Dividing fractions

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Dividing fractions

• A presentation by Mr. Firester for his classes

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What does fractional division mean?

• Division means something goes into something else
• Think of whole numbers. How many times does 2 go
  into 8?
• We can think of groupings of 2. How many
  groupings of 2 are there in 8?

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Lets test this out

• Use your color tiles to set up groups to solve
  the following problem 7 divided by 3
• Draw a picture of your groupings
• What did you do with the remainder?

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How to handle the remainder

• Your groupings are 3 each.
• You only have one color tile remaining.
• What part of the original grouping is the 1 tile?
• Thats right. The extra tile is 1/3 of the
  grouping.
• Therefore there are 2 1/3 groupings of 3 in 7.

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Extending the model to fractions

• Now suppose that each of the color tiles
  represented a quarter.
• Suppose I had 7/4. How many times would 3/4 go
  into 7/4?
• If you are doing this with color tiles, you are
  probably thinking that making the groupings is
  the same as repeatedly subtracting 3 tiles.

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What is division?

• Division can be thought of as a process of
  grouping
• It can also be thought of as a process of
  repeated subtraction
• Division is the opposite of multiplication
• If multiplication is repeated addition, then
  division must be repeated subtraction

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Now lets consider the following example
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Experiment for yourself

• Use your color tiles.
• Set up groupings of 4ths. Youll need two full
  groups and then 3 tiles by themselves.
• Since ½ is the same as 2/4, start to regroup or
  take away groups of 2/4.
• How many groupings do you have?
• What will you do with the remainder?
• Draw a picture of what you did.

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Your drawing might look like this
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• So far we have learned that division is repeated
  subtraction or regrouping.
• We have used color tiles to demonstrate division
  examples.
• We can draw a picture with groupings that have
  been circled.
• In the case of dividing by ½, we decided to
  change the ½ to 2/4 so that we were looking at
  the same denominators and same size pieces.

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Now its your turn.

• You will work on the following examples.
• For each example, try to use your color tiles to
  re-enact the problem.
• Draw a picture to show what you did.
• See if you can come up with a rule that explains
  how to do division by mixed numbers.

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Closing- Some rules for dividing by fractions or
mixed numbers

• You should begin by changing the mixed numbers to
  improper fractions.
• If the denominators are the same, just divide the
  numerators. Any remainder is a fraction
  represented by the remainder over the
  denominator.
• If the denominators are not the same, them
  convert them to common denominators.

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Before you begin, understand

• In division, the first number being divided is
  the numerator. The second number after the
  division symbol is the denominator.
• Any division example can be represented as a
  complex fraction where the fraction bar means
  division.
• Ask yourself, how many times must I take away the
  denominator from the numerator.
• In cases where the numerator is greater than the
  denominator, the answer will be a proper
  fraction.

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Lets compare this methodology to the algorithm
you have learned. Here are the steps

• Change both mixed numbers to improper fractions.
• Invert the denominator (second number).
• Multiply the tops (numerators). Multiply the
  bottoms (denominators).
• Reduce to simplest terms (or cross cancel before
  multiplying).
• Change back to a mixed number.

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How does these compare?

• The new rules uses a common denominator to reduce
  the fraction you get.
• In the original algorithm (set of rules) you
  didnt know why you had to invert and multiply.
• In the new method, you never multiply.
• The new method uses common denominators to
  simplify the problem.