Understanding Fraction Representations Group 2 Session 2

1. Understanding Fraction Representations Group 2 Session 2 Amy LeHew Elementary Math Facilitator Meeting February2013

2. Cut a sentence strip so that it is 5/6 units long. Bring the ends of the sentence strip together to fold the strip of paper in half. Label this point.

3. 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.

4. 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.

5. 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. a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. 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. c. 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. d. 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.

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

7. 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)).

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

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

10. 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.”

11. 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

12. What did this look like in CMS • Lay out 2-3 examples of number line work from your school. Label them 3rd, 4th, or 5th Gallery Walk

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

14. 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

15. 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?

16. 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.

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

18. Construct a Number Line Using your blank number line sheet and Cuisenaire rods, label the lines as indicated below 1stline in Halves 2ndline in Fourths 3rdline in Sixths 5th line in Twelfths 0 1

19. 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?

20. From Area to Linear • How might this impact student’s understanding of fractions as numbers on a number line? (using the Cuisenaire rods to identify fractions on a number line) • 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

21. Building Up from a Unit Fraction • http://www.telegraph.co.uk/news/newsvideo/weirdnewsvideo/9246505/Americas-tallest-man-gets-measured-for-size-21-trainers.html

22. Inch Bricks

23. Reading Group Amy plans to build an enclosure for her great danes that is 1/8 of a mile wide and 1/4 of a mile long. Her daughter argues, “that won’t be enough room! And proves her point by showing that…

24. Carefully cut out a strip of paper that has a length of 5/6 - Bring the ends of the strip together to fold the strip of paper in half. How long is half of the strip? Use your strip to mark this point on the number line. - What two numbers can you multiply to find the length of half the strip? Write an equation to show this.

25. Unfold your paper strip so that you start with 5/6 again. Now fold the strip of paper in half and then in half again. • How long is half of half of the strip? Use your strip to mark this point on the number line. ii. What numbers can you multiply to find the length of half the strip? Write an equation to show this http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/000/000/965/original/illustrative_mathematics_965.pdf?1343856895

26. How many 1" x 1" square tiles can you fit into a 12" by 8" gridHow many 1" x 1" square tiles can you fit into 3/4 of a 12" by 8" gridHow many 1/2" x 1/2" square tiles can you fit into a 7 1/2" x 9 1/2" gridHow many 1/4" x 1/2" rectangular tiles can you fit into a 10 1/2" x 8 1/2" grid

27. 5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. • Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) • Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.