4.1 Basic School (grades 5–9; students aged 10–15)
4.1.2 Algebra for students of ages 10–12 (grades 5–6)
The material that usually pertains to grades 5–6 is labeled in curricula as “Elements of Algebra.” The objectives that its study is meant to meet are defined differently by different teams of contributors.
The main purpose of studying algebra, as defined by the authors of the textbooks by Vilenkinet al.(2007, 2008), is the formation of basic formal-operational skills. Of the basic content described above,
this series of textbooks selects the following items for this stage of education: literal expressions, numerical values of literal expressions, using literal notation to indicate the properties of arithmetic operations, transformations of literal expressions (multiplying a sum by a number, factoring out common factors, combining like terms, simplifying products, and removing parentheses that have a plus or minus sign in front of them), solving equations, and using algebraic methods to solve word problems.
In their methodological approaches, the authors rely substantially on the fact that the students have already acquired some experience in working with letters in elementary school. Therefore, they begin using letter symbolism quite extensively in the very first classes of fifth grade without any kind of special discussion about mathematical language and, in particular, about the meaning of literal expressions.
The course makes no explicit distinction between concrete, visual arithmetic and formal, abstract algebra. One might say that in some sense the students study “algebraized arithmetic.” Each time new types of numbers are introduced and new computational algorithms are examined, the students also carry out assignments that involve manipulating literal expressions and solving equations, in which this new knowledge is used. Below, we offer examples of this concurrent development of the arithmetic and algebraic components of the course, reflected in exercises from the textbook by Vilenkinet al.(2008).
Arithmetic Algebra
I. Ordinary fractions Compute using the
distributive property:
8115 ã429 +8115 ã679. (p. 88)
Simplify the expression
5 18x+
5
12x−14x
. Simplify the expression
13
15m− 34m+121 mand find its value whenm=212;614.
Solve the equation
7
12m+ 23m−14m=7. (p. 89)
Arithmetic Algebra Divide: 35 ÷259 ;3397 ÷1315.
(p. 98)
Represent the following quotient in the form of a fraction: mn ÷ak;b÷nc. (p. 98)
Solve the equation (a)yữ112 =213ã13; (b)312
2 3x+ 47
=213. (p. 100) II. Decimal fractions
Find the value of the expression
102,816ữ(3.2ã6.3)+3.84.
(p. 67).
Simplify the expression
3.7x+2.5y+1.6x+4.8y. (p. 76) Solve the equation
9.5x−(3.2x+18x)+3.75=6.9.
(p. 78) Compute
0.2ã6.2ữ0.31−56ã0.3
2+1114 ã0.22ữ0.01 . (p. 112)
Find the value of the expression (a) 2yx− 2xy whenx=18.1−10.7, y=35−23.8;
(b) 5.7−a4.5 +2.8+a4.4 when a=217+145. (p. 112) III. Ratios and proportions Find the ratio of 0.25 to
0.55. (p. 118)
The length of a rectangle isacm and its width isbcm. The length of another rectangle ismcm and its width isncm. Find the ratio of the area of the first rectangle to the area of the second rectangle. Find the value of the obtained expression ifa=6.4, b=0.2,m=3.2,n=0.5. (p. 123) IV. Positive and negative numbers
Perform the following operations:
−6ã4−64ữ(−3.3+1.7).
(p. 198)
Find the value of the expression (3m+6m)÷9, ifm= −5.96.
Solve the equation−47y= 218. (p. 198)
Arrange the terms in a convenient order and find the value of the expression
−6.37 + 2.4 − 3.2+ 6.37 − 2.4. (p. 208)
Simplify the expression
6.1−k+2.8+p−8.8+k−p. (p. 208)
Remove the parentheses and find the value of the expression
−6.9−(4.21−10.9). (p. 216)
Simplify the expression
−a−(m−a+p);
m−(a+m)−(k+a). (p. 217) Write down the difference of the two expressions−p−aandk−a, and simplify it. (p. 217)
Solve the equation
7.2−(6.2−x)=2.2. (p. 217)
The solutions to the equations reproduced in this table are based on arithmetic techniques: students solve them by relying on facts about dependencies between the components of operations, which are expressed in rules for finding unknown terms, minuends, divisors, and so on. At the same time, these types of equations have a fairly high level of difficulty.
Note that the subject of equations involves not only using algo- rithms, but also using the algebraic method to solve word problems.
Students solve a considerable number of word problems by forming equations. The problems’ algebraic component is developed in parallel with the formation of students’ operational abilities and is connected with the content of arithmetic problems. We will illustrate this by providing examples of problems solved by sixth graders:
(1) In order to make sour cherry jam, one must combine two parts cherries with three parts sugar (in mass). How many kilograms of sugar and how many kilograms of cherries must be used in order to obtain 10 kg of jam if its mass is reduced by 1.5 times during cooking?
[Equation:3x+2x=10ã1.5, wherexis the mass of one part in kilograms.]
(2) Three boxes contained 76 kg of sour cherries. The second box had twice as many sour cherries as the first, while the third contained 8 kg more sour cherries than the first. How many kilograms of sour cherries were in each box?
[Equation:x+2x+(x+8)=76, wherexis the mass of sour cherries in the first box, in kilograms.]
(3) The arithmetic mean of four numbers is 2.75. Find these numbers if the second is 1.5 times greater than the first, the third is 1.2 times greater than the first, and the fourth is 1.8 times greater than the first.
[Equation:(x+1.5x+1.2x+1.8x)÷4=2.75, wherexis the first number.]
(4) A father is313 times older than his son, while the son is 28 years younger than his father. How old is the father and how old is the son?
Equation:31
3x−x=28, wherexis the son’s age.
This organic integration of arithmetic and algebra concludes with a certain systematization of the algebraic material: an examination of strictly algebraic questions — removing parentheses, the coefficient, like terms, and solving equations. The solving of equations is now grounded in the use of rules for equivalent transformations of equations (the word “equivalence” — which in Russian textbooks is reserved for logical equivalence only — is, of course, not used at this stage).
Here, the students deal with formal algebra, and the level of the transformations presented to them is quite high.
In this way, these textbooks achieve rather close integration of arithmetic and algebraic material. However, teaching experience points to a number of negative consequences arising from such early and insistent “algebraization.” First, this approach to some extent hinders the formation and development of practically oriented arithmetic skills, such as the use of percentages in real-life situations. While students formally assimilate the central topics of arithmetic — fractions and decimals — their computational skills suffer. A considerable percentage
of students is unable to compare fractions or put them in ascending order, to shift from one form of fractional notation to another. This is revealed by both national and international studies. Thus, many students have difficulty with the following types of problems:
• Which of the following numbers is the smallest: 16, 23, 13, 12.
• Which of the following numbers is contained between the numbers 0.07 and 0.08? 0.0075, 0.6, 0.075, 0.75.
• Find the ratio of the numbers 0.5 and 0.3.
Setting the formation of formal-operational skills pertaining to the transformation of literal expressions as a central objective, the authors rise to a sufficiently high level of such transformations, exceeding the capacities of a considerable number of 12-year-old children.
Schoolchildren are not always able to handle much easier problems than those which they solve in class (see the table above). For example:
• Solve the equation 12x=6.
• Which of the following expressions is equal to the sum a +a + a + a?
(1)a+4, (2)a4, (3)4a, (4)4(a+1).
As a consequence, the textbooks of the following stage (Makarychev et al., 2009a, 2009b, 2009c) do not begin at the level set by the textbooks of Vilenkinet al.In terms of the transformations of algebraic equations that the students are asked to carry out and the equations that they are asked to solve, the first classes in algebra at the following stage of education (grade 7) do not constitute a natural continuation of what has come before; in these classes, everything begins anew.
The key feature of the second set of textbooks for this stage of schooling (Dorofeev, Sharyginet al., 2007a, 2007b) stems from the emphasis that they place on the arithmetic and algebraic components of the course: the balance in them has shifted in favor of the former.
A greater role is now played by arithmetic, the study of number systems, computational algorithms; most importantly, the course relies extensively on using arithmetic methods to solve word problems, which is seen as an effective way to facilitate the students’ logical development.
At the same time, the approach to presenting algebraic material is fundamentally altered as well. The quantity of formal “algebraic” work is substantially reduced; the very purpose of studying this material is
different. One might say that at the center of attention is the role of letters as elements of mathematical language. First of all, the letter acts as the “name” of any number in some set. This is underscored in formulations that use quantifying phrases such as “for any. . .” and
“for all. . ..” Consider the following example of a text that is read by students in fifth grade:
You know the commutative property of addition: when the places of terms are switched, the sum does not change. In accordance with this property, for example,
280+361=361+280, 0+127=127+0.
Using letters, the commutative property can be written in the following way:
For any numbersaandb,a+b=b+a.
This literal equality, which expresses a general property of the addition of numbers, has replaced for us an infinite number of number equalities (Dorofeev, Sharyginet al., 2007a, p. 82).
Similar arguments are presented in introducing literal notation for the commutative property of multiplication, the associative property of addition and multiplication, and so on.
The letter may also act as a proper noun. For example,πis a quite definite number, about which the students so far know only that it is a number of a new kind, which is neither an integer nor a fraction, and that it may be expressed approximately in decimals. Special letters are “assigned” to the sets of natural numbers, integers, and rational numbers —N,Z, andQ, respectively.
Students learn the rules for writing literal expressions, in particular the role of parentheses as a “grouping” sign. Classroom activity is mainly aimed at getting the students to learn and grasp the significance of and reasons for introducing letters, and to practice “translating”
from Russian into mathematical language. Several examples:
1. Write in the form of a mathematical sentence:
(a) the numberkis less than 5; (b) the absolute value of the number mis greater than 1; (c) the square of the numberais equal to 4.
(Dorofeev, Sharyginet al., 2007b, p. 244)
2. The following examples illustrate a certain rule. Formulate this rule and write it down using letters:
(a) 7ã0=0,15.3ã0=0, 25ã0=0;
(b) 4+(−4)=0;0.3+(−0.3)=0; 13+
−13
=0. (Dorofeev, Sharyginet al., 2007b, p. 245)
3. In order to write “long” expressions, mathematicians often use an ellipsis. For example, the expression1ã2ã3ã. . .ã50 means the product of all natural numbers from 1 to 50. Write down the following in the form of a mathematical expression:
(a) the product of all natural numbers from 1 to 100;
(b) the product of all natural numbers from 1 ton; (c) the sum of all natural numbers from 1 to 100;
(d) the sum of all natural numbers from 1 to n. (Dorofeev, Sharyginet al., 2007b, p. 245)
4. Write down the following problem in the form of an equation and solve it:
Tanya thought of a number, multiplied it by 15, and sub- tracted the result from 80. She obtained 20. What number did Tanya originally think of? (Dorofeev, Sharygin et al., 2007b, p. 259)
Algebraic “technique” — the transformation of literal expres- sions — belongs to the next educational stage and begins to be studied systematically in grade 7. But at the stage of grades 5–6, the study of number systems and computational algorithms is organized in such a way as to create a substantive foundation for the study of algebraic transformations in the future: students learn the properties of arithmetic operations as an apparatus for the transformations of numeric expressions. Thus, in the fifth-grade course, students examine the possibility of using the rules of addition and multiplication in order to substitute numeric expressions with simpler expressions whose value may even be found mentally. The problems presented to the children are simple and understandable; the work they do is substantive, motivated, and easy to appreciate. At the same time, the students perform quite serious manipulations with numeric expressions: they write down numeric sequences, group terms and factors in a convenient
manner, factor out common factors in numeric sums and products, and so on. Two examples:
Example 1. Students are asked to find the value of the product 4ã7ã11ã25 (Dorofeev, Sharyginet al., 2007a, p. 84).
They reason in the following manner: the product of 4 and 25 equals 100, and multiplying by 100 is easy, and therefore let us group the factors in the following way:
4ã7ã11ã25=(4ã25)ã(7ã11)=100ã77=7700. Example 2. Students are asked to find the value of the fraction
1 3−15 2 3−12
(Dorofeev, Sharyginet al., 2007b, p. 11).
To find the value of this expression, the students can perform three operations: find the value of the fraction’s numerator, find the value of the fraction’s denominator, and divide the former by the latter. But they can also employ a different approach: using the “basic property of frac- tions” (the fact that multiplying the numerator and the denominator of a fraction by the same number produces a fraction that is equal to the original fraction), they can manipulate the given “multistory” fraction and obtain the answer much more easily and quickly. The students’
reasoning is approximately as follows: let us multiply the numerator and the denominator of the fraction by a “convenient” number to get rid of the fractions in the numerator and the denominator. In the given case, this number can be, for example, 30:
1 3 −15
2
3 −12 = 30ã1
3 −15 30ã2
3 −12= 10−6 20−15 = 4
5.
Of course, this solution is presented as an alternative to the first.
Although it is demonstrated to all students, the teacher emphasizes that it makes sense to proceed in this way if the intermediate computations can be performed mentally.
Performing transformations of this kind constitutes a good, sub- stantive form of practice, which prepares the students for learning to carry out transformations of literal expressions, which, as has already been noted, are a topic of study at the subsequent stage (grades 7–9) — as is solving equations by using transformations. At this stage, however,
the aim of this activity is not so much the development of a skill as the simple process of carrying out such transformations.