Why Fractions?

1. Why Fractions? • “Understanding fractions is one of the most important outcomes of mathematics education because of the pervasive use of fractions throughout mathematics.” Dr. Randall Charles (primary author of enVisionMATH) in Math Across The Grades, 2011

2. The National Mathematics Advisory Panel (NMAP) (2008) • Offers recommendations for how we can best prepare elementary and middle school students for success in algebra, a gateway to mathematics in high school and beyond. http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf

3. Why is this an issue? • Only 24% of age 13 and 17 students identified 2 as the estimated sum for 12/13 + 7/8, with a greater percentage identifying 19 or 21 as the estimated sum (NAEP, 1978). • Only 50% of 8th grade students successful in ordering 2/7, 1/12, and 5/9 from least to greatest (NAEP, 2004). • Only 29% of age 17-year old students translating 0.029 as 29/1000 (NAEP, 2004).

4. Understandings - NOT • Not viewing fractions as numbers at all, but rather as meaningless symbols that need to be manipulated in arbitrary ways to produce answers that satisfy a teacher. • Focusing on numerators and denominators as separate numbers rather than thinking of the fraction as a single number. • Confusing properties of fractions with those of whole numbers

5. Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. – Use equal-sharing activities to introduce the concept of fractions. Use sharing activities that involve dividing sets of objects as well as single whole objects. – Extend equal-sharing activities to develop students’ understanding of ordering and equivalence of fractions. – Build on students’ informal understanding to develop more advanced understanding of proportional reasoning concepts. Begin with activities that involve similar proportions, and progress to activities that involve ordering different proportions.

6. An early example: If we cut this cake so that you and two friends could share it, what would the slices look like? How can we talk about and write how much of the cake you will each get?

7. Just this once… • Would you rather share your favorite pizza with 3 other people or with 7 other people?

8. How can you share 8 cookies with the four children in your family? Can you make a drawing to show me how you would do this? • What if you had 10 cookies, now how would you share them with the 4 children in your family? • If you had 13 cookies to share among 4 friends, how many cookies would each person get? Would it be more or less than if you shared 12 cookies.

9. CCSS Grade 1 – Geometry • Partition circles and rectangles into two and four equal shares, describe halves, fourths, Grade 2 - Geometry • Partition circles and rectangles into two, three, or four equal shares, describe halves, thirds, fourths, …. Describe the whole as two halves, three thirds, four fourths. Recognize that equal shares need not have the same shape.

10. Developing understanding of fractions, especially unit fractions Grade 3 • Developing understanding of fractions, especially unit fractions Grade 4 • Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers Grade 5 • Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions) Grade 6 • Connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; • Completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers

11. 3.NF.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. Today’s CCSS Focus Burns, Pasvogel

12. ½ of 20 = 10 ½ of 100 = 50 What does ½ traditionally look like? Burns, Pasvogel

13. Now what does ½ look like on a number line? Burns, Pasvogel

14. What number is halfway between 0 and 1? Some students may initially be surprised that there are numbers between 0 and 1. Burns, Pasvogel

15. What number is halfway between 0 and one-half? What other ways might you see one-half expressed? What number is one-fourth more than one-half? One-sixth 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? Which number is closest to 0? Which number is closest to 1? What would you call a number halfway between 0 and one-twelfth? Questions to help students reason about fractions as numbers Burns, Pasvogel

16. Realizing that ¼ lies between 0 and ½ on the number line reinforces the relationship between halves and fourths. What number is halfway between 0 and ½? Burns, Pasvogel

17. Students may initially say there are several numbers here: 2/4, 3/6, and 6/12. This is an excellent opportunity to introduce the idea that although these look like different numbers, they are actually different ways to name the number, much like “one hundred” can also be called “ten tens.” This is also an opportunity to discuss hat names for the same number have in common. What other ways might you see ½ expressed? Burns, Pasvogel

18. This question can help students begin to realize about relative value of different fractions and compute without the need or converting to numbers with like units (common denominators). What number is ¼ more than ½? 1/6 more than ½? Burns, Pasvogel

19. This question encourages students to compare fractions to the unit. What number is 1/6 less than 1? Burns, Pasvogel

20. This question exposes students to fractions greater than one and can support their understanding that 4/3 is the same as 1 1/3. What number is 1/3 more than 1? Burns, Pasvogel

21. This question provides another chance for students to encounter equivalents. They can also begin to represent why there is no sixth equivalent to 1/12 or 3/12 (or 5/12, 7/12, 9/12, 11/12) What number is halfway between 1/12 and 3/12? Burns, Pasvogel

22. This provides another example of when “the larger denominator, the smaller fraction” is true. Which number is closest to 0? Burns, Pasvogel

23. This can help students see that knowing both a numerator and a denominator is necessary to understanding a fraction’s value. It can also provide a very reliable and frequently sufficient way to compare fractions, without needing to find common denominators and create equivalent fractions. Which number is closest to 1? Burns, Pasvogel

24. This question asks students to extend their understanding and provides a foundation for helping them reason about fraction multiplication, that is, why does ½ x 1/12 =1/24? What would you call a number halfway between 0 and 1/12? Burns, Pasvogel

25. The concept of the unit fraction is the quantity you get when you divide a whole into b equal parts. The unit fraction is written 1/b The quantity b is derived from how many equal partitions make the whole Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. (pg 26 of Iowa Common Core. www.corecurriculum.ioaw.gov) The BIG Ideas of A Unit Fraction Burns, Pasvogel

26. Burns, Pasvogel

27. 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. Introducing 3.NF.2b Burns, Pasvogel

28. Explain your answer. 5/6 or 7/8 Think, Ink, Pair, Share Indentify the larger fraction Burns, Pasvogel

29. 5/6 or 7/8 “I know that 5/6 is larger than 7/8 because sixths are bigger than eighths. The smaller denominator means the fraction is larger.” What do you think of this explanation?What important idea did this student use to solve the problem?Does this reasoning make sense? Why or why not? Circle the large fraction and explain your answer. Burns, Pasvogel

30. 3/4 or 5/12 “Five is more pieces than 3 pieces so 5/12 is more than ¾.” What do you think of Sarah’s explanation?What important idea about fractions did this student use to solve the problem?Does this student’s reasoning make sense to you? Why or why not? Circle the large fraction and explain your answer. Burns, Pasvogel

31. The smaller the denominator, the larger the fraction. The larger the denominator, the smaller the fraction. You can’t compare fractions with different denominators. Fractions are always less than 1. To compare two fractions, you only need to look at the numerators (or denominators). Finding a common denominator is the only way to compare fractions with different denominators. Exploring Misconceptions Burns, Pasvogel

32. Listing the language to use Listing the language not to use in the “no” circle Using the Language of the Standard Burns, Pasvogel

33. Fraction Track from Illuminations Site Burns, Pasvogel

34. Our Ultimate Goal … …to develop mathematically proficient students. Burns, Pasvogel

35. http://www.nctm.org/uploadedFiles/Math_Standards/Principles_and_Standards_for_School_Mathematics/E-Examples/Grades_3–5/fraction1_SQTM.movhttp://www.nctm.org/uploadedFiles/Math_Standards/Principles_and_Standards_for_School_Mathematics/E-Examples/Grades_3–5/fraction1_SQTM.mov Author's Name IM&E CCSSM National PD

36. Author's Name IM&E CCSSM National PD