As well known investigation at GCSE level called frogs.
Many of you might well have attempted that particular
investigation and want to start this particular section by
having a look at that investigation, because it's
going to show us why we might want to use brackets.
Now, how does the investigation work? Well, what we have to do
is try and interchange these coins at this side the pound
coins. With the 10 P coins at this site and we have to do it
by using one of two kinds of move, we can either slide into
an empty space, or we can hop or jump over a coin of the opposite
kind. So here a pound coin jumped over the 10 P coin. We
can never go back. We must keep
going. Forwards and so these 3 coins have with those rules to
interchange with these 3 coins over here. So let's just see
what happens. Let's see if we can keep count of the number of
moves, not only keep count of the number of moves, but the
number of different kinds of moves hops remember all slides.
That was a slide.
That's a hop, and the object is to do it in the minimum number
of moves. Now when I do it, it will be the minimum number of
moves. Don't worry about that, but we need to keep count of how
many hops, how many slides and how many moves altogether so.
Let's begin and will just keep a tally here of hops and slides.
So we begin one slide.
Over there one hop.
One slide. And now with that we want one.
Two hops
There one slide.
Now one. 2 three hops.
A slide gets us that one home.
And then two hops a slide gets us that one home.
And then Hopkins that one home and then a slide gets that one
home. So let's just talk these up 123456 slides.
369 hops so altogether a grand total of
15 moves. OK, when you were doing this investigation at
school, or possibly you may not even have done it. What you
would have done would be draw up a table so we want the number of
coins. On
each side. Hops?
Slides.
Moves Got our table there. So number of coins on
each side. We could have one 2, three which is the case. We did
four or five.
Hops? Well, let's fill in this one for this table here. We had
nine hops, six slides, and that gave us a total of 15 moves. Now
we could do it for one.
On each side, just do it
quickly. One slide.
One hop, one slide. So altogether two slides and one
hop. And we can do it again.
Quickly for these.
A slide.
A hop.
A slide, so that's two slides all together in one hop, and now
we have another two hops.
A slide. A hop.
And the slide. So altogether we
had there. Four hops and for slides. I'm not going to go
ahead and do it for 4 coins. It's quite easy to do and the
results that we get will be 16 and 8 and 25 and.
10 Giving us a total number of moves of
three, 824 and 35.
Now the object with most of these investigations is to try
and arrive at a prediction. What would happen if we had any
number of coins? Can we say what the result would be if we had 10
coins on each side? If we had 50 coins on each side, 'cause this
is the power that maths gives us the power to model it?
In symbols and be able to use those symbols to make our
predictions, so that's what we're looking at. Can we make
that prediction? Well, let's have a little look. Want to
concentrate here on this column and here on this column.
I actually need a few more columns because I want to look
at this set of numbers and I want to look at how this set of
numbers is made up or created. So what I'm going to do is now
we've got the data that we need. I'm going to turn over
the sheet and I'm going to rewrite this table on a
separate sheet and then we can analyze it without having all
these bits and pieces around the edge. So we pick up the
coins.
Put them to one side.
Turn over the sheet an begin again. So here we've got the
coins and the number of coins on each side. 12345 and we
have the number of hops that was one 4 nine 1625. Then
we have the number of slides.
And that was 2468
ten. And then the total number of moves which we
got from the total of the hops and the slides. So
that's three 815-2435.
Now what we said was we want it to be able to make a prediction
no matter how many coins we've got on each side. We want it to
be able to say how many moves it was going to take us to
interchange those two sets of
coins. Let's have a look at this
set of numbers. And let's try and relate it to this set of
numbers. Well, if we look at three an 15.
15 is 3 times by 5.
Just a lucky guess. Well, let's have a look at the next one.
4 and 24 will 24 is 4 times by 6.
An again 5 and 35 that's five times by 7, so eight is 8
the same relationship with two? Yes, it is. That's two times by
4. And even for this top one we've got one times by three.
So let's look at these numbers here. These two columns. Here we
have this number. The number of
coins. The number of coins and here we seem to have two more
than the number of coins, so if this was an what we seem to be
saying is that here we've got N.
Multiplied by, I'm not going to write anything in there at the
moment, and plus two now it's the whole of this that we want
to multiply by N everything, not just the N, not just the two,
but all of it, because when we do one times by three, we
multiply all of the three by one. When we do the two times by
4, we multiply all of the four by the two, and so on. So this
has to be.
Help together in a bracket so that we can show we are
multiplying everything inside the bracket by what is outside
the bracket. So let's just leave that there for the moment.
Let's have a look now at these sets of numbers.
And what we do know is that if we add these sets of
numbers together, we do get the total number of moves.
Can we see any kind of
relationship between? These numbers the number of coins and
these two was a fairly obvious one. This is 1 squared. This is
2 squared, 3 squared, 4 squared and five squared. In other
words, it's this number multiplied by itself. So if we
end we can see we've got end
squared. If we look down here.
Each one of these relates in the same way four is 2 * 2 six is
2 times by three, 8 is 2 times by 410, two times by 5, so again
2 times by 1, two times by two 2 times by 3, two times by 4, two
times by 5. This is our N again, so we must have two times by N
or two N.
Now the number of moves can't change simply because we've
written it down differently.
We know the total number of moves is the sum of these two
terms. That must be the same as that the number of moves
simply can't change just because it written down differently.
Doesn't mean that the value changes, and So what we must
have is that the total number of
moves. Is equal to both of those expressions, and so they must
both be equal. So that shows us two things. First of all, it
shows us why we need brackets to keep things together as entities
as a whole.
It also shows us how we can remove brackets because we
multiply that with that to give us the end squared and we can
multiply that with that to give us the two N.
So bearing these things in mind, what we're going to have a look
at is multiplying out brackets and removing brackets from
expressions. Now we've seen why we need it now. We need to be
able to work both ways across this equal sign. We need to be
able to work from the brackets to the expression and from the
expression to the brackets. Now we're only going to go one way
in this particular clip. That's from the brackets to the
multiplied out. Or expanded xpression coming back this way
is factorizing and we look at that in another section.
So let's take some examples 3 times by X +2.
Everything inside has to be multiplied by what is outside,
so we need 3 times by X and we need 3 times by two
which is 6. The plus sign, because that's plus there.
X times X minus Y everything inside must be multiplied by
what is outside, so we've X times by XX squared X times by
Y is X why we are taking the wire way so we've got X
times by minus Y is going to give us minus XY.
Minus 3A squared.
And then in the bracket, 3 minus B. Now we're multiplying
everything inside by what's outside. What's outsiders got
this minus sign attached to it? So we gotta take care. So we
have minus 3A squared times by 3 - 9 A squared.
Minus 3A squared times minus B, so the minus minus together
gives us a plus and we have 3A squared B.
We can do this across a number of terms, so if we minus two X
let's say X minus Y minus Z.
So we've minus two X times by X. That's that times by that minus
two X squared we've.
Minus two X times Y minus Y. That gives us +2 XY
and then minus two X times Y minus said plus 2X Zedd.
OK, those are reasonably straightforward and you can find
plenty of those in textbooks and other sources for you to
practice. Let's now have a look at what happens if we want to
multiply out expressions where there are two brackets
multiplying each other.
So let's take X +5 times by X
plus 10. What now?
Well, what I want just to go back to is a simple one like
let's say 3 times by X plus 10.
If you remember what we did, we said, well,
we'll take the three.
And multiply it with the X and will take the three an. We will
multiply it by the 10.
So that this was the entity we were multiplied by. Let's think
of this X +5 as being the three, so for this one we would do 3
times by X. So for this one will do X +5 times by X.
So the X +5 is behaving exactly like the three.
Now here we would do 3 times by 10 and rather than right it is
30. I will write it as three times by 10.
So this is taking the place of the three, so it's going to be
X +5 times by 10.
Here we would have a plus sign, so here we
will have a plus sign.
Now, the fact that I've got the X and the 10 at the end doesn't
make any difference. I'm still multiplying them together so the
same rules apply. I still have to multiply everything inside
this bracket by what is outside this bracket. So now I need that
with that, which gives me an X squared and I need that with
that, which gives me 5X. I need that with that, which gives me
10X and I need that with that, which gives me.
50 and so we tidy up the middle
bit. 5X plus 10X is
15 X. And there we have.
Our expression.
OK.
This. Here. Is an extra step we can manage without
it provided we keep track of what we've done.
Notice I've got an arrow from here going to each of these.
Alternatively, if you've got an arrow going from that extra that
X and an arrow going from that
extra that 10. So every term is got sort of two links associated
with it. So let's have a look at doing that without going into
this step. Now we've seen how we can do it can, we shorten the
process a little bit.
So we've X +5 times by
X plus 10.
So we've got to multiply everything in this bracket by
everything in this bracket. So remember we had to have X by X,
so that's their X squared. Then we have to have X times by 10,
so that's there 10X. Now we need to make the same links here, so
that's X by 5.
Giving us 5X and then 5 by 10, giving us 50 and so we end up
with exactly the same expression as we had before.
Gain, we've got two hours going into this bracket. Two
arrows going into this bracket and then coming back. We've
got the same from 10. There are two arrows linking back.
We can use this process for brackets like this that leaders
to quadratic expressions an for multiplying any pair of brackets
together, and in fact the arrows help us to see that we've
completed the multiplication 'cause they help us to see that
if we've got two terms here, then there are two arrows
linking this first term, one with each term there, and so on.
So let's have another look at some examples of this working
our way. Down some quite complicated ones, but beginning
with some simple ones, very much like we've just had a look at.
So first of all, we got X takeaway 7 times by X takeaway
10 or X minus Seven times by X
minus 10. Let's expand this as we've done before, so X times my
X that gives us X squared.
And X times by now this is minus 10, so it gives us minus 10X.
Minus 7 times by X minus 7X and minus Seven times by minus
10. Plus 70 and we can simplify that X
squared minus 17X plus 70.
Take X plus six times by X minus 6.
Everything in this bracket must be multiplied by everything in
this bracket, so we have X times by X. That's how X squared.
X. Times by minus six gives us minus 6X.
Plus six times by X their loss
6X. And then six times by minus 6 - 36 and now again we need
to simplify this. Looking at these ex is so with X squared.
Now we have to take away 6X and add on 6X. The net effect is
nothing so there are no excess minus 36, so we don't always get
this middle term sometimes and this is an example notice it
structure. Certain symmetry about it, a 6 and a six hour
plus under minus.
And the ex is will disappear.
Take another one, 2X minus 3X plus one. This time it's 2X not
just X, but again, everything in here must be multiplied by
everything in there. So we have two X times by X. That gives
us two X squared.
2X times by one that gives us 2X.
Minus three times by X that's
minus 3X. And minus three times by one is minus 3.
Here we've got some excess. We must simplify them. We've got to
gather them together so we have two X squared.
Plus 2X minus three X. We're adding on 2X taking away 3X.
We still have their foreign X to take away, minus three.
Now let's have a look at just two or three more of
these. If we take 3X minus two and three X +2.
Have we seen something like this before? A minus sign in a plus
sign a two and a 2A3X and 3X. Not just an expert of 3X.
Are we going to get the same
sorts of things? Symmetry suggests that we might, so let's
have a look.
3X times by three X. That will give us nine X squared 3 * 3
X times by X.
3X times by +2 is plus 6X minus two times
by three. X is minus
6 X. And minus two times by +2 is minus four. Gotta
tidy up these middle terms.
Nynex squared. Plus 6X minus 6X no access, and we left with
minus four on the end. So yes, it was another example of that
kind of expression where the middle term is going to
disappear. Let's Make it a little bit harder and
let's have a look at something with three terms
in this bracket and only two in this, so we make
that One X +2, but let's put in X squared.
Minus two X cubed +8.
Got three terms here. We need to keep very careful track and make
sure that we multiply everything in here by everything in there.
So let's begin.
X squared times by X? That's X cubed X squared
times by two. That's two
X squared. Minus 2X now minus two X cubed. Now times by
X is minus 2 X till the
4th. And minus two X cubed times by two is minus 4X cubed,
and now we've got to do the 8.
So we've got 8 times by X Plus 8X and we've got 8.
Times by two. So that's plus 60.
What we can see is that from each of these terms we've got
two arrows coming.
And that's what we must have. 'cause we got two terms here so
we can be pretty happy that in fact, we've done what we set out
to do. And now we need to simplify. And it's usually
better to write down expressions with the highest power in X
first. This here is the highest power in X. It's X to the power
4. So we write that one down first, minus 2X to the power 4.
What else we got? We got any ex cubes. Well yes we have. We've A
plus X cubed an A minus 4X cubed. So that's going to give
us minus three X cubed.
Sometimes helps just to put a little tick on top of them, just
to show that, yeah, we've actually dealt with those X
squared's only one that's there plus two X squared. So we've
dealt with that one plus 8X. We've dealt with that 1 + 16
we've dealt with that one, so there our answer.
Let's see if we can just take this a little bit further by
multiplying together 2 quadratic expressions.
Normally come up with something as difficult as this, or perhaps
as tedious, but it's a good exercise to see if you've
actually got the hang of what you're supposed to be doing. So
we've got to multiply everything in this bracket by everything in
that bracket. There are three terms here, so if we've done it
right, we ought to have three arrows coming from each of
these, associating with each of these. If you think about that
ought to give us 9 terms.
In the same way that this one.
Three terms here, 2 terms that gave us six terms
altogether. 123456 before we simplified it, so we're going
to have nine terms here.
Let's work our way through it.
X squared times by X squared X to the 4th.
X squared times by X Plus X cubed.
X squared times by minus 6 - 6 X squared.
Finished with that one on to this One X times by X squared
plus X cubed.
X times by X Plus X squared X times by minus 6 - 6 X. Now this
one here minus two times by X squared getting a bit fraught
with the brackets here with the arrows linking them, but
nevertheless we can still use it to help us check. So minus two
times by X squared minus two X
squared. Minus 2 times by X so that gives us minus 2X and
then finally minus two times by minus six gives us plus 12.
This now needs tidying up, so we start with the highest powers
1st and write down the highest power, then the next, then the
next and so on down the line we take them off as we did here so
we can show that we've coped with them. We've taken them into
account, so our highest term is this one. Here X to the power
four, and there are no others. Now we want the ex cubes. Well,
there's one there, and there's one there. So that's plus 2X
cubed, so we can.
Take off those two X squared's.
Minus six X squared.
Plus, X squared minus two X squared, so we've got minus six
X squared minus two X squared. That's minus eight X squared,
and then we're adding on One X squared, so altogether that's
minus Seven X squared, and we can take them off. We've
accounted for them.
Minus six X minus two X the X terms next altogether that's
minus 8X. And then at the end we just got this number 12 that
we have not accounted for, and so that's plus 12.
So that's multiplying out
brackets. Every pair of brackets that needs to be multiplied out,
or indeed if it's three sets of brackets, you do 2 together, and
then the third would. The resulting expression is done
with the third one one. I want to have a look at now is just
simply removing brackets. Sometimes we get collections of
expressions nested in various sets of brackets.
So let's have a look at the sorts of horrible, awful
expressions that people write down in textbooks and say have a
go at these and remove the
brackets. So here we are, whole set of letters jumbled together
with brackets in and what we're being asked to do is just remove
these brackets and simplify. So the first thing is remove the
brackets, get rid of them. So let's have a look. OK, now we're
taking away B minus C, so we've got to take away everything
that's in the brackets, and we've got to take away the B and
then takeaway minus see which of course is the same as adding see
on. Plus a no problem there plus B because we've got to add on
everything that's in this bracket. So plus be plus minus C
which is just minus C.
Plus B and then again. We've gotta take away everything
that's in this bracket. So we gotta take away, see an then
takeaway? Minus A and that's the same as adding a on.
Now we have to simplify, so we have to look for all the days,
all the bees and all the Seas. So let's take the ace Pursuivant
A&A&A and they're all, plus. So that's three a we can just take
them off to show that we've dealt with them.
Bees minus B Plus B plus
B. Minus B Plus B know bees and then another be to add
on plus be. So we've dealt with all the bees.
Plus C minus C minus C Plus C minus C, no seas and the
sea still to take away minus C.
So that was fairly straightforward. No nesting of
brackets, so let's have a look at one where we've actually got
some brackets nested so minus.
5X inside that bracket minus 11 Y.
Minus 3X. Minus.
5 Y minus three X minus six Y.
Now that looks pretty awful. What's a brackets?
And if we look, we've got a pair of brackets here covering
everything. We've got an expression in a pair of
brackets. There got another pair of brackets here and inside.
We've got another pair of brackets. Where do we start?
Looks a bit tricky, but what you do is you start right inside the
brackets so you look inside each set of brackets and you say what
do I need to do to get rid of one set of brackets and usually
go right inside the most complex
bit? And this bit here is by far the most complex bit. So let's
begin by going right inside and looking at this.
Really nested embedded set of brackets and get rid of them to
begin with. So if we're going to do that, we write the rest
down just as it was.
We won't touch it at all.
We're just going to look at this bit. We're taking away
everything there, so we're taking away 3X and we're
taking away minus six Y, so altogether. That's takeaway
3X, and then the minus minus is plus six Y. And then that
Square bracket and that one.
It's beginning to look a little bit simpler. We've now got this
big set of square brackets outside, but we've got that one
there and that one there, and we can deal with both of these
together. Be'cause, they're no different from what we've got up
there, really. So equals minus 5X minus 11 Y and then minus
minus is A plus 3X minus five Y
minus. Minus 3X is A plus, 3X minus plus six Y, so we're going
to have minus six Y and restore the Big Square bracket.
I think we want to tidy this up. We've got lots of axes and wise
floating about so really we've got to tighten this up, so let's
have a look. And we'll do it simply inside the bracket.
So we don't need to think at the moment about the effect of this
minus sign with just looking at that bit there. That's inside
the bracket. So we 5X3X and 3X, that's 11X.
Minus 11 Y minus five Y minus six Y all together we
are taking away 22Y and close the big bracket and now we've
got to remove this final bracket so we have minus 11X plus
minus takeaway minus 22 Y.
So what? Looked quite an awful lump of algebra has now come
down to be something that's really quite small. An quite
manageable will just take one more 'cause it's helpful to get
this focus on getting right inside those brackets and what
you will notice, I think, is that we tend to use different
sets of brackets, different types of brackets.
To draw our attention tool, the different lumps of algebra that
we've got to deal with.
So here in this expression we've got three sets of brackets, the
curly brackets, the square brackets, and the round
brackets, and notice I've worked my way in, so I'm now right
inside, right in the middle of this expression. So this is the
one I'm going to go for first. Everything else for the moment
is going to stay the same, not going to touch anything else,
just this round bracket here. So I've got to multiply.
By two so 2 * 10 A is 20 A and two times by minus B
is minus 2B. Put my square bracket in. Put my curly bracket
here. What now? I think it would be a good idea to
try and tidy this up before we went any further, so we've got
5A Minus Square bracket. I've got 6A plus 28, that's 26 a
takeaway to be.
Close up the brackets at the
end. Equals 3B, Now I've gotta go for the square bracket. Now
these are inside the curly brackets so it's the square
brackets we tackle. I've made that look like a six. Let's make
it more clearly. AB minus.
5A. Take away, take away everything inside this
bracket. So takeaway 26 a takeaway minus 2B so it's the
same as adding on to be and keep the curly brackets. Now
we can simplify this.
3B minus.
5A takeaway 26. So that's take away altogether or minus 21 A
plus 2B. Close the bracket.
And now come to the final step. Almost the final step. Anyway,
we can get rid of these curly brackets 3B. Take away
everything in the curly brackets. Minus minus gives us
plus 21 A minus +2 BS. That's taking away to be and then. Now
we just need to simplify this and what we've got is to be
taken 3 be sorry 3B takeaway to beat is B.
Plus 21 a so we were able to reduce that.
Huge lump of algebra with three sets of brackets to a very
simple expression. And again, notice where did we start right
in the middle with the most nested set of brackets that we
could find right in the middle. And we simplified that. And then
we gradually worked our way
outwards. Dealing with each set of brackets intern and
simplifying as we went along so as to make the job easier
for ourselves and be more sure that we've got the right
answer at the end.