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Students in the middle grades should
learn algebra both as a set of concepts and competencies tied to the representation
of quantitative relationships and as a style of mathematical thinking
for formalizing patterns, functions, and generalizations. In the middle
grades, students should work more frequently with algebraic symbols than
in lower grades. It is essential that they become comfortable in relating
symbolic expressions containing variables to verbal, tabular, and graphical
representations of numerical and quantitative relationships. Students
should develop an initial understanding of several different meanings
and uses of variables through representing quantities in a variety of
problem situations. They should connect their experiences with linear
functions to their developing understandings of proportionality, and they
should learn to distinguish linear relationships from nonlinear ones.
In the middle grades, students should also learn to recognize and generate
equivalent expressions, solve linear equations, and use simple formulas.
Whenever possible, the teaching and learning of algebra can and should
be integrated with other topics in the curriculum.
The study of patterns and relationships in the middle grades should focus on patterns that relate to linear functions, which arise when there is a constant rate of change. Students should solve problems in which they use tables, graphs, words, and symbolic expressions to represent and examine functions and patterns of change. For example, consider the following problem:
Charles saw advertisements for two cellular telephone companies. Keep-in-Touch offers phone service for a basic fee of $20.00 a month plus $0.10 for each minute used. ChitChat has no monthly basic fee but charges $0.45 a minute. Both companies use technology that allows them to charge for the exact amount of time used; they do not "round up" the time to the nearest minute, as many of their competitors do. Compare these two companies' charges for the time used each month.
Students might begin by making a table, picking convenient numbers of minutes, and finding the corresponding costs for the two companies, as shown in figure 6.8a. Using a graphing calculator, students might then plot the points as ordered pairs (minutes, cost) on the coordinate plane, obtaining a graph for each of the two companies (see fig. 6.8b). Some students might describe the pattern in each graph verbally: "Keep-in-Touch costs $20.00 and then $0.10 more per minute." Others might write an equation to represent the cost (y) in dollars in terms of the number of minutes (x), such as y = 20.00 + 0.10x.
 
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Before the students solve the problem, a teacher might ask them to use their table and graph to focus on important basic issues regarding » the relationships they represent. By asking, "How much would each company charge for 25 minutes? For 100 minutes?" the teacher could find out if students can interpret and extend the patterns. Since the table identifies only a small number of distinct points, a teacher could ask why it is legitimate to connect the points on the graph to make a line. Students might also be asked why one graph (for ChitChat) includes the origin but the other (for Keep-in-Touch) does not (see fig. 6.8b). Most students will recognize that the ChitChat graph includes the origin because there is no charge if no calls are made but the Keep-in-Touch graph includes (0, 20) because the company charges $20.00 even if the telephone is not used.
Many students will naturally seek a formula to express these patterns, but questions such as the following would be a good catalyst for others: How can you find the cost for any number of minutes for the Keep-in-Touch plan? For the ChitChat plan? What aspects of the stated price schedule are indicated in the graph? How? Students are likely to observe the constant difference between both the successive entries in the table and the coordinates of the points for each company along a straight line. They may explain the pattern underlying the function by saying, "Whenever you talk for one more minute, you pay $0.10 more (or $0.45 more), so the points go up the same amount each time." Others might say that a straight line is reasonable because each company charges a constant amount for each minute. Teachers should encourage students to explain their observations in their own words. Their explanations will provide the teacher with important insights into the students' thinking, particularly how well they recognize and represent linear relationships.
A solution to the stated problem
requires comparing data from the two companies. A teacher might want to
ask additional questions about this comparison: Which company is cheaper
if you use the telephone infrequently? If you use it frequently? If you
cannot spend more than $50.00 in a month but you want to talk for as many
minutes as possible, which company would be the better choice? Considering
questions such as these can lay the groundwork for a pivotal question:
Is there a number of minutes that costs the same for both companies? Such
questions could give rise to many observations. For example, most students
will notice in their table that something important happens between 50
and 60 minutes, namely, using ChitChat becomes more expensive than using
Keep-in-Touch. From the graph, some students may observe that this shift
occurs at about 57 minutes: Keep-in-Touch is the cheaper company when
a customer uses more than 57 minutes in a month. Experiences such as this
can lay a foundation for solving systems of simultaneous equations.
The problem could also easily be extended or adapted in ways that would draw students' attention to important characteristics of the line graph for each company's charges. For example, to draw attention to the y-intercept, students could be asked to use a graphing calculator to examine how the graph would be affected if Keep-in-Touch increased or decreased its basic fee or if ChitChat decided to begin charging a basic fee. Students' attention could be drawn to the slope by asking them to consider the steepness of the lines using a question such as, What happens to the graph for Keep-in-Touch if the company increases its cost per minute from $0.10 to $0.15? Through experiences such as these, students should develop a general understanding of, and » facility with, slope and y-intercept and their manifestations in tables, graphs, and equations.
The problem could also easily be extended to nonlinear relationships if, for instance, the companies did not charge proportionally for portions of minutes used. If they rounded to the nearest minute, then the cost for each company would be graphed as a step function rather than a linear function. In another variation, a nonlinear pricing scheme for a third company could be introduced.
Another important topic for class
discussion is comparing and contrasting the merits of graphical, tabular,
and symbolic representations in this example. A teacher might ask, "Which
helps us see better the point at which the two companies switch position
and Keep-in-Touch becomes the more economical—a table or a graph?"
"Is it easier to see the rate per minute from the graph or from the equation?"
or "How can you determine the rate per minute from the table?" Through
discussion, students can identify the strengths and the limitations of
different forms of representation. Graphs give a picture of a relationship
and allow the quick recognition of linearity when change is constant.
Algebraic equations typically offer compact, easily interpreted descriptions
of relationships between variables.
Working with variables and equations is an important part of the middle-grades curriculum. Students' understanding of variable should go far beyond simply recognizing that letters can be used to stand for unknown numbers in equations (Schoenfeld and Arcavi 1988). The following equations illustrate several uses of variable encountered in the middle grades:
27 = 4x + 3
1 = t(1/t)
A = LW
y = 3x
The role of variable as "place
holder" is illustrated in the first equation: x is simply taking
the place of a specific number that can be found by solving the equation.
The use of variable in denoting a generalized arithmetic pattern is shown
in the second equation; it represents an identity when t takes
on any real value except 0. The third equation is a formula, with A,
L, and W representing the area, length, and width, respectively,
of a rectangle. The third and fourth equations offer examples of covariation:
in the fourth equation, as x takes on different values, y
also varies.
A problem such as the one in figure
6.9 (adapted from Educational Development Center, Inc. 1998, p. 41) could
give students valuable experience in deciding whether two expressions
are equivalent. A teacher might encourage students to begin solving this
problem by drawing several more boxes of various sizes so they can look
for a pattern. Some students will probably note that the caramels are
also arranged in a rectangular pattern, which is narrower and shorter
than the rectangular arrangement of chocolates. Using this observation,
they might report, "To find the length and width of the caramel rectangle,
take 1 off the length and 1 off the width of the chocolate rectangle.
Multiply the length and width of the caramel rectangle to find the number
of caramels." If L and W are the dimensions of the
array of chocolates, and C is the number of caramels, then this
generalization could be expressed symbolically as C = (L – 1)(W – 1).
Other students might find and use the number of chocolates to find the
number of caramels. For example, for a 3
5 box of chocolates,
they might propose starting with the 15 chocolates, then taking off 3
because "there is one less column of caramels" and then taking off 5 "because
there is one less row of caramels." This could be expressed generally
as C = LW – L – W. Although
both expressions for the number of caramels are likely to seem reasonable
to many students, they do not yield the same answer. For the 35 array, the first
produces the correct answer, 8 caramels. The second gives the answer 7
caramels. Either examining a few more boxes with different dimensions
or reconsidering the process represented by the second equation would
confirm that the second equation needs to be corrected by adding 1 to
obtain C = LW – L – W + 1. The algebraic
equivalence of (L – 1)(W – 1) and LW – L – W
+ 1 can be demonstrated in general using the distributive property of
multiplication over subtraction.
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Through a variety of experiences
such as these, students can learn the strengths and limitations of various
methods for checking the equivalence of expressions. In some instances,
the equivalence of algebraic expressions can be demonstrated geometrically;
see the "Geometry" section of this chapter for a demonstration that (a + b)2 = a2 + 2ab + b2.
Most middle-grades students will
need considerable experience with linear equations before they will be
comfortable and fluent in transforming or solving them. Although students
will probably acquire facility with equations at different times during
the middle grades, by the end of grade 8, students should be able to solve
equations like 84 – 2x = 5x + 12 for
the unknown number, to recognize as identities such equations as 1 = t(1/t)
(when t is not 0), to apply formulas such as V = 
r2h,
and to recognize that equations such as y = –3x
+ 10 represent linear functions that are satisfied by many ordered
pairs (x, y). Students should be able to use equations of the
form y = mx + b to represent linear relationships,
and they should know how the values of the slope (m) and the
y-intercept (b) » affect
the line. For example, in the "cellular telephone" problem discussed earlier,
they should recognize that y = 0.10x + 20 and y
= 0.45x are both linear equations, that the graph of the
latter will be steeper than that of the former, and that the former intersects
the y-axis at (0, 20) rather than at the origin.
Students' facility with symbol
manipulation can be enhanced if it is based on extensive experience with
quantities in contexts through which students develop an initial understanding
of the meanings and uses of variables and an ability to associate symbolic
expressions with problem contexts. Fluency in manipulating symbolic expressions
can be further enhanced if students understand equivalence and are facile
with the order of operations and the distributive, associative, and commutative
properties.
A major goal in the middle
grades is to develop students' facility with using patterns and functions
to represent, model, and analyze a variety of phenomena and relationships
in mathematics problems or in the real world. With computers and graphing
calculators to produce graphical representations and perform complex calculations,
students can focus on using functions to model patterns of quantitative
change. Students should have frequent experiences in modeling situations
with equations of the form y = kx, such as relating
the side lengths and the perimeters of similar shapes. Opportunities can
be found in many other areas of the curriculum; for example, scatterplots
and approximate lines of fit can model trends in data sets. Students also
need opportunities to model relationships in everyday contexts, such as
the "cellular telephone" problem.
Students also should have experience in modeling situations and relationships with nonlinear functions, such as compound-interest problems, the relationship between the length of the radius of a circle and the area of the circle, or situations like the one in figure 6.10. If students have only a few points to examine, it can be difficult to see that » the graph for this problem is not linear. As more points are graphed, however, the curve becomes more apparent. Students could use graphing calculators or computer graphing tools to do problems such as this.
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When doing experiments or dealing
with real data, students may encounter "messy data," for which a line
or a curve may not be an exact fit. They will need experience with such
situations and assistance from the teacher to develop their ability to
find a function that fits the data well enough to be useful as a prediction
tool. In their later study of statistics, students may learn sophisticated
methods to determine lines of best fit for data. In the middle grades
when students encounter a set of points suggesting a linear relationship,
they can simply use a ruler to try several lines until they find one that
appears to be a good fit and then » write
an equation for that line. An example of this sort of activity, related
to a scatterplot of measurements, can be found in the "Data Analysis and
Probability" section of this chapter. With a graphing calculator or computer
graphing software, students can test some conjectures more easily than
with paper-and-pencil methods.
In their study of algebra, middle-grades students should encounter questions that focus on quantities that change. Recall, for example, that ChitChat charges $0.45 a minute for phone calls. The cost per minute does not change, but the total cost changes as the telephone is used. This can be seen quite readily from the two graphs in figure 6.11. The meaning of the term flat rate can be seen in the cost-per-minute graph, which shows points along a horizontal line at y = 0.45, representing a constant rate of $0.45 a minute. The total-cost graph shows points along a straight line that includes the origin and has a slope of 0.45.
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Students may be confused when they first encounter two different graphs to represent different relationships in the same situation. Teachers can assist students in understanding what relationships are represented in the two graphs by asking such questions as these: When the number of minutes is 4, what do the values of the corresponding point on each graph represent? When the number of minutes is 8? Why is the y-value of the cost-per-minute graph constant at 0.45? How much does the total-cost graph increase from 5 minutes to 6 minutes? Why? How would each graph change, if at all, if the cost per minute were changed to $0.20?
A slight modification of the problem,
such as the addition of a third telephone company with a different pricing
scheme, can allow the analysis of change in nonlinear relationships:
p. 229
Quik-Talk advertises monthly cellular phone service for $0.50 a minute for the first 60 minutes but only $0.10 a minute for each minute thereafter. Quik-Talk also charges for the exact amount of time used. »
A teacher might ask students to graph the rates of change in this example. Figure 6.12 shows the cost-per-minute graph and a total-cost graph for Quik-Talk's pricing scheme. Students can answer questions about the relationships represented in the graphs: Why does the cost-per-minute graph consist of two different line segments? How can we tell from the graph that the pricing scheme changes after 60 minutes? Why is part of the total-cost function steeper than the rest of the graph?
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Comparing the cost-per-minute graph to the total-cost graph for phone calls can help students develop a clearer understanding of the relationship between change (cost per minute) and accumulation (total cost of calls). These concepts are precursors to the later study of change in calculus.
Students' examination of graphs of change and graphs of accumulation can be facilitated with specially designed computer software. Such software allows students to change either the number of minutes used in one month by dragging a horizontal "slider" (see fig. 6.13) or the cost per minute by dragging a vertical slider. They can then observe the corresponding changes in the graphs and in the symbolic expression for the relationship. Technological tools can also help students examine the nature of change in many other settings. For example, students could examine distance-time relationships using computer-based laboratories, as discussed in the "Measurement" section of this chapter. Such experiences with appropriate technology, supported by careful planning by teachers and interactions with classmates, can help students develop a solid understanding of some fundamental notions of change.
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