WEBVTT
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Intro to Fractions.
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I realize that fractions are not necessarily everyone's favorite topic.
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I hope to convince you during the course of the
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rest of this module that you can actually do fractions.
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You can make sense of this topic.
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Let's start at the beginning.
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Here's a number line with a few fractions.
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Of course most of the fractions fall between the integers.
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Notice that fractions can be positive or negative.
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Also, there are an infinite number of fractions between any two integers.
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So between, say, one and two, there's an infinity of fractions.
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We won't spend a lot of time on this idea, but it's just an
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important perspective to have on some of
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the things the test will ask about fractions.
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Notice that any negative fraction can be written
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with the negative sign in a few different places.
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For example, we could have the negative sign
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out in front of the fraction, negative one fifth.
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We could also write the negative in front of the one, so negative one divided by 5.
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We could also write 1 divided by negative 5.
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So, all three of those are perfectly equivalent and interchangeable.
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And we could switch back and forth freely between them.
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Sometimes people get really locked up, they think that, that
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negative sign has to sit in one place and can't move.
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Well no, all three of these are equivalent,
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you're free to move that negative sign around.
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So, the, the test will actually expect you to have that fluency.
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So let's talk about some terms.
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Suppose we have the fraction 3 over 16, what do we call
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the place where the three is sitting and where the 16 is sitting?
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The top part, the upstairs of a fraction is the numerator.
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This fraction has a numerator of 3.
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The bottom part, the downstairs, of a fraction is the denominator.
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This fraction has a denominator of 16.
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Two ways of thinking about a fraction, well first
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of all we can think of it as division.
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Two sevenths means two is divided by
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sevens, so there's an actual arithmetic operation occurring.
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A completely different way is pieces of pie.
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If a pie, a hypothetical pie is cut into seven
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equal pieces, then two sevenths means two of those seven pieces.
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It would mean this much of the circle.
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So notice that the first one, you could
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call it an arithmetic way of thinking about fractions.
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The second one is more a visualization.
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And it's actually very important to cultivate both of
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these, because these employ different sides of your brain.
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And if every time you look at a fraction and
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you think both of the division as well as the
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diagram, then you're going to be using both sides of
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your brain, and you're gonna be understanding fractions much more deeply.
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For fractions such as 35 over 5 it's useful to think
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about the fraction as division because that division we can actually perform.
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35 divided by 5 equals 7.
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Notice that 35 over 5, that, in every way,
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is an equivalent way to write the number 7.
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When we write 35 over 5, we are writing the number seven in kind of a hidden form.
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This leads to the idea of equivalent fractions.
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Two fractions are equivalent if they have the same numerical
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value even though they have different numerators and different denominators.
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So, two thirds equals ten fifteenths.
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Those fractions are equal.
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We are totally allowed to put an equal sign
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between them, which of course is a very powerful statement.
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We're saying that those two have exactly the
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same, mathematical value, even though they look very different.
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In fact, one way to see what's going on here,
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we can write the 10 and the 15 as products.
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2 times 5 over 3 times 5.
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In other words, the numerator and the denominator of
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the original fraction, both got multiplied by the same factor.
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In fact we can expand that idea.
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If we are given one fraction, we can always find an equivalent
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fraction by multiplying the numerator, and the denominator by the same factor.
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So for example, we start out with three eighths
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we could multiply the numerator and the denominator by 4.
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This would give us twelve over thirty two an equivalent fraction.
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Essentially, we're multiplying the fraction by 4 over 4 which equals 1.
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As we can multiple both numerator and denominator by the same factor, we can
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also factor out the same positive integer
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from both the numerator and the denominator.
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So for example, suppose we have 6 over 42, well we could multiply.
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We can express both of those as products, 6 times 1 over 6 times 7.
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And then, we can get, we can factor out
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those six, and cancel them, and get down to 1/7ths.
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The sixes cancel.
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Notice that canceling is a form of division.
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That's very important.
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Canceling is a form of division.
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Some folks have a have misconceptions about cancelling.
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Cancelling the common number in the numerator
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denominator does not merely quote go away.
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A lot of people have this sloppy way of talking about it.
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The sixes go away.
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That's not a very healthy way to think about this, it leads to mistakes.
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For example, suppose we had 8 over 40, we wanted to simplify it.
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We'll clearly, we can divide, we can cancel the eights.
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So 40 divided by 8 is 5.
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That gives us the denominator.
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Well people were stuck on the, the concept of eights go away.
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Well then they wonder what happens in the numerator.
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What if the eight goes away, what's left in that numerator?
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Instead of thinking at of it in terms of the eights go away, we cancel the eights.
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That is to say we divide by the eights, and 8 divided by 8 is 1.
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The eights don't go away.
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They divide, 8 over 8 to equal 1.
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This is a crucially important idea for anyone who
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previously has been using the go away understanding of cancelling.
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The equivalent fraction with the lowest possible integer value of
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numerator and denominator is the fraction written in lowest terms.
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So for example, we start up with 72 over 96, both of which are divisible by 6.
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We can divide that to 12 over 16.
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Both of that are divisible by 4.
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We can divide down to 3 over 4.
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We cannot divide any further.
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That fraction is in lowest terms.
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Once the fraction is in lowest terms we cannot simplify any further.
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Lowest terms is also known as simplest form.
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Make a habit of always, always, always
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writing all your fractions in simplest form.
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Well why?
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First of all it will make all your calculations
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much easier because you'll be dealing with smaller numbers.
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On the GRE multiple choice, the answer choice
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listed will almost always be in simplest form.
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So you'll have to simplify the simplest form in
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order to find something that matches the given answer choice.
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On the numerical entry, you can enter a non
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simplified fraction, as long as it fits in the box.
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Still it's a good idea to practice simplifying though, because simplifying
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along the way may make your calculation easier so there it still will be helpful.
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Practice writing these factions in lowest terms.
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Pause the video here and then we'll talk about these.
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.
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Okay, here are the answers.
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The last big fraction topic for this video is
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how to handle fractions that are greater than one.
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If we have a fraction greater than one, we have a choice about its form.
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We can write the fraction as either an improper fraction,
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that is to say a fraction in which the numerator is larger than the denominator-
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That's what it means to be an improper
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fraction, the numerator is larger than the denominator.
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Or, we can write it as a mixed numeral, that
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is to say, something that is part integer and part fraction.
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So for example, we could say five thirds equals one and two thirds.
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Five thirds is an improper fraction.
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One and two thirds is the mixed numeral of the same numerical value.
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And that's why we can put an equal sign between them.
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Notice when a mixed numeral is written correctly the
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fraction part is always a fraction less than one.
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Any part greater than one is put into the integer.
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Also notice that by convention we write the integer and the fraction parts of a
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mixed numeral right next to each other,
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but the understood operation between them is addition.
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So when we write, four and three fifths, what we actually mean, we don't
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have to write it, but what that actually means is 4 plus three fifths.
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This makes it easy to change from a mixed numeral to improper fraction form.
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Write the mixed numeral as addition and then multiply the integer by
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c over c, where c is the denominator of the fraction part.
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So, for example, we'll start with four and three fifths.
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We'll write that as 4 plus three fifths.
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Now we're going to multiple that 4 by 5 over 5, and of
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course we're allowed to multiply by 5 over 5 because that equals 1.
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And that will give us 20 over 5, plus 3 over 5, and those add to 23 over 5.
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That is the improper fraction form that is
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equal to the mixed numeral four and three fifths.
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Sometimes a test problem will give all the
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answers in either mixed numeral or improper fractions form.
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So, either one could appear on the test.
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Notice that if you have a choice, improper fractions are almost
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always more efficient for calculations of all sort, than are mixed numerals.
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Unless the problem forces you to use mixed numerals,
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you'll be better off changing everything to improper fractions.
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In summary, we talked about the basic fraction terms and how to think
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about fraction, the division way of looking at it versus the pie method.
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We talked about the important idea of equivalent fractions.
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We talked about how cancelling is a form of division,
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it doesn't mean go away, it means we're actually performing division.
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We talked about the importance of learning how to
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put fractions in lowest terms, also known as simplest form.
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And we talked a little bit about the idea of mixed numerals and improper fractions