1. Structures of Addition & Subtraction and Multiplication & Division Math 412 January 28, 2008 See Chapter 10, Van de Walle (2007)

2. Helping students to Understand Two basic tools for developing understanding: – Word Problems – Concrete Models

3. Join Problems Join: Result Unknown – Dylan had 3 pennies and his mother gave him 2 more pennies. How many pennies does he have in all? Join: Change Unknown – Dylan had 3 pennies and his mother gives him some more pennies. Now he has 5 pennies. How many pennies did his mother give him? Join: Initial Unknown – Dylan had some pennies and his mother gave him 2 more pennies. Now he has 5 pennies. How many pennies did he have to start? Initial Change Result

4. Separate Problems Separate: Result Unknown – Gillian had 10 shells in her bucket. She gave her brother 3 shells. How many does she have left? Separate: Change Unknown – Gillian had 10 shells in her bucket. She gave some to her brother. Now she has 7 shells left. How many did she give to her brother? Separate: Initial Unknown – Gillian had some shells in her bucket. She gave 3 to her brother. Now she has 7 shells left. How many did she start with? Initial Change Result

5. Part-Part-Whole Problems Part-Part-Whole: Whole Unknown – There were 7 boys and 6 girls at a party. How many children in all were at the party? Part-Part-Whole: Part Unknown – There were 13 children at a party. If 7 of them were boys, how many were girls? Part Whole

6. Compare Problems Compare: Difference Unknown – Angela has $35 and Kelly has $42. How much more money does Kelly have? Compare: Larger Unknown – Angela has $35. Kelly has $7 more than her. How much money does Kelly have? Compare: Smaller Unknown – Kelly has $7 more than Angela. If Kelly has $42, how much does Angela have? Larger Smaller difference

7. Structures of Multiplication and Division Equal Groups: Whole Unknown (Multiplication) Equal Groups: Size of Groups Unknown (Partition Division) Equal Groups: Number of Groups Unknown (Measurement Division) Comparison Product Unknown (Multiplication) Comparison: Set Size Unknown (Partition Division) Comparison: Multiplier Unknown (Measurement Division) Combinations: Product Unknown Combinations: Size of a Set Unknown

8. Multiplication can be: Repeated Addition (sets of equal amounts) Rate (jumps on a number line, speed,…) Arrays (area models)

9. Division can be: Repeated Subtraction (sets) – The Number of Sets Fair-Sharing – The Number in Each Set Rate Arrays ( the missing dimension)