1. Fractions 3-6 Central Maine Inclusive Schools October 18, 2007 Jim Cook

2. Workshop Goals  What should students know and be able to do?  What common difficulties do students have with fractions?  What instructional practices can help students understand fractions?

3. Understanding the meaning of fractions  Too much school instruction on fractions is with computation. When instruction focuses on the meaning of fractions, it is often too brief and superficial. As a result, students are forced to learn rules and procedures for computations without a sound understanding of what they’re operating on.

4. What are appropriate goals for fraction instruction?  Name fractional parts of regions and sets  Find fractions on a number line  Represent fractional parts using standard notation (proper and improper fractions, mixed numbers) and also with concrete and pictorial representations  Understand equivalence  Compare and order fractions  Make reasonable estimates with fractions  Compute with fractions  Solve problems with fractions

5. MLR goals for students  1997 MLR Grade 3 Read, write model, and compare simple fractions with denominators of 2, 3, 4  2007 MLR Grade 3 Students recognize, name, illustrate, and use simple fractions  Recognize, name, and illustrate fractions with denominators from two to ten  Recognize, name, and illustrate parts of a whole  Compare and order fractions with like numerators or with like denominators

6. MLR goals for students  1997 MLR Grade 4 Read, compare, order, classify, and explain simple fractions through tenths Solve real-life problems involving addition and subtraction of simple fractions  2007 MLR Grade 4 Students understand, name, illustrate, combine, and use fractions  Add and subtract fractions with like denominators and use repeated addition to multiply a unit fraction by a whole number  List equivalent fractions  Represent fractions greater than one as mixed numbers and mixed numbers as fractions  Connect equivalent decimals and fractions for tenths, fourths, and halves in meaningful contexts

7. MLR goals for students  1997 MLR Grade 5 Read, compare, order, use, and represent simple fractions (halves, thirds, fourths, fifths, and tenths with all numerators) Compute and model addition and subtraction with simple fractions with common denominators Create, solve, and justify the solution for multi- step, real-life problems involving addition and subtraction with simple fractions with common denominators

8. MLR goals for students  2007 MLR Grade 5 Students understand, name, compare, illustrate, compute with, and use fractions  Add and subtract fractions with like and unlike denominators  Multiply a fraction by a whole number  Develop the concept of a fraction as division through expressing fractions with denominators of two, four, five, and 10 as decimals and decimals as fractions

9. MLR goals for students  1997 MLR Grade 6 Read, compare, order, use and represent fractions (halves, thirds, fourths, fifths, sixths, eighths and tenths with all numerators) Compute and model all four operations with common fractions Create, solve, and justify the solution for multi-step, real-life problems with common fractions

10. MLR goals for students  2007 MLR Grade 6 Students add, subtract, multiply, and divide numbers expressed as fractions, including mixed numbers Students understand how to express relative quantities as percentages and as decimals and fractions  Use ratios to describe relationships between quantities  Use decimals, fractions, and percentages to express relative quantities  Interpret relative quantities expressed as decimals, fractions, and percentages

11. Developing the meaning of fractions  Fractions have four basic interpretations Measure  Part of a region  Part of a set  Location on a number line Quotient  1/3 is what you get when you divide 1 pizza between three people Ratio  The ration of pizzas to people is 1 to 3, 1:3, 1/3 Operator  There are 1/3 as many pizzas as people

12. Developing the meaning of fractions  Students should understand all 4 interpretations and how they are interrelated. Present fractions using all four interpretations.  Using manipulatives is important  Understanding fractions as parts of regions may be easiest

13. Fractions as parts of regions  NCTM lesson: Fun with Fractions  http://illuminations.nctm.org/Lesson Detail.aspx?id=U113

14. Wipe-Out  Version I  Version II

15. Set model—fractions as parts of sets  Get six green triangles Put them into two equal groups Put them into three equal groups  Fraction Line-Up

16. Number Line model—fractions as locations on a number line  Find the number line master in your packet

17. Fraction Representations  Using a variety of representations and asking students to switch between them enhances understanding. Real objects Manipulatives  Fraction circles  Fraction rectangles  Pattern blocks Drawings Words Symbols

18. Fraction Representations  Ask students to make connections “How are these different representations alike?”

19. Fraction Representations  Ask students to consider negative examples. “Why is this not 1/3?” “Why is this not ¼?”

20. Fraction Representations  Have students generate their own fractional parts Make a fraction kit.

21. Fraction Kit Activities  Cover Up  Uncover Version I Version II  What’s missing  Comparing pairs

22. Fraction Equivalence  Students should have strong conceptual understanding of equivalent fractions based on lots of experience. They can then relate that understanding to numerical methods for generating equivalent fractions. Simplifying fractions Generating fractions with common denominators

23. Comparing and Ordering Fractions  Strong concepts of the relative size of fractions, based on experience with physical objects and drawings, supports students’ number sense.  Estimating with fractions depends on ideas about the relative size of fractions.  Without sufficient experience with physical objects, students make errors, often using whole-number thinking when working with fractions.

24. Comparing and Ordering Fractions  Students should consider these cases: Same denominator Same numerator Fractions with different numerators and denominators  Students might relate fractions to a benchmark like ½.

25. Fractions as Quotients  Try sharing 12 cookies between 4 people.  Try sharing 3 cookies between 4 people. Use the cookie masters in your materials. Students have more success making thirds, fifths, etc. if they use toothpics.  Try sharing 7 cookies between 3 people. Help students connect mixed numbers and improper fractions.

26. Lesson Ideas for mixed numbers  Ask students to share cookies between different numbers of people. Use cookie cutouts and glue. Don’t use a numbers that are multiples. Do several examples.  Share between three, four, and six people.

27. Operating on Fractions  NCTM recommends using simple denominators that can be visualized concretely or pictorially and are apt to occur in real-world settings.  Emphasis in instruction must shift from learning rules for operations to understanding fraction concepts. Begin by asking students to use fraction pieces to add fractions.

28. Adding Fractions  Pick 2 Make a train with two pieces on your whole strip that are not the same color. Build another train the same length using pieces that are all the same color. Record. Try to build other one-color trains the same length. For each, record.

29. Adding Fractions  Help students make the connection between the procedures for adding fractions and their experience with manipulatives. They might even invent rules for adding fractions!

30. Multiplying Fractions  Use physical objects and drawings to develop meaning.  Use whole number meaning for multiplication. Repeated addition Use the commutative property and “of”  Use rectangles  Help students develop the rules for themselves

31. Dividing Fractions  Relate dividing fractions to dividing whole numbers 6 ÷ 2 = 3  “How many times can I subtract 2 from 6?”  “How many 2’s are in 6?”  Check by multiplying  ÷ means “into groups of.” 6 ÷ ½ =

32. Dividing Fractions  Use fraction pieces ¾ ÷ ½ = “how many ½’s are in ¾?”

33. Dividing Fractions  True or False?  “You can multiply the dividend and divisor both by the same number, and the answer stays the same.”  “If you divide a number by one, you get the same number.”