Ohio Resource Center > Record > Building Bridges to Algebraic Thinking

ORC Resource Number #1112

Building Bridges to Algebraic Thinking

http://illuminations.nctm.org/LessonDetail.aspx?ID=L247

PROFESSIONAL COMMENTARY

This site shows how single-solution arithmetic problems can be generalized to algebraic problems. The process of extending a specific problem to a general relationship helps students make the transition from arithmetic to algebra, using variables to describe the general relationship. Numerical values are organized into tables and graphs, which are then used to derive algebraic representations. Problem situations modeled by linear, quadratic, and exponential functions are discussed. Internet extensions are provided, and graphing calculator activities are suggested. This resource is adapted from an article published in the February 1997 issue of Mathematics Teaching in the Middle School. (author/sw)

CAREER APPLICATION

Career-technical teachers who are seeking to instill a mathematical understanding of algebraic principles will find this site particularly helpful. Through these lessons, students will come to an appreciation of how mathematics is used to solve real problems. Two links present problems which are extended practices of the same principles.

OHIO STANDARDS

Mathematics Academic Content Standards

Patterns, Functions and Algebra Standard

Benchmarks (5 - 7)

Describe, extend and determine the rule for patterns and relationships occurring in numeric patterns, computation, geometry, graphs and other applications.

Represent, analyze and generalize a variety of patterns and functions with tables, graphs, words and symbolic rules.

Use variables to create and solve equations and inequalities representing problem situations.

Use symbolic algebra to represent and explain mathematical relationships.

Use rules and variables to describe patterns, functions and other relationships

Use representations, such as tables, graphs and equations, to model situations and to solve problems, especially those that involve linear relationships.

Benchmarks (8 - 10)

Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.

Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.

Grade Level Indicators (Grade 5)

Justify a general rule for a pattern or a function by using physical materials, visual representations, words, tables or graphs.

Use calculators or computers to develop patterns, and generalize them using tables and graphs.

Use variables as unknown quantities in general rules when describing patterns and other relationships.

Create and interpret the meaning of equations and inequalities representing problem situations.

Model problems with physical materials and visual representations, and use models, graphs and tables to draw conclusions and make predictions.

Grade Level Indicators (Grade 6)

Represent and analyze patterns, rules and functions, using physical materials, tables and graphs.

Use words and symbols to describe numerical and geometric patterns, rules and functions.

Produce and interpret graphs that represent the relationship between two variables.

Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations.

Grade Level Indicators (Grade 7)

Represent and analyze patterns, rules and functions with words, tables, graphs and simple variable expressions.

Generalize patterns by describing in words how to find the next term.

Recognize a variety of uses for variables; e.g., placeholder for an unknown quantity in an equation, generalization for a pattern, formula.

Grade Level Indicators (Grade 8)

Relate the various representations of a relationship; i.e., relate a table to graph, description and symbolic form.

Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems.

Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems.

Grade Level Indicators (Grade 9)

Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations.

Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations.

Grade Level Indicators (Grade 10)

10.

Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions.

Mathematical Processes Standard

NATIONAL STANDARDS

Principles and Standards for School Mathematics

Algebra Standard

Understand patterns, relations, and functions

Expectations (3 - 5)

describe, extend, and make generalizations about geometric and numeric patterns;

represent and analyze patterns and functions, using words, tables, and graphs.

Expectations (6 - 8)

represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;

relate and compare different forms of representation for a relationship;

Expectations (9 - 12)

understand relations and functions and select, convert flexibly among, and use various representations for them;

Represent and analyze mathematical situations and structures using algebraic symbols

Expectations (3 - 5)

describe, extend, and make generalizations about geometric and numeric patterns;

represent and analyze patterns and functions, using words, tables, and graphs.

Expectations (6 - 8)

represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;

relate and compare different forms of representation for a relationship;

develop an initial conceptual understanding of different uses of variables;

use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;

Expectations (9 - 12)

understand relations and functions and select, convert flexibly among, and use various representations for them;

Use mathematical models to represent and understand quantitative relationships

Expectations (3 - 5)

describe, extend, and make generalizations about geometric and numeric patterns;

represent and analyze patterns and functions, using words, tables, and graphs.

Expectations (6 - 8)

represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;

relate and compare different forms of representation for a relationship;

develop an initial conceptual understanding of different uses of variables;

use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;

model and solve contextualized problems using various representations, such as graphs, tables, and equations.

Expectations (9 - 12)

understand relations and functions and select, convert flexibly among, and use various representations for them;

Representation Standard

RESOURCE TYPE

Content Resource

STANDARDS ALIGNMENT

Grades 5–10

CAREER FIELDS

General Career Skills

TOPICS

Mathematics --
Algebra;
Nonlinear functions;
Variables;
Linear functions;

FOUND IN

Standards First

KEYWORDS

quadratic function;
exponential function;
generalized arithmetic

Publisher: National Council of Teachers of Mathematics

Author: Roger Day and Graham Jones

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