Developing Algebraic Reasoning in CMP
Early Experiences with Algebraic Reasoning
Developing Symbolic Reasoning
Traditionally, an Algebra 1 course focuses on rules or specific strategies for solving standard types of symbolic manipulation problems-usually to simplify or combine expressions or solve equations. For many students, symbolic rules for manipulation are memorized with little attempt to make sense of why they work. They retain the ideas for only a short time. There is little evidence that traditional experiences with algebra help students develop the ability to "read" information from symbolic expression or equations, to write symbolic statements to represent their thinking about relationships in a problem, or to meaningfully manipulate symbolic expressions to solve problems.
In the United States, algebra is generally taught as a stand-alone course rather than as a strand integrated and supported by other strands. This practice is contrary to curriculum practices in most of the rest of the world. Today, there is a growing body of research that leads many United States educators to believe that the development of algebraic ideas can and should take place over a long period of time and well before the first year of high school. Developing algebra across the grades and integrating it with other strands helps students become proficient with algebraic reasoning in a variety of contexts and gives them a sense of the coherence of mathematics. Transition to High School in Implementing CMP
Developing Algebraic Reasoning in Connected Mathematics
The Connected Mathematics program aims to expand student views of algebra beyond symbolic manipulation and to offer opportunities for students to apply algebraic reasoning to problems in many different contexts throughout the course of the curriculum. The development of algebra in Connected Mathematics is consistent with the recommendations in the NCTM Principles and Standards for School Mathematics 2000 and most state frameworks.
Algebra in Connected Mathematics focuses on the overriding objective of developing students' ability to represent and analyze relationships among quantitative variables. From this perspective, variables are not letters that stand for unknown numbers. Rather they are quantitative attributes of objects, patterns, or situations that change in response to change in other quantities. The most important goals of mathematical analysis in such situations are understanding and predicting patterns of change in variables. The letters, symbolic equations, and inequalities of algebra are tools for representing what we know or what we want to figure out about a relationship between variables. Algebraic procedures for manipulating symbolic expressions into alternative equivalent forms are also means to the goal of insight into relationships between variables. To help students acquire quantitative reasoning skills, we have found that almost all of the important tasks to which algebra is usually applied can develop naturally as aspects of this endeavor. (Fey, Phillips 2005)
There are eight units which focus formally on algebra. Titles and descriptions of the mathematical content for these units are:
Variables and Patterns
Introducing Algebra
Representing and analyzing relationships between variables, including tables, graphs, words, and symbols
Moving Straight Ahead
Linear Relationships
Examining the pattern of change associated with linear relationships; recognizing, representing, and analyzing linear relationships in tables, graphs, words and symbols; solving linear equations;
writing equations for linear relationships
Thinking With Mathematical Models
Linear and Inverse Variation
Introducing functions and modeling; finding the equation of a line; representing and analyzing inverse functions
Looking for Pythagoras
The Pythagorean Theorem
Exploring square roots; exploring and using the Pythagorean Theorem, making connections in the coordinate plane among coordinates, slope, and distance
Growing, Growing, Growing
Exponential Relationships
Examining the pattern of change associated with exponential relationships; comparing linear and exponential patterns of growths; recognizing, representing, and analyzing exponential growth and decay
in tables, graphs, words and symbols; developing rules of exponents
Frogs, Fleas, and Painted Cubes
Quadratic Relationships
Examining the pattern of change associated with quadratic relationships and comparing these patterns to linear and exponential patterns, recognizing, representing, and analyzing quadratic functions in
tables graphs, words, and symbols; determining and predicting important features of the graph of a quadratic functions, such as the maximum/minimum point, line of symmetry, and the x-and y-intercepts;
factoring simple quadratic expressions
Say It With Symbols
Making Sense of Symbols
Writing and interpreting equivalent expressions; combining expressions; looking at the pattern of change associated with an expression; solving linear and quadratic equations
Shapes of Algebra
Linear Systems and Inequalities
Exploring coordinate geometry; solving inequalities; solving systems of linear equations and linear equalities.
Early Experiences with Algebraic Reasoning
Even though the first primarily algebra unit occurs at the start of seventh grade, students study relationships among variables in grade 6.
There also are opportunities in 6th and in 7th grade for students to begin to examine and formalize patterns and relationships in words, graphs, tables, and with symbols.
- In Shapes and Designs (Grade 6), students explore the relationship between the number of sides of a polygon and the sum of the interior angles of the polygon. They develop a rule for calculating the sum of the interior angle measures of a polygon with N sides.
- In Covering and Surrounding (Grade 6), students estimate the area of three different- size pizzas and then relate the area to the price. This problem requires students to consider two relationships: one between the price of a pizza and its area and the other between the area of a pizza and its radius. Students also develop formulas and procedures-stated in words and symbols-for finding areas and perimeters of rectangles, parallelograms, triangles, and circles.
- In Bits and Pieces I, II and III (Grade 6), students learn, through fact families, that addition and subtraction are inverse operations and that multiplication and division are inverse operations. This is a fundamental idea in equation solving. They use these ideas to find a missing factor or addend in a number sentence.
- In Data About Us (Grade 6), students repre- sent and interpret graphs for the relationship between variables, such as the relationship between length of an arm span and height of a person, using words, tables, and graphs.
- In Accentuate the Negative (Grade 7), students explore properties of real numbers, including the commutative, distributive, and inverse properties. They use these properties to find a missing addend or factor in a number sentence.
- In Filling and Wrapping (Grade 7), students develop formulas and procedures-stated in words and symbols-for finding surface area and volume of rectangular prisms, cylinders, cones, and spheres.
Developing Functions
In a problem-centered curriculum, quantities (variables) and the relationships between variables naturally arise. Representing and reasoning about patterns of change becomes a way to organize and think about algebra. Looking at specific patterns of change and how this change is represented in tables, graphs, and symbols leads to the study of linear, exponential, and quadratic relationships (functions).
Linear Functions
In Moving Straight Ahead, students investigate linear relationships. They learn to recognize linear relationships from patterns in verbal, tabular, graphical, or symbolic representations. They
also learn to represent linear relationships in a variety of ways and to solve equations and make predictions involving linear equations and functions. Problem 1.3 illustrates the kinds of questions
students are asked when they meet a new type of relationship or function-in this case, a linear relationship. In this problem students are looking at three pledge plans that students suggest for a
walkathon.
Moving Straight Ahead. p. 9
Whereas many algebra texts choose to focus almost exclusively on linear relationships, in Connected Mathematics students build on their knowledge of linear functions to investigate other patterns of change. In particular, students explore inverse variation relationships in Thinking With Mathematical Models, exponential relationships in Growing, Growing, Growing, and quadratic relationships in Frogs, Fleas, and Painted Cubes. Examples are given below which illustrate the different types of functions students investigate and some of the questions they are asked about these functions. By contrasting linear relationships with exponential and other relationships, students develop deeper understanding of linear relationships.
Inverse Functions
In Thinking With Mathematical Models, students are introduced to inverse functions.
Thinking With Mathematical Models. p. 32
Exponential Functions
In Growing, Growing, Growing, students are given the context of a reward figured by placing coins called rubas on a chessboard in a particular pattern, which is exponential. The coins are
placed on the chessboard as follows.
Place 1 ruba on the first square of a chessboard, 2 rubas on the second square, 4 on the third square, 8 on the fourth square, and so on, until you have covered all 64 squares. Each square should have twice as many rubas as the previous square.
In this problem students use tables, graphs, and equations to examine exponential relationships and describe the pattern of change for this relationship.
Growing, Growing, Growing. p. 7
Quadratic Functions
In Problem 1.3 from Frogs, Fleas and Painted Cubes, students use tables, graphs, and equations to examine quadratic relationships and describe the pattern of change for this relationship.
Frogs, Fleas and Painted Cubes. p. 10
As students explore a new type of relationship, whether it is linear, quadratic, inverse, or exponential, they are asked questions like these:
- What are the variables? Describe the pattern of change between the two variables.
- Describe how the pattern of change can be seen in the table, graph, and equation.
- Decide which representation is the most helpful for answering a particular question. (see Question D in Problem 1.3 Frogs and Fleas and Painted Cubes above)
- Describe the relationships between the different representations (table, graph, and equation).
- Compare the patterns of change for different relationships. For example, compare the patterns of change for two linear relationships, or for a linear and an exponential relationship.
Developing Symbolic Reasoning
After students have explored important relationships and their associated patterns of change and ways to represent these relationships, the emphasis shifts to symbolic reasoning.
Equivalent Expressions
Students use the properties of real numbers to look at equivalent expressions and the information each expression represents in a given context and to interpret the underlying patterns that a symbolic
statement or equation represents. They examine the graph and table of an expression as well as the context the expression or statement represents. The properties of real numbers are used extensively
to write equivalent expressions, combine expressions to form new expressions, predict patterns of change, and to solve equations. Say It With Symbols pulls together the symbolic reasoning
skills students have developed through a focus on equivalent expressions. It also continues to explore relationships and patterns of change. Problem 1.1 in Say It With Symbols introduces
students to equivalent expressions.
Say It With Symbols. p. 6
In Problem 2.1 students revisit Problem 1.3 from Moving Straight Ahead (see above) to combine expressions. They also use the new expression to find information and to predict the underlying pattern of change associated with the expression.
Say It With Symbols. p. 24
Solving Equations
Equivalence is an important idea in algebra. A solid understanding of equivalence is necessary for understanding how to solve algebraic equations. Through experiences with different functional
relationships, students attach meaning to the symbols. This meaning helps student when they are developing the equation-solving strategies integral to success with algebra.
In CMP, solving linear equation is an algebra idea that is developed across all three grade levels, with increasing abstraction and complexity. In grade six, students write fact families to show the inverse relationships between addition and subtraction and between multiplication and division. The inverse relationships between operations are the fundamental basis for equation solving. Students are exposed early in sixth grade to missing number problems where they use fact families. Below is a description of fact families and a few examples of problems where students use fact families to solve algebraic equations in grades 6 and 7. These experiences precede formal work on equation solving.
In Bits and Pieces II (Grade 6), Bits and Pieces III (Grade 6), and Accentuate the Negative (Grade 7), students use fact families to find missing addends and factors.
Bits and Pieces II. p. 22
Bits and Pieces III. p. 28
Accentuate the Negative. p. 30
In Variables and Patterns (Grade 7), students solve linear equations using a variety of methods including graph and tables. As students move through the curriculum, these informal equation- solving experiences prepare them for the formal symbolic methods, which are developed in Moving Straight Ahead (Grade 7), and revisited throughout the five remaining algebra units in eighth grade.
Moving Straight Ahead. p. 85
Say It With Symbols (Grade 8), pulls together the symbolic reasoning skills students have developed through a focus on equivalent expressions and on solving linear and quadratic equations.
Say It With Symbols. p. 42
Shapes of Algebra (Grade 8), explores solving linear inequalities and systems of linear equations and inequalities. By the end of Grade 8, students in CMP should be able to analyze situations involving related quantitative variables in the following ways:
- identify variables
- identify significant patterns in the relationships among the variables
- represent the variables and the patterns relating these variables using tables, graphs, symbolic expressions, and verbal descriptions
- translate information among these forms of representation
Students should be adept at identifying the questions that are important or interesting to ask in a situation for which algebraic analysis is effective at providing answers. They should develop the skill and inclination to represent information mathematically, to transform that information using mathematical techniques to solve equations, create and compare graphs and tables of functions, and make judgments about the reasonableness of answers, accuracy, and completeness of the analysis
