Fractions: Getting the Whole Picture

1. Fractions: Getting the Whole Picture Fraction Hot Topic Workshop November 1, 2012 Complete “Fractions of Words” sheet

2. Fraction Understandings • What misconceptions do students have about fractions? • Why are fractions so difficult for students to understand? • What are the important fraction concepts that students need to understand?

3. Fractions Researchers have concluded that fractions is a complex topic and causes more trouble for elementary and middle school students than any other area of mathematics. Why do you think this is the case? What makes fractions difficult for students?

4. Reasons for Difficulties in Learning Fractions • Material is being taught: • too abstractly • too procedurally • outside meaningful contexts • through rote memorization of procedures • more attention on algorithms and less attention on number sense and reasoning • without connections • with limited models

5. EOG Weight Distribution

6. Fractions can make SENSE!

7. What is a FRACTION?

8. Visualizing Fractions • In your mind, picture three quarters. • What image did you create? • Create a different picture of three quarters. • What image did you create?

9. Visualizing Fractions • What image did you create? three quarters

10. Fraction Understandings • What early experiences do students have with fractions? • There’s a quarter moon tonight. • You can have half of my cookie. • It’s a quarter past one. • The recipe calls for two-thirds cup of sugar. • The dishwasher is less than half full. • I earned half a dollar.

11. Fractions in the Real World • Find examples of fractions in the real world • Illustrate or take pictures of examples • Fraction Scavenger Hunt

12. Meanings of Fractions • Values • Fractions are rational numbers that can be counted and ordered • They can represent parts of a region or a set or a point on the number line • Operators • One can find a fractional part of a value • A fraction can represent a division problem • Ratios • A comparison of two quantities

13. Fractions are NUMBERS Name an amount (quantity), a part of a specified whole Name a point on a number line An infinite amount of fractions exists between any two whole numbers Can be counted

14. Partitioning • Partitioning is KEY to understanding and generalizing concepts related to fractions

15. Partitioning in Grades 1 & 2 First Grade Geometry 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

16. Partitioning in Grades 1 & 2 Second Grade Geometry 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. 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.

17. Second Grade • Ms. Nim gave her students a picture of a rectangle. Then she asked them to shade in one half of the rectangle. Which one shows one half?

18. Second Grade • Which pictures show one half of the shape shaded?

19. Partitioning in Grade 3 Third Grade Geometry 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as ¼ of the area of the shape.

20. Relationship of a Fraction to its Whole • Fractions are defined in relation to a whole • Need to understand what the fraction is “of” • The whole can be • One object • A collection of multiple objects • A quantity

21. Models & Representations • Area/Region Models • Linear/Measurement Models • Set Models • Symbols (with meaning) Models introduced in 3rd grade Model added in 4th grade 3 71 4 8 2

22. A Fraction Represents… Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; Understand a fractiona/b as the quantity formed by a parts of size 1/b

23. From Models to Symbols • Top Number (numerator) • The counting number. It tells how many shares or parts of a certain size are being counted. Numerator -- Latin word meaning number • Bottom Number (denominator) • Tells fractional part being counted • If a 4, counting fourths. If a 6, counting sixths… • The number of equal parts into which the whole is partitioned; parts or shares of the whole Denominator -- Latin word meaning namer • How many of what type of parts how many what

24. 1 Unit Fraction • The amount formed by 1 of the parts when a whole is divided into b equal parts; 1/b of a whole

25. Two Fifths • What does the denominator represent? • 1 whole object is split into 5 equal parts • Each part is ⅕of 1 whole object • What does the numerator represent? • 2 parts of 1 whole object, where the size of each part is ⅕ ⅕ ⅕ ⅕ ⅕ ⅕

26. Unit Fractions • A unit fraction is a proper fraction with a numerator of 1 and a whole number denominator • is the unit fraction that corresponds to or to or to • As there are 3 one-inches in 3 inches, there are 3 one-eighths in

27. Unit Fractions 1 b Unit fractions are the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of whole numbers Unit fractions are formed by partitioning a whole into equal parts and naming fractional parts with unit fractions 1/3 +1/3 = 2/3 1/5 + 1/5 + 1/5 = ? We can obtain any fraction by combining a sufficient number of unit fractions

28. Unit Fractions • The numerator 3 of ¾ shows that 3 is the number you get by combining 3 of the 1/4 ’s together when the whole is divided into 4 equal parts • A fraction such as 5/3 shows combining 5 parts together when the whole is divided into 3 equal parts – best shown on a number line

29. Unit Fractions • Decompose the following fractions in as many ways as you can

31. Unit Fraction Counting • Fractional Parts Counting • Display pie-piece – tell what fraction this represents of the whole and count as a class • Example: each pie-piece is one-third What is another way we can say eight-thirds?

32. Problem Solving • Some girls were sharing some bananas so that each person got the same amount. Each girl got ¼ of a banana. How many bananas and how many girls could there have been? • How would students solve this problem? • What models/representations would they use?

33. Fractions • Third Grade • Halves, thirds, fourths, sixths, eighths • Fourth Grade • Halves, thirds, fourths, fifths, sixths, eighths, tenths, twelfths, hundredths • Fifth Grade • No limits specified

34. Naming Fractions Students need many opportunities to model and name non-unit fractions This process begins with regional representations and is extended to include number lines The advantage of regional representations is their familiar contexts (pizza slices, sections of apples, cake etc.) for students.

35. Area/Region Models • Region is cut into smaller parts • Examples: pattern blocks, grid paper, geoboards, circles, rectangles, triangles

36. REGIONS as Models for Fractions

37. Identify the Thirds Which of these regions is divided into thirds? Why or why not?

38. Identify the Fourths

39. Area Models • Geoboard Models • Divide the geoboard into half • Make halves on the geoboard and record all of your ideas on dot paper How do you know the two parts are halves? How do you know the two parts are equal? • Repeat with fourths and eighths How will the size of each piece change? • Extension: Combine fractions to create a design

40. Fractional Parts of a Whole (Region) How many ways can you show fourths on a geoboard?

41. Fractional Parts of a Whole (Region) • Does region show fourths of the square? • How do you know?

42. Fractions and Geoboards Videos: Annenberg Learner - Learner Express http://www.learner.org/vod/vod_window.html?pid=905 http://www.learner.org/series/modules/express/pages/ccmathmod_07.html

43. Fractional Parts of a Whole If the yellow hexagon represents one whole, how might you partition the whole into equal parts? Name the fractional parts with unit fractions.

44. Fractional Parts of a Whole Name the unit fractions that equal one whole hexagon

45. Fractional Parts of a Whole • Two yellow hexagons = 1 whole • How might you partition the whole into equal parts? Name the unit fraction for one triangle; one hexagon; one trapezoid and one rhombus

46. Fractional Parts of a Whole • One blue rhombus = 1 whole • What is the value of the red trapezoid, the green triangle and the yellow hexagon? • Show and explain your answer

47. Caution • Be sure to identify the whole or whole unit. The unit is not consistent and may change.

48. Want half of a candy bar?

49. Identifying Fractional Parts • What part is red? Blue? Green? Yellow?

50. Area Models • Pattern Blocks • Extensions: • Create a pattern block design. Assign one block the value of 1 and find the value of your entire picture. • Change the whole. How does this change the value of your design? • Challenge students to create a design with a predetermined value. (Ex. – If the hexagon is 1, create a design with a value of 24 1/3.)