Lesson video

Right, well done for loading this video today.

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

Please feel free to pause bits and rewind bits to help you as you are learning this new topic.

Anything that I can do to help, I'll try to add in as we work our way through.

I really do hope there's bits of these algebra that you're really, really gonna enjoy.

Lots of chances for you to explore things and to develop your algebra reasoning, and thinking skills.

Right, let's get started then.

Right, hi, guys.

So today, we're gonna look at checking your understanding of expressions and equations.

By the end of this lesson, you'll be able to generalise a situation using words to express what is happening.

So the main keyword we're gonna look at today then is this idea of generalising.

So to generalise is to formulate a statement or a rule that applies correctly to all relevant cases.

What that means is we're gonna have a look at some different cases, some different patterns, some different scenarios, and you're gonna be able to put into words what is happening in a way that means that that will apply to any case to do with that thing, so all relevant cases.

I'll give you an example.

So to find the number of hours in any number of days, we can multiply the number of days by 24.

So because we know there's 24 hours in a day, if I want to know how many hours there are in 2 days, I know that I can do 2 times 24.

If I then wanted to know how many hours in 5 days, I'd do 5 times 24, and that's gonna work for any number of days, so that's what we mean by all relevant cases.

So what we've done here is we've generalised this rule, we've come up with a generalisation.

So we're gonna start by identifying patterns in scenarios, and then we'll work on describing our own rule for a pattern.

Identifying patterns in scenarios, so here's a scenario.

Alex is going hiking, he needs to pack 3 snacks for each day he is away.

So what we're gonna do is we're gonna use this rule to work out how many snacks he needs for any length hike.

So just start thinking about, if I told you he went hiking for 2 days, if I told you he went hiking for 5 days, what would you do to work out how many snacks he needs? Can you come up with a rule? I'm gonna do this by drawing a table.

So I've got the number of days, and then the number of snacks.

So for 1 day, he'll need 3 snacks.

For 2 days, he'll need 6 snacks.

For 3 days, he'll need 9 snacks, and for 4 days, he'll need 12 snacks.

Can we see what is happening now then what is the rule to get from the number of the days to the number of snacks? Well, each time, we are multiplying by 3.

So if I then said to you, how many snacks would you need for 5 days? You know you'd do 5 times 3.

If I said, how many snacks would you need for 10 days, you know you'd do 10 times 3.

So this idea of multiplying by 3 will work for any number of days, so that's our generalised rule.

So if we write that in words, we could say to work out the number of snacks he needs, we multiply the number of days by 3.

Okay, so Alex now says, "For my hike, I need to pack 10 snacks." So I'd like you to have a quick think about his statement.

Can he be correct? Can you explain your thinking? Maybe you can get some mathematical language in there, as well.

Have a think.

Right, so you might come up with something like this.

So his answer should be in multiple of 3.

Because he needs 3 snacks per days, we should have all our answers as multiples of 3.

10 isn't a multiple of 3.

So you might have noticed that for 3 days, he'd need 9 snacks, and for 4 days, he'd need 12 snacks.

So this idea of having 10 snacks is not gonna work with our rule.

Okay, we're gonna change the scenario slightly now.

So Alex now decides he should pack 1 spare snack, as well.

So still, 3 per day.

We're just gonna stick an extra 1 in just in case.

Izzy says, "Now you can multiply the number of days by 3, and then add 1 to work out how many snacks you need." See what you think about Izzy's statement.

Sam says, "You can actually work this out quicker by multiplying the number of days by 4." Right, think about those two statements, who is correct? Can you explain your thinking? Right, again, I'm gonna put this into table for us to think about.

So what I've got is the number of days.

The first time, if he goes for 1 day, he needs 3 snacks, but now he's gonna put in an extra 1.

If he goes for 2 days, he'll need 6 snacks, and he is gonna put in an extra 1.

For 3 days, he'll need 9 snacks, and put in an extra 1, and so on.

So what Izzy said is true, that's how he's going to work out the number of snacks.

Let's see if Sam's statement works.

So Sam reckons that you could just multiply by 4.

Well, that looks to work for 1 day, but let's look at 2 days.

Is 2 times 4 gonna give you 7? No, it's not.

Is 3 times 4 gonna give you 10? No, it's not.

So we can't make that easier.

Multiplying by 3 and adding 1 is not the same thing as multiplying by 4, okay? And that's actually quite a useful distinction, the more we are looking at patterns.

Right, have a little bit of a check then.

So Alex needs to pack 2 pairs of socks for each day of his hike.

Which of the following statements are true? I'll read them out, and then let's have a think.

So A, multiplying the number of pairs of socks by 2 tells us how many days Alex is hiking.

Multiplying the number of days by 2 tells us how many pairs of socks he needs.

For every additional day hiking, Alex needs to pack 2 additional pairs of socks.

Have a think about those statements, which ones do you think are true? Right, fantastic.

If you think got that the right way round, it's that second one.

It's the number of days by 2 that gives us the number of socks he needs, okay? Or number of pairs of socks he needs.

So make sure you get that the right way round.

And this second one's, this last one, for every additional day hiking, so if he went on an extra day hike, he would need 2 additional pairs of socks, okay? 'Cause he needs 2 per day.

Right, time for you to have a practise then.

A sandwich is made up of 2 slices of bread, 4 slices of cucumber, and 3 slices of cheese.

What I want you to do is fill in this table to show the ingredients needed for different numbers of sandwiches.

So I filled in some bits for you.

You need to fill in the remainder.

Whilst you're doing that, have a think about what a sensible column heading for the first column would be, 'cause you see I haven't put a heading there, that's 'cause you are gonna think about what a sensible column heading is.

Right, give that a go, and then we'll come together for the next set.

Okay, so with same scenario, so two sandwiches made up of two slices of bread, four slices of cucumber, and three slices of cheese.

This time, you need to have got filling in the gaps in these statements.

The third one might take a little bit of thinking about.

So the third question, if I know the number of slices of cheese, how could I work out the slices of cucumber? You might wanna look back at your table to help you.

Off you go, and then we'll look at the answers.

Fantastic, well done.

So looking at our table then, you'll see the missing gap.

We've got 8 for the first one for cucumber, we've got 6 for the second one for bread, and 12 for the second one for cucumber.

The third row should be 4 in that first column.

And then a missing number of 12 in the cheese column.

And then the one that starts off with a 7, you should have 14, 28, and 21.

And the bottom row should go 20, 40, 80, and the cheese was already filled in for you 60.

So that first column is representing the number of sandwiches we need.

So something like number of sandwiches would've been a really good column heading.

Right, let's look at these statements then.

So the number of slices of bread needed for any sandwich is found by multiplying the number of sandwiches by 2.

So because we want 2 slices of bread per sandwich, we're gonna do the number of sandwiches times 2 to get the number of slices of bread.

And being able to put that into words is really important.

Right, the second one, the number of slices of cucumber needed for any sandwich can be found by multiplying the number of sandwiches by 4.

This time, we need 4 slices of cucumber per sandwich.

So we're gonna take the number of sandwiches, and times that by 4.

Right, well done if you've got this last one.

So we're now looking at the relationship between the bread and the cucumber.

So not the sandwiches in the cucumber, but the bread in the cucumber.

So the number of slices of cucumber needed for any sandwich could be found by multiplying the number of slices of bread by 2.

We need 2 slices of bread per sandwich, and we need 4 slices of cucumber per sandwich.

So we can think of it as every slice of bread, you need two slices of cucumber.

Or times in the number of slices of bread by 2 to get the number of slices of cucumber.

Right, this last one to think about then.

So if I know the number of slices of cheese, how can I work out the cucumber? There are different ways of doing this.

I think the easy way of doing it is working out the number of sandwiches, and then getting the slice of the cucumber.

So if I know how many slices of cheese I need, and there's 3 per sandwich, if I divide by 3, that'll tell me how many sandwiches I had.

Once I know how many sandwiches I had, I can times by 4 to get the slices of cucumber.

When you work with fractions, there are other ways of going about that, but it's a nice idea to wait back to the sandwiches before you then work up to the cucumber, and there's nothing wrong with that at all.

Right, well done for all that thinking.

We're gonna have a look at the next part of our lesson now.

Right, so this time we're gonna have a look at some patterns and you are gonna have a go at describing a rule.

You might wanna make sure that you have a pen and paper for this one, 'cause we're gonna have a look at drawing some patterns and some different ways of going about explaining the patterns we have drawn.

So Sam, Alex, and Izzy are drawing the pattern below.

So I just want you to have a look at this pattern.

I want you to think about how would you draw this pattern.

You might wanna pause the video, and have a go at drawing it yourself, thinking about how you actually construct that pattern.

Right, we're now gonna watch the different ways that Sam, Alex, and Izzy draw the pattern.

I want you to think about describing their methods in words.

They might have done it the same way as you.

They might have done it a different way to you, but what we're looking at is we're looking at how the patterns they have drawn are different.

They're all trying to draw the same pattern, but how the methods they go about it are different, and how we can put that in words.

We're coming back to that idea of a generalised rule.

So Sam draws the pattern this way.

Right, describe in words how they have drawn this.

Pause the video and have a think about how you would explain what Sam has done.

Right, how would Sam draw this pattern with 10 squares? Can you use your rule that you've just written to explain what they would do for 10 squares? And lastly, how could we generalise how many lines make up this pattern for any number of squares? So now, think about how many lines make up this entire pattern.

What if I wanted to have an extra square? What if I wanted to have 10 squares? How could we make a rule for how to draw this pattern for any number of squares? Okay, so you might have said something like this.

So Sam has drawn five C shapes, then a vertical line at the end.

For 10 squares, you might have said Sam would have to draw 10 C shapes, and then put a vertical line at the end.

Right, if we try and put this into a generalised rule then, what we'd have is we'd have 3 lines for each of the squares needed.

So 3 lines to make up that C shape that makes up the most of the square plus 1 for the end.

So I've written that as three lines for each square plus one for the end.

And then I could apply that to any number of squares I wanted.

Right, let's look at the way that Alex draws this pattern.

Okay, describe in words then how Alex has drawn his pattern.

Thinking about how he would draw this pattern for 10 squares, and then how you could generalise this as a rule.

You might wanna think this time as to whether this will look different to Sam's or not.

Have a think, and then we'll look through what you've said.

Okay, so I might say something like this.

Your answer might have been similar.

So Alex has drawn 5 horizontal lines for the top.

Then he drew 5 horizontal lines underneath, and then he had to draw in the 6 vertical lines.

If he was doing this with 10 squares, then he'd need 10 horizontal lines at the top, 10 underneath, and then 11 vertical lines.

Right, how could Alex generalise how many lines make up this pattern? So you might have said something like there's 2 horizontal lines for each square.

Okay, you might have split that up even more and said 1 horizontal line at the top for each square, 1 horizontal line at the bottom for each square.

But I'm trying to keep this as succinct and as easy to read and to write as I can.

So I've gone with 2 horizontal lines for each square.

Then the vertical lines is the number of squares add 1.

If I need to draw 5 squares, I need 6 vertical lines.

If I need to draw 10 squares, I need 11 vertical lines.

So it's the number of squares I want, add 1 to give me the vertical lines.

Put that together, I've got 2 horizontal lines for each square.

And then the number of vertical lines is the number of squares add 1.

You might have come up with an even better way of explaining that.

Okay, let's compare this to Sam's then.

So just like Sam, this equals 3 lines per square, the 2 horizontal ones we talked about, and then the first vertical one.

So 3 lines per square.

But we know we need that extra vertical line at the end, don't we? Okay, and if we think about the way Alex did it, he drew the horizontal lines for all his squares, but when he drew his vertical lines, he needed an extra one than the number of squares he had.

Now, this is actually exactly the same as what Sam did.

Right, we're gonna look at the last way of drawing this, which is Izzy's way of drawing the pattern.

Right, describe in words how she has drawn this.

Again, thinking about how she would draw this pattern with 10 squares, and what would the generalised rule look like, and could you make this look similar to, or is it different from what Alex and Sam did? Pause the video, and have a think.

You could always rewind and have a look at the pattern again, and then we'll come back together and look at your ideas.

Okay, so you might have had something like, Izzy has drawn a square, and then 4 backwards C shapes.

You might have come up with an even better way of explaining that, but she started with a square, and then drew 4 extra backwards C shapes.

If she was doing this with 10 squares, she would draw 1 square and then 9 backwards C shapes.

So how many lines make up this pattern? It would be 4 lines for the first square.

Then this bit was quite hard to put into words.

So 4 lines for the first square, and then 1 less than the number of squares, 'cause if she's already drawn 1 square, she needs 1 less.

So 1 less than the number of squares multiplied by 3 for the C shapes, 'cause you need 3 lines for each C shape.

Putting that together, then you've got 4 lines for the square.

Then 1 less than the number of squares multiplied by 3 for the C shapes.

Now this is the same as Sam's, except the extra vertical line is kind of drawn first when we draw the square.

And then 3 lines per square after that.

You might already have an idea as to which of those explanations you think is the nicest.

I think Sam's was quite easy to explain.

I think Alex's was quite a nice way of drawing it.

Okay, but they all actually end up being the same thing.

We've just drawn them in slightly different ways.

Right, time for you to have a check then.

So how could you calculate the number of lines needed for each pattern? You've got some descriptions, you've got some generalised rules on the left, and you've got some patterns on the right.

What you need to describe is the number of lines needed to make up each pattern.

Have a go at matching them up, and then we'll check them.

Fantastic.

So that first one, multiply the number of squares by 3, and then add 1 that's actually really similar to the pattern we just looked at with Sam, Izzy, and Alex.

So that's that bottom pattern.

It's the number of squares you need multiplied by 3.

And then you need that extra line at the end to finish off the third square.

The second one is multiply the number of squares by 4.

Well, that's the top one, 'cause each square needs 4 lines in that pattern.

And the last one, multiply the number of squares by 5, and add 1.

Well, if you see the squares, each square has 4 sides, but it has 2 lines in the middle.

So it's actually quite similar to that bottom 1 again.

But it's got 2 extra lines in the middle.

So each square has 5 lines, and then you need the 1 on the end to finish off that pattern.

Fantastic, so now, you're gonna have a go at this practise.

So Sam is drawing the letter C in different size squares by shading individual sections.

Below is the example in a 3 by 3 square and a 5 by 5 square.

So notice, they shade all the way up to the end of the square for the bars on the C at the top and the bottom.

And it's only 1 square thick all the way round.

That one there's called a 3 by 3 square, 'cause it's 3 squares across the top and 3 down the side.

So although, it's 9 squares in total, we call it a 3 by 3, because it's 3 across the bottom and 3 up the side.

What I would like you to do is see if you can work out how many sections would be needed to draw a seat in a 7 by 7 square.

So how many would you shade in a 7 by 7 square? You could use square paper to help you for that one.

B, how many sections would be shaded to draw a C in a 50 by 50 square? You're definitely not gonna want to draw that one, so have a think about any patterns you spot.

Then, you're gonna describe the rule for finding the number of shaded sections needed for any size square.

So can you explain it in a way that we can apply it to any size square we want? And finally, a C shape is made of 73 shaded sections.

Can you then work out what size square it is? Give those a go, and we'll look at the next set.

Right, well done with that one.

So Sam makes their letter C into a letter O.

So but low is another example in a 3 by 3 and a 5 by 5 square.

So now, they have now turned their letter C into a letter O by shading in extra squares.

So your questions now then, how many more sections would they need to add to their C to turn it into an O? And I want you to think about that in a 3 by 3, a 5 by 5, and a 7 by 7.

So you could use your previous drawing if you drew for the last question to work out how many more sections would need to be shaded for that 7 by 7.

How many sections would be shaded overall, so not just how many more, just how many sections overall, to draw an O in a 23 by 23 square.

So you might wanna start thinking about your patterns.

C, describe the rule for finding the number of shaded sections needed in any sized square.

And the last one, an O is made of 40 shaded sections.

What size square is it in? Again, lots of good thinking required for this question.

So give it a go, and we'll run through the answers.

Lots of great thinking involved in that one, so well done, guys.

So how many sections would be needed in a 7 by 7 square? You might have drawn it to help you out, and you might have noticed that there'd be 7 at the top, 7 are on the bottom, but only 5 in the middle.

So you end up with 19 sections.

For your 50 by 50 square, you wouldn't have wanted to draw that one out.

So you needed to think about what is happening with this pattern.

So again, lots of ways to think about it.

You might have noticed that it's 50 along the top, 50 along the bottom, and then only 48 down the side.

Or you might have done 3 lots of 50 for the top, the side, and the bottom, and then taken away the 2 corners, because you've counted those twice.

Either way, you should get an answer of 148.

So describe the rule then.

We've already touched on that by thinking about our 50 by 50 square.

You might have said something like 3 times the number of squares in 1 row minus 2, that's like that second way I was talking about before.

You might have noticed that it's the top, add the side at the bottom, which gives you 3 lots of the number of squares in 1 row, but then you've counted the corners twice.

So you would need to minus 2.

Or you might have done 2 times the numbers of squares in 1 row.

So you might have done the top and the bottom, which would be 2 times the number of squares in 1 row.

And then you'd have to add 2 less than the number of squares in 1 row.

So if it was 50 and 50, you'd then add 48, you'd add 2 less.

Right, the last one, what size square would give you 73 shaded sections? So it is a 25 by 25.

Okay, you could have done that by thinking about adding in those two corners.

Again, you know how if you did it that first way, you minus 2 at the end, you might have thought, "Well, if it's 73 shaded sections, that's 75 included the corners, which is 25, 25, and 25." Right, turning it into an O then, so how many more sections did Sam need? Well, they needed 1 more square for the 3 by 3.

They needed 3 more squares for the 5 by 5, and they needed 5 more squares for the 7 by 7.

So it's actually 2 less than the number of squares in 1 row that we need to add on.

How many sections would be shaded overall to draw an O in a 23 by 23 square? Several ways to think about this.

You could have done it by doing 23 at the top, 23 at the bottom, and then 21 and 21 at the 2 sides, because we've already got the top and the bottom shaded.

Or you might have done 4 lots of 23 to get the 4 sides, but then you'd need to take away 4 for the 4 corners, because you've counted them all twice.

So you end up with 88, whichever method you use for that one.

Describing the rule then, so we talked about this a minute ago, so one way you could do it is 4 times the number of squares in 1 row, so you can do all the sides together.

So that's 4 times the number of squares in 1 row.

But like I just said, you've counted all the corners twice then, so you need to minus 4.

Or you could think about the top and the bottom separately to the two sides.

So you've got the number of squares in 1 row times 2 to get the top and bottom.

And then you need 2 less than the number of squares in 1 row for the 2 sides.

So if you put that together, you get 2 times the number of squares in 1 row for the top and bottom.

Add 2 times the numbers of squares in 1 row, subtract 2 for the 2 sides.

Again, that was quite a tricky one to put into words.

You might have done an even better job than me.

If you did find it a little bit tricky to put that into words, try saying it out loud.

Try explaining it to somebody else, and then sometimes, it's easier to get that written down.

And a 40 shaded sections, then what size square is it in? Well done if you didn't fall into the trap, and you notice it would be an 11 by 11.

If you put 10 by 10, I can understand why.

Because to make a square, 10 times 4 is 40.

But remember that those corners, okay, we end up counting twice if we use that method.

So for 40 shaded sections is gonna be an 11 by 11, 11 at the top, 11 along the bottom, and then 9 down each side, 'cause we've already shaded the corners.

Some amazing thinking there today.

And this is all work that's gonna help you build your algebra skills as you move through algebra.

So we've looked at checking your understanding of expressions and equations, and we've seen that rules can be used to work out solutions to a scenario.

We've also looked at where there is a pattern, we can generalise a situation and use this to find any value in a pattern.

And lastly, we looked at patterns being described in words, so that that can help us identify rules.

And remember, we said sometimes, just saying things out loud can help us put things into words, and then that will then help us when we are moving through the algebra.

Right, loads of fantastic thinking in that lesson today, and I hope that really helps you develop your algebra skills as you move through with your mathematics.

Fantastic, well done.

It'd be really nice to see you again soon.