In first grade, students build a foundation for arithmetic by developing a robust understanding of addition and subtraction. Mathematics educators have developed a classification system of 15 different types of problems in which addition and subtraction are used:

 1) An initial quantity increases and the final value is unknown.

 Example: Two bunnies sat on the grass.Three more bunnies hopped there. How many bunnies are on the grass now?

 2) An initial quantity increases and the amount of change is unknown.

 Example: Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two?

3) An initial quantity increases and the initial amount is unknown.

 Example: Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?

 4) An initial quantity decreases and the final value is unknown.

 Example: Five apples were on the table. I ate two apples. How many apples are on the table now?

5) An initial quantity decreases and the amount of change is unknown.

Example: Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?

 6) An initial quantity decreases and the initial amount is unknown.

 Example: Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?

 7) A quantity is made up of two parts and the size of the whole is unknown.

 Example: Three red apples and two green apples are on the table. How many apples are on the table?

8) A quantity is made up of two parts and the the size of a part is unknown.

 Example: Five apples are on the table. Three are red and the rest are green. How many apples are green?

 9) A quantity is made up of two parts and the sizes of both parts are unknown.

 Example: Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?

 10) One quantity is more than another and the difference is unknown.

 Example: Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

11) One quantity is less than another and the difference is unknown.

Example: Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?

 12) One quantity is more than another and the larger quantity is unknown.

 Example: Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?

 13) One quantity is less than another and the larger quantity is unknown.

 Example: Lucy has three fewer apples than Julie. Lucy has two apples. How many apples does Julie have?

 14) One quantity is more than another and the smaller quantity is unknown.

 Example: Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?

 15) One quantity is less than another and the smaller quantity is unknown.

 Example: Lucy has three fewer apples than Julie. Julie has five apples. How many apples does Lucy have?

It is important to allow students many opportunities to model the different problem types – with manipulatives and by drawing pictures and bar diagrams – so that they concentrate on the actions and relationships in the problems and not rely on “key” or “clue” words.

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