Fractions: Computations

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Fractions Computations Operations

• Teaching for Conceptual Understanding

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Introduction

• Represent the following operation using each of
  the representations
• Describe what the solution means in terms of each
  representation you use.
• Pictures
• Manipulatives
• Real World Situations
• Symbolic
• Oral/Written Language

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Fractional Parts Counting
5 Fourths
3 Fourths
10 Fourths
10 Twelfths
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Discussion

• What does the bottom number in a fraction tell
  us?
• What does the top number in a fraction tell us?

Misleading notion of fractions Top number tells
how many. Bottom number tells how many parts
to make a whole.
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For example

• If a pizza is cut in 12 pieces, 2 pieces makes
  1/6 of the pizza
• Here, the bottom number does not tell us how many
  parts make up the whole!

• Better Idea!
• We can assume the top number counts while the
  bottom number tells what is being counted.
• ¾ is a count of three things called fourths

Using this notion, what is another way to
say/write Thirteen sixths? Explain your rationale.
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Activity

• Using the manipulatives, your task is to find a
  single fraction that names the same amount as
• must be able to provide an explanation for your
  result.
• Then, determine the mixed number for 17/4 and
  provide a justification for your result.

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Parts Whole Tasks
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If 12 counters are ¾ of a set, how many counters
are in the full set?
If 10 counters are five-halves of a set, how many
counters are in one set? (What must be half of
one set?)
If purple is 1/3, what strip is the whole? If
dark green is 2/3, what strip is the whole? If
yellow is 5/4, what strip is 1 whole?
Purple
Dark Green
Yellow
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Dark Green
If the dark green is the whole, what fraction is
the yellow strip?
Yellow
Dark Green
If the dark green strip is one whole, what
fraction is the blue strip?
Blue
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Getting to Conceptual Understanding

• A father has left his three sons 35 camels to
  divide among them in this way
• One-half to one brother, one-third to another
  brother, and one-ninth to the third brother.
• How many camels does each brother receive?
  Explain your solution to this problem.

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Benchmarks of Zero, One-Half, and One
Sort the following fractions into three groups
close to zero, close to ½, and close to one.
Provide rationale and explanation for your
choices.
Close Fractions Name a fraction that is close to
one but not more than one. Name another fraction
that is even closer to one. Explain why you
believe this fraction is even closer to one than
the first. Show using manipulatives.
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Partner/ Group Activity
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Exploring Ordering Fractions lt or gt

• Task Which fraction in each pair is greater?
  Give an explanation for why you think so?
• What are some of the usual approaches that
  students would choose to use in this activity?
  Why do you think this is?

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Conceptual Thought Patterns

• More of the same size parts to compare 3/8 and
  5/8
• - students will choose 5/8 as larger because 5gt3
  right choice, wrong reason
• - comparing 3/8 and 5/8 should be like comparing
  3 apples and 5 apples

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• Same number of parts, but parts of different
  sizes
• to compare ¾ and 3/7
• Misconception students will choose 3/7 as the
  larger because 7 is more than 4 and the top
  numbers are the same
• However, if a whole is divided into 7 parts, the
  parts will be smaller than if divided into only 4
  parts thus, ¾ is larger
• Like comparing 3 apples with 3 melons same
  number of things but melons are bigger

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• More or less than one-half or one whole
• comparing 3/7 and 5/8
• 3/7 is less than half of the number of sevenths
  needed to make a whole, so 3/7 is less than
  one-half
• thus, 5/8 is more than one-half and is therefore
  the larger fraction

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• Distance from one-half or one whole
• compare 9/10 and ¾
• Misconception 9/10 is bigger because 9 and 10
  are the bigger numbers
• 9/10 is larger than ¾ because although each is
  one fractional part away from a whole, tenths are
  smaller than fourths and so 9/10 is closer to one
  whole.

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Addition and Subtraction Explorations

• Paul and his brother were each eating the same
  kind of candy bar. Paul had ¾ of his candy bar.
  His brother had 7/8 of a candy bar. How much
  candy did the two boys have together?
• Using drawings /or manipulatives, solve this
  problem without setting it up in the usual manner
  and finding common denominators?
• Can you think of two different methods?

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One possible Solution
We could take a fourth from the 7/8 and add it to
the ¾ to make a whole. That would leave 5/8.
Thus, the total eaten would be 1 5/8 of candy bar.
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Using Cuisenaire Rods for Fraction Computation

• Jack and Jill ordered two identical sized pizzas,
  one cheese and one pepperoni. Jack ate 5/6 of a
  pizza and Jill ate ½ of a pizza. How much pizza
  did they eat together?
• What would we expect our students to show that
  would demonstrate their conceptual understanding?

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Solution

• Find a strip for the whole that allows both
  fractions to be modeled.

Dark Green
Whole
Yellow
5/6
Light Green
1/2
Dark Green
Yellow
Light Green
Red
A Red is 1/3 of the Dark Green, so the solution
is 1 1/3.
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Multiplication of Fractions Beginning Concepts

• Using visual representations/manipulatives, model
  the following computation
• There are 15 cars in Michaels toy car
  collection. Two-thirds of the cars are red. How
  many red cars does Michael have?

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• Using visual representations/manipulatives, now
  model these following problems
• You have ¾ of a pizza left. If you give 1/3 of
  the left-over pizza to your brother, how much of
  the whole pizza will your brother get?
• Someone ate 1/10 of the cake, leaving only 9/10.
  If you eat 2/3 of the cake that is left, how much
  of the whole cake have you eaten?
• Gloria used 2 ½ tubes of blue paint to paint the
  sky in her picture. Each tube holds 4/5 ounce of
  paint. How many ounces of blue paint did Gloria
  use?

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When pieces must be subdivided into smaller unit
parts, the problems become more challenging

• Zack had 2/3 of the lawn left to cut. After lunch
  he cut ¾ of the grass that he had left. How much
  of the whole lawn did he cut after lunch?
• Bill drank 1/5 of his pop before lunch. After
  lunch he drank 2/3 of what was left. How much pop
  did he drink after lunch?

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Division of Fractions Beginning Concepts

• Think about the following problem
• Cassie has 5 ¼ yards of ribbon to make three
  bows for birthday packages. How much ribbon
  should she use for each bow if she wants to use
  the same length of ribbon for each?
• What types of solutions would we anticipate our
  students to come up with?
• How could we model the solution using
  manipulatives/multiple representations?
• How many different ways?

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• In the following problem, the parts must be split
  into smaller parts
• Mark has 1 ¼ hours to finish his three
  household chores. If he divides his time evenly,
  how many hours can he give to each?

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Questions/Discussion

• Inverse vs. Reciprocal
• 4/5 - representations
• Think of the many forms that even the symbolic
  can be represented decimals, rates, ratios,
  etc.
• Other? More???

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• Applying this Reasoning
• Solving Problems

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• A A student is sorting into stacks a room full
  of food donated by the school for the local food
  bank. He sorted of it before lunch and then
  sorted of the remainder before school ended.
  What part (fraction) of all the food will be left
  for him to sort after school?

1 3
3 4
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• B Use manipulatives to solve this problem.
  Mark ate half of the candies in a bag. Leila ate
  2/3 of what was left. Now there are 11 candies in
  the bag. How many were in the bag at the start?
• C Bills Snow Plow can plow the snow off the
  schools parking lot in 4 hours. Janes plowing
  company can plow the same parking lot in just 3
  hours. How long would it take Bill and Jane to
  plow the schools parking lot together?

Think of the math content involved with this
problem Think of some Before activities that
could be used. What would the debrief look/sound
like in the classroom after the task was complete?
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Teaching Through Problem Solving

• D Mrs. Get Fit teaches Math and Phys-ed. To
  incorporate Math into the Phys-ed class, she
  divided the class into eight groups. There are
  three students in each group. The first person in
  the group runs ¼ of a lap of the track, the
  second person runs 1/6 of the track, and the
  third person runs 1/3 of a lap of the track. How
  many laps of the track are run in total by all
  eight teams combined?