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G'day, Im Dr Peter Price of Classroom Professor. Welcome to this week's video in the Free Math
Worksheet Series. This topic this week is on "Improper Fractions and Mixed Numbers";
it comes from this eBook called "Bring It On! Book 3: Fractions Worksheets". So this
is from the most advance set of worksheets that we have in the Developing Number Fluency
Series and it's for Grade 5 or in Australia Year 6. So it's an advance topic, it's looking
at fractions, we know that students find fractions difficult, we know there are dozens and dozens
of subtopics, if you include decimal fractions and percentages and ratios, there's just a
mass of different questions that we ask our students to deal with, to tackle, and sometimes
they find them very difficult. I've taught young adults that find fractions difficult,
I've spoken to adults who aren't at university, you know, people generally find fractions
difficult. So let's take our time, let's develop the ideas, not exactly slowly but methodically
and in a way that students understand what's going on. Fractions topics of all, lend themselves
to symbol manipulation, or as I usually call it, "Mindless Symbol Manipulation", in other
words, don't think of what it all means, just do this to all the numbers. We will ultimately
get to that point and you'll see I'll talk about that in a moment on the board, but early
on especially we want to help the students understand why it's that way, and if possible
use materials as I've got on the white board to help the students see how the numbers relate
to each other, how they connect together and how it is that we get the answers that we
get. I've seen far too many adult students at university who just don't get it and are
confused and frustrated about it, not to spend a little bit of time just recommending patience
and you know, care in helping the students understand what it means. Alright, so we're
going to have fractions like this one, now I should say I only have a limited number
of pieces so these magnetic pieces are beautiful but I don't have lot and lots and lots of
them, so I'm having to restrict the examples that I put on the board, but I'm sure you'll
forgive me for that, and you'll see what I mean anyway. So here's an example, we have
"1 3/8" so this is our initial fraction, which is a mixed number of course, mixed because
we have a whole number and we have a fractional part. We're going to be referring to eights
and awful lot, because in this example, that's the denominator and that tells us the size
of the pieces, so that has a significant impact obviously on the answer that we're going to
get. So we're going to record this as an improper fraction, and of course that improper fraction
will be in eights, so the question to our students will be, "How many would that be
if it was all in eights?" Alright, so that's where we're going with this, I'll just mention
in passing we're going to call this "3/8" even though it's a bit harder to get your
tongue around the "eights" rather than "Three over eight", which doesn't help anybody, it's
not the correct name for the fraction and if you say "Three over eight" it does nothing
to trigger an understanding what size the pieces are, three over eight is like a spelling
exercise. Alright, so "1 3/8 is how many eights?" now the symbolic method which I'm sure you
know is to multiply the 1 by the 8 and add 3, there's our answer, "Boom!" However, that
does, as I've said, it doesn't help students to understand where that comes from, so we're
going to want to help our students see, "How we get an answer, what on earth are we doing
here, how does it make sense?" So let's, first of all refer to the number of eights in a
whole, so we can put the eights on the board, of course we can count them. It's... I hesitate
to say it's obvious, I don't think it is obvious to students, I remember teaching elementary
or primary level students years ago, and we were talking about sixths and I said, "How
many sixths are there in a whole?" and basically no one in the class knew the answer and I
thought, "Why not, it's so obvious" and I hadn't made that connection, so we want to
help the students make that connection they're called "Eights" "Why?" Because there are eight
of them, "Why do we write an 8 here?" Because that's the number of pieces in the whole,
they're all equal; they are all called "Eights", there are eight of them. Ok, so we've established
it, now if I had different materials I would put these on top, to show that those 8 eights
equal the same as the whole. But these magnetic pieces don't actually stick like that so,
you'll have to take my word for it. So, there's 1 3/8, "How many are we going to have?" I'm
not going to draw all over the magnetic piece but obviously there are 8 here and 3 there,
we're slowing this down just for emphasis, I suppose the answer is going to be "11".
I would like to help the students see that this is 8/8 and this is 3/8, and of course
that makes "11/8", and notice I'm saying "Eights" all the time, that is quite deliberate, because
that's the name of the fraction, it refers to the size of the fraction and we want to
continually emphasize that. And the fact that we're referring to both of these terms, then
we can add them, if this is some other sort of fraction or indeed if it was just a whole,
you'll have to do something else to it, but if they're all eights we can add 8/8 and 3/8
just as we can have 8 oranges and 3 oranges and we can add them all and we get oranges.
Ok, so that's the basic idea, ok, so that's not terribly difficult, let's do another example
only this time we'll go on the opposite direction. So, this time we're going to use some sixths.
So moving on to our second example, this time we're going to start with an improper fraction
and convert it into a mixed number, as I said before I don't have a large number of these
pieces, in fact I only have enough sixths to make up 1 whole, so I've drawn around them,
let me label them as well, so these are all "1/6", I would want the students to see that
these are the same size, I don't want to try and pretend that I'm doing anything untoward,
let's move the eights out of the way, alright. So here we have a number of sixths and we're
going to find out what this comes to, what this is equal to as a mixed number. So we've
got, 1 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 sixths "What would that number be?" So
over here we're going to write "13/6 = _" now it will be a whole number of course, and
a number of sixths and so we want our students to work out what the answer will be. Now in
the worksheet it refers to a division method which of course is the standard way of working
this out. So we're going to divide the 13 by sixth, which is a division fact with a
remainder, 13 / 6 is 2 remainder 1, it's going to be "2 1/6", ok so that's out of the way,
knew that's what the answer was going to be, but "How can we help our students understand
that?" rather than just jump straight into symbols, because I'm convinced that at this
point the people get confused. Not because that's a terribly difficult example or that
they couldn't do a page full of these in their workbook or their worksheets, but because
they'll get it mixed up with other processes and they will forget and then, you know, in
years to come they'll go, "I can't remember how to do that and it's all very confusing."
So I would rather help the students understand what it all means. So let's use the pieces
that we do have here, and put them together and say to the students, "What do you see?"
"How many sixths do I have here?" obviously, its "6 sixths", so 6 sixths is a whole, so
we might record that, "6/6 = 1 whole". Alright, "How many sixths do we have here?" "13" "How
can we use that information to help us figure out all of this and work out how may wholes
and sixths we have here?" Now I know that if you were to do this with your students,
by now half the class will be jumping out of their seats saying, "That's too easy",
that's if they're that enthusiastic, jumping out of their seats saying, "I can do it! I
can do it! I know what the answer is! I know what the answer is!" But we want to help them
understand why it will take a bit of time and even though they can see what it is in
this example, we're going to give them more examples where it will take more, you know
more effort and there is more chance of getting it wrong, so let's just take our time. So
6 sixths a whole, there's one whole, "Do we have enough here to make another whole?" Now
that's a very similar question the one we might ask using base ten numbers when we're
adding and we add together for example 6 ones and 7 ones and we get "13", we done immediately
write 13 down, we say, "Oh look, do you have enough there to make into a group of 10?"
if you have 13 ones, of course you do, so you regroup. This is a lot like regrouping,
"Do we have enough here to make another whole?" Yes we do, we can make 6 of these into another
whole, obviously I can't move the ones that I've drawn, so I'm going to draw a rough loop
around them and say, "That's another whole as well" so there's one whole here, one whole
here. It's also "6/6" so let's say that, "1=6/6, 1= 6/6 and 1/6 left over", so here's our 2
and here's 1/6. I apologize for my board writing, I have injured my shoulder and it's rather
sore. Ok, so that's the process, it is going to take some time, it is going to require
some thinking on the part of the students, I was going to say, "Exploring" it's not necessarily
exploring, in the sense of discovering a whole lot of things new, but it is discovering how
it works, so we want them to get to the point where they can look at an example like that
and go, "I can do that it's easy, it's just division, I can divide 13 by 6, I'll get 2
and one left over and blah blah blah". But we want them to be able to always picture
it in their head and mind, so if we say 17 fourths or 17 quarters, then we want them
to go, "Ok, I know 4 fourths or 4 quarters is a whole, so if I divide the 17 by 4 that
gives me 4 wholes, because I know 4 fours a 16, there's 1 left, it's going to be 4 one
fourth or one quarter". So we want that train of thinking, that train of logic to be available
to the students so that they're not simply remembering some routine, but they're thinking
about what it all means and they can get the answer correct. Ok that's it, we've come to
the end of the video, I look forward to talking to you next time.