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Algebra is one of the five content strands of *Principles and Standards* and is increasingly appearing as a strand on state standards lists. Exactly what this means from a state viewpoint is not at all uniform so you may be influenced in how you approach this topic by what your state requires. One goal of this chapter is to help develop a useful concept of what algebra or algebraic thinking is about at the K to 8 level.

In my search for an appropriate perspective on K-8 algebra the writings of the late James Kaput consistently offered what I believe is the most useful organization of algebraic thinking is found in the five forms of algebraic reasoning that he described (T-79). Kaput emphasized that these are not separate components of the curriculum. He saw the making of generalizations and a meaningful use of symbolism as supporting all of algebraic reasoning and mathematics in general. These ideas are addressed in the first section of the chapter with specific attention to the equal sign, variable, and equation solving. The next two components - structure, and patterns and functions - Kaput saw as separate strands to be addressed specifically. In this chapter you will find a full section on helping students make structure explicit and beginning to prove generalizations based on structure. The remainder and bulk of the chapter develops ideas of patterns and functions. Kaput's fifth form of algebraic reasoning, mathematical modeling, is "a web of languages and permeates all the others" (Kaput, 1999, p. 135). While modeling is explicitly mentioned at the end of the chapter, much of this aspect of algebraic reasoning is found in the sections on functions.

And so, although Kaput's description of algebraic thinking guided the organization of this chapter, it is not explicitly found in the headings used throughout. You may have to help your students see how these components build together as you work through the chapter.

Out of an exploration of open sentences an understanding of variable can and should develop. In this chapter I have restricted the uses of variable to two: as an unknown value and as a quantity that varies. The correct use of the equal sign with variables that vary are essential symbolisms used throughout the remaining components of algebraic reasoning - especially in the areas of functions and mathematical modeling. The use of variables as representing specific unknown values leads to a need to solve equations and techniques there of.

Growing patterns now command a larger presence in the intermediate and middle-grades curriculum. Although mathematically interesting themselves, I see growing patterns as a vehicle for developing early concepts of function. The discussion distinguishes between recursive relationships and functional relationships. It also introduces the notion of different representations for the functional relationships found. These relationships are explored further as the concept of function is expanded in the rest of the chapter.

In your exploration, technology should be used whenever possible. The graphing calculator is more than adequate and I strongly suggest that every student have access to one if possible even if just for this chapter.

Real-World Contexts. Several other categories have "real" contexts. What sets these apart is that all the examples result in clearly defined formulas. This is in contrast to scatter plots of real data. Proportional Situations. Of course these are also real world but it is useful to separate these out because they make an important connection to the key idea of proportional reasoning. Note again that all proportional relationships have linear graphs that pass through the origin. This was also noted in Chapter 18. Formulas and Max/Min Problems. Here the connection is to measurement and provides some interesting examples of functions that students can get their hands on. Scatter Plot Data. The idea of a best-fit line is developed completely in Chapter 22. It can just as well be introduced here. A best-fit line is a functional relationship that approximates real-world phenomena that do not strictly obey the relationship. If found with a graphing calculator, then the equation and chart are also immediately available, even for non-linear relationships. An important idea here is that the function found as the best-fit line is a mathematical model for the real-world situation. It is important for students to see that mathematical models often do not reflect the wide degrees of variation that occur in real relationships. Contrast these situations with those in the section on Real-World Contexts. Fun Experiments. There are no new ideas here, only a collection of fun things to do in the classroom. Depending on the situation, there may or may not be a nice theoretical model to match the experimental data. Generalizations About Functions Rate of change has been an underlying concept in all discussions from growing patterns forward. In this section, this idea is developed more explicitly as students begin to explore the rise and fall of graphs without equations. The slope concept is then used to create the general category of linear functions. Slope is defined in the usual manner found in early algebra courses. Here the intent is to keep the discussion informal but you can make it as technical as you wish.