In high school, students' understanding of number is the foundation for their understanding of algebra, and their fluency with number operations is the basis for learning to operate fluently with symbols. Students should enter high school with an understanding of the basic operations and fluency in using them on integers, fractions, and decimals. In grades 912, they will develop an increased ability to estimate the results of arithmetic computations and to understand and judge the reasonableness of numerical results displayed by calculators and computers. They should use the real numbers and learn enough about complex numbers to interpret them as solutions of quadratic equations.
High school students should understand more fully the concept of a number system,
how different number systems are related, and whether the properties of
one system hold in another. Their increased ability to use algebraic symbolism
will enable them to make generalizations about properties of numbers that
they might discover. They can study and use vectors and matrices. They
need to develop deeper understandings of counting techniques, which further
develops the conceptual underpinnings for the study of probability.
High school students should become increasingly facile in dealing with very large and very small numbers as part of their deepening understanding of number. Such numbers occur frequently in the sciences; examples are Avogadro's number (6.021023) in chemistry or the very small numbers used in describing the size of the nucleus of a cell in biology. As citizens, students will need to grasp the difference between $1 billion, the cost of a moderate-sized government project, and $1 trillion, a significant part of the national budget.
They need to become familiar with different ways of representing numbers. As part of their developing technological facility, students should become adept at interpreting numerical answers on calculator or computer displays. They should recognize 1.05168475E-12 as a very small number given in scientific notation, 6.66666667 as the approximate result of dividing 20 by 3, and ERROR as a response for either an invalid operation or a number that overflows the capacity of the device.
Students' understanding of the mathematical development of number systemsfrom
whole numbers to integers to rational numbers and then on to real and
complex numbersshould be a basis for their work in finding solutions
for certain types of equations. Students should understand the progression
and the kinds of equations that can and cannot be solved in each system.
For example, the equation 3x = 1 does not have an
integer solution but does have a rational-number solution; the equation
x3 = 2 does not
have a rational-number solution but does have a real-number solution;
and the equation x2 + 4 = 0
does not have a real-number solution but does have a complex-number solution.
Whereas middle-grades students should have been introduced to irrational numbers, high school students should develop an understanding of the system of real numbers. They should understand that given an » origin and a unit of measure, every point on a line corresponds to a real number and vice versa. They should understand that irrational numbers can only be approximated by fractions or by terminating or repeating decimals. They should understand the difference between rational and irrational numbers. Their understanding of irrational numbers needs to extend beyond and .
High school students can use their understanding of numbers to explore new systems, such as vectors and matrices. By working with examples that include forces or velocities, students can learn to appreciate vectors as a means of simultaneously representing magnitude and direction. Using matrices, students can also see connections among major strands of mathematics: they can use matrices to solve systems of linear equations, to represent geometric transformations (some of which can involve creating computer graphics), and to represent and analyze vertex-edge graphs.
Properties that hold in some systems may not hold in others. So teachers and students should explicitly discuss the associative, commutative, and distributive properties, and students should learn to examine whether those properties hold in the systems they study. The exploration of the properties of matrices may be particularly interesting, since the system of matrices is often the first that students encounter in which multiplication is not commutative.
In grades 912, students can use algebraic arguments in many areas, including in their study of number. Consider, for example, a simple number-theory problem such as the following:
What can you say about the number that results when you subtract 1 from the square of an odd integer?
It is easy to verify that the number that results is even, and that it
is divisible by 4. But if students try a few examples, such as starting
with 3 or 5, they may note that the results they obtain are divisible
by 8. They might wonder if this property will hold in general. Proving
that it does involves finding useful representations. If students decide
to express an arbitrary odd integer as 2n + 1 and the resulting
number as A, some quick computations show that A = (2n
+ 1)2 1 = 4n2
+ 4n = 4(n)(n + 1). The observation
that either n or (n + 1) must be even gives an additional
factor of 2, showing that A must be divisible by 8. Working
such problems deepens students' understanding of number while providing
practice in symbolic representation, reasoning, and proof.
As high school students' understanding of numbers grows, they should learn to consider operations in general ways, rather than only in particular computations. The questions in figure 7.1 call for reasoning about the properties of the numbers involved rather than for following procedures to arrive at exact answers. Such reasoning is important in judging the reasonableness of results. Although the questions can be approached by substituting approximate values for the numbers represented by a through h, teachers should encourage students to arrive at and justify their conclusions by thinking about properties of numbers. » For example, to determine the point whose coordinate is closest to ab, a teacher might suggest considering the sign of ab and whether the magnitude of ab is greater or less than that of b. Likewise, students should be able to explain why, if e is positioned as given in figure 7.1, the magnitude of is greater than that of e. Listening to students explain their reasoning gives teachers insights into the sophistication of their arguments as well as their conceptual understanding.
- The graphs of the functions f(x) = x, g(x) = , h(x) = x2, and j(x) = x3 are shown [in fig. 7.2]. Identify which function corresponds to which graph and explain why.
- Given f(x) = 30/x2 and a > 0, which is larger: f(a) or f(a + 2)? Explain why.
Students should also extend their understanding of operations to number systems that are new to them. They should learn to represent two-dimensional vectors in the coordinate plane and determine vector sums (see fig. 7.3). Dynamic geometry software can be used to illustrate the properties of vector addition. As students learn to represent systems of equations using matrices, they should recognize how operations on the matrices correspond to manipulations of such systems.
The organized lists and tree diagrams that students will have used in the elementary and middle grades to count outcomes or compute probabilities can be used in high school to work on permutations and combinations. Consider, for example, the task of determining how many » two-person committees can be chosen from a group of seven people. Students should learn that the tree diagram they draw to represent the number of possibilities has a multiplicative structure: there are seven main branches, representing the first choice of committee member, and six branches off each of those, representing the choice of the second member. They also need to understand that this method of enumerating committees results in "double counting": each committee of the form (person 1, person 2) is also represented as (person 2, person 1). Hence the number of two-person subcommittees is
The students should also understand and be able to explain why the number of
two-person committees is the same as the number of five-person committees
that can be chosen from a group of seven people. This kind of reasoning
provides the conceptual underpinnings for work in probability.
Students should be able to decide whether a problem calls for a rough estimate, an approximation to an appropriate degree of precision, or an exact answer. They should select a suitable method of computing from among mental mathematics, paper-and-pencil computations, and the use of calculators and computers and be proficient with each method. Electronic computation technologies provide opportunities for students to work on realistic problems and to perform difficult computations, for example, computing roots and powers of numbers or performing operations with vectors and matrices. However, students must be able to perform relatively simple mental computations as the basis for making reasonable estimates and sensible predictions and to spot potential sources of error. Suppose, for example, that a student wants to know the cube root of 49, enters the expression 49^1/3 into a calculator, and the number 16.3333333 is displayed in response. The student should note immediately that there has been an error, because the cube root of 49 should be between 3 and 4, and 16 16 16 is much larger than 49. To have the calculator compute the cube root, the student would need to have entered 49^(1/3).
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