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Understanding Fraction Representations Session 2

Understanding Fraction Representations Session 2. Amy LeHew Elementary Math Facilitator Meeting October 2012. Which is Closer to 1?. 4 / 5 OR 5 / 4. 4 / 5 OR 5 / 4. Solutions?. Visual Fraction Models.

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Understanding Fraction Representations Session 2

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  1. Understanding Fraction RepresentationsSession 2 Amy LeHew Elementary Math Facilitator Meeting October 2012

  2. Which is Closer to 1? 4/5 OR 5/4

  3. 4/5 OR 5/4 Solutions?

  4. Visual Fraction Models Last time we looked closely at types of visual fraction models and considered the connection between the task and visual model used to solve it.

  5. In CMS What visual fraction model is used by most students in our schools? What visual fraction model is used the least?

  6. Numbers and Operations with Fractions Take a look at the 3rd grade fraction standards • Highlight the phrase “visual fraction model” everywhere it appears. • Underline the phrase “number line” everywhere it appears.

  7. Develop understanding of fractions as numbers. 1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

  8. Numbers and Operations with Fractions Turn and make a statement to your partner about something you learned while highlighting/underlining.

  9. Number Lines Support understanding of important properties of fractions • Numerical unit • Relationship between whole numbers and fractions • Density of rational numbers (infinite # between any two) • A number can be named infinitely many ways http://lmr.berkeley.edu/docs/Pt3-Ch13NCTM%20Yearbook07-4.pdf

  10. Area Models • Most widely used fraction model? Area • Partition pizzas, brownies, etc. • Limitations of area models • Count pieces without attending to the whole (don’t distinguish between fractional part of a set from continuous quantity (area)).

  11. How can number lines help students understand fraction concepts that are often obscured by area models?

  12. Most students were successful identifying whole numbers on a variety of number lines.

  13. Very few were successful in identifying fractions on a number line Of the students who were successful, they counted “two-sixths” for the point depicted in the figure below: it is “six spaces and that’s two.”

  14. Most had difficulty generating equivalent names for the same point • One-third is three pieces and one thing • There isn’t another fraction name • Don’t know

  15. Which is Larger? 11 15 or 9 How do you know?

  16. Always True? Sometimes True? “The smaller the denominator, the bigger the fraction”

  17. Classroom Scenario Read the classroom scenario on pages 20-22 • When you finish, reflect silently on the following: • Do student in your building consider fractions as numbers? Do teachers? • What other generalizations do students and teachers make about fractions? Are some of these generalizations helpful?

  18. Always True? Sometimes True? Consider the following two statements. Declare Always True, Sometimes True If sometimes true, show an example and a counter example. 1. The larger the denominator, the smaller the fraction. 2. Fractions are always less than one. 3. Finding a common denominator is the only way to compare fractions with different denominators.

  19. What’s the Research? Which is larger5/6 OR 7/8 ? 7/8 Since 8 is bigger than 6 and 7 is bigger than 5 5/6 The smaller the number the bigger the pieces. 5/6 The 1/6 piece is bigger than 1/8 5/6 If the denominator is smaller, the piece is bigger This one (7/8 ) because it has more pieces

  20. How do we ensure students make meaning of the numerator and denominator? • How can we make sure students think of fractions as numbers

  21. Construct a Number Line Using your blank number line sheet and Cuisenaire rods, label the first line in halves 2nd line in Thirds 3rd line in Fourths 4th line in Sixths 5th line in Twelfths 0 1

  22. Answer With A Partner What number is halfway between zero and one-half? What number is one-fourth more than one-half? What number is one-sixth less than one? What number is one-third more than one? What number is halfway between one-twelfth and three-twelfths? What would you call a number that is halfway between zero and one-twelfth?

  23. From Area to Linear • How might this impact student’s understanding of fractions as numbers on a number line? (using the Cuisenaire rods to make a number) • How does a number line diagram help students make meaning of the numerator and denominator? • This activity found on page 23-26 • Look at page 27+ for using 2 wholes

  24. The Tortoise and the Hare Time has run out for the tortoise and the hare! The Hare jumped five-eighths of the way The tortoise inched along to three-fourths of the way. Who is winning?

  25. The Tortoise and the Hare How would constructing a number line diagram to solve this task help students make meaning of the numerator and denominator?

  26. Open Number Lines With fractions? Yea or Na

  27. DO NOT ADD!!! Use < = > to compare the following sum: (do not add) *You must represent your thinking on a number line diagram. 1/2 + 1/4 ______ 1/3 + 1/5

  28. Illustrative Mathematics Project 3.NF Closest to 1/2 3.NF Comparing sums of unit fractions 3.NF Find 1 3.NF Find 2/3 3.NF Locating Fractions Greater than One on the Number Line 3.NF Locating Fractions Less than One on the Number Line 3.NF Which is Closer to 1?

  29. Next Month • Bring 3rd grade unit 7: Finding Fair Shares • Bring 4th grade unit 6: Fraction Cards and Decimal Squares

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