Proper fractions

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Proper fractions

• The value of the numerator is less than the value
  of the denominator.
• Proper in this case does not mean correct or best.

2
Improper fractions

• The value of the numerator is greater than or
  equal to the value of the denominator.

3
Mixed numbers

• Meaning of

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Writing mixed numbers as improper fractions

• The algorithm that is taught in schools obscures
  the meaning. This is true for many algorithms,
  which are efficient ways of carrying out
  operations.

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Write mixed number as improper fraction and vice
versa
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Operations with fractions

• Addition
• Subtraction
• Multiplication
• Division

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Adding and subtracting fractions
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1/2 1/3
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Multiplying fractions

• Repeated addition model
• Area model

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Multiplication of fractions

• Fraction as operator
• The multiplication algorithm is best explained by
  the area model.

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2/3 of 2 1/2
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Mixed number times mixed number
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Dividing fractions

• Division of fractions is most easily understood
  as repeated subtraction.

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11 divided by 1 1/2
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Multiplicative Inverses

• We know that division is the inverse of
  multiplication.

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Multiplicative inverses

• The multiplicative inverse of a is 1/a
• The multiplicative inverse of a/b is b/a

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Dividing fractions

• Because division is the inverse operation of
  multiplication, dividing a number by a fraction
  is equivalent to multiplying the number by the
  multiplicative inverse, called the reciprocal, of
  the fraction.

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Exploration 5.12

• Drawn to scale
• Part 1 Use reasoning not algorithms to answer 1
• Part 2 Choose a model from the list that was not
  represented in the problems and make up a story
  problem using the fraction ¾. Are there any
  models that are not possible with fractions?
  Explain.

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Operations with fractions

• Addition
• Subtraction

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Operations with fractions

• Multiplication

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Operations with fractions

• Division

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Exploration 5.13

• Begin in class and finish for homework 5.13
  Part 1 2-7 Part 2 Choose one of the models
  from the list that was not illustrated in the
  problems in Part 1 and write a story problem
  using the fraction ¾. Also, are there any models
  that are not possible with fractions? Explain.
• Homework problems from the textbook
• pp. 303-305 3b,d,e,f, 13, 21, 22, 25
• Note that in 3, you should not use algorithms
  to calculate the result use reasoning to decide
  the answer to the question.

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Extra Practice

• 1. You have from 1000 - 1130 to do a project.
  At 11, what fraction of time remains? At 1120,
  what fraction of time remains?
• Use a diagram to explain how you know. Are there
  certain diagrams that are more effective?
  Discuss this with your group.

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Extra Practice

• 2. Is 10/13 closer to 1/2 or 1?
• Use a diagram to explain how you know. Are there
  certain diagrams that are more effective?
  Discuss this with your group.

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Extra Practice

• 3. If a/b 3/4, will the value of (a x)/(b
  x) be less than, equal to, or greater than 3/4.
• Use a diagram to explain how you know. Are there
  certain diagrams that are more effective?
  Discuss this with your group.

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Exploration 5.14

• Read the directions carefully and do 1
• Discuss with your partner
• Do 2
• Discuss with your partner
• Do 3

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Homework for Wednesday

• Exploration 5.14
• Read section 5.3 in your textbook
• Do problems pp. 305-307 30, 31, 33, 36, 41,
  44a,c,d,h,k, 45a, 48