|
On Algebra Education in Russian Schools
151
and so on. Several examples:
• Compute:
53
2
−27
2
79
2
−51
2
. (Makarychev, 2009a, p. 167)
• Compute using the formula (a − b)(a + b) = a
2
− b
2
:
(a) 201
·199; (b) 1.05 · 0.95. (Makarychev, 2009a, p. 164)
• Find 10 factors of a number equal to 97
2
− 43
2
. (Dorofeev,
Suvorova et al., 2005, p. 218)
• Check the following equalities:
1
2
−
1
3
=
1
2
· 3
,
1
3
−
1
4
=
1
3
· 4
,
1
4
−
1
5
=
1
4
· 5
,
1
5
−
1
6
=
1
5
· 6
.
Continue this sequence of equalities. Write down the correspond-
ing literal equality and prove it. (Dorofeev, Suvorova et al., 2009a,
p. 20)
• Prove that if the side of a square is increased 10 times, then its
area will increase 100 times. How many times will the volume of a
cube increase if its edge is increased n times? (Dorofeev, Suvorova
et al., 2005, p. 162)
Equations and systems of equations. In parallel with the topic
of transformations, the textbooks also develop the related topic of
equations, inequalities, and systems. The textbooks contain no direct
answers to the following question: What is an equation? The term itself
is familiar to students from the course in mathematics for grades 5–6.
In seventh grade, the students are given a word problem, which is
then used as a basis to formulate an equality that contains an unknown
magnitude, indicated by a letter. The students are reminded that such
an equality is called an “equation,” and that in order to obtain the
answer to the problem this equation must be solved. For example,
the concept of an equation was introduced in the following way in a
class observed by the authors of this chapter, in which the textbook
of Dorofeev, Suvorova et al. (2005) was used (the lesson also aimed
to demonstrate conclusively to the students the advantages of the
algebraic method over the arithmetic one).
Initially, students were given the following problem:
A family has two pairs of twins, born three years apart. In 2002, all
of them together turned 50. How old was each twin in 2000?
March 9, 2011
15:2
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch04
152
Russian Mathematics Education: Programs and Practices
This problem was solved by the students not independently, but in a
group discussion organized by the teacher. On the teacher’s suggestion,
the students solved this problem arithmetically. Using their combined
effort, the children solved the problem, but said that it was difficult.
After this, the teacher suggested solving it algebraically, noting that
to do so it would be necessary to translate the conditions given in the
problem into the language of mathematics. The students reasoned as
follows:
Let x stand for the age of the younger twins in 2000. Then the older
twins were x + 3 years old in that year. Two years later, the younger
twins were x + 2 years old, while the older ones were x + 5 years old.
Thus, we have an equation: (x + 2) + (x + 2) + (x + 5) + (x + 5) = 50.
After simplifying, we obtain the equation 4x + 14 = 50, and hence
x = 9. Thus, the younger twins were 9 years old in 2000, and the
older twins were 12 years old. The students felt that this solution was
considerably easier.
In this way, a word problem provides the motivation for subsequent
formal activity in studying equations and learning algorithms for solv-
ing them. A significant portion of this part of the course involves study-
ing equations with one variable. These are integral equations, which,
as a result of transformations, are reduced to linear equations of the
type ax + b = 0, or to quadratic equations of the type ax
2
+ bx + c = 0.
A considerable portion of this part of the course is devoted to
the study of quadratic equations, which is a tradition in Russian
schools. The quadratic trinomial and quadratic equations serve as
a means of introducing the students to certain mathematical ideas.
They contain material that is conceptually rich and convenient for
organizing cognitive activity, and at the same time corresponds to the
capacities of students at this age. The central topic here is the derivation
of the quadratic formula; some of its variations are sometimes also
examined — the formula for the roots of an equation with an even
second coefficient, and the formula for the roots of a reduced quadratic
equation (a = 1). Along with learning the algorithm for solving
equations by using the quadratic formula, the students carry out
elementary investigations; for example, they solve problems of the
March 9, 2011
15:2
9in x 6in
Russian Mathematics Education: Programs and Practices
b1073-ch04
Do'stlaringiz bilan baham: | 
