STANDARD 13 - ALGEBRA
Standard 13 - Algebra - Grades K-2
Students can develop a strong understanding of algebraic concepts and processes from consistent experiences in classroom activities where a variety of manipulatives and technology are used. The key components of this understanding in algebra, as identified in the K-12 Overview, are: patterns, unknown quantities, properties, functions, modeling real-world situations, evaluating expressions and solving equations and inequalities.
Students begin their study of algebra in grades K-2 by learning about the use of pictures and symbols to represent variables. They look at patterns and describe those patterns. They begin to look for unknown numbers in connection with addition and subtraction number sentences. They model the relationships found in real-world situations by writing number sentences that describe those situations. At these grade levels, the study of algebra is very much integrated with the other content standards. Children should be encouraged to play with concrete materials, describing the patterns they find in a variety of ways.
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 11 (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 students grow older and 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. The students draw a picture of the situation, and try to decide how many marbles would balance the box by physically removing two marbles from each side of the balance. In a problem involving an inequality, students might be asked to find out how many books Jose has if he has more than three books but fewer than ten. Situations from the classroom and the students' real experiences should provide ample opportunities to construct and solve such open sentences.
As operations are developed, students need to 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 understand 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. 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 form a foundation for more sophisticated mathematical models.
Standard 13 - Algebra - Grades K-2
Indicators and Activities
The cumulative progress indicators for grades K-2 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in kindergarten and in grades 1 and 2.
Experiences will be such that all students in grades K2:
1. Understand and represent numerical situations using variables, expressions, and number sentences.
2. Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and number sentences, and translate from one to another.
3. Understand and use properties of operations and numbers.
4. Construct and solve open sentences (example: 3 + = 7) that describe reallife situations.
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