Pre-Algebra and Algebra Instruction and Assessments

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Algebra and
NCTM Focal Points

Overview of the Focal Points

PreK through grade 8

• Important math topics / content in
mathematics by grade

• Cohesive clusters of related ideas /
concepts / skills that form foundation for
higher level mathematics
– NOT discrete topics that teachers teach
and students master

Algebra Learning Expectations
Across States: Grade 4

State Algebra Total
California 7 43
Texas 4 32
New York 5 56
Florida 10 89
Ohio 6 48
Michigan 0 56
New Jersey 6 56
North Carolina 3 26
Georgia 3 45
Virginia 2 41

Algebra in Prekindergarten

• Patterns: recognize and duplicate
sequential patterns

Algebra in Kindergarten

• Identify, duplicate, and extend:
Number patterns
– Sequential and growing patterns with

• Preparation for creating rules that
describe relationships
– E.g., each new pattern grows by one

Algebra in Grade 1

• Understand connection between
counting and operations of addition and
(e.g., adding two is the
same as “counting on” two )

• Use properties of commutativity and
associativity to add whole numbers

• Begin to relate addition and subtraction
as inverse operations

Algebra in Grade 2

• Use number patterns to extend
understanding of properties of numbers
and operations

– E.g., skip counting builds foundation of
understanding multiples and factors

Algebra in Grade 3

• Understand meaning of multiplication
and division (equal “jumps,” equal-sized
groups, area models)

• Creation and analysis of patterns and
relationships involving multiplication and

Algebra in Grade 4

• Students understand number patterns
involving all operations and nonnumeric
growing or repeating patterns

– Can use rules to describe a sequence of
numbers or objects
– E.g., 2, 4, 8, 16, 32

Algebra in Grade 5

• Use patterns, models, relationships to
write and solve simple equations and

E.g., 2, 4, 8, 16, _
_ = 16*2 = 32.

• Create graphs of simple equations
x: 1 2 3 4 5
y: 2 4 8 16 32

Algebra in Grade 6

• Algebra is a Curriculum Focal Point for
the first time on its own

• Write and evaluate mathematical
expressions; use expressions and
formulas to solve problems.

• Expressions in different forms can be

• Understand that variables represent
numbers whose precise values are not

Algebra in Grade 7

• Algebra is in all 3 Curriculum Focal

Algebra in Grade 8

• Analyzing and representing linear
functions, solving linear equations , and
systems of linear equations

• Also beginning to explore nonlinear

Algebra Instruction

Algebra and the Elementary

• Algebra has often been characterized as
developmentally constrained due to its
inherent abstractness (e.g., Kieran, 1981,
1985; Vergnaud, 1985)

• Research in the former Soviet Union
suggested that young children could
generalize arithmetic , moving from particular
to generalized numbers, learning to use
variables and covariation in word problems,
and focusing on the concept of function
(Davydov, 1991, Bodanskii, 1991)

Algebra and the Elementary

• Recent research suggests that
inappropriate instruction may have had
a decisive role in the poor results from
early studies of algebraic reasoning
among adolescents in the U.S. (Booth,
1988; Schliemann & Carraher, 2002).

Critical Topics for Teaching
and Learning Algebra

1. Variables and constants
2. Decomposing and setting up word
3. Symbolic manipulation
4. Functions
5. Inductive reasoning and mathematical

Milgram (2005)

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