a look at division.

So using division, I want you to look at these two calculations.

3/5 divided by 2/3 and 3/5 multiply by 3/2.

Let's have a look at 3/5 divided by 2/3 first.

Well, what I'm going to do is identify 3/5 of this shape.

You can see that in blue.

Then I'm gonna identify 2/3 of the shape using horizontal strips.

Here is 2/3 which I've highlighted in a lighter blue.

Now all I've done is pull out that 2/3 and now we're going to count how many 2/3 fit into that 3/5.

Well, if I count, it's 9/10.

9/10 of that 2/3 fit into the 3/5.

Now let's work out that calculation, 3/5 times 3/2.

Well, that works out to be 9/10 as well.

So what do you notice? We have 3/5 divided by 2/3 is 9/10, and 3/5 multiplied by 3/2 is 9/10.

What do you notice? Well, hopefully you spot dividing by a number and multiplying by the reciprocal of that number will always give you the same result regardless if it's fractions or not.

Why does this work? Well, I'm going to look at an example so we can understand why it works.

Same again, I've got a diagram, and I've split the diagram into 2/5.

You can see that using the columns, I've got 2/5 highlighted, and I'm gonna divide it by 3/7.

You can look at the rows and see I've got 3/7 highlighted.

I know the answer will be 14/15.

Let's examine this a little bit more.

Identifying the areas of each section, you might be able to see, this is a length of two and that's a length of five.

This is a length of three and this is a length of seven.

So pulling out that area which represents our 3/7, you can actually see that is an area of three multiply by five.

And then we count how many lots of that area, which is the 3/7, goes into our area, which is 2/5.

In other words, how much of this rectangle fits into this rectangle, and that's what the calculation actually shows.

This can then be worked out as 2/5 multiply by 7/3, which then shows when dividing by a fraction we're multiplying by the reciprocal.

So just remember, division can be rewritten as multiplication by the reciprocal.

And this works all the time and can make calculations, particularly involving fractions, more efficient.

So now let's have a look at a check.

I'm going to do the first part and I'd like you to do the second part.

We're asked to work out 2/5 divided by 4/9 and show all our possible working out.

Well, we know 2/5 divided by 4/9 is the same as 2/5 multiplied by the reciprocal of our 4/9, which is 9/4.

Working this out, we have an answer of 18/20, which then can be simplified to 9/10.

Now it's time for your check.

See if you can give this a go.

Press pause for more time.

Great work.

Let's see how you got on.

5/6 divide by 3/4 is the same as 5/6 multiply by 4/3.

This works out to be 20/18, which then simplifies to give me 10/9.

Very well done if you've got this, and if you converted it into a mixed number, you would've got 1 1/9.

Fantastic work.

Now let's have a look at adding or subtracting fractions.

We can use the identity property of multiplication to write equivalent fractions.

In other words, we know any number multiply by itself is always one.

For example, 2/3 multiply by one is 2/3.

From here, we actually know any fraction can be written as one.

2/2, 5/5, 11/11.

So that means if we combine both of these facts together, we can write an equivalent fraction.

For example, 3/8 multiply by 5/6.

We have different denominators.

So let's look at that denominator of six and eight.

Do you know what the lowest common multiple is of six and eight? Well, it's 24.

So what I'm going to do is make my denominator 24 by multiplying 3/8 by 3/3, and I'm going to make my denominator of 24 by multiplying 5/6 by 4/4.

Now I can simply add because I've got a common denominator of 24, so my calculation of 3/8 and 5/6 is exactly the same as 9/24 add 20/24, which then works out to be 29/24.

I want to convert it into a mixed number.

That would be 1 5/24.

Now it's time for another check, but looking at addition and subtraction of fractions.

See if you can give this a go, give your answer as a mixed number, where appropriate, and ensure you simplify.

Press pause for more time.

Great work.

Let's see how you got on.

Well, for a, you should have got 3/20, for b, 1 1/40, and for c, 29/22, which is 1 7/22.

Great work.

And here's the working out if you wanna press pause and have a little look.

Well done.

Fantastic work, everybody.

We've done a lot of work on fractions.

So let's do our task.

Work out the simplified answers to the following.

Give your answer as a mixed number where appropriate.

Press pause for more time.

Well done.

Let's move on to question two.

Great question because we're looking at area and perimeter.

So work out the area and perimeter of the following, giving your answer as a simplified fraction and a mixed number where appropriate.

Press pause if you need more time.

Fantastic work, everybody.

Let's move on to these answers.

Here are our final answers.

Press pause if you need more time to mark.

Well done.

For question 2a, let's work out the area and perimeter of our rectangle.

Well, the area can be found by multiplying the length by the width, giving me 3/20 metres squared.

The perimeter are two lots of that length and width.

So you add the 3/4 and the 1/5, and then multiply by two to give me 19/10 or 1 9/10 metres.

Very well done if you've got this one.

Question 2b was an excellent question.

We had to use Pythagoras' theorem in order to find the length of the hypotenuse so we can work out the perimeter.

So using Pythagoras' theorem, it's the square root of 3/8 all squared, and the 1/2 all squared, which gives me the length of the hypotenuse to be 5/8.

From here, the area is simply 1/2 times the base times the height, which is 3/32 centimetres squared.

And the perimeter is the sum of the three lengths, which works out to be 3/2 centimetres.

And converting it into a mixed number, 1 1/2 centimetres.

Fantastic work if you've got this.

So now let's move on to the last part of our lesson, using a calculator efficiently.

Scientific calculators are fantastic tools, and the Casio fx-991 ClassWiz can do so much, including simplifying, adding, subtracting, multiplying, and dividing fractions.

For example, let's input 7/8 subtract 4/11.

To do this, we simply press seven, the fraction button, eight, scroll to the right to make sure that you come out of that fraction function.

Subtract our four, press that fraction button again, 11.

You can then press OK or Execute.

And the answer is 45/88, easy.

The Casio ClassWiz also allows you to change the format of an answer, too.

So when an answer is found, pressing the format button allows you to change how you want to see the answer.

And when an answer is found, pressing the format allows you to change how you want to see your answer.

So let's have a look at an example.

Let's change the improper fraction of 25/3 into a mixed number.

To do this, press 25, the fraction button, three, and then Execute or OK.

So you'll see 25/3 displayed on your screen.

Now to convert it into a mixed number, press format, then scroll down using the down cursor and find Mixed fraction.

Then press Execute.

And what you'll see is you'll see 25/3 converted to 8 1/3.

So now what I'd like you to do is, using your calculator, convert the following improper fractions to mixed numbers.

See if you can give it a go.

Press pause for more time.

Great work.

Let's see how you got on.

Well, we should have had 16 4/5 for 84/5, 13 1/12 for 157/12, and 8 1/6 for 98/12.

Well done.

So the Casio ClassWiz allows you to input and operate on mixed numbers, too.

So let's change a mix number into a decimal and improper fraction.

The mixed number button is accessed by pressing the Shift and then the fraction button.

So you'll see it come up on a screen with three little squares.

Now from here, you can insert the integer and then the proper fraction.

So let's insert 5 7/8 subtract 3 1/2.

Press Shift and that fraction button.

So then you can insert 5 7/8.

Don't forget to scroll to the right to leave the fraction function.

Then you press subtract and then press the Shift and that fraction button again and insert the 3 1/2.

So you'll see your calculation looks exactly the same on your calculator.

Press OK or Execute, and it gives you the final answer of 19/8.

And to convert it into a mixed number, press format, scroll down to Mixed fraction, press execute, and then you'll see your final answer to be 2 3/8.

Scientific calculators are fantastic.

Now it's time for your task.

What I want you to do is work out the following calculations using your calculator, and give your answer as a mixed number.

See if you can give it a go.

Press pause for more time.

Well done.

Let's move on to question two.

Can you work out that hidden digit using your calculator? Press pause for more time.

Great work.

Let's go through these answers.

Well, for question one, a, b, and c, you should have had these.

For 1d, here are just some examples where you could get an answer of 3 1/2.

They're an infinite number.

Massive well done for creating your own questions where the answer is 3 1/2.

For question two, here are our missing values.

Really well done as that was quite tricky.

Great work, everybody.

So in summary, there are a few different ways to add, subtract, multiply, and divide decimals.

And we've looked at the most efficient approaches using the column method, the associative law, equivalent fractions, and short division.

We have reviewed adding and subtracting fractions using equivalent fractions, multiplying fractions, and dividing fractions using multiplication by the reciprocal.

And finally, knowing functions and applications of a scientific calculator allows us to input calculations and change the format efficiently.

Great work, everybody.

Well done.

I've finished the video