Algebra Strand

The Algebra Strand: Developing Algebraic Reasoning in Connected Mathematics

Connected Mathematics 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 situation are understanding and predicting patterns of change in variables. The letters, symbolic equations or 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 discovered that almost all of the important tasks to which algebra is usually applied can develop naturally as aspects of this endeavor. (Fey, Phillips 2005)

Overall Goals for CMP Algebra Strand

Connected Mathematics develops four mathematical strands: Number and Operation, Geometry and Measurement, Data Analysis and Probability and Algebra. The mathematical learning goals below signify what students should be able to do in Algebra by the end of eighth grade.

Patterns of change--functions

• Identify and use variables to describe relationships between quantitative variables in order to solve problems or make decisions

• Recognize and distinguish among patterns of change associated with linear, inverse, exponential and quadratic functions

Representation

• Construct tables, graphs, symbolic expressions and verbal descriptions and use them to describe and predict patterns of change in variables

• Move easily among tables, graphs, symbolic expressions and verbal descriptions

• Describe the advantages and disadvantages of each representation and use these descriptions to make choices when solving problems

• Use linear and inverse equations and inequalities as mathematical models of situations involving variables

• Use appropriate representations to solve problems

Symbolic Reasoning

• Write equations to model problem situations

• Connect solving equations in one variable to finding specific values of functions

• Solve linear equations and inequalities and simple quadratic equations using symbolic methods

• Use the distributive and commutative properties to write equivalent expressions and equations

• Find equivalent forms of many kinds of equations and expressions, including factoring simple quadratic expressions

• Solve systems of linear equations symbolically

Algebra Units

There are 7 Algebra Units in CMP, 8 if we include Looking For Pythagoras, which is largely algebraic in focus, though the main topic is a geometric idea. In addition there are units which develop algebraic thinking, though their focus is largely on number, or geometry. For example, Accentuate the Negative introduces negative numbers, integers and rational numbers, order of operations and some number properties; Comparing and Scaling develops proportional reasoning and students learn to solve proportions is various ways in this unit. These topics are often considered part of an Algebra curriculum.

1. Variables and Patterns (7th grade)

2. Moving Straight Ahead (7th grade)

3. Thinking with Mathematical Models (8th grade)

4. Looking for Pythagoras (8th grade)

5. Growing, Growing, Growing (8th grade)

6. Frogs, Fleas and Painted Cubes (8th grade)

7. Say it with Symbols (8th grade)

8. Shapes of Algebra (8th grade)

The formal study of algebraic ideas starts in grade 7 units, but students begin to study relationships among variables in grade 6. There are opportunities in 6th and 7th grade for students to examine and formalize patterns and relationships in words, graphs, tables, and with symbols, before Variables and Patterns is studied.

Every important idea addressed in the CMP Algebra Strand is carefully placed to make initial development appropriate to student developmental level, and also to connect productively to other units already studied. For example, the idea of "Solving Equations" comes in formally with Moving Straight Ahead, but prior to this Unit students have been writing number sentences in all the fraction units in 6th grade (Bits and Pieces I, II, III), and solving for unknowns, by using fact families. This "fact family" strategy is continued in Accentuate the Negative, in which students solve equations where the solution may be a negative rational number. In Variables and Patterns students write equations for relationships between 2 variables, and tailor these 2-variable equations to reflect a 1-variable equation whose solution will answer a particular question, which they then solve using table and graph methods. By the time that students reach Moving Straight Ahead they are well prepared to write and solve linear equations, using tabular, graphical or symbolic methods.

It is important to note that many goals are revisited in later units, in the same grade level or later, either within classroom problems or in the Connections problems in the ACE homework assignments. For example, "Solving Equations" is introduced formally in Moving Straight Ahead but is revisited in Thinking With Mathematical Models, where the equations are now both linear and non-linear. In Growing, Growing, Growing students solve exponential equations; and in Frogs and Fleas students solve quadratic equations. In Say it with Symbols students solve some complex linear equations as well as quadratic equations. They apply the techniques they have learned for solving linear equations to solving linear inequalities and 2-variable systems of equations in Shapes of Algebra. Meanwhile, units that are not algebraic in focus are interspersed between these algebraic units, and connections and distributed practice of algebraic ideas continue to occur.

In order to have a clearer idea of the particular goals for each unit, the Mathematical Help section lists the goals for each unit.

Algebra Content in CMP Algebra Strand and Algebra Content of Traditional Algebra 1 Class

Parents and teachers sometimes attempt to compare what is done in an Algebra 1 class with the Algebra strand in CMP, usually with a view to making the transition to high school smooth for CMP students. With this in mind, the following table has been created, using the mathematical goals for each Investigation in CMP's Algebra units. These goal statements have been simplified to make the language more familiar to parents and guardians, and to make it easier for parents/guardians to relate these goals to the section headings in any Algebra 1 textbook. The order of the CMP units has been retained, since careful sequencing, continual refinement of ideas, and connections within and across strands are important characteristics of the CMP curriculum.

The CMP Algebra Strand includes most of the ideas in a typical Algebra 1 course, and some which are not usually included in Algebra 1.