# Pre-Algebra and Algebra Instruction and Assessments

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2006

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made to the content or appearance of this product.**

**Algebra and**

NCTM Focal Points

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

shapes

• 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

subtraction (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

division

**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

inequalities

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

equivalent

• Understand that variables represent

numbers whose precise values are not

yet specified

**Algebra in Grade 7**

• Algebra is in all 3 Curriculum Focal

Points

**Algebra in Grade 8**

• Analyzing and representing linear

functions, solving linear equations , and

systems of linear equations

• Also beginning to explore nonlinear

functions

## Algebra Instruction

**Algebra and the Elementary
Grades**

• 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
Grades
**

• 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

problems

3. Symbolic manipulation

4. Functions

5. Inductive reasoning and mathematical

induction

Milgram (2005)

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