Lesson video

Well done for choosing to load this video and learn with us today.

My name is Ms. Davies.

Please feel free to pause the video and rewind as necessary, and I'll do my best to help you with some of the concepts that we are coming across today.

There's some really fun bits of maths involved, so I really, really hope there's bits that you will enjoy.

So let's get started.

All right, welcome to our lesson on generalised algebraic statements.

We're gonna be looking at lots of different rules and patterns today, and by the end of the lesson, you'll be able to use relationships that have been generalised using algebraic statements.

So we're bringing together the words and the relationships with our algebra skills and writing algebraic statements.

So there's some keywords here that we're gonna be using lots during the lesson.

If you're not really confident with them, you might wanna just pause the video, read through, okay? Because we're gonna use them as we go through and you wanna make sure that you are happy with what it is we are referring to.

So we're breaking the lesson down into two parts.

So we're gonna look at using algebra to describe a relationship.

And then we're gonna look at writing and interpreting formulae.

So using algebra to describe a relationship, and we're gonna have a look at some patterns.

We're gonna have a look at scenarios so you might want to have something that you can write and draw with so you can think about how these patterns develop.

So we can use algebra to describe patterns and relationships.

So I'd like you to have a look at the patterns that have been drawn below.

They're all drawn using straight lines, okay? In this case they're represented as purple.

So I've drawn pattern one, pattern two, pattern three, and you might wanna sort of think about how this pattern is developing what's happening, okay? What would the next one look like, okay? And any kind of rules and things that you notice between them.

So pause the video and have a think about how you would draw the next pattern.

You might want to physically have a go at drawing it, okay? Or you might wanna think about how you change pattern three into pattern four.

And then I want you to work out how many lines we need to make up the fifth pattern, okay? You can do the fourth pattern first and then move on to the fifth pattern.

Or could you find a way of getting straight to the fifth pattern? Again, you might wanna draw some of the things out.

All these shapes are made using straight lines.

We wanna know how many of those lines are needed.

Right, well done.

Hopefully you had a go at exploring some of those, maybe drawing some things out so you can think about how this pattern is developing.

So each rhombus, so those quadrilaterals we have in each pattern, each rhombus needs four lines plus one to connect it to the previous one, okay? So with one exception that we'll talk about in a minute, each thrombus that you are going to draw, you'll need to add on an extra line to connect it to the previous one.

Essentially what that means is, per rhombus, you're gonna need five lines.

I've highlighted them so you can see.

So you can see your rhombus and your previous one connecting.

So that's the five, okay? Notice where the five is appearing in our pattern.

If we were to draw pattern four, we'd add on an extra five, wouldn't we? Right, one thing to notice though is that first rhombus in every single pattern, not just the first pattern, every pattern, that first rhombus doesn't need a line to connect it to the previous one 'cause it's the first one.

So what that means, although there's five per rhombus, there's only four for the first one.

So you can do five lots of each rhombus, but you need to take away one for that first rhombus.

So we could write that as five times the number of rhombuses subtract one.

What that means is, if I ask for the number of lines in the fifth pattern, you can do five multiplied by five minus one.

Or if I wanted the number of lines in the 10th pattern, you could do five times 10 minus one.

And the important thing about generalising a relationship is that you can use that to work out any scenario in that pattern, okay? So it should apply to all relevant cases.

Let's think about how we can use our algebra skills then to write that in a nice succinct way.

So that's five times N, where N is the number of rhombuses take away one, or five N subtract one.

So we could do exactly the same thing with scenarios.

So Jacob wants to calculate the number of seconds it takes him to walk to score every day.

He's gonna do that by measuring how many minutes it takes him, and then change that to seconds.

So we know that there are 60 seconds in a minute.

So how could we write that relationship? Have a little bit of a think.

How would you write the relationship of 60 seconds in a minute? Okay, so let's think about what calculations we would do.

So to calculate the number of seconds, we would multiply the number of minutes by 60.

So that means the relationship can be written as S equals 60M where S is the number of seconds, and M is the number of minutes, okay? The S doesn't represent seconds.

The S represents the number of seconds, okay? It's really important to remember that these letters are representing numbers.

It's really important again to make sure you're writing down what they are representing.

S is the number of seconds, M is the number of minutes.

So let's return to Jacob walking to school.

So it takes him 15 minutes, so how could we work this out in seconds? Well, if we've got our formula S equals 60M, we can do 60 times the number of minutes.

So 60 times 15 to get 900 seconds.

And again, that rule is gonna work for any number of minutes that it takes him.

So it's a generalised rule that works for every case in this scenario.

So all relevant cases in this scenario can be worked out using that rule.

Right, have a go yourself then.

So, "True or false, in this pattern each diagonal has two dots.

The expression for finding the number of dots in any pattern is two times the number of diagonals or 2D." So is that the expression for finding the number of dots in any pattern? Have a look at the patterns down the side and think about whether they're true or false.

And then we'll think about justifications for our answers.

Right, there's quite a lot to think about in this question.

So each diagonal has two dots, straight away I'm not sure I agree with that.

If you look at that first one, that first diagonal only has one dot.

And then if you look in the second one, one of the diagonals has two dots, but the other one only has one.

So the expression for finding the number of the dots can't be two times the number of diagonals because that first diagonal only has one dot, okay? So well done if you spotted that that was false.

Right, we've got two justifications for our answers.

So one is the first diagonal only has one dot, so it should be 2D takeaway one.

Second one says each new diagonal adds two each time, so it should be D plus two.

Which of those do you think is correct? Right, well I dunno if you spotted that it's that first one, it's two dots per diagonal.

It's just we've got that case where the first one is only one dot, so you just need to remember to take off one.

So 2D subtract one, would be that rule.

It's true that each diagonal adds two each time.

However, what we're looking for is we're looking for a relationship between the diagonals and the dots, okay? So you want to be able to work out how many dots there are if there were 10 diagonals or 20 diagonals.

So we're looking for that relationship, rather than just spotting the pattern of what's happening each time, okay? So that first rule is gonna allow us to find out the number of dots for any number of diagonals.

Brilliant, time to have a practise then.

So for each of these questions, there's a piece of information, and then you need to pick the correct relationship.

So for this first one, there are seven days in a week.

What I want to know is, which of the following shows that relationship? So is it 7D equals W, or 7W equals D? In each question, I've told you what those letters are representing.

Give it a go and then we'll look at the next bit.

Right, well done with those worded examples.

We're now gonna have a look at some patterns.

So what I'd like you to do is write an expression for the number of dots needed for any number of squares.

So you're looking at how many dots per square.

Question five, you need to write an expression for the number of dots needed to make any number of Hs.

So you need to look at the relationship between the number of Hs and the number of dots.

So for example, the first H, first pattern has one H and it has seven dots.

The second pattern has two Hs, how many dots does it have? Third one has three Hs, how many dots does that have? And can you get a relationship between the number of Hs and the number of dots? Give it a go and we'll see if we agree.

Right, well done.

So hopefully you thought about which variable you're multiplying by seven, okay? And decided that it should be seven times the number of weeks will give you the number of days.

If it was five weeks, so you'd want to know how many days that was, you'd do seven times five.

That would tell you how many days.

For the second one, there's 12 months in a year.

So if you wanted the relationship between the number of months and the number of years, if you did 12 times the number of years, that would give you the number of months, okay? So if somebody was three years old, if you did 12 times three, that would tell you how many months old they were.

So 12 times three would be 36 in that case, and that'll give you a number of months.

Two correct answers for the bottom one.

So, "Laura is three years younger than her dog, which of the following shows the relationship?" So because Laura is three years younger than her dog, Laura's age plus three will give her dog's age.

You could have drawn a bar model for that to help you if you wanted.

The other statement that is true is that her dog's age takeaway three is Laura's age.

So that first one and that last one are the same thing.

So looking at the patterns then.

So this one, it's five times the numbers of squares is the number of dots.

So you can use any letter I went with S.

So five S where S is the number of squares.

Well done if you also clarified what your letter was representing.

The second one was a little bit trickier, so amazing if you managed to get this one first time.

What might have helped you was to start writing out the number of dots for each pattern, and thinking about how you get from one pattern to the next, what sort of extra each time, okay? And that can help you draw that relationship between the number of Hs and the number of dots.

So for this one, it's four dots per H plus the three to start with.

So if you kind of ignore the first line on the first H, then you are just adding three, you're adding four dots for each H that you want to add on, okay? So it's four for each H plus three to start with.

So four H plus three, where H is the number of H shapes.

That was a super tricky question.

So well done if you managed to get that.

Brilliant, so now I'm gonna have a look at writing and interpreting formulae.

We've looked a little bit at some formulae, but we're gonna go into more detail about how to write them, and how to interpret them, and use them.

So really, really practical use of maths.

So, "A car is pulling a caravan up a hill." I'm gonna write an equation, and this is equation is gonna be about the distance in metres of the vehicles from the bottom of the hill.

So what we're gonna do is, we're gonna measure how far the vehicles are from the bottom of the hill, okay? And that distance is gonna be represented by this equation.

So the equation is S equals T, subtract 0.

8.

I want you to have a little bit of a think about what you think S stands for and what you think T stands for.

I want you to pause the video and really think about how you would word this.

You might wanna try some things out, okay? In this equation, what do you think that S and that T stands for? We're gonna look at the rest of the equation in a second as well, but any ideas that you're coming up with looking at that, jot them down.

Brilliant, so there maybe loads of things that you talked about and thought about whilst looking at that equation.

This is what I came up with.

So you might've said something similar to me.

That S is the distance between the caravan and the bottom of the hill.

And T is the distance between the car and the bottom of the hill.

The reason I've gone for that way round is that to get S, we're doing T subtract 0.

8.

So I'm thinking that T must represent the distance between the car and the bottom of the hill, because the car's gonna be further up than the caravan 'cause the car's pulling the caravan.

One thing you might have thought about or you might want to think about moving forward is that we've got two objects here.

So it might depend what part of the caravan and the car you are measuring from.

Are you measuring from the front of the car and the front of the caravan? Or the back of the car and the front of the caravan? Or are you measuring from the wheels, okay? That is gonna change your answer.

So it is important that we are happy with that S and T as to exactly what it is representing.

Brilliant, so now let's think about this 0.

8.

What do you think this 0.

8 represents? What it might help to do is to pick some values for S and T thinking about what you've decided S and T represent, and then think about what that 0.

8 might mean in your context.

It may change depending on what you've thought about in terms of S and T.

So don't worry if you don't have the same answers as somebody else that you are learning with.

Or if you don't say exactly the same thing to what I do in my answers.

Give it a go and we'll talk about it together.

Brilliant, so what we're saying is that your answer might depend upon which part of the vehicles we measured to.

So for example, you might have said that the 0.

8 is the length of the towbar, which is a really nice description for what 0.

8 is.

But in that case we need to have been measuring to the back of the car and to the front of the caravan as our distance.

Otherwise you might have said something like, 0.

8 is the difference between the centre of the car and the centre of the caravan, or the front of the car and the front of the caravan.

For what we're doing at the moment and just playing around with formulae, it doesn't matter massively, but it does go to show why it's really important that we define what those variables are representing in our context.

Brilliant, so then we're looking at picking some values for S and T.

So you can pick any value for T, remember T's how far up the hill the car is.

And then to get S, all you need to do is T minus 0.

8.

For example, if we pick T to be five metres, to calculate S, we would do five subtract 0.

8, which give us 4.

2.

So we looked at one scenario there.

We can essentially write formula for any scenario.

So let's think about this one.

So we've got the cost of a hotel.

So the cost for a hotel is 100 pound per night plus 80 pound booking fee.

Let's think about how we could work out the cost of staying for three nights.

So what would we do if we wanted to stay for three nights? Well, we do a hundred multiplied by three and then we'd add on our booking fee, so add on 80.

So if I wanted to stay any number of nights, what would our formula look like? So remember, we need to use a letter to represent our variable.

So our variable is the number of nights.

So how could we use a letter to represent our variable and put this formula together? Have a go yourself and then see if you've got the same as me.

Brilliant, so we're gonna do 100 multiplied by the number of nights plus 80, or 100N plus 80.

If we're writing that as a formula, we'll have that as C equals 100N plus 80.

So we've got C equals 100N plus 80.

So what does C and N represent? Pause the video and write down what you think they would represent.

Fantastic, so C is the cost in pounds of staying in the hotel, whereas N is the number of nights.

Hopefully you've written something similar to what we've got there.

We can now use this formula to work out how much it would cost to stay for any number of nights.

So for six nights for example, we'd do 100 times six plus 80 and that would give us our 680.

Fantastic, so let's look at this example.

The time to cook a turkey is 40 minutes per kilogramme plus 45 minutes.

Which of the following are true? A two kilogramme turkey would take 125 minutes.

The formula for the time to cook the turkey is A equals 40 plus 45 B.

Or for each additional kilogramme of mass, the turkey needs 45 more minutes.

Have a think about those, see if you can decide which are true.

All right, well done if you spotted that top one would be true.

So 40 minutes per kilogramme, so 40 times two, and then add on your 45.

The second one isn't true, we should have 40B plus 45 because it's 40 minutes per kilogramme.

So 40 times the number of kilogrammes plus 45, and therefore the bottom one's incorrect as well 'cause it's 40 minutes per kilo.

So each additional kilogramme is going to be 40 minutes extra, not 45 minutes extra.

Brilliant, time to have a practise then.

So, "Izzy is doing a sponsored swim.

For every length of the pool she swims. Aisha will donate two pounds," so it's two pounds to every length of the pool.

Izzy's grand is being very generous and will donate 60 pounds no matter how many lengths she swims. I want you to look at these questions.

So the first one, you need to form an expression for the amount of money Izzy will raise.

Think about what an expression looks like.

B, you're gonna write a formula and I'm being picky about what variables you should use.

So write a formula for the amount C, that she would raise if she swam N lengths.

C, use your formula to work out how much money she would raise if she swam 20 lengths.

Then the final question, say she raises 124 pounds, can you now turn your formula into an equation for that situation? So if she raised 124 pounds, what would your equation look like? Give those four a go and then we'll look at the next set.

Brilliant, so now Alex is doing some knitting so that he can sell some scarfs and hats.

So it's gonna cost him a certain amount of money for the wool, and then he is gonna charge the money for his time.

So every scarf costs him three pounds worth of wool.

Each hat costs him two pounds worth of wool.

Then he's gonna charge just a 20 pound flat fee for his time.

So it doesn't matter how many scarves or hats he sells, he's just gonna charge an extra 20 pound.

So, which formula is gonna represent the cost of making any number of items? So you've got three choices.

Decide which of those is the correct formula.

Then I want you to think about what the C represents in the formula and what the N represents in the formula.

And then just say that Alex's costs come to 70 pounds.

What would an equation look like for that individual case in your scenario? Right, Some good thinking needed for that question.

So give it a go and then we'll come back together.

Brilliant, so let's go back to Izzy and her sponsored swim.

So an expression, any letter could be used, but I've gone with N 'cause I know I'm gonna use that for my next question.

So 2N plus 60.

Right, I was picky about the letters for the second question.

So a formula for amount C, she would raise if she swam N lengths.

So that's C equals 2N plus 60.

And then C, usual formula to work out how much money she would raise if she swam 20 lengths.

So we need to do two times 20 plus 60, and that gives you 100 pounds.

And the last one, we want an equation if she raised 124 pounds.

So we've got 2N plus 60 equals 124.

And that's an equation now for a specific amount of money.

Brilliant, so let's look at this formula for Alex's knitting.

So, we need two pounds for each hat and three pounds for each scarf.

It doesn't matter which way round that we write that.

And we need a 20 pound overall fee, okay? So 2C plus 3D, plus 20 will give us that formula as long as we write it as N equals 2C plus, 3D plus 20.

All right, B, so what does C represent in the formula? Well, because we've written it as two C, the C must be the number of hats knitted.

Make sure you haven't got them the wrong way round.

So, C is the number of hats, N is the overall cost in pounds.

'Cause in our formula, see we haven't put any pounds or pence symbols, any units.

So it's important now that we say that that's in pounds.

If Alex's cost comes to 70 pounds, you could write that as 2C, plus 3D, plus 20, equals 70.

So let's have a look at what we've learned today.

So we've looked at how expressions can be used to describe patterns.

We've also looked at how equations can be used to show relationships.

And we did those with patterns where we looked at one square was five dots, or one rhombus was five lines, four lines plus the one that connected it.

We also looked at relationships between worded examples, so the days and weeks, okay? And making sure that we get our relationship written the right way round by thinking about how you can calculate the number of something given the number or something else.

And finally, we can write formulae to represent scenarios and we can use these to calculate the value of one variable if we know the values of other variables.

So we started to use our formulae to input values and see then what happened to that formula.

Right, well done today guys.

So thank you for joining us as we work through this lesson.

It'd be really nice if you would join us for some other lessons at a later date.