Lesson video

Hi everyone, Ms. Jones here, and today's lesson is all about exploring equality.

So thinking about what equality means, especially obviously in a maths context.

But before we can begin, make sure that you have a nice, clear space, avoiding any distractions.

You try and find a nice quiet space to work and that you have a pen and some paper, because it's really important about writing down some notes and doing the calculations with pen and paper.

Pause the video here to make sure you've got all of that ready.

Okay.

The first thing I would like you to do is think of a number, multiply it by two, add four, divide it by two.

And finally, subtract your original number.

Once you've done that with one number, I'd like you to try this again with a different starting number and again, and see what you notice.

For extra points, if you manage to prove what you notice is always going to be true with a bit of algebra, that would be just incredible, but don't worry if that sounds a bit complicated.

How could you improve this student's response? So he's written his calculation.

How could you improve how he's written that? There's some problems with what he's written there.

So pause the video to have a go at those three things.

So hopefully no matter what starting number you use, you arrived at two.

This is always going to be the case, because if we look at that with a bit of algebra, we think of a number and we call it A for example, we multiply it by two, we get 2A, we add four, we get 2A add four.

We divide all of that by two.

And we get A add two, and if we subtract our original number, which was A, we end up with two.

So it works for every single number.

Amazing job if you manage to get to that point with the algebra.

The problem with that student's response is that it needs to be separated into different stages.

Because, eight multiplied by two is not, if we pause it there, is not equal to 16 add four.

Eight multiply by two is equal to 16, 16 add four is equal to 20.

So it's really important that we don't just write all of our workings out in one line without separating them out, because we're going to end up writing things that aren't true.

18 multiplied by two does not equal 16 add four, which does not equal 20 divided by two.

So well done if you managed to work out that mistake there.

You can use a known equation to form related equations.

When forming a related equation, you should check that both sides balance.

So for example, we have an equation here.

Three multiplied by nine equals 10 add 17.

That is true.

This person is saying I added eight to each expression.

And when he did that, he's got three multiplied by nine, now add eight.

And that still equals to 10 add 17 add eight, because they're still balanced.

Whatever he's done to one side he's done to the other.

This person say I multiplied each expression by 10.

So we have 10, lots of three, lots of nine, and we have 10 lots of 10 add 17.

It's really important that we know how that's been written, because if I just wrote 10 add 17, and then just multiplied by 10, that is not multiplying the whole thing by 10.

Because according to our order of operations and our priority, we would have to do 17 multiplied by ten first.

And then add 10.

That's not correct.

We want to multiply the whole thing by ten so we need to make sure we've got those brackets there.

Using a bar model can help you visualise equality.

So we have here as an example, two variables are related by the equation N equals 2M.

And this has been represented by this bar model.

We've got an N here, which is the same as two Ms. And those Ms are the same value so we need to make sure they're the same size.

This person is saying that 2N equals 4M.

So if we accurately and with the ruler, of course, ignoring my terrible drawing, if we extended that to be two Ns, and we then have one, two, three, four, Ms. Remember those should be equal, not like mine.

He is in fact correct.

Two Ns do equal four M.

And actually what all he's done there is multiplied both sides by two.

And you can see he's preserved equality because whatever he's done to one side he's done to the other.

So it works and it's balanced.

This person has said 2N equals 2M add N.

So she's still got two Ns.

We've got two Ms and this time another N.

And so we imagine this is just one N.

That works because what she's done is she's added N to both sides of her equation, and it's still going to be equal.

She's preserved equality.

What other equations linking N and M, using perhaps this bar model to help you, but based on this equation, can you find? So we've already multiplied both sides by two and we've added N to both sides.

And we can see that that has preserved equality.

What other equations linking N and M can you find? Pause the video to have a go at that? So here are some examples.

Amazing job if you spotted one of the examples that you came up with here and well done if you've got any extras as well.

So you can see here, we don't just need to multiply by two.

We could multiply both sides by nine.

Here we've got three Ns equals 2M add N, so we've added another N to this one.

Well done if you managed to spot those.

Pause the video to have a go at your independent task.

You were asked in the first question to decide if each of the following are true or false.

Five subtract two equals two add three equals five multiply by two equals 10.

So, first of all, I'm thinking four subtract two equals two.

So that does not equal 10.

So I'm straight away thinking here that is not going to be correct, because we shouldn't be writing it out like that remember.

The second one, seven subtract three is four.

Is that the same as four subtract zero? Yes.

Is that the same as three subtract negative one? Yes.

So that one is true.

For question two, you were given an equation 2B equals A and you were asked to complete the following.

So if 2B equals A, 2B add three equals A add three.

Cause if I add a three to one side, I need to do the same to the other to preserve that equality.

And here were the rest of the answers.

Well done if you got those correct.

In the Explore task, use the cards to form equations related to A equals B.

So if I know that A equals B, which other ones do I know will equal each other? So straight away, I can see that A add two, if I add two to this side, I've got to add two to the other side.

Be looking for B add two somewhere.

It's worth noting here that we're not just looking at pairs.

Actually some of them are in threes and there are no odd ones out.

So there are none that don't much anything on there.

So you may have to think about using a bit of substitution and thinking a little bit outside the box for some of them.

Pause the video to have a go at that.

So here are your answers.

We had B equals A or A equals B, so therefore A add two equals B add two.

A divided by two equals B divided by two.

Hopefully you all got that, but that is also the same as 0.

5B because that is the same as saying half lots of B or B divided by two.

4B equals 4A.

Hopefully we all got those two, but actually that's also equal to A add 3B, because if you think about A is B, B add 3B is 4B.

A squared equals B squared, which is also equal to AB.

Because if we say we've got A times multiplied by B, B is A so we could also say we've got A multiply by A, which is A squared.

So that's why some of those were in trios.

And that's why we had to do a bit of substitution there, but that's, that's why they're all equal.

So amazing job if you got all of those and well done if you just got the pairs.

That's still a really good job, but extra special well done if you've got all three of them.

Great job today.

Keep up the great work and have a go at the quiz at the end to make sure that you've checked your understanding and you get all of this and you're doing really well with it.

So really good job today.

I'll see you next time!.