Lesson video

Hi, I'm Miss Kidd-Rossiter and I'm going to be taking today's lesson on combining algebraic relationships.

It's going to build on all the algebra work that you've done so far.

Before we get started, make sure you've got a pen and something to write on, and that you're in a nice, quiet place free from distractions.

If you need to pause the video now to get any of that sorted, then please do, but if not, let's get going! So, try this activity then.

Antoni has made an equilateral triangle using sticks.

He wants to make a square, a pentagon, a hexagon, an octagon, et cetera using the same sticks.

He says, "The perimeter of the triangle is 4p + 3 cm." Can you write expressions for the perimeters of the other shapes that he wants to make? If you feel confident with this activity get going now, so pause the video.

If not, I'll give you a little bit of support.

Excellent.

If you need a little bit of support that's absolutely fine, that's what I'm here for.

So we've got a key word here, equilateral.

What does equilateral mean, can you tell me now? Excellent.

It means that the three sides of this triangle are the same length.

So that means that each stick must have the same length.

So I think it would be easiest for me, first of all, to figure out what the perimeter of the hexagon would be.

How many of these sticks will I need for a hexagon? Tell me now.

Excellent, I'll need six sticks, won't I.

So if I've got three sticks that have a perimeter of 4p + 3, then that means that six sticks will be double that, won't it.

So it will be 8p + 6.

So, could you figure out the length of one stick, and then could you use that to help you to find the perimeters of the other shapes? Pause the video now and have a go at this task.

Excellent work, let's go through this together then.

So, the square then.

In order to work out a square, a pentagon, an octagon, or any shape that hasn't got sides that are a multiple of three, then I will need to know what one stick is, won't I.

If three sticks are 4p + 3 centimetres, then that means that one stick must be 4p + 3 all divided by 3.

So that means that one stick will be 4/3 of p, or 4p/3, whichever way you prefer to write it, plus one.

So the square will have a perimeter of four of those sticks, so four times p + 1 which would give us p + 4.

The pentagon would be 5 lots of those sticks, so 5 lots of p + 1, which gives us p + 5.

The hexagon we already did on the previous slide, that one was slightly easier because we just doubled the perimeter here, and then the octagon would be 8 lots of p + 1, which would be p + 8.

And remember those are all in centimetres, I should have added those in as I went along.

So, let's have a look at the Connect part of today's lesson then.

So we are given this statement here, this is an equation, so 3x + 2 = 4a + 3.

And I'm going to show that using a bar model.

So here's my 3x + 2, and I'm told that that's equal to 4a + 3.

Now I don't know the size of x, and I don't know the size of a, but I know that my 4a + 3 is equal to my 3x + 2, so that means that it must be the same length when I represent it in my bars.

So, first thing I'm going to do is I'm going to look at doubling my 3x + 2.

So if I double my 3x + 2, what would I get? Excellent, I would get 6x + 4.

Is 6x + 4 = 4a + 3? No, it's not.

So if I've doubled the 3x + 2, what do I need to do to the 4a + 3? Excellent, I double it.

So there's another 4a + 3, so if I've doubled 4a + 3, what do I get? Excellent, 8a + 6.

So I know that if 3x + 2 = 4a + 3, then 6x + 4 must equal 8a + 6.

Let's have a look then if we're adding on.

So I'm going to add three to my top equation, so 3x + 2 + 3, what does that give me, 3x + 2 adding on 3, what's that? Excellent, 3x + 5.

Is that still equivalent to 4a + 3? No it's not, so what do I need to do to 4a + 3 to make it equivalent? Excellent, add three as well.

So 4a + 3 + 3, which gives us 4a + 6.

So if I know that 3x + 2 = 4a + 3, I also know that 3x + 5 = 4a + 6.

What about if we're taking away? So I'm going to take away two from each side now, so I've taken away two from there, so 3x + 2 and then I take away two, what does that give me? Excellent, just 3x.

Is that equivalent to 4a + 3? No, so what do I need to do to make sure that it is equivalent to 4a + 3? Excellent, also take away two.

So 4a + 3 - 2 gives me what? Excellent, 4a + 1.

So I can say that 3x = 4a + 1.

And you see all the different equations I'm getting here from my starting point.

Finally then, let's have a look at what happens if we divide by 4.

So I can show that here.

I'm looking for this part here, or any of the equivalent parts.

So 3x + 2 divided into four equal pieces will give me x + 2/4, and what do we know that 2/4 is equivalent to? Excellent, half.

So x + 1/2.

And that's equivalent to what? Excellent.

4a + 3 all divided by 4, and that gives me 4a/4 is what? Excellent, just a, and then add 3/4 which will leave 3/4.

So we know then that x + 1/2 is equal to a + 3/4.

So these are all equations that we can find out from our starting point.

So this time, we're given two equations.

Pause the video now and write the two equations for me.

Excellent.

So the first one is the same as we had on the last slide, so 3x + 2 = 4a + 3.

And the second one is 5x + 2 = 6a + 5.

Now if we know this, then we can also add together these two diagrams. So we could add together 3x + 2 and 5x + 2 to get what? Excellent, it would be 8x + 4, wouldn't it, because 3x + 2 + 5x + 2, we collect our like terms to get 8x + 4.

So if we add those two together, what do we know that that will be equivalent to? Excellent, it will be equivalent to these two added together, won't it.

So 4a + 3 + 6a + 5 gives us 10a + 8.

So we can write another equation based off these two, that we know that 8x + 4 is equivalent to 10a + 8.

We could also look at subtracting one from the other.

So if we took away the 3x + 2 from the 5x + 2, we would be left with 2x wouldn't we, because we've eliminated three of the x's and we've eliminated the add two, and if we subtracted the 4a + 3 from the 6a + 5, then we would be left with 2a + 2.

So again, in a similar fashion, we can say that 2x = 2a + 2.

You are now going to apply your learning to the independent task.

So pause the video here, navigate to the independent task, and when you're ready to go through some answers, resume the video.

Good luck! Excellent work, let's go through these then.

So we're given that 2a + 5 = b, and 3a = b - 1.

So we are asked to complete 4a + 10.

So we can see that this equation here has been multiplied by two to give us 4a + 10, so that means that the whole equation has been multiplied by two, so our answer here must be 2b.

5a + 5.

Well we can see that we've added the first equation to the second equation to get 5a + 5, so that means that we need to do b + b - 1, which we should simplify to give us 2b -1.

a - 5, well you can see that this time I've taken this equation from this equation, so when I've done that I've done 3a - 2a which has given me the a, and then taken away the 5, so I need to do b - 1 - b, which gives me when I simplify it just -1.

So you can see here that that's already my answer for the next one, so I can put that in.

And then 7a + 10, this one's a little bit trickier because I've combined the two equations, but I've also multiplied one by two.

So I've multiplied the first equation by two to give me 4a + 10 = 2b, and then I've added it to the second equation.

So 4a + 10 + 3a gives me 7a + 10, so that means I need to do 2b + b - 1, which when I simplify it gives me b - 1.

Excellent.

Moving on then, we're given these equations this time, so 22x - 20y, this full equation has been multiplied by two, so that would give me 36z.

7x - 15y, well here I've done the first equation take away the second equation, so I would get 18z - 45.

To get an answer of 135 I must have multiplied this equation here by three, so that gives me 12x + 15y.

To get 18z + 45, I must have added the two equations together, so 11x + 4x = 15x, and -10y + 5y gives me -5y.

And then finally 45 - 18z, well I've done this equation, take away this equation, so that means I've done 4x + 5y - , so that gives me 4x - 11x + 5y + 10y, because I'm taking away a negative.

So that gives me -7x + 15y.

Now if you write that the other way around, if you write 15y - 7x = 45 - 18z, that's absolutely fine as well, well done.

Let's move on to the explore task now then.

So use the equations in the cards above, so these three here, the top three, to find ways of completing the equations in the cards below, so these six down here, and then when you've done that, what are the other equations could you write? Pause the video now and have a go at this task.

Excellent.

There were lots of different ways to answer these, so well done.

6x + 8 I can see is two lots of my first equation, so that must mean that it's 4y.

8x + 4 I can see is my first and my second equation added together, so that must give me 4y + 8.

20x - 1 was a little bit more tricky.

What I did was I multiplied this full equation here by four, so I got that 20x = 8y + 32, and then I took away one, which would give me 8y + 31.

Then, 8x + 1, I linked that one to this one here, because 8x + 4 I know is 4y + 8, so that means that 8x + 1 must be 4y + 5.

8 + x then, the first thing I did was I rearranged this equation here to give me that 8 = 5x - 2y, which means that 8 + x must be 6x - 2y.

And then for this one here, the last one, I rearranged this equation to get 2x = 3y - 21, so that means that 2x - 7 must equal 3y - 28.

What other equations did you write? Did you manage to think of different ways to write those ones? Excellent work on that.

That's the end of today's lesson, so thank you very much for all your hard work.

Don't forget to go and take the end of lesson quiz, so that you can show me what you've learned, and hopefully I'll see you again soon.

Bye!.