Today’s Goals

1. Grade Four:Fractions and DecimalsUnderstanding Common Core Fraction Expectations In 4th Grade- Nathaniel Alexander ElementaryMonday 12/17/2012 3:30 pm -- 5:30 pmThursday 1/10/2013 3:30 pm -- 5:30 pmThursday1/31/2013 3:30 pm -- 5:30 pmUnit 6Fraction Cards and Decimal Squares

2. Today’s Goals • Honor the challenge in this work and set the tone for teachers as learners • Build conceptual knowledge of fractions, and acknowledge most of us come with procedural • Become proficient with the work in Investigation 1 • Know how and where to highlight the standards for students.

3. Let’s Start with some MATH! Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4. Solve: 5/6 x 2/3 It’s easy… • Multiply the numerators 5x2 =10 • Multiply the denominators 6x3=18 • Now you must reduce 10/18. divide 10 and 18 by 2 and you get 5/9 Draw a representation and write a short story (scenario) to go with it. Do we need to invert? Or is that with division?

5. How challenging was it to…? …Solve the problem? …Draw a picture/representation? …Write a word problem? 5/6 + 2/3 5/6 x 2/3

6. Most of us were taught to solve fraction problems using a procedure. Stolen Opportunity! What learning did that take from us?

7. Highlighting the Common Core • What do the Common Core State Standards have to say about HOW students demonstrate their understanding of fractions? • On your standards, highlight the phrase “using (a) visual fraction model(s)” everywhere you see it.

8. What makes work with fractions and decimals so difficult for students? (p. 139-140)

9. Investigation 1Fractions of an Area: Halves, Fourths, and Eighths • Halves, fourths, and eighths • Thirds and sixths • Fractions of a set In grade 4, expectations are limited to fractions with denominators….?

10. One-fourth of a sandwich • At your table… • Create 4 DIFFERENET representations of ¼ of a sandwich. (On the left side of your poster).

11. One-fourth of a Sandwich • How do you know this is ¼? • How could you PROVE it?

12. One-fourth of a Sandwich • How do you know these fourths are equal?

13. Why Bother? Common Misconception • 2.G.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. RECOGNIZE THAT EQUAL SHARES OF IDENTICAL WHOLES NEED NOT HAVE THE SAME SHAPE. “If they don’t look the same, they aren’t equal” COMMON MISCONCEPTION

14. One-fourth of a Sandwich • If the blue is ¼, then what is the white? Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

15. Page 29 • Where is the opportunity to be mindful about standards 4.NF.3 a and b?

16. One-eighth of a Sandwich • At your table… • Create 4 DIFFERENET representations of 1/8 of a sandwich.

17. How did you find Eighths?

18. Relationships are KEY! • Using fourths to find eighths

19. Fractions that are Equal

20. Thirds and Sixths Chart: Fractions That are Equal • Look at the bottom of page 34: Discussion- How are Thirds and Sixths Related? Read to the bottom of page 35. • What is the math focus for discussion? • How is the idea of equivalent fractions introduced?

21. Thirds and Sixths Chart: Fractions That are Equal • Look at the bottom of page 34: Discussion- How are Thirds and Sixths Related? Read to the bottom of page 35. If you skipped this discussion, what standard would students miss? 4.NF.1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

22. Fractional Parts of Groups • I have a crate of 24 oranges. ¼ go to Mr. Freed. The rest go to Ms. Lee. • What fraction of the oranges will Ms. Lee get? • How many oranges will Mr. Freed get? • How many oranges will Ms. Lee get?

23. Read Standard 4.NF.3d • Look at the 3 possible student responses on page 39. Which student best illustrates this standard? • Why?

24. Give a situation where… ?????? ¼ is greater than ½

25. Write fractions to show all of the parts of the rectangle = 1

26. Combinations to 1 How could students prove whether the following equation is true or false? + + + = 1 But I thought students didn’t have to add with unlike denominators! Read teacher note p. 56

27. Next Time • Bring some student examples of SAB 14

28. True or False • Students must find common denominators to add fractions. • Students in 4th grade only add and subtract with common denominators.

29. Making Fraction Cards 1/2

31. Big Picture Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.