People tend to learn by identifying patterns and generalizing or extending them to some conclusion (which may or may not be true). A major emphasis in the mathematics curriculum in the early grades should be the opportunity to experience numerous patterns. The development of algebra as a language should build on these experiences. The ability to extend patterns falls under Standard 14 (Patterns and Functions), but having students communicate their reasoning is also an algebra expectation. Initially, ordinary language and concrete materials should be used for communication. As patterns become more complex, students should develop the ability to use tables and pictures or symbols (such as triangles or squares) to represent numbers that may change or are unknown (variable quantities).

The primary grades provide an ideal opportunity to lay the foundation for the development of the ability to represent situations using equations or inequalities (open sentences) and solving them. Students can be asked to communicate or represent relationships involving concrete materials. For example, two students might count out eight chips and place them on a mat. One of the students then places a margarine tub over some of the counters and challenges the other student to figure out how many chips are hidden under the tub. A more complex situation might involve watching the teacher balance a box and two marbles with six marbles, drawing a picture of the situation, and trying to decide how many marbles would balance the box by physically removing two marbles from each side of the balance. A situation involving an inequality might ask students to find out how many books Jose has if he has more than three books but fewer than ten books. Situations from the classroom and students' real experiences should provide ample opportunities to construct and solve such open sentences.

As operations are developed, students should examine properties and make generalizations. For example, giving students a set of problems which follow the pattern 3+4, 4+3, 1+2, 2+1, etc. should provide the opportunity to develop the concept that order does not affect the answer when adding (the commutative property). After students see that these properties are not necessarily true for all operations (e.g., 5-2 is not equal to 2-5), the teacher should mention that the properties are important enough to be given names and begin using the names. However, the focus of this work should be on using the properties of operations to make work easier rather than on memorizing the properties and their names.

Students in grades K-2 spend a great deal of time developing meaning for the arithmetic operations of addition, subtraction, multiplication, and division. As they work toward understanding these concepts, they focus on developing mathematical models for concrete problem situations. The number sentences that they write to describe these problem situations are the basic foundation for more sophisticated mathematical models.

Experiences will be such that all students in grades K-2:

A. understand and represent numerical situations using variables, expressions, equations and inequalities.

• Students represent a problem situation with an open sentence. For example, if there are 25 students in the class and Marie brought 26 cookies for snack, how many will be left over? (26 - 25 = ?) Another example might be that we have 10 cups left in the package and there are 25 children in the class, so how many more cups do we need to get? (10 + ? = 25)
• Students are given a piece of paper with the base of a pan balance drawn on it, a paper pan balance, and a fastener. They make a working replica of a pan balance. Throughout the year, the teacher has them place replicas of objects on both sides to represent numerical situations and then place the balance in the proper position. For example, the teacher might have students place two apples plus three apples on one side and four apples on the other. The students should show the balance tilted so that the side with four is higher than the side with five.
• Students make a table relating the number of people and the number of eyes. They use a symbol such as a stick figure to represent the number of people and a cartoon drawing of an eye to represent the number of eyes and then express the relationship between them.
  

• Students in groups are given a container approximately one-third full of water. Students measure the height of the water using Unifix cubes. They add five marbles to the container and measure the height of the water. (It is helpful if the container is the right size for adding five marbles to make the water go up exactly one Unifix cube.) The students continue adding marbles and measuring the height of the water until they have added twenty-five marbles. Students describe the relationship between the number of marbles added and the height of the water.
• Students look at a series of pictures which form a pattern. They draw the next shape, describe the pattern in words, and explain why they chose to draw that shape.
• Using a calculator, students play "Guess my rule." The lead student enters an expression such as 5+4 and presses the "=" key; she shows only the answer to her partner. The second student enters several numbers, one at a time, pressing the "=" key after each number. When the second student thinks they know the pattern (in this case, adding 4), they make a guess. The rule is written in words and then using a picture or symbol for the variable (the number which the second student enters).
• Placing four different-colored cubes in a can, students predict which color would be drawn out most if each child drew one without looking. The teacher helps the students keep track of their results by making a chart with the colors on the horizontal axis and the number of times a color is drawn on the vertical axis. As students pull cubes, a post-it or an "x" is placed above the color drawn, forming a frequency diagram. After several draws, the students describe the patterns they see in the graph.

• Students are given five computational problems to solve. They are permitted to use the calculator on only two of them. Two of the problems are related to another two by operation properties (e.g., 3 + 2 and 4 + 6 are related to 2 + 3 and 6 + 4 by the commutative property) and the last involves a property of number such as adding 0. Students share their thought processes in a followup discussion.
• The teacher has a box containing slips of paper with open sentences such as 25 - 8 = [] or 15 + [] = 23. Students draw out a slip and tell or write a story which would involve a situation modeled by the sentence.
• Students understand that, since the order of the numbers when adding them is not important, they can solve a problem like 3 + 8 by counting up from 8.

• Kindergarten students play the "hide the pennies" game. The first player places a number of pennies (say 7) on the table and lets the other player count them. The first player covers up a portion of the pennies, and the second player must determine how many are covered. They may represent the situation with markers or pictures to help them. Older students playing this game write a number sentence that describes the situation.
• Students are given a box of unknown weight. They are told that the box and two "weights" would exactly balance seven "weights." They draw a picture of the situation and then use the actual objects to determine the "weight" of the box.