TIPM3 Grades 4-5

1. TIPM3Grades 4-5 April 12, 1011

2. Agenda Announcements, College Credit Analyzing Student Work Multiplication of Fractions Fraction Games

3. Analyzing Student Work Solve the problem yourself and compare your solution with the people at your table. Decide on what you will accept as a correct answer. Will you give partial credit? Design a rubric. Does the student draw a picture to represent the solution? Does the student write an equation? If the answer is partially correct or incorrect, can you determine what misconception the student may have? What instructional strategies could you try to help the student?

4. Teachers’ MKTMathematical Knowledge for Teaching What is the difference between knowing mathematics and knowing how to teach mathematics? Dan Meyers talks about MKT

5. Teachers’ MKTMathematical Knowledge for Teaching Do CPS Quiz

6. Mathematical Knowledge Pedagogical Content Knowledge KCS – Knowledge of Content and Students What teachers know about their students and the content KCT – Knowledge of Content and Teaching Which representations that are clearest for kids Subject Matter Knowledge • CCK – Common Content Knowledge • Knowledge held commonly by others, not special to teachers • SCK - Specialized Content Knowledge • Special knowledge i.e. radicals

7. Teachers’ MKTMathematical Knowledge for Teaching What specific issues of mathematical language language rise? Where does the teacher slow down and spend time and why might that be? What issues are not emphasized or taken up, and why might that be? What issue related to children’s understanding arise in the development of the definition of a fraction? How are they managed?

8. Multiplying Fractions Algorithms for multiplication of common fractions are deceptively easy for teachers to teach and for children to use, but their meanings are elusive. Children who are taught rules for performing computation with these algorithms can multiply fractional numbers with ease. However, if they compute by rules alone, they will understand little of the meanings behind the computation. Wu, Zhijun. (2001) “Mutliplying Fractions.” Teaching Children Mathematics (November, 2110): 174-177.

9. Multiplying Fractions Multiplication of fractions challenges students to examine many of their ideas that they have developed about multiplication from their work with whole numbers. The challenge is not on computation, but rather is one of conceptualization. Wu, Zhijun. (2001) “Mutliplying Fractions.” Teaching Children Mathematics (November, 2110): 174-177.

10. Multiplying Fractions In early years, multiplication with whole numbers is taught as repeated addition. If this is the only link between multiplication and addition, students’ concept of multiplication will be limited. Some real life situations represented by multiplication with mixed numbers, common fractions or decimal fractions cannot easily be interpreted with the repeated addition model.

11. Multiplying Fractions Graeber, Anna O., and Patricia F. Campbell. “Misconceptions about Multiplication and Division.”Arithmetic Teacher 40 (March 1993): 408–11. Students’ development of the meaning of multiplication with fractional numbers should emerge from experience with genuine problems. (Graeber andCampbell, 1993) Allow students to explore situations and make conjectures. (NCTM, 2000) Problem situations should be appropriately desi9gned to provide context in which students can solidify their existing knowledge, extend what they know, and further develop generalized ideas about the operation. (Wu, 2001)

12. Mathematical Tasks Should: Build on students’ prior knowledge Scaffold students’ thinking Provide an appropriate amount of time Model high-level performance Sustain pressure for explanation and meaning

13. A Mathematical Task For You A cake recipe calls for 2/3 cup of flour. How much flour is needed for 4 cakes? Ask yourself: • What does the problem seem to involve? • What is your reaction to it? • Do you have a sense about the kind of solution it might have? • Does anyone have a question for the whole group? Draw at least one and if possible, two models for this problem.

14. Another Task for You Eva takes 4 pints of water for a hike up a mountain trail and back. She thinks she will need to drink 2/3 of the water on the way up. How many pints does Eva think she will drink on the way up? Ask yourself: What does the problem seem to involve? What is your reaction to it? Do you have a sense about the kind of solution it might have? Does anyone have a question for the whole group? Draw at least one and if possible, two models for this problem.

15. Building an Understanding of and Fluency with Multiplication and Division of Fractions • What are the differences and similarities in the previous two problems? • Help students see the relationships of multiplication of fractions and decimals with multiplication of whole numbers. • Help students distinguish between situations associated with multiplication of fractions and decimals with those associated with division of fractions and decimals.

16. Building an Understanding of and Fluency with Multiplication and Division of Fractions Commutative Property of Multiplication 4 x 2/3 and 2/3 x 4 have the same value. These expressions have different interpretations 4 x 2/3 can be interpreted as 4 groups of 2/3 or 4 times as many, 4 times as far, 4 times as heavy, etc.

17. Building an Understanding of and Fluency with Multiplication and Division of Fractions In contrast, 2/3 x 4 can be interpreted as 2/3 of a group of 4 Or 2/3 times 4 The chart summarizes important aspects

20. A Possible Answer 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1 whole I used fraction bars to show 2 thirds for the amount of flour in each cake. Then I made 3 more groups of 2 thirds bars to have a total of 4 groups of 2 thirds. 1/3 1/3 1 whole 1 whole Then I used whole bars to see that 8 thirds is equal to two wholes and 2 thirds, or 2 and 2/3. So 2 2/3 cups of flour are needed for the whole cake.

21. Another Possible Answer I drew a rectangle on grid paper to represent one whole. I divided it into 3 equal parts to represent thirds and I shaded 2 parts to represent 2/3.

22. Another Possible Answer I repeated the diagram until I had 4 groups of 2/3.

23. Another Possible Answer I repeated the diagram until I had 4 groups of 2/3.

24. Anne’s Work I repeated the diagram until I had 4 groups of 2/3.

25. Another Possible Answer There are 3 thirds in one whole, so I outlined groups of three to show wholes. There were two wholes and two thirds left over so 4 x 2/3 = 2 wholes and 2 thirds left over.

26. Reflect Can the previous two models easily be used to model multiplication problems involving two fractions or mixed numbers? What are the limitations of these models? These two models bridge what is known about whole numbers to fractions. What model might be able to be applied to a greater variety of problem types and more readily lead students to generalizable symbolic representation?

27. Multiplying Fractions van Essen, Gerard, and Christiaan Hamaker. “Using Self-Generated Drawings to Solve Arithmetic Word Problems.”Journal of Educational Research 83 (July/August 1990): 301–12 Some mathematical concepts are abstract concepts that have no single, clear, concrete referent. Students must encounter and reflect on these at an abstract level. (Wu, 2001) Drawing a picture can be an intermediate step between a mental representation and a physical representation. (Gerard and Hamkaker, 1990)

28. Try This Before the new semester, all the notebooks at the local store are discounted by ¼. A notebook originally costs$o.96. How much do you save on one notebook if you buy it today? Ask yourself: What does the problem seem to involve? What is your reaction to it? Do you have a sense about the kind of solution it might have? Does anyone have a question for the whole group?

29. A Solution $0.24 $0.24 $0.24 The bar represents the original cost of the notebook, $0.96. The whole is $0.96 and ¼ is discounted, so the whole is partitioned into four equal-sized sections. When $0.96 is subdivided into four equal parts, each part is $0.24 . $0.96 ÷ 4 = $0.24 Before the new semester, all the notebooks at the local store are discounted by ¼. A notebook originally costs$o.96. How much do you save on one notebook if you buy it today?

30. Do you multiply or divide? $0.24 $0.24 $0.24 Is the number sentence ¼ x 0.96 or 0.96 ÷ ¼? Before the new semester, all the notebooks at the local store are discounted by ¼. A notebook originally costs$o.96. How much do you save on one notebook if you buy it today?

31. Try This • Ask yourself: • What does the problem seem to involve? • What is your reaction to it? • Do you have a sense about the kind of solution it might have? • Does anyone have a question for the whole group? Julie bought 4/5 of a yard of material for her class project. Later, she found that she only needed ¾ of of the material. How much material did Julie use for her project?

32. Julie’s Project Julie bought 4/5 of a yard of material for her class project. Later, she found that she only needed ¾ of of the material. How much material did Julie use for her project?

33. Julie’s Project Julie bought 4/5 of a yard of material for her class project. Later, she found that she only needed ¾ of of the material. How much material did Julie use for her project? 3/4 of 4/5 = 3/5

34. Try This • Draw a grid model for 2/3 x 3/5 and 2 and 2/3 x 3 and 1/2