In grades 5-8, the mathematics curriculum should include explorations of algebraic concepts and processes so that students can-

• understand the concepts of variable, expression, and equation;
• represent situations and number patterns with tables, graphs, verbal rules, and equations and explore the interrelationships of these representations;
• analyze tables and graphs to identify properties and relationships;
• develop confidence in solving linear equations using concrete, informal, and formal methods;
• investigate inequalities and nonlinear equations informally;
• apply algebraic methods to solve a variety of real-world and mathematical problems.

Focus

The middle school mathematics curriculum is, in many ways, a bridge between the concrete elementary school curriculum and the more formal mathematics curriculum of the high school. One critical transition is that between arithmetic and algebra. It is thus essential that in grades 5-8, students explore algebraic concepts in an informal way to build a foundation for the subsequent formal study of algebra. Such informal explorations should emphasize physical models, data, graphs, and other mathematical representations rather than facility with formal algebraic manipulation. Students should be taught to generalize number patterns to model, represent, or describe observed physical patterns, regularities, and problems. These informal explorations of algebraic concepts should help students to gain confidence in their ability to abstract relationships from contextual information and use a variety of representations to describe those relationships.

Activities in grades 5-8 should build on students' K-4 experiences with patterns. They should continue to emphasize concrete situations that allow students to investigate patterns in number sequences, make predictions, and formulate verbal rules to describe patterns. Learning to recognize patterns and regularities in mathematics and make generalizations about them requires practice and experience. Expanding the amount of time that students have to make this transition to more abstract ways of thinking increases their chances of success. By integrating informal algebraic experiences throughout the K-8 curriculum, students will develop confidence in using algebra to represent and solve problems. In addition, by the end of the eighth grade, students should be able to solve linear equations by formal methods and some nonlinear equations by informal means.

Discussion

Understanding the concept of variable is crucial to the study of algebra; a major problem in students' efforts to understand and do algebra results from their narrow interpretation of the term. Many students assign a numerical value to a letter from the start; others ignore the letter, and still others treat the letter as shorthand for an object (b means boy rather than number of boys). Students need to be able to use variables in many ways. Two particularly important ways in grades 5-8 are using a variable as a placeholder for a specific unknown, as in n + 5 = 12, and as a representative of a range of values, as in 3t + 6. Students who work with computers are likely to encounter the replacement use of variables. Students need to tell from the context how a variable is being used.

Giving students opportunities to explore interesting problems, applications, and situations does not guarantee that they will make the appropriate connections; it is inevitable that some students might lose sight of the important mathematical ideas that underlie any activity. They need to be encouraged and helped to reflect on their explorations and summarize concepts, relationships, processes, and facts that have emerged from their discussions.

The following example illustrates how students can develop a sophisticated understanding of how algebra can be used to model situations and how the algebraic model is related to other models or representations:

Working with square tiles, students can explore the question, "Can you add tiles to this figure [see fig. 9.1] to make a new figure with a perimeter of 18 units?" (Tiles must touch each other along an entire edge.)

Fig. 9.1. Tile shapes

Students can discover many interesting facts and relationships in exploring this problem. They can discover that adding a tile to fill in a corner where it will touch other tiles along two edges does not change the perimeter at all; that adding a tile that touches another tile along one edge changes the perimeter by exactly two units; and that adding a tile so that it touches three edges actually reduces the perimeter. The students can write algebraic expressions to summarize their discoveries, for example, p + 2 or p - 2 for adding tiles that touch one or three edges, respectively.

Once the class has found different ways to add tiles to make a new figure with a perimeter of 18, students can explore other problems, such as determining the fewest number of tiles that can be added. Students discover that at least three tiles must be added before the perimeter will total 18 units. A question about the greatest number of tiles that can be added to reach a perimeter of 18 units leads to an interesting discovery: A rectangle that is 4 tiles by 5 tiles uses the most tiles. This raises the question, What other rectangles have perimeters of 18 units? Collecting and organizing the class data yields a table of values for further investigation.

Fifth-grade students might cut physical representations of the rectangles from grid paper. These can be stacked on a second sheet of grid paper to produce a physical graph of the relationship between length and width (see fig. 9.2).

Fig. 9.2. Rectangles

In later grades, students can locate points representing the length versus the width of each rectangle (see fig. 9.3). Teachers can ask such questions as, Does it make sense to connect the points in the graph? How would you interpret the points where the line intersects the horizontal axis? If you can use fractional measures for the sides of the rectangle with a perimeter of 18 units, what width would you expect to find for a length between 4 and 5 units? What would you say about its area? Students can make a graph to picture the relationship between the length and the area (see fig. 9.4).

Fig. 9.3. Width versus length

Fig. 9.4. Area versus length

Other questions to explore include, What happens to the curve at the top of the graph? Can you find the rectangle with the maximum area for a fixed perimeter of 18 units? These questions can help students discover that the rectangle with the maximum area for a fixed perimeter is a square. Here the students have explored in a concrete way an idea that is fundamental to calculus: finding maximum and minimum values.

From these concrete, numerical, and graphic models emerges an algebraic model of the relationships, 2L + 2W = 18 or W = 9 - L and the rule for area of a rectangle, A = LW. Thus A = L(9 - L) or A = 9L - L². Connecting A = 9L - L squared to the graph of the data points for different values of L is a very important result of this exploration and discussion.

Relating models to one another builds a better understanding of each. Every step in these examples helps students develop an understanding of symbolic representation: exploring a concrete situation to determine patterns, constructing a table of data, looking for ways to generalize the situation described by the table, asking questions about how the variables are related, making a graphical representation, and looking for maximum and minimum points, or points where the graph intersects the axes. Different problem situations call for different approaches. Appropriate situations for exploration can arise from mathematical or real-world contexts. In either case, problem solving of this sort enables students to develop confidence in their ability to use algebraic ideas.

Many informal ways of studying algebraic expressions and solving linear equations can arise in such contexts. In the number trick illustrated in Standard 2 (see fig. 2.1 ), tiles represent the "any number" notion of variable and sets of beans represent known numbers. Variables can be introduced as names for the tiles. Tile and bean combinations can be transformed: Pick a tile, add 5, double the result. Students should act out these directions and be able to describe them in words and symbols: n + 5; 2 x n + 10. The association of language, materials, and actions builds intuitions of algebra. Such algebraic and "tile" expressions can be assigned different values by asking each student in a group to give a value for "any number" (the tile) and then evaluate the expression. This can lead to guess-and-test equation solving with students challenged to find the value of one tile if two tiles plus 10 is 20.

A computer also can be very effective in exploring relationships between two expressions in a real-world context. In a problem about newspaper deliveries, P represents the number of papers delivered. One paper pays 50 cents a day plus 8 cents for each paper delivered; another pays 10 cents for each paper delivered. The computer can generate a set values for P and related values for 8 x P + 50 and 10 x P, leading to the discussion and solving of 8 x P + 50 = 10 x P, or 8 x P + 50 < 10 x P in informal ways.