The algebraic idea of the "unknown" quantity appears in school curriculum as early as kindergarten when children consider questions such as "1 plus 2 buttons equals how many buttons?" By second grade, students are asked to consider problems such as how many boys were at a party if 9 girls were there and 15 people attended. Or they might ponder how many cookies were on the platter at the beginning of the party if the boys ate 30 and the girls ate 14. These are called "unknown partner" and "unknown start" problems.
Things You'll Need
 Pencil and paper
 Document camera or overhead projector
 Overhead transparencies
 White board and dry erase markers

Tell students that when there is a missing quantity in a math problem, such as "8 plus 12 equal how much," the "how much" part is called an unknown. Use the document camera, overhead projector or white board to offer a few similar problems. Ask them to identify the unknown total.

Explain that sometimes the total is known, but another number in the problem is unknown. Tell a brief story: "20 students went on a field trip. 8 brought water bottles and the others drank from water fountains. How many had to use water fountains?" Ask them what partner number to combine with 8 to make 20. Offer several more questions of this kind, which the publisher Houghton Mifflin Harcourt refers to as an "unknown partner: put together" problem at its Education Place website.

Move on to an "unknown partner: take apart" problem. Make up another story, such as "There were 16 members at the chess club meeting, but 3 had to go home early. How many were still playing chess?" Offer several additional problems following this pattern. Ask students to explain how they find their answers. Expect a number of explanations, including counting down (counting backwards) from 16.

Try some unknown start problems, including this one at the Education Place website: "Greta's chicken laid some eggs. Then the chicken laid 7 more. Now Greta has 13 eggs. How many eggs did the chicken lay at the start?" The addition of 7 eggs to the unknown starting figure makes this a "change plus" pattern.

Stretch student thinking with a "change minus" pattern in which a quantity is subtracted from an unknown starting sum to arrive at a known sum. An example might be "Clare gave Paola 5 candies. She still has 7. How many did she start with?"

Draw pictures to illustrate how to visualize the problems. Also teach students how to use the "math mountains" solution technique presented in the "Math Matters" newsletter of North Carolina's Wendell Creative Arts & Science Magnet Elementary. Draw three boxes in a pyramid shape. Mark the top box "total." One of the two other boxes can either be an "unknown start" figure or an "unknown partner" depending on the problem presented. This makes the solution more visual.
References
 Photo Credit several apples image by Maria Brzostowska from Fotolia.com