New Jersey Mathematics Curriculum Framework

## STANDARD 13 - ALGEBRA

 All students will develop an understanding of algebraic concepts and processes and will use them to represent and analyze relationships among variable quantities and to solve problems.

## Standard 13 - Algebra - Grades K-2

### Overview

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.

• 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: 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 read The Doorbell Rang by Pat Hutchins. They act out the story and realize that many different combinations of students can share 12 cookies equally.

• 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.

2. Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and number sentences, and translate from one to another.

• Students in groups are given a container to which they add water until its height is 5 centimeters, measured with Cuisennaire rods. They add marbles to the container until the height of the water is 6 centimeters. They continue adding marbles, recording each time the number of marbles it takes to raise the water level one centimeter. They describe the relationship between the number of marbles added and the height of the water.

• In regular assessment activities, 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 tries to guess the rule by entering different numbers, one at a time, pressing the = key after each number. The calculator, after each = is pressed, should show the sum of the entered number and the second addend (in this case, 4). (Some calculators perform this function differently; see the user's manual for instructions.) When the second student thinks she knows the pattern (in this case, adding 4), she makes a guess. The pattern is written in words and then as a rule using a picture or symbol forthe variable (the number which the second student enters).

• Placing four differentcolored cubes in a can, students predict which color would be drawn out most often if each child draws one cube 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 select cubes, an "x" is placed above the color drawn, forming a frequency diagram. After several turns, the students describe the patterns they see in the graph.

• Students read Ten Apples Up on Top! by Theo Le Sieg and discuss the mathematical comparisons and equations that appear in the story.

3. Understand and use properties of operations and numbers.

• 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 follow up discussion.

• The second grade 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 discover that, since the order of the numbers when adding them is not important, they can solve a problem like 3 + 8 by starting with 8 and counting up 3, as well as by starting with 3 and counting up 8.

• In their math journals, students write their reactions to the following situation:

Sally just used her calculator to find out that 324 + 486 was equal to 810. In another problem, she must find the answer to 486 + 324. What should she do? Why?

4. Construct and solve open sentences (example: 3 + = 7) that describe reallife situations.

• 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. Some second-grade students are ready to write a number sentence that describes the situation.

• Students are given a bag with Unifix cubes. They are told that the bag and 2 cubes balance 7 cubes. They use a balance scale to find how many cubes are in the bag.

### References

Hutchins, Pat. The Doorbell Rang. Mulberry Books, 1986.

Le Sieg, Theo. Ten Apples Up on Top! New York, NY: Random House, 1961.

### On-Line Resources

http://dimacs.rutgers.edu/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.