1. Content Session 11 July 15, 2009

2. Georgia Performance Standards Instruction and assessment should include the use of manipulatives and appropriate technology. Topics should be represented in multiple ways including concrete/pictorial, verbal/written, numeric/data-based, graphical, and symbolic. Concepts should be introduced and used in the context of real world phenomena. (emphasis added)

3. Georgia Performance StandardsProcess Standards 5 Students will represent mathematics in multiple ways. • Create and use representations to organize, record, and communicate mathematical ideas. • Select, apply, and translate among mathematical representations to solve problems. • Use representations to model and interpret physical, social, and mathematical phenomena.

4. Representations • Tools for teachers to • present problems • explain ideas • demonstrate procedures • etc. • Tools for students to • organize, record and communicate ideas • solve problems • model various phenomena

5. Representations include • concrete/pictorial • verbal/written • numeric/data-based • graphical • symbolic

6. Representation “fluency” means: • students are familiar with a variety of representations - i.e., students are able to create and interpret a variety of representations • students are able to freely move across a variety of representations • students are able to select an appropriate representation

8. Visual Representations • Pictures • Diagrams • Graphs

10. For the school sports festival, two fourth grade teams are making 40 posters altogether. Team A will make 8 more posters than Team B. How many posters will each team make?

11. How might you represent this problem visually?

13. M2N2 • M2N2. Students will build fluency with multi-digit addition and subtraction. b. Understand and use the inverse relation between addition and subtraction to solve problems and check solutions.

15. M1N3 Students will add and subtract numbers less than 100, as well as understand and use the inverse relationship between addition and subtraction. h. Solve and create word problems involving addition and subtraction to 100 without regrouping. Use words, pictures and concrete models to interpret story problems and reflect the combining of sets as addition and taking away or comparing elements of sets as subtraction.

19. Can you represent this problem? • Carrie had some candies. Her friend, Kim, gave her 8 more candies. Now, Carrie has 14 candies. How many candies did Carrie have at first?

20. Can you represent this problem? Carrie had some candies. Her friend, Kim, gave her 8 more candies. Now, Carrie has 14 candies. How many candies did Carrie have at first?

21. Can you represent this problem? Carrie had some candies. Her friend, Kim, gave her 8 more candies. Now, Carrie has 14 candies. How many candies did Carrie have at first?

22. Can you represent this problem? Carrie had some candies. Her friend, Kim, gave her 8 more candies. Now, Carrie has 14 candies. How many candies did Carrie have at first?

23. Can you represent this problem? 14 = ? + 8; ? = 14 - 8 Carrie had some candies. Her friend, Kim, gave her 8 more candies. Now, Carrie has 14 candies. How many candies did Carrie have at first?

24. Can you represent this problem? • Juan has some marbles. Tony has 8 more marbles than Juan does. If Tony has 14 marbles, how many marbles does Juan have?

25. multiplication & divisiondouble number lines • There are 4 apples on each plate. If there are 6 plates, how many apples are there altogether?

26. multiplication & divisiondouble number lines • There are 4 apples on each plate. If there are 6 plates, how many apples are there altogether?

27. multiplication & divisiondouble number lines • There are 4 apples on each plate. If there are 6 plates, how many apples are there altogether?

28. There are 24 children in a classroom and 6 large round tables. How many children should be seated at each table if there must be the same number of children at each table?

29. There are 24 children in a classroom and 6 large round tables. How many children should be seated at each table if there must be the same number of children at each table?

30. There are 24 students in a class. A hexagonal table can seat 6 students. How many hexagonal tables do we need to seat all students?

31. There are 24 students in a class. A hexagonal table can seat 6 students. How many hexagonal tables do we need to seat all students?

32. Can you represent these problems? Problem 1 Paul is building a book case. If each shelf can hold 15 books and there are 5 shelves in the book case, how many books can be placed on in the book case? Problem 2 Willie has a board that is 32 feet long. If he cuts the board into 4 equal length pieces, how long will each piece be? Problem 3 Carlos bought 8 packages of gum. If each package has 6 pieces of gum, how many pieces of gum did Carlos buy altogether? Problem 4 Lynn bought 28 chocolate bars. They were in packages of 4. How many packages did Lynn buy?

33. How can you represent this problem in a diagram? Book 5B

35. How about this problem?

37. A little more challenging problem

38. A percent problem

44. Concluding Thoughts • We can help students develop representation fluency. • Developing representation fluency takes time. • We must make developing representations a specific focus of our instruction.