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*UNDERSTANDING* Arithmetic Word Problems

Traditionally, students were taught “key” words to solve arithmetic word problems: words and phrases like “altogether” and “in all” for addition problems and “left” and “less than” for subtraction problems. An immediate objection to this approach is that it doesn’t model solving real life arithmetic problems which aren’t typically accompanied with key words. Another is that key words have little descriptive value but are included in arithmetic word problems primarily to make them read correctly. Finally, key words are unreliable: They don’t always go with the operations with which they’re “supposed” to go, and they don’t always appear in problems.

Consider the four problems below, all of which begin with the same sentence, all of which end with “How many?” and none of which contain any of the usual key words for the operations that are used to solve them:

• Kipp collects 4 brands of baseball cards: Fleer Ultra, Studio, Topps and Upper Deck. He has 2 packs of Fleer Ultra, 3 of Studio, 3 of Topps and 4 of Upper Deck. How many packs of baseball cards does he have? (2 + 3 + 3 + 4 = 12)

• Kipp collects 4 brands of baseball cards: Fleer Ultra, Studio, Topps and Upper Deck. He has 12 packs. Three are Topps. How many of the other brands does he have? (12 – 3 = 9)

• Kipp collects 4 brands of baseball cards: Fleer Ultra, Studio, Topps and Upper Deck. He has 3 packs of each brand. How many packs does he have? (4x3=12)

• Kipp collects 4 brands of baseball cards: Fleer Ultra, Studio, Topps and Upper Deck. He has 12 packs. How many of each brand does he have if he has the same number of packs per brand? (12 ÷ 4 = 3)

A student relying on key words to solve these problems would be guessing. The *real* key to solving arithmetic word problems is UNDERSTANDING—being able to identify what’s going on and having a literal feel for what’s taking place.

In everyday math, addition, subtraction, multiplication and division are used to solve not four different types of problems, but *eight*—two for each operation:

**Addition:** “combining” and “counting on”*

**Subtraction:** “take away” and “comparing to find the difference”

**Multiplication:** “grouping/repeated addition” and “combinations”

**Division:** “grouping/repeated subtraction” and “sharing”

*Some authors list “counting on” under subtraction since it is often used to work problems where the difference between numbers is small, like the amount of change from a $10 bill for a $7 purchase: A clerk counts “eight-nine-ten” for a count of 3 as they give out the change.