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Understanding Fractions: Exploring Different Meanings and Representations

Learn about fractions, their different meanings, and how they can be represented using models like number lines, pattern blocks, and more. Explore the concept of fractions as parts of a whole, measures, quotients, and decimal numbers. Suitable for 5th graders and beyond.

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Understanding Fractions: Exploring Different Meanings and Representations

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  1. Content Session 2 July 8, 2009

  2. In a recent basketball game, Diana made 6 of 8 free throws in the first half. In the second half, she made 8 of 10 free throws. • What fraction of free throws did Diana make in the first half? • What fraction of free throws did she make in the second half? • What fraction of free throws did she make in the whole game?

  3. A 5th grader announces, “See, we can just add the numerators and the denominators together to add fractions!”

  4. When do you use fractions?

  5. Fractions are difficult in part because there are different meaning / interpretations of fractions. To understand fractions means, in part, to be able to freely move across these different meaning.

  6. What is a fraction? • Part of a whole • M1N4 c: Identify, label, and relate fractions (halves, fourths) as equal parts of a collection of objects or a whole using pictures and models. • M2N4 a. Model, identify, label, and compare fractions (thirds, sixths, eighths, tenths) as a representation of equal parts of a whole or of a set. b. Know that when all fractional parts are included, such as three thirds, the result is equal to the whole.

  7. What is a fraction? • Measures • M3N5 c. Understand the fraction a/b represents a equal sized parts of a whole that is divided into b equal sized parts. d. Know and use decimal fractions and common fractions to represent the size of parts created by equal divisions of a whole.

  8. What is a fraction? • Quotient • M5N4 a. Understand division of whole numbers can be represented as a fraction (a/b= a ÷ b).

  9. Another reason fractions are difficult: • A single number may be labeled with multiple fractions • Which form of fraction is appropriate depends on the context • Grade 4: being aware some fractions represent the same number • Grade 5: being able to create equivalent fractions

  10. Decimal Numbers • Another form of representing fractions with powers of 10 as the denominators • Extension of the decimal numeration system

  11. Unitary Perspective of Numbers • 30 is 3 tens • 30000 is 3 ten-thousands • 3/5 is 3 1/5-units • 3/8 is 3 1/8-units • 0.3 is 3 0.1-units • 0.003 is 3 0.001-units

  12. Representing Fractions & Decimal Numbers • Number Lines: different kinds of numbers (whole numbers, fractions, and decimal numbers) can be represented on the same model • The National Math Panel recommends an increased attention on number lines (with fractions).

  13. Show ¾ using: • Pattern blocks • Cuisenaire rods • Connecting cubes

  14. What’s wrong (if any)? • The green piece shows 1/3.

  15. Three Common Attributes • Area • Length • Counts

  16. Fraction Representations

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