Lesson video

Thank you for choosing to learn using this video today.

My name is Ms. Davis and I'm gonna help you as you work your way through this lesson.

Feel free to pause the video and work through things at your own pace.

Lots of exciting things to look at, so let's get started.

Welcome to our lesson on checking understanding of algebraic notation.

By the end of the lesson, you'll be able to represent situations using algebra.

There's a few keywords that we're gonna use today.

The first one is an expression.

An expression contains one or more terms where each term is separated by an operator.

Now, this is different to an equation 'cause an equation is used to show two expressions that are equal to each other, so an equation will have an equal sign.

We're also gonna talk about integers.

So an integer is any positive or negative whole number or zero as well counts as an integer.

There's some examples on your screen.

So in this lesson, we're gonna explore lots of different interesting and exciting problems. This lesson is split into two parts.

We're gonna start by talking about some missing number problems, which I hope you're gonna really enjoy.

And then we're gonna move on to expressing the problems algebraically.

And then this is a skill that's gonna open up all sorts of extra problems and things that you can solve, which you might have found difficult to do without those algebra skills, so really exciting lesson today.

So missing number problems occur often in mathematics and in real life situations.

You often don't know that you are using algebra and you are using it in this missing number context because you're using your logic skills instead.

So Aisha is going on holiday and is allowed up to 20 kilogrammes of luggage.

When she stands on the scales with her suitcase, it says 67 kilogrammes.

She then weighs herself as 49 kilogrammes.

Will her suitcase be allowed on the plane? So this is a really useful method if ever you're trying to weigh a suitcase.

So let's pull this problem apart.

What does the missing number represent in this problem? What is it that we are missing? Pause the video and think about your answer.

Right, you might have said that it's the weight of the suitcase.

That's what she's trying to work out and that's what she doesn't know at the moment.

What could we do then to calculate the missing value? We're gonna look at it together.

You might already have an idea.

So we can pick out the key pieces of information.

What information are we using to work this out? Well, it's the fact that with her suitcase, the scales say 67 and without the scales say 49.

I'm gonna pop that on a number line.

We want to know the difference between them.

So what we're gonna do is we're gonna do 67, subtract 49, and that gives us 18 kilogrammes.

And then the really important thing to do when you're solving problems is to just look back at the question and see if you've answered it.

So the question was, is her suitcase allowed on the plane? Yes, it'll be allowed on the plane because it is less than 20 kilogrammes.

So we're gonna have a look at a slightly trickier problem now.

Alex has five blocks of different heights.

He lines them up from smallest to largest.

Each block is two centimetres taller than the previous block.

If he stacks the two smallest blocks on top of each other, they are the same height as the largest block.

I want to know how tall a tower of all five blocks will be.

We're gonna work through this problem all together.

If you'd like to have a go at solving it yourself first, pause the video now and see what you come up with.

If you're not sure, then we're gonna talk about the different steps as we go through, so no problem.

So let's think about how we can start to solve this problem.

We could start by picking out key information.

So we know that each block is two centimetres taller than the previous block and we know that the sum of the two smaller blocks is the larger block.

One thing we can do, which is often really helpful, is drawing a diagram.

So there's my diagram, I've got my five blocks, each one two centimetres taller than the previous one.

It's important to remember that this is only a guide.

I can't draw it to scale 'cause I dunno how tall the blocks are.

That's the problem I'm trying to solve.

So when you're using a diagram, it is helpful for you to start visualising things, but you might need to change your diagram as more information comes to light.

What you could also do is you could just try a number.

So let's just say the smallest block is three centimetres.

So you'd have three and the next one would be five and seven and nine, then 11.

If that was true, the two smaller blocks would be three plus five, which is eight.

Now, remember, the two smaller blocks have to equal the largest.

That's not the case if the smallest is three.

So we know that the smaller block must be bigger than three.

Let's see if we can find a logical way to answer this problem.

So what is a piece of information that we don't know, which would help us solve this? Pause the video and have a think.

Right, we want to know the height of one of the blocks, maybe the smallest block and then we can work out the others from the smallest block.

What we can do then is we can call the smallest blocks question mark and we can look at how the other heights are related to that smaller one.

So the next one is a question mark plus two and then a question mark plus two plus two.

Then a question mark plus two, plus two plus two.

And finally a question mark plus two, plus two plus two plus two.

What we've done now is we've written the heights of all the blocks in relation to that first one.

Remember that second piece of information.

If I stand the two smallest blocks on top of each other, which I've done on the screen, it's gotta be the same height as the largest.

Now, if you've drawn a diagram, you might have to redraw your diagram now so that that works.

I would like you to pause the video now and see if you can work out the height of the smallest block.

You might even be able to answer the question, what is the height of a tower made of all five blocks? I wonder whether you managed to solve that one.

So looking at our diagram, that second one, we can see that one of the question mark blocks must be equal to three of the two blocks.

So that's 2, 4, 6.

That smallest block must be six.

We can try it out.

If that's six, the next one would be eight.

Six, add eight is 14 and the largest block is six, add two, add two, add two, add two, which is also 14.

So that does now work.

How tall would a tower of all five blocks be? It's 50 centimetres, fantastic.

We have just solved a really complex problem using a set of really simple steps.

So let's think about what elements helped to solve that problem.

Well, we were told how each of the heights were related to each other.

We were also given a statement of equality.

We were told that the two smaller blocks were equal to the largest block.

The methods that we used, we drew a diagram.

That was very helpful.

We tried a number, might not have been particularly helpful, but it did give us a bit of an idea of what sort of value we were looking for, and we could test our answer at the end to see if it worked.

And then lastly, we picked an unknown value.

We thought about what it was that we needed to know and we wrote the other values in relation to it and at the moment we used a question mark.

So let's see if we can apply any of those methods to investigate another missing number problem.

So Izzy has two sticks.

When she balances them on top of each other, their total height is 70 centimetres.

When she stands them next to each other, one is eight centimetres taller than the other.

How long is each stick? Like before, if you'd like to pause the video and see if you can work it out now, that's absolutely fantastic.

If you'd like to watch the first step, we'll do that together.

So we've picked out our key information and let's have a look at drawing a diagram.

So the two sticks must be equal to 70 and one is eight centimetres taller than the other.

We could try a value like we did before.

Let's say the smaller one was 20.

The larger one would have to be 28, and together they would be 48 centimetres.

That doesn't work.

We're trying to get 'em to be 70 centimetres, so they must be longer than 20 and 28.

Again, you might want to pause the video now and have a go at solving this problem.

Fantastic.

I'd like you to hold onto your answer for a little bit and we're gonna have a look at some other people attempting the problem.

So Andeep and Sophia have a go at solving this problem too.

I'd like you to have a look at their working, think about whether you approached it in a similar way, and then think about what they should do next.

So what Andeep's done is he's taken the diagram that we drew before and he has combined the two bits.

So we know that the green one is eight centimetres taller than the purple one.

So when we put them together, it's like having two of the purple one plus an eight centimetres.

So Andeep has said the two purple sections are worth 62.

I wonder if you did something similar.

What he needs to do now is he needs to half the 62 to work out what each purple one would be.

So the smaller rod is 31 and the large one is 39.

Well done if you've got that using your method two.

Let's now have a look at Sophia's method.

Now, Sophia's actually done exactly the same thing but has used letters instead of diagrams. She's written an equation.

In fact, she's written four equations.

So her top equation, green equals purple plus eight.

That was just the piece of information from the problem.

Second one, green plus purple equals 70.

Again, that's the same piece of information from the problem.

And then she said, well, hang on, if green is the purple plus eight, then green plus purple is the purple plus eight plus purple again.

So she's just changed.

Instead of writing G for green, she's written purple plus eight, because we know those two things are the same.

She then knows that two of the purples equals 62, so exactly the same as Andeep just with letters and equations instead of a diagram, and she gets the same answer.

Have a think about whether your method was similar to either of those two.

So, we have looked at a couple of problems now.

We're now gonna have a look at equations where we have more than one unknown.

So Laura has collected 120 coins in a video game.

It costs 30 coins to buy a pet and 10 coins to play a mini game.

How could she spend all her coins, so none are wasted? Again, if you'd like to try it, feel free to pause the video and give it a go before I give you some clues.

So we could write this problem as an equation now.

We could have 30p where p is the number of pets 'cause it's 30 coins per pet, plus 10g where g is the number of games 'cause it's 10 coins per game, equals 120.

So we've now got an equation with two unknowns.

If you haven't already, pause the video and see if you can find values for p and g which work.

Lovely.

Let's have a look at the way I did it and then we'll see if we've got the same answers.

So I tried some values for p and g.

So I said if p was one, that would've cost her 30 coins.

That means she has 90 left over, so can play nine games.

Then I tried two pets.

She bought two pets, that would cost her 60 coins.

She'd have 60 left over , so she could play six mini games.

And then I've done the same for a few other values.

Why do p and g have to be positive integers? In this case, they represent the number of times something is happening, so they can't be negative values and they can't be decimals.

I've actually missed out one example here.

I wonder if you found it.

If she bought no pets, then she could have played 12 mini games.

I wonder if you got all five of those options.

Time for a check.

So Izzy has two sticks.

One is exactly double the length of the other.

When she stands on top of each other, they are 24 centimetres high.

What is the length of each stick? Before you work out the problem, I'd like you to tell me which diagram best represents this problem.

Pause the video and think about your answer.

Well done If you picked C because one stick in exactly the double the length of the other, you can write it as two times the other one, so in total, we have the first stick and then the second stick, which is twice as tall.

I'd like you now to use the diagram to solve this problem.

Pause the video.

So if we do 24 divided by three, that'll tell us the height of the smaller stick.

So the smaller stick is eight and the larger stick is 16.

And then we can just check that that does add up to 24 and it does.

A chance to practise then.

For each question, you need to have a go at solving the problem.

You could use a diagram to help you or remember some of the other methods we've used like trying a value.

When you're done, we'll have a look at the next set.

Well done on those first two problems. Same again with the second two.

This time, we've got a rectangle and we're trying to work out its area.

You may wish to draw a rectangle and think about how you could use your question mark to label the sides.

With d, there's some fractions work as well.

Make sure you are laying out your working out really clearly so we can follow what you are doing.

When you finished, we'll have a look at the next set.

The last set of problems then.

Again, make sure you read the problems carefully.

For b, I'd like you to write your answer with a full sentence.

Off you go.

Well done guys.

Some of those problems were quite tricky.

I hope you enjoyed solving them and then felt really satisfied when you found a value that worked.

I'm gonna talk you through how I did this one.

So currants and sultanas have to be the same.

So I've drawn a bar with a question mark in each, but raisins is gonna be double, so I've bought drawn two bars with a question mark in each.

All four of those bars then must equal 420 grammes.

My calculation was 420 divided by four and that tells me what the currants are.

The sultanas are the same and the raisins are double.

You get 105 grammes, 105 grammes and 210 grammes.

Well done if you got it.

For b, so I drew a bar to represent the puppy's age.

If Jacob was six years old when the puppy was born, he's always going to be the puppy's age plus six.

So I've drawn another bar with a question mark and a six on top of it.

Now, if Jacob's double the puppy's age, that means that question mark plus six must be the same as two question marks, 'cause double is the same as the puppy plus six.

That means that Jacob would be 12 and the puppy would be six.

For the second part, if Jacob's three times the puppy's age, that means the question mark plus six must be the same as three question marks.

This time, each question mark must be three.

So Jacob would be nine and the puppy would be three.

And then Jacob is three times the puppy's age and he is also six years older.

Lovely, looking at our rectangles.

So I've drawn a rectangle.

If the length is double the width, then I can write two question marks on the length and one on each width, remembering rectangles have four sides, so I need to do the same on the other side.

The perimeter is 24, I can do 24 divided by my six question marks, so I know that each question mark is four.

That means the shorter length is four and the longer length is double, so the longer length is eight.

Remembering that I can calculate the rectangle by doing the length multiplied by the width, I get 32 centimetres squared.

Well done if you put your units in.

And for d, so the length of rectangular cupboard is one metre longer than its width.

So if I call the width question mark, then the length is question mark plus one.

If they all have to add up to nine, take away the two ones that I know.

So the remaining question marks have to add up to seven.

The seven divided by four and I've just left that as seven over four because I want my answer as a fraction.

So the width is seven over four, the length is seven over four plus one, and I've written that as 11 over four.

So a bit of fraction skills.

Seven over four multiplied by 11 over four gives me 77 over 16.

If I wanted to change that into a mixed number, I could have written it as four and 13 16th.

Either is absolutely fine as a fraction.

And then our units are metres squared 'cause it's our area.

And finally, well done if you spotted that it was four sour sweets and two toffees and that's actually the only option that worked because they can only be whole numbers of sweets.

For Sam, well done If you spotted that a and b could be negative numbers or decimals.

So actually, there are an infinite number of answers.

Sam has found two that work but there are plenty more if we can use negative numbers or decimals.

I hope you enjoyed stretching your brain on some of those problems. We are now gonna have a go at expressing problems algebraically.

So we can form expressions for real life situations using letters to represent unknowns or variables.

So this year, Aisha's dad is exactly four times her age.

How could we write an expression for his age? Pause the video.

What would you write? So we could use any letter to represent Aisha's unknown age.

I'm gonna go with a.

Her dad would then be a multiplied by four.

Well done if you remembered that we write that as 4a.

Aisha's aunt is three years younger than Aisha's dad.

How could we write an expression for her age? What would you put? Lovely.

We could write her age as 4a minus three.

Okay, Aisha's cousin is twice her age plus four.

How would you write an expression for their age? Lovely, we could write their age as 2a plus four.

So Aisha says, if I add four to my age, then double it, I get my sister's age.

Jacob reckons that means your sister and cousin are the same age.

Is Jacob correct? Pause the video, think about your answer.

This is a really important point.

So well done if you said that, no he's not correct.

Let's look at why.

Adding four to Aisha's age would give a plus four, and then her sister is double that.

So that's double the whole of a plus four.

We need to write brackets around the a plus four because it's that whole expression that is being doubled.

So we can write that as two, bracket, a plus four.

That was not the same as her cousin.

They are different things.

The brackets change the priority of operations.

So for Aisha's cousin, we're only doubling Aisha's age and then we're adding four.

But for Aisha's sister, we're doubling the whole thing in the bracket.

If you wants to use your expanding bracket skills, two lots of a plus four is equivalent to 2a plus eight.

Aisha's dog is a quarter of her age.

Andeep says you can write that as a divided by four.

Jacob, as a quarter a.

Lucas, as a over four.

Whose answer is best? Pause the video, what do you think? All three answers are correct.

It's preferable when working with algebra to use the fractional notation instead of just using a division symbol.

Jacob's and Lucas's answers then are probably the best way of writing it, but those are exactly the same.

So either Jacob's or Lucas's is the way we are gonna write division moving forwards.

Time for you to have a check.

So I'd like you to match up the statements within the expression that represents it.

Read them carefully, off you go.

Well done.

Let's have a look at our answers.

So the top one matches with four lots of b plus three and that needs to be in brackets 'cause the whole thing's being multiplied by four.

The second one, four plus a number, then multiplied by three.

So we can put the four plus a number in a bracket and we want the whole thing multiplied by three.

So the third one, we four lots of a number, add three, so it's that top one.

And lastly, the bottom one, a number multiplied by three, then added to four is that third one down.

Priority of operations says that we multiply by three first anyway, so it doesn't matter what order we write it.

The b multiplied by three would be done first, then add four.

We would probably write that as four plus 3b using our algebra skills 'cause we don't like to write the multiplication symbols where we can help it.

Have a look now at this second set.

Match up the statements with an expression again.

Off you go.

Lovely.

We're gonna look at these individually this time, just to make sure we're happy with this notation.

So a third of a number, you can write that as a over three, or you could have written that as a third a.

But out of those options, a over three.

Four lots of a number divided by three.

That's the same as four thirds of a.

Multiplying by four and dividing by three, it doesn't matter what order.

you could equally write that as 4a over three.

The third one, a quarter of a number.

So one quarter multiplied by a or you could have written that as a divided by four.

Remember, multiplying by quarter and divided by four are the same thing.

And lastly, a number divided by four, then multiplied by three.

Doesn't actually matter what order we divide and multiply by.

So we can have 3a over four or you could have three quarters of a.

In a bar model, if a bar representing one expression is the same length as another bar, then the expressions are equal.

That means we can form an equation.

So for this bar model, we could write the equation, 12 plus 2x equals 4x plus five.

We can write an equation for any scenario where we know things are equal.

Let's look at some.

Jacob says I'm thinking of a number.

When I add 10, then divide by five, I get the number six.

Let's look at how we'd form an equation.

He's thinking of a number, so I've picked the letter b.

He adds 10, then he wants to divide the whole thing by five.

So let's use that fractional notation and his answer is six.

We've now formed an equation.

In the fractional notation, the numerators and the denominators are grouped.

So we don't actually need brackets around the b plus 10.

We know we've gotta do that first because it's the entire numerator of our fraction.

Right, then Jacob says, but when I divide by five, then add 10, I get the number 14.

Pause the video, how would this look different? Alright, so this time, divided by five first, I should have b over five and then I want to add 10 and that equals 14.

Check you're happy with the difference between those two.

Lucas and Andeep each have some sweets.

When Lucas gives five of his sweets to Andeep, they both have the same amount.

How could we form an equation for this scenario? Pause the video, give it a go.

So let's call the number of sweets Lucas has b and the number of sweets Andy has a.

After Lucas gives Andeep five, you could write Lucas' as b minus five and Andeep's as a plus five.

Lucas has just lost five and and Andeep's gained five.

Those things are now equal.

We have now formed an equation for that scenario.

Can you think of how many sweets they could have started with? There's loads of answers for this one.

It's any number where Lucas has 10 more than Andeep.

So if Lucas had 20 and Andeep had 10, then when Lucas gives five to Andeep, they both have 15.

Where an equation has more than one unknown number, it is possible for there to be more than one option for values that make it true.

And we saw some of those earlier.

A quick check.

I would like you to pick a number from the box for c and a different number from the box for d, so that the equation c plus 3d equals 16.

There is more than one answer.

See how many you can find.

Lovely, c could have been one and d could have been five.

I wonder if you found that one, or c could have been 13 and d could have been one.

Or, c could have been 16 and d could have been zero.

There are other options but not from the box that was there.

True or false? If a and b are integers, there are five possible answers for the value of a and b in the equation 4a plus 2b equals 20.

Pause the video, read it again.

What do you think? Lovely, that one is false.

Which of these justifications is the right reason? Lovely, it's the fact that integers include negative numbers as well.

There are therefore an infinite number of and answers for that one.

Time for your final practise.

I would like you to match the bar models to the equations.

Off you go.

Well done, this time, I'd like you to match the statements to the equations.

Think carefully about your priority of operations.

Fantastic, so that first one, you could have written 4x plus five equals a.

There are other options, but not that were given to you.

Second one, 4x plus a equals five.

Third one, 2a plus five equals 4x plus three.

D, a plus five equals a plus 4x and e, a plus 2x plus three equals a plus x plus five.

And the second set, so a number minus two, then divided by five is the bottom one.

B, a number divided by five, then minus two, so that should be the top one.

C, two divided by five less than a number.

That was probably the trickiest.

That is the second one down, two over x minus five.

D, two divided by a number, then minus five, so that's going to be the third one down.

And e, a number subtracted from two, then divided by five.

So that's the fourth one down, two minus x over five.

Well done.

I hope you enjoyed solving those problems in the first part of the lesson and I hope you feel that your algebra skills are a lot better after that second part.

What we've looked at today then.

With enough information, we can find missing numbers in context.

We can solve missing number problems by using diagrams or equations.

Missing number problems and real-life situations can be expressed using algebra.

And then we looked at some problems where we had two unknowns and that means possible pairs of numbers can be found to make that statement true, right? I would like you to remember all those algebra skills next time you are trying to solve a problem and see if they help speed up the process, make it easier to find out those missing numbers.

I look forward to seeing you again.