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G'day, I'm Dr Peter Price of Classroom Professor. Welcome to this video in the Free Math Worksheets
Series. This week's topic is one of the most challenging that I've talked about in this
series; it's a topic involving common "Common Fractions" and we're going to be converting
them to "Decimals" by looking at how many hundredths they're equal to. So the worksheets come from
this book, entitled "Classroom Professor Gadgets: Equivalent Fractions" so the gadgets is a
set of software that we have available, but the worksheets are available separately or
together with the software. And the idea is that we're giving the students tools to understand
the fractions, I don't know how you feel about it, but I for one am a little bit tired of
exercises to help students find answers in a sort of mechanical way, without any thinking
at all, without any conceptual development, without understanding what the numbers mean
or what the fractions are, just simply, "Do this, do this, do this.. Oh look, magical!
There's the answer!" So I think that is a terrible way to teach mathematics and I'd
rather it was never taught that way ever, anyway, don't have time to go on and on about
it, but our approach is to help students understand the size of the numbers, understand the connections
and lead them if possible to discover the relationships. So I'm not trying to pedantically
say, "Here it is, follow this, this is a method, blah, blah, blah" there's plenty of time for
the method down the track, but early on, sorry that's an Australian expression, further on
the students' development there will be time to say, "Now you understand it, here's a method
and we'll practise the method and everybody will get the method" but we shouldn't start
with the method at the beginning. Alright, so further development into that idea, I'll
put into a PD Course about teaching fractions, so that's for a time in the future. So the
worksheets themselves, there are four worksheets in the set, as usual, let me put that closer
to the camera, they look a lot like that, so there's a large example in the middle and
we've called this an "Investigation", so as I just said, we want the students to effectively
discover the relationship rather than being told what it is. So, this comes after looking
at equivalent fractions between different common fractions, so we've, in the book and
with the software, we can help students understand that a half is equal to 3/6 or 4/8 or 5/10
and so on, not because we say so, not because we've got some mechanical method to find it,
but because we're looking at the picture, and the software does this by showing two
different diagrams and you can see that they're both the same and they have different symbols
and so on, but we're going to extend that now and look at hundredths. So let's look
at a fraction and in the first worksheet you'll see that we have 3/4 as a fraction and we're
going to convert that into hundredths. So what we want the students to do is to work
out "What equivalent fraction is there for 3/4, where the denominator is 100?" or in
other words, "How many hundredths are the same as 3/4?" Now I'm not going to try and
draw hundredths on the board, that's on the beauties of the software, that it will show
you hundredths you know, in a blink, because the software is designed to do that. But if
we just use a physical model like this and say, "Can you think about how many hundredths
there would be?" If there were 100 pieces, let's say, "Look at this one, if that was
divided into 100 equal pieces called "Hundredths" how many of those hundred pieces will fit
on those three there?" So the students should be able to see that if we divided this hundred
into four equal parts, a fourth or a quarter of a hundred is "25", so we would have 25/100
that will be equal to this quarter or fourths and another 25 here and another 25 there,
sorry, I said "Three Quarters" several time earlier, that's what we call it in Australia,
and the UK and British Commonwealth countries, I know in the US you call them "Fourths" so,
"Quarters or Fourths", they're synonyms, mean the same thing. So we're going to be able
to help our student see that a fourth or a quarter is 25/100, if we've got 3 quarters
it looks like this, we use a whole range of resources and exercises and questions for
the students, let me use a different colour, of course the answer here is going to be "75".
You'll notice I haven't talk about a method for multiplying and dividing and working that
out, we'll do that as I said, later on in the students development, when your students
are ready for that, then obviously we'll talk about the fact that these two numbers have
both been multiplied by the same number and in this case of course it's "25", but I'd
immediately say, "Why 25?" "Where does that come from?" and refer back to this model here,
as I said if we saw this as 100 that would have 25 pieces, 25 hundredths making up the
fourth or quarter. Alright, so that's not too difficult, relatively easy, we can help
the students see that if we're making an equivalent fraction in hundredths using a starting fraction
like fourths or quarters, it's not too difficult, mainly because 4 is a factor of 100 or we
can divide 100 by 4. "What would we have however, if we had 1/3?" and that's the next exercise
on the worksheet, so that's the one that's an investigation here in the middle of the
page with a large circles, 1/3 on the left hand side and then a diagram in the worksheet
showing 100 pieces, "How many hundredths are there going to be?" Now I would prefer to
take a bit longer, obviously we don't have a lot of time on the video, and on the worksheet
we get to this point pretty quickly, but you might want to do more exercises thinking about
different fractions equal to hundredths before you get to the 1/3 example, obviously this
is a tricky example and the hardest thing about it is, we can't divide 100 by 3 evenly,
we get some you know, irregular sort of answer. But this is what we're trying to do, we're
trying to say, "1/3 is equal how many hundredths?" you'll recognise straight away that equal
sign isn't going to help us because we don't have answer up her or at least we don't have
a whole number answer. So we're actually going to use a different symbol and we think this
is appropriate, now you'll have to look at your curriculum to see when students are ready
for the symbol "Approximately Equal to", but in this context this is exactly the symbol
that we need, it's not going to be an exact number, I realised, if you're thinking, well
we could, we could write "33 and 1/3" yes, I know we could, but we're trying to find
a whole number that's close, so you know, bear with me. So we're going to say, "1/3
is around about how many?" using the diagram on the worksheet, because we've got all the
100 pieces actually marked, there it is, on the worksheet here, we could invite our students
to colour in the right number that they think, so you know, colour in as many as you need
to in order to have it equal the 1/3, and of course they'll be colouring in 33, they
should be able to see 33 is the closest, 34 is too big, and so on. We could talk about
the fact that when it was fourths we can divide the hundred by 4 to find out there were 25,
we can try dividing 100 by 3, so there's a number of ways of getting to the answer, but
what we're going to say is, "This is approximately equal to 33". With a little bit more discussion,
with more thinking on the part of the students and again not because you've told them the
answer, but because they've had a chance to think, they might be able to see that there's
actually 1/3 more than that, ok. Now the next step, and it's the same on each of these worksheets
this week, we're going to write that down as a decimal fraction, so we'll write, "1/3
again, equals 0." and of course again this is when it gets tricky, I hope you can see
that what we're trying to do here is develop thinking and understanding, as I've already
said several times, rather than some mechanical method. So we're not going to immediately
say, "Oh let's do some division and workout the answer" we're not going to jump to a calculator,
I know a calculator will give us the answer, but we want to understand where that answer
comes from. So what we're going to do here is say, "Well this is 33/100, it's approximately"
"What does 33/100 look like if we write it as a decimal?" and again we should change
this symbol to an approximately equal sign as we have in the worksheet, and we can see
1/3 is approximately equal to "0.33", now that's not the exact answer, there's a lot
more that we could talk about and when students are older, because we think this is probably
around about, let's say, now I'm guessing here, you'd have to check the curriculum,
Grade 5 or Year 6, something like that, and older. We don't have to get in to the heavy
detail of what it is exactly, we're trying to introduce this as an approximation, the
other thing to say here is, that's close enough for most everyday purposes, if we're dividing
by 3 and we say, "That's 33 hundredths, that's pretty well close enough for what we need".
Ok, so there are more exercises like this, it's a difference to all the worksheets, so
a lot of the other worksheets that I've talked about, having a page full of exercises, and
what we're trying to develop there is fluency, and speed, and memorization, and you know,
getting the answers right quickly and so we have the Ten Minutes a Day Series for students
to be timed on their work, this is not a timed worksheet, this will take quite a bit of discussion,
and thinking, and talking and playing with ideas, it would be the sort of thing you could
do with a group of students and have them investigate it together, and you can you know,
tweak and add different examples and that sort of thing. So a little bit different,
a little bit challenging for the students, I'd love to hear your feedback, if you'd like
to let me know what you think, but that's it for this week and I'll talk to you next
time.