Structures of Addition & Subtraction and Multiplication & Division

Math 412 January 28, 2008

See Chapter 10, Van de Walle (2007)

Helping students to Understand

Two basic tools for developing understanding:– Word Problems– Concrete Models

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

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

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

Part

Whole

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

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

Multiplication can be:

Repeated Addition

(sets of equal amounts) Rate (jumps on a number line, speed,…)

Arrays (area models)

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Division can be:

Repeated Subtraction (sets)– The Number of Sets

Fair-Sharing– The Number in Each Set

Rate

Arrays ( the missing dimension)

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