ALGEBRA Concepts Welcome back, students!

Standards  Algebra is one of the five content strands of Principles and Standards and is increasingly appearing as a strand on state standards lists..

Ohio State Standards Reminder: Ohio’s academic content standards in mathematics are made up of six standards.  Number, Number Sense and Operations Standard  Measurement Standard  Geometry and Spatial Sense Standard  Patterns, Functions and Algebra Standard  Data Analysis and Probability Standard  Mathematical Process Standard

Generalizations and Symbolism  A correct understanding of the equal sign is the first form of symbolism that is addressed.  The development follows many of the ideas found in the work of Tom Carpenter and his colleagues in the book Thinking Mathematically (Heinemann, 2003).  The failure of the curriculum to construct an accurate understanding of the equal sign has been well documented in the research dating back to the 1970s and yet little has been done to correct this failing.  Students from grade 1 through middle school continue to think of = as a symbol that separates problem from answer. The ideas in the first section of the chapter in your text (chapter #15 in the 6 th edition) should not be viewed as belonging only in the early childhood curriculum. The methods and ideas are appropriate for and adaptable to any level.

Generalization and Symbolism (cont’d)  Out of an exploration of open sentences an understanding of variable can and should develop.  In this chapter the uses of variable have been restricted 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.

Making Structure Explicit  A recent focus of research in algebraic reasoning is on structure or generalizations based on properties of our number system.  Generalization is again the main interest. For example, when students note that 4 x 7 = 7 x 4, what helps them to understand that this is true for all numbers?  Further, how do students go about "proving" that properties such as the commutative property or relationships on odd and even numbers are always true?  By helping students make conjectures about the truth of equations and open sentences, they can then be challenged to make decisions about the truth of these conjectures for all students.

Patterns  Extending, inventing and observing patterns, and being able to match patterns formed of different physical materials yet logically alike (isomorphic), is certainly doing mathematics.  To the extent that these patterns can be generalized and described symbolically (A-B- B-A-B-B-…) place them squarely in the realm of algebraic reasoning.  Examining pattern in numeric situations such as number sequences or the hundreds chart are another form of early pattern exploration.

Patterns  Although mathematically interesting themselves, growing patterns are used as a vehicle for developing early concepts of function.  Growing patterns now command a larger presence in the intermediate and middle-grades curriculum.  The discussion distinguishes between recursive relationships and functional relationships. It also introduces the notion of different representations for the functional relationships found.

Functions and Representations of Functions There are five different representations that can be used to help make the function concept meaningful to students  Contextual representation (pattern)  Table representation  Symbolic equation representation  Graphical representation  Equations to represent functions-- modeling (language)  It is important that students see functions in all of these representations and are able to see how each is a different way of seeing the same functional relationship.

Sources of Functional Relationships in the Classroom  Not only do different contextual situations provide interest, each different context is another example of mathematizing our world - mathematical modeling - the fifth type of algebraic thinking

Real World Concepts  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.

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  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.

Application Activity #1  Website: Traffic Jam Using the Java applet, investigate the "Traffic Jam" task. What was your strategy for having the people exchange places in the smallest number of moves? What pattern can you find to determine the number of moves for any number of people? Traffic Jam

Application Activity #2  Website: Understanding Distance, Speed, and Time Relationships Using Simulation Software Explore the e-example, "Understanding Distance, Speed, and Time Relationships Using Simulation Software." Set up several trials using the simulation applet. What big ideas about functions and representing change over time would you anticipate that students might learn while working on this activity? Understanding Distance, Speed, and Time Relationships Using Simulation Software

Let’s Have a Great Rest of the School Year!