• STANDARD 13 — ALGEBRA

STANDARD 13 — ALGEBRA
K-12 Overview
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.
Descriptive Statement
Algebra is a language used to express mathematical relationships. Students need to understand how
quantities are related to one another, and how algebra can be used to concisely express and analyze those
relationships. Modern technology provides tools for supplementing the traditional focus on algebraic
techniques, such as solving equations, with a more visual perspective, with graphs of equations displayed on
a screen. Students can then focus on understanding the relationship between the equation and the graph, and
on what the graph represents in a real-life situation.
Meaning and Importance
Algebra is the language of patterns and relationships through which much of mathematics is communicated.
It is a tool which people can and do use to model real situations and answer questions about them. It is also a
way of operating with concepts at an abstract level and then applying them, often leading to the development
of generalizations and insights beyond the original context. The use of algebra should begin in the primary
grades and should be developed throughout the elementary and secondary grades.
The algebra which is appropriate for all students in the twenty-first century moves away from a focus on
manipulating symbols to include a greater emphasis on conceptual understanding, on algebra as a means of
representation, and on algebraic methods as problem-solving tools. These changes in emphasis are a result
of changes in technology and the resulting changes in the needs of society.
The vision proposed by this Framework stresses the need to prepare students for a world that is rapidly
changing in response to technological advances. Throughout history, the use of mathematics has changed
with the growing demands of society as human interaction extended to larger groups of people. In the same
way that increased trade in the fifteenth century required businessmen to replace Roman numerals with the
Hindu system and teachers changed what they taught, today's education must reflect the changes in content
required by today's society. More and more, the ability to use algebra in describing and analyzing real-world
situations is a basic skill. Thus, this standard calls for algebra for all students.
What will students gain by studying algebra? In a 1993 conference on Algebra for All, the following points
were made in response to the commonly asked question, “Why study algebra?”
New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra — 405
 C Algebra provides methods for moving from the specific to the general. It involves discovering the
patterns among items in a set and developing the language needed to think about and
communicate it to others.
C Algebra provides procedures for manipulating symbols to allow for understanding of the world
around us.
C Algebra provides a vehicle for understanding our world through mathematical models.
C Algebra is the science of variables. It enables us to deal with large bodies of data by identifying
variables (quantities which change in value) and by imposing or finding structures within the data.
C Algebra is the basic set of ideas and techniques for describing and reasoning about relations
between variable quantities.
Standard 8 (Numerical Operations) addressed the need for us to rethink our approach to paper-and-pencil
computation in light of the availability of calculators; the need to examine the dominance of paper-and-pencil
symbolic manipulation in algebra is just as important. The development of manipulatives, graphing
calculators, and computers have made a more intuitive view of algebra accessible to all students, regardless of
their previous mathematical performance. These tools permit and encourage visual representations which are
more readily understood. No longer need students struggle with abstract concepts presented with very few
ties to real-life situations. Rather, the new view of algebra offers real situations for students to examine, to
generalize, and to represent in ways which facilitate the asking and answering of meaningful questions.
Moreover, inexpensive symbolic processors perform algebraic manipulations, such as factoring, quickly and
easily, reducing the need for drill and mastery of paper-and-pencil symbol manipulation.
K-12 Development and Emphases
Algebra is so significant a part of mathematics that its foundation must begin to be built in the very early
grades. It must be a part of an entire curriculum which involves creating, representing, and using quantitative
relationships. In such a curriculum, algebraic concepts should be introduced in conjunction with the study of
patterns and developed throughout each student’s mathematical education. The earlier students are exposed
to informal algebraic experiences, the more willing they will be to use algebra to represent patterns.
The concept of representing unknown quantities begins with using symbols such as pictures, boxes, or
blanks (i.e., 3 + ~ = 7). It is vital that students recognize that the symbol that is used to represent an
unknown quantity has meaning. The only way this can be accomplished is to consistently relate the use of
unknowns to actual situations; otherwise, students lack the ability to judge whether their answers make sense.
As students develop their understanding of arithmetic operations, they need to investigate the patterns which
arise. Some of these patterns (which are commonly called properties) should be initially expressed in words.
As the students develop more facility with variables, the properties can be expressed in symbolic form.
In the middle grades, problem situations should provide opportunities to generalize patterns and use
additional symbols such as names and literal variables (letters). This development should continue
throughout the remainder of the program, ensuring that the relationship between the variables (unknowns)
and the quantities they represent is consistently stressed. Middle school students should extend their ability
to use algebra to generalize patterns by exploring different types of relationships and by formalizing some of
406 — New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra
those relationships as functions. They should explore and generalize patterns which arise from nature,
including non-linear relationships. As students move into the secondary grades, the graphing calculator and
graphing software provide tools for examining relationships between x-intercepts and roots, between turning
points and maximum or minimum values, and between the slope of a curve and its rate of change. As the
student continues through high school, similar experiences should be provided for other functions, such as
exponential and polynomial functions; these functions should be introduced using situations to which
students can relate.
The use of algebra as a tool to model real world situations requires the ability to represent data in tables,
pictures, graphs, equations or inequalities, and rules. Through exploration of problems and patterns, students
are provided with opportunities to develop the ability to use concrete materials as well as the representations
mentioned above. Having students use multiple representations for the same situation helps them develop an
understanding of the connections among them. The opportunity to verbally explain these different
representations and their connections provides the foundation for more formal expressions.
A fundamental skill in algebra is the evaluation of expressions and the solution of equations and
inequalities. This process will be easier to understand if it is related to situations which give them meaning.
Expressions, equations, and inequalities should arise from students’ exploration in a variety of areas such as
statistics, probability, and geometry. Elementary students begin constructing and solving open sentences.
The use of concrete materials and calculators allow them to explore solutions to real-life situations.
Gradually, students are led to expand these informal methods to include graphical solutions and formal
methods. The relationship between the solutions of equations and the graphs of the related functions must be
stressed regularly.
IN SUMMARY, there are algebraic concepts and skills which all students must know and apply confidently
regardless of their ultimate career. To assure that all children have access to such learning, algebraic thinking
must be woven throughout the entire fabric of the mathematics curriculum.
NOTE: Although each content standard is discussed in a separate chapter, it is not the intention that each
be treated separately in the classroom. Indeed, as noted in the Introduction to this Framework, an
effective curriculum is one that successfully integrates these areas to present students with rich and
meaningful cross-strand experiences.
New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra — 407
 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.
408 — New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra
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.
New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra — 409
 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 K-2:
1. Understand and represent numerical situations using variables, expressions, and number
sentences.
C 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)
C 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.
C 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.
&+&º'+'+'+'
2. Represent situations and number patterns with concrete materials, tables, graphs, verbal
rules, and number sentences, and translate from one to another.
C 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.
C 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.
C 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 for the variable (the number which the
second student enters).
410 — New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra
 C Placing four different-colored 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.
C 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.
C 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.
C 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.
C 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.
C 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 real-life situations.
C 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.
C 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
New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra — 411
 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.
412 — New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra
 Standard 13 — Algebra — Grades 3-4
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.
In grades K-2, students use pictures and symbols to represent variables, generalize patterns verbally and
visually, and work with properties of operations. Although the formality increases in grades 3 and 4, it is
important not to lose the sense of play and the connection to the real world that were present in earlier grades.
As much as possible, real experiences should generate situations and data which students attempt to
generalize and communicate using ordinary language. Students should explain and justify their reasoning
orally to the class and in writing on assessments using ordinary language. When introducing a more formal
method of communicating, such as the language of algebra, it is helpful to revisit some of the situations used
in previous grades.
Since algebra is the language of patterns, the mathematics curriculum at this level needs to continue to focus
on patterns. The use of letters to represent unknown quantities should gradually be introduced as a
replacement for pictures and symbols. The use of function machines permits the introduction of letters
without the need to move to formal symbolic algebra. Since they have had the opportunity to experience real
function machines such as the calculator or a gum bank, where one penny yields two pieces of gum, the
notation of function machines should make sense.
Here the box is thought of as the function machine times 2 which takes in a number “a” and produces a
number “b” which is twice “a.” Students can use such symbols to communicate their generalization of
patterns. They put two or more machines together making a composite function; for example, they can follow
the times 2 machine with an add 3 machine. They determine not only what each input produces but also what
input would produce a given output.
Students should continue to communicate their generalizations of
patterns through ordinary language, tables, and concrete materials.
Graphs should be introduced as a method for quickly and
efficiently representing a pattern or function. They should develop
graphs which represent real situations and be able to describe
patterns of a situation when shown a graph. For example, when
given the graph at the right which shows the relationship between
the number of bicycles and wheels in the school yard, they should
be able to describe the relationship in words.
New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra — 413
Students in grades 3 and 4 should continue to use equations and inequalities to represent real situations.
While variables can be introduced through simple equations such as 35 ÷ n = 5, students should be viewing
variables as place holders similar to the open boxes and pictures they have already used. At these grade
levels, they need not use variables in more complicated situations. Given a situation such as determining the
cost of each CD if 5 of them plus $3 tax is $23, they should be permitted to represent it in whatever way they
feel comfortable. Students should be able to use, explain, and justify whatever method they wish to solve
equations and inequalities. Some may continue to use concrete materials for some situations; they might
count out 23 counters, set aside 3 for the tax, and divide the remainder into 5 equal piles of 4. Others might
try different numbers until they find one that works. Some students may write
23 ! 3 = 20 and 20 ÷ 5 = 4. Still others may want to relate this to function machines and figure out what had
to go in for $23 to come out. It is important for students to see the diversity of approaches used and to
discuss their interrelationships.
Students should continue to examine the properties of operations and use them whenever they would make
their work easier. There are some excellent opportunities for providing a foundation for algebraic concepts in
these grades. For example, explaining two-digit multiplication by using the area of a rectangle (see figure
below illustrating 13 x 27) provides the student with a foundation for multiplication of binomials, the
distributive property, and factoring. While the teacher at this grade level should focus on the development of
the multiplication algorithm, the teacher of algebra several years later will be able to build on this experience
of the student.
414 — New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra
 Standard 13 — Algebra — Grades 3-4
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by
activities which illustrate how it can be addressed in the classroom in grades 3 and 4.
Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such
that all students:
1. Understand and represent numerical situations using variables, expressions, and number
sentences.
C Students do comparison shopping based on items that are for sale in multiples. For instance:
If chewing gum is sold at 3 packs for 85 cents (p = 85/3), is that a better or a worse buy
than a single pack for 30 cents (p = 30)? They sort their examples into groups where the
multiple buy is a better deal, the same deal, or a worse deal than the single package deal.
C One student has been folding origami cranes to send to Hiroshima for Peace Day in August.
He brings the 47 cranes that he has folded so far to class and asks for help to fold many
more. The class decides to have each of the 26 students fold one crane each week for the rest
of the school year. The teacher asks groups of students to find a way to determine how many
cranes will be in the collection after some given number of weeks. She starts off the
discussion by having students list the numbers for the first few weeks:
47,
47 + 26,
47 + 26 + 26,
47 + 26 + 26 + 26,
and so on.
They figure out whether they can reach 500 cranes by the end of the year.
2. Represent situations and number patterns with concrete materials, tables, graphs, verbal
rules, and number sentences, and translate from one to another.
C Students compare two allowance plans. Plan A provides an allowance of $5 the first week
and adds $1 each week. Plan B starts with 1¢ and doubles the allowance each week. Using
calculators, students make tables listing the number of the week, the amount of allowance
under Plan A and the amount of the allowance under plan B. They complete several rows of
the table so that they understand what is happening with each plan and see that Plan B soon
overcomes Plan A. They might want to try to use physical objects such as centimeter cubes
to demonstrate the behavior of the two plans visually.
C Each student is given an even number of square tiles and asked to use them all to make a
rectangle with two columns. Students are asked to notice that the heights of the rectangles
are different for different starting numbers of tiles. They collect the data into a table, giving
each student’s name, the number of tiles used, and the height of the rectangle. They
understand that the number of tiles is the area and can figure out the height of the rectangle if
New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra — 415
 they know the number of tiles that are used — that is, they can verbalize that the height of
the rectangle is half the total number of tiles.
C Students read 1 Hunter by Pat Hutchins, wherein a determined hunter looks and looks and
looks for animals but sees none, even though the reader can clearly see 2 elephants, 3
giraffes, 4 ostriches, and so on, up to 10 parrots. They are asked how many animals in all
the hunter was unable to see. Students use graphs, concrete materials, pictures, and number
sentences to express their understanding of the situation.
C Students play Guess My Rule by suggesting inputs and having the rule-maker (the teacher
or a student) put the corresponding outputs into a table like this one:
Input Output
3 7
1 4
16 19
. .
. .
Students should always be challenged to show they understand the rule by giving a verbal
explanation of it. Partially filled-in Guess My Rule tables are a good assessment technique
to evaluate the students’ inductive reasoning power and their level of comfort with arithmetic
operations.
3. Understand and use properties of operations and numbers.
C When the students are introduced to two-digit by
two-digit multiplication, they begin with a
problem of finding the area of a rectangular field
which is 37 feet by 44 feet. They know they need
to multiply the numbers to find the area, but they
don’t know how to multiply without calculators.
The teacher draws a rectangle and uses a line to
divide the width into two regions which are 30 feet
and 7 feet. She does the same with the length,
cutting it into lengths 40 feet and 4 feet. This
divides the rectangle into four smaller rectangles
(30×40, 30×4, 7×40, 7×4) all of which are
multiplications the students can do.
C Lea and Suzanne discovered a method for multiplying even numbers by six easily. Their
method, applied to the example 6×24, is:
Cut the other even number in half 12
Add a zero 120
Add the number 120+24=144
When they told their classmates their discovery, they were stumped when they were asked
why it worked. The teacher, grasping the teachable moment, divided the class into groups
and challenged them to do a few examples using the girls’ method and try to figure out and
explain why it worked.
416 — New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra
 C Students and teacher together work through Robert Froman’s book, The Greatest Guessing
Game: A Book about Dividing to reinforce their notions of division.
C Students explain that they solved a problem like 300 ! 56 mentally by first subtracting 50
and then subtracting 6, since that is the same as subtracting 56. They also do 25×7×4 by
first multiplying 25×4 and then multiplying by 7. Such simplifications will give a good
foundation for later work in algebra.
4. Construct and solve open sentences (example: 3 + ~ = 7) that describe real-life situations.
C In an assessment situation, groups of students are asked to describe in words the situation of
four people sharing a five dollar bill found on the way to school, and then to transform it to
symbolic form using pictures, symbols or letters.
C Students want to help the New Jersey environment and raise money at the same time. They
discover that in two bordering states (New York and Delaware), plastic soda bottles can each
be turned in for a 5¢ refund. They write an equation which represents the amount of money
they will receive for b bottles. Students answer questions such as How much money will we
get for 25 bottles? and How many bottles will we need to make $10?
C Students are presented with a function machine representing the situation of buying music
tapes for $5 each through the mail and paying a $3 shipping and handling charge for the
order. They answer questions such as How much would it cost for 5 tapes? and How
many tapes were bought if the bill was $43?
References
Fromer, Robert. The Greatest Guessing Game: A Book About Dividing. New York, NY:
Thomas Y. Crowell Publishers, 1978.
Hutchins, Pat. 1 Hunter. New York, NY: Greenwillow Books, 1982.
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.
New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra — 417
 Standard 13 — Algebra — Grades 5-6
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.
It is important that students continue to have informal algebraic experiences in grades 5 and 6. Students have
previously had the opportunity to generalize patterns, work informally with open sentences, and represent
numerical situations using pictures, symbols, and letters as variables, expressions, equations, and inequalities.
At these grade levels, they will continue to build on this foundation.
Algebraic topics at this level should be integrated with the development of other mathematical content to
enable students to recognize that algebra is not a separate branch of mathematics. Students must understand
that algebra is an expansion of the arithmetic and geometry they have already experienced and a tool to help
them describe situations and solve problems.
Students should use algebraic concepts to investigate situations and solve interesting mathematical and real
world problems. There should be numerous opportunities for collaborative work. Since algebra is the
language for describing patterns, students should have regular and consistent opportunities to discuss and
explain their use of these concepts. They should write generalizations of situations in words as well as in
symbols. To provide such opportunities, the activities should move from a concrete situation or
representation to a more abstract setting. Students at this level can begin using standard algebraic notation to
represent known and unknown quantities and operations. This should be developed gradually, moving
them from the previous symbols in such a way that they can appreciate the power and elegance of the new
notation.
Students need to learn how variables are different from numbers (a variable can represent many numbers
simultaneously, it has no place value, it can be selected arbitrarily) and how they are different from words
(variables can be defined in any way we want and can be changed without affecting the values they represent).
Students need to see variables (letters) used as names for numbers or other objects, as unknown numbers in
equations, as a range of unknown values in inequalities, as generalizations in pattern rules or formulas, and as
characteristics to be graphed (independent and dependent variables).
An algebraic expression involves numbers, variables, and operations such as 2b, 3x ! 2, or
c ! d. In fifth and sixth grade, students should begin to become familiar with the common notational shortcut
of omitting the operation sign for multiplication, so that when b=3, 2b equals 6 and not 23. Thus they
recognize that there are slightly different rules for reading expressions involving variables than those
involving only numbers.
Students in grades 5 and 6 should focus on understanding the role of the equal sign. Because it is so often
used to signal the answer in arithmetic, students may view it as a kind of operation sign — a “write the
418 — New Jersey Mathematics Curriculum Framework — Standard 13 — Algebra
answer” sign. They need to come to see its role as a relation sign, balancing two equal quantities. Students
should develop the ability to solve simple linear equations using manipulatives and informal methods. With
the appropriate background, students at grades 5 and 6 have the ability to find the solution of an equation,
such as 7 for x+5=12, whether they use manipulatives, a graph, or any other method. It is imperative that in
the discussion of the solution of an equation, the many methods in obtaining that solution are described.
Students in grades 5 and 6 should use concrete materials, such as algebra tiles, to help them develop a visual,
geometric understanding of algebraic concepts. For example, students can represent the expression 3x ! 2 by
using three strips and two units. They should make graphs on a rectan