4.2 High School (grades 10–11; students aged 16–17)
4.2.4 The final attestation in algebra for 11th graders
Upon completing their mathematics education, all graduates of sec- ondary general education schools in Russia must take the Uniform State Exam (USE). Without going into a detailed discussion on the structure and aims of the USE here,1 we will confine ourselves to describing several problems from the USE just on the two topics examined in this chapter.
The transformation of algebraic expressions.The “difficult” part (C) of the exam contains no problems specifically on this topic, although the solutions to problems in other topics require sufficient proficiency in carrying out transformations of algebraic expressions. The following problem is typical of an easier section:
Find the value of the expression
2√ 72 14 .
As can be seen, the students’ ability to operate with roots of degree n is tested on quite primitive examples. The problems have a purely technical character.
Equations and inequalities.The problems in this “easy” section on a different topic also presuppose command of the standard algorithm.
The knowledge provided by the basic course is sufficient to solve them.
Consider the following example:
Find the root of the equation√
−72−17x= −x. If the equation has more than one root, indicate the lesser of them.
1Editorial note: For a more detailed treatment of the USE, see Chapter 8 of this volume.
The following problem may serve as an example from a more difficult section of the exam (in this section, this problem is about average in difficulty):
Solve the inequality√ 7−x <
√x3−√6x2+14x−7 x−1 .
Formally, this problem is only for advanced-course graduates, since the basic course does not address solving irrational inequalities.
Although the probability that this problem will be solved by basic-level graduates is by no means zero — if it does not scare them off immediately, this inequality can be solved by graduates who have completed the basic program (it is not so difficult to see that one must solve the inequality8x−x2−7< x3−6x2+14x−7 on the interval (0, 7]) — the general tendency to learn standard algorithms by rote and to apply them “head-on” can do the student a disservice. Thus, the student who, after studying the basic course decides to do the “college”
part of the exam as well, can turn out to be psychologically unprepared for such work.
The “difficult part” of the exam contains many problems pertaining to material that is shared by the basic and advanced courses. Their formulations are thus understandable to all students. However, without a developed capacity for mathematical thought, without the skills associated with advanced mathematical activity, it is unrealistic for students to hope to solve them. To some extent, it may be said that students who have completed the basic course, but who by the time they graduate from high school have decided for one or another reason to go on to colleges that require applicants to take a USE in mathematics, are thus given a chance to display their giftedness.
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