Entry Task for Rational Number

1. Entry Task for Rational Number • Represent 3/5 in as many ways as possible • List concerns you see in your classroom with regards to rational numbers

2. 4 out of 3 people have trouble with fractions

3. Rational NumberWHAT WE CAN LEARN FROM STUDENT WORK Northwest Mathematics Conference Bellevue, Washington October 12, 2007 presented by Mary Holmberg Lynda Eich

4. Mathematics is: • Mathematics isa language and science of patterns • Mathematical content must be embedded in the mathematical processes • For all students to learn significant mathematics, content must be taught and assessed in meaningful situations

5. Rational Number – What we can learn from student work GOALS: • Define rational number • Analyze student work • Connect research to conceptual understanding of fractions

6. Why rational number is so difficult “Of all the topics in the school curriculum, fractions, ratios, and proportions arguably hold the distinction of being • the most protracted in terms of development, • the most difficult to teach, • the most mathematically complex, • the most cognitively challenging, • the most essential to success in higher mathematics and science…”

7. Why Student Work? • Diagnostic Tool • Common Errors • Incomplete Knowledge • Incorrect Knowledge • Formative Assessment

8. Rational Number A rational number – “fraction” is a number that can be expressed as a ratio of two integers. Different ways to interpret rational numbers: • Part- whole meaning • Quotient meaning • Ratio meaning • Operator meaning Students must understand the different interpretations of rational numbers as well as how the different interpretations interrelate

9. Process to Review Student Work • Do the problem • Analyze student work • Discussion of student work • Review research

10. Algorithm vs. Concept Explain or show how to change 3 2/5 to an improper fraction using words, numbers, and/or pictures. Explain or show how to change 17/4 to mixed number using words, numbers, and/or pictures.

11. Questions for Analyzing Student Work What did the student do? What does the student understand? (evidence) What questions would you want to ask the student to learn more about their thinking? What does the student have yet to learn? What would be your next learning goal for this student?

12. Research • Video clip • Let students develop rules - “There is absolutely no reason ever to provide a rule about multiplying the whole number by the bottom number and adding the top number. Nor should students need a rule about dividing the bottom number into the top to convert fractions to mixed numbers. These rules will readily be developed by the students but in their own words and with complete understanding. “Van de Walle p. 69 • Changing mixed numbers to improper fractions

13. Cake Servings Problem Wanda really likes cake. She decides that a serving should be of a cake. She has 4 cakes. How many servings does she have? (Schifter, Bastable, Russell, 1999, p. 69)  Explain how you got your answer using words, numbers, and/or pictures.

14. Questions for Analyzing Student Work What did the student do? What does the student understand? (evidence) What questions would you want to ask the student to learn more about their thinking? What does the student have yet to learn? What would be your next learning goal for this student?

15. Research • Delay using algorithm procedures until students are ready • Guidelines for developing computational strategies: • Begin with simple contextual tasks • Connect the meaning of fraction computation with whole -number computation • Let estimation and informal methods play a big role in development of strategies • Explore each of the operations using models

17. Ordering Fractions Brandon was give the task to order the following fractions on a number line starting with the greatest fraction: 3/8 4/7 5/10 What is the greatest fraction? Explain how you know using words, numbers, and/or pictures.

18. Questions for Analyzing Student Work What did the student do? What does the student understand? (evidence) What questions would you want to ask the student to learn more about their thinking? What does the student have yet to learn? What would be your next learning goal for this student?

19. Research • Concepts vs. rules • Develop number sense with fractions • Which fraction is greater? 4/5 or 4/9 4/7 or 5/7 3/8 or 4/10 5/3 or 5/8 2/4 or 14/28 • Conceptual thought patterns for comparison • More of the same-sized parts • Same number of parts but parts of different sizes • More and less than one-half or one whole • Distance from one-half or one whole • Equivalence • Numerical transformations

20. Field Trip Problem The school cafeteria made seventeen submarine sandwiches to share among four groups of students. Since there weren’t the same number of kids in each group, the sandwiches were divided according to the picture below. Each group shared the sandwiches equally without any leftovers. Several of the kids complained that it hadn’t been fair—that some kids got more to eat. (Fosnot, Dolk, 2002, p. 3) • Determine what portion of a sandwich each student will get in each group.

21. Questions for Analyzing Student Work What did the student do? What does the student understand? (evidence) What questions would you want to ask the student to learn more about their thinking? What does the student have yet to learn? What would be your next learning goal for this student?

22. Research Development of student’s reasoning about fractions • Making meaning by linking quotients to divided quantities • Fair Sharing/Partitioning • Starts in kindergarten and continues • Real life situation • Importance skills are developed • Exploring the mathematical properties of fractions as number

23. Rational Number – What we can learn from student work

24. Bridge to Practice • Pick an idea that came up today that you found particularly interesting. What is your current thinking about this idea? • Where are you and/or your school now with regard to this idea? • What are one or two things that you will go back and pursue to move yourself and/or your school along with this idea?

26. Resources • Cuoco, Albert A., and Curcio, Frances R., The Roles of Representation in School Mathematics, 2001 Yearbook, Reston: The National Council of Teachers of Mathematics, Inc., 2001.  • Curcio, Frances R., and Bezuk, Nadine S., Understanding Rational Numbers and Proportions, Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8, Reston: The National Council of Teachers of Mathematics, Inc., 1994.  • Fosnot, Catherine Twomey, and Dolk, Maarten, Young Mathematicians At Work, constructing Fractions, Decimals, and Percents, Portsmouth: Heinemann, 2002.  • Gregg, Jeff, and Underwood, Diana, “Measurement and Fair-Sharing Models for Dividing Fractions,” Mathematics Teaching in the Middle School, May 2007, 490-496.  • Kilpatrick, Jeremy, Swafford, Jane, and Findell, Bradford, Adding It Up, Helping Children Learn Mathematics., Washington, DC: National Academy Press, 2001. • Lester, Frank K, Jr., Second Handbook of Research on mathematics Teaching and Learning, Charlotte, NC: Information Age Publishing, Inc., 2007 • Litwiller, Bonnie, and Bright, George, Making Sense of Fractions, Ratios, and Proportions, 2007 Yearbook, Reston: The National Council of Teachers of Mathematics, Inc., 2002. • Rhynard, Karen, and Aurand, Eric, “Why Are Fractions so Difficult?” Texas Mathematics Teacher, Spring 2004, 14-19.  • Van de Walle, John A. and Lovin, LouAnn H., Teaching Student-Centered Mathematics, Grades 5-8, Boston: Pearson Education, Inc., 2006, 66 – 105.  • Watanabe, Tad, “Representations in Teaching and Learning Fractions,” Teaching Children Mathematics, April 2002, 457-463.