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Proper fractions

This article explores the concepts of proper fractions, improper fractions, and mixed numbers, and discusses their meanings and operations. It also provides explanations using diagrams and encourages reasoning over algorithms. Additional practice questions are included for better comprehension.

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Proper fractions

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  1. 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. Unit Fractions Unit fractions are fractions whose numerator is 1: 1 1 1 1 1 2 7 24 100 8

  4. Mixed numbers • Meaning of

  5. 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.

  6. Write mixed number as improper fraction and vice versa

  7. Operations with fractions • Addition • Subtraction • Multiplication • Division

  8. Adding and subtracting fractions

  9. 1/2 + 1/3

  10. Multiplying fractions • Repeated addition model • Area model

  11. Multiplication of fractions • Fraction as operator • The multiplication algorithm is best explained by the area model.

  12. 2/3 of 2 1/2

  13. Mixed number times mixed number

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

  15. 11 divided by 1 1/2

  16. Multiplicative Inverses • We know that division is the inverse of multiplication.

  17. Multiplicative inverses • The multiplicative inverse of a is 1/a • The multiplicative inverse of a/b is b/a

  18. 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.

  19. Exploration 5.12 • “Drawn to scale” • Part 1 Use reasoning not algorithms to answer #1 • Part 2 Write justifications for the following: • #1: 3, 6, 8, 13, 16 • #2: 1, 2, 7, 8, 9, 13, 15, 16

  20. Extra Practice • 1. You have from 10:00 - 11:30 to do a project. At 11, what fraction of time remains? At 11:20, 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.

  21. 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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