Although algebra is a word that has not commonly been heard in grades 35 classrooms, the mathematical investigations and conversations of students in these grades frequently include elements of algebraic reasoning. These experiences and conversations provide rich contexts for advancing mathematical understanding and are also an important precursor to the more formalized study of algebra in the middle and secondary grades. In grades 35, algebraic ideas should emerge and be investigated as students
In grades 35, students should investigate numerical and geometric patterns
and express them mathematically in words or symbols. They should analyze
the structure of the pattern and how it grows or changes, organize this
information systematically, and use their analysis to develop generalizations
about the mathematical relationships in the pattern. For example, a teacher
might ask students to describe patterns they see in the "growing squares"
display (see fig. 5.3) and express the patterns in mathematical sentences.
Students should be encouraged to explain these patterns verbally and to
make predictions about what will happen if the sequence is continued.
In this example, one student might notice that the area changes in a predictable wayit increases by the next odd number with each new square. Another student might notice that the previous square always fits into the "corner" of the next-larger square. This observation might lead to a description of the area of a square as equal to the area of the previous square plus "its two sides and one more." A student might represent his thinking as in figure 5.4.»
As they study ways to measure geometric objects, students will have opportunities to make generalizations based on patterns. For example, consider the problem in figure 5.5. Fourth graders might make a table (see fig. 5.6) and note the iterative nature of the pattern. That is, there is a consistent relationship between the surface area of one tower and the next-bigger tower: "You add four to the previous number." Fifth graders could be challenged to justify a general rule with reference to the geometric model, for example, "The surface area is always four times the number of cubes plus two more because there are always four square units around each cube and one extra on each end of the tower." Once a relationship is established, students should be able to use it to answer questions like, "What is the surface area of a tower with fifty cubes?" or "How many cubes would there be in a tower with a surface area of 242 square units?"
In this example, some students may use a table to organize and order their data,
and others may use connecting cubes to model the growth of an arithmetic
sequence. Some students may use words, but others may use numbers and
symbols to express their ideas about the functional relationship. Students
should have many experiences organizing data and examining different representations.
Computer simulations are an interactive way to explore functional relationships
and the various ways they are represented. In a simulation of two runners
along a track, students can control the speed and starting point of the
runners and can view the results by watching the race and examining a
table and graph of the time-versus-distance relationship. Students need
to feel comfortable using various techniques for organizing and expressing
ideas about relationships and functions.
In grades 35, students can investigate properties such as commutativity, associativity, and distributivity of multiplication over addition. Is 35 the same as 53? Is 1527 equal to 2715? Will reversing the factors always result in the same product? What if one of the factors is a decimal number (e.g., 1.56)? An area model can help students see that two factors in either order have equal products, as represented by congruent rectangles with different orientations (see fig. 5.7). »
At this grade band the idea and usefulness of a variable (represented
by a box, letter, or symbol) should also be emerging and developing more
fully. As students explore patterns and note relationships, they should
be encouraged to represent their thinking. In the example showing the
sequence of squares that grow (fig. 5.3), students are beginning to use
the idea of a variable as they think about how to describe a rule for
finding the area of any square from the pattern they have observed. As
students become more experienced in investigating, articulating, and justifying
generalizations, they can begin to use variable notation and equations
to represent their thinking. Teachers will need to model how to represent
thinking in the form of equations. In this way, they can
» help students connect the ways they are describing their
findings to mathematical notation. For example, a student's description
of the surface area of a cube tower of any size ("You get the surface
area by multiplying the number of cubes by 4 and adding 2") can be recorded
by the teacher as S = 4n + 2.
Students should also understand the use of a variable as a placeholder
in an expression or equation. For example, they should explore the role
of n in the equation 8015 = 40n and be
able to find the value of n that makes the equation true.
Historically, much of the mathematics used today was developed to model real-world situations, with the goal of making predictions about those situations. As patterns are identified, they can be expressed numerically, graphically, or symbolically and used to predict how the pattern will continue. Students in grades 35 develop the idea that a mathematical model has both descriptive and predictive power.
Students in these grades can model a variety of situations, including geometric patterns, real-world situations, and scientific experiments. Sometimes they will use their model to predict the next element in a pattern, as students did when they described the area of a square in terms of the previous smaller square (see fig. 5.3). At other times, students will be able to make a general statement about how one variable is related to another variable: If a sandwich costs $3, you can figure out how many dollars any number of sandwiches costs by multiplying that number by 3 (two sandwiches cost $6, three sandwiches cost $9, and so forth). In this case, students have developed a model of a proportional relationship: the value of one variable (total cost, C) is always three times the value of the other (number of sandwiches, S), or C = 3 S.
In modeling situations that involve real-world data, students need to
know that their predictions will not always match observed outcomes for
a variety of reasons. For example, data often contain measurement error,
experiments are influenced by many factors that cause fluctuations, and
some models may hold only for a certain range of values. However, predictions
based on good models should be reasonably close to what actually happens.
Students in grades 35 should begin to understand that different
models for the same situation can give the same results. For example,
as a group of students investigates the relationship between the number
of cubes in a tower and its surface area, several models emerge. One student
thinks about each side of the tower as having the same number of units
of surface area as the number of cubes (n). There are four sides
and an extra unit on each end of the tower, so the surface area is four
times the number of cubes plus two (4 n + 2).
Another student thinks about how much surface area is contributed by each
cube in the tower: each end cube contributes five units of surface
area and each "middle" cube contributes four units of surface area. Algebraically,
the surface area would be 2 5 + (n 2) 4.
For a tower of twelve cubes, the first student thinks, "4 times 12, that's
48, plus 2 is 50." The second student thinks, "The two end cubes each
have 5, so that's 10. There are 10 » more
cubes. They each have 4, so that's 40. 40 plus 10 is 50." Students in
this grade band may not be able to show how these solutions are algebraically
equivalent, but they can recognize that these different models lead to
the same solution.
Change is an important mathematical idea that can be studied using the tools of algebra. For example, as part of a science project, students might plant seeds and record the growth of a plant. Using the data represented in the table and graph (fig. 5.9), students can describe how the rate of growth varies over time. For example, a student might express the rate of growth in this way: "My plant didn't grow for the first four days, then it grew slowly for the next two days, then it started to grow faster, then it slowed down again." In this situation, students are focusing not simply on the height of the plant each day but on what has happened between the recorded heights. This work is a precursor to later, more focused attention on what the slope of a line represents, that is, what the steepness of the line shows about the rate of change. Students should have opportunities to study situations that display different patterns of changechange that occurs at a constant rate, such as someone walking at a constant speed, and rates of change that increase or decrease, as in the growing-plant example.
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