Developing Meanings for Operations

Addition and Subtraction

Problem Structures

• Join Problems
• starting amount
• change amount
• result

Frank has 10 bananas.  Tom gave him some more. Frank now has 15 bananas.  How many bananas did Tom give Frank?

• Separate Problems
• initial amount is the whole
• separate parts

Sue had 17 dogs. She gave 6 to Sam.  How many dogs does Sue have now?

• Part-Part-Whole Problems
• two parts that are combined into one

Kevin has 6 pennies and Tony has 7 quarters.  How many coins do they have together?

• Compare Problems
• comparison of two quantities

Gina has 12 frogs and Andrea has 8 frogs.  How many more frogs does Gina have?

Multiplication and Division

Problem Structures

• Equal-Group Problems
• When the number and group size is known - multiplication
• When the number of sets or the size of sets is unknown - division

Sara has 4 bags of potatoes.  There are 10 potatoes in each bag.  How many potatoes does she have?

• Comparison Problems
• Multiple copies of the other
• One set is a multiple of the other

Tom found 24 Easter eggs.  He found 6 times as many as Lou.  How many eggs did Lou find?

• Combination Problems
• Counting the number of possible pairings

Kathy bought 10 pairs of pants and 5 shirts. All of the pieces of clothing can be worn together.  How many different outfits does Kathy have?

• Area and Other Product-of-Measures Problems
• The product is a different type of unit from the other two factors

Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7^th ed.). Boston, MA: Pearson.