In the selection of the substantive material, we primarily focused on the content of mathematics studied in elementary school in the Russian Federation (1 - 4 classes). At the same time, we were presented to important and some tendencies of world and domestic education, which obliges to consider the content of education in a competence aspect. Thus, the developed tool is made with the calculation of the next perspective.

The main content of mathematics in elementary school is grouped around the concept of a natural number. This includes the entire traditional arithmetic material concerning both the formal side of the concept of the number (positional recording of numbers, standard action algorithms on numbers, the procedure for performing actions, the properties of actions) and the meaningful, associated with the score and measurement of values \u200b\u200b(most of the material related to The concept of magnitude is mastered through the solution of so-called text tasks). The geometric material is also largely devoted to the measurement and calculations (length and area of \u200b\u200bindividual figures). In addition, an initial acquaintance with ordinary fractions is given, but the main study of ordinary and decimal fractions falls on the main school (5, 6 classes). With the introduction of new standards, a material is included on data analysis, but in a very small amount.

The section "Numbers and calculations" includes everything that is associated with the formal side of the concept of natural number. At the heart of the formal operating with natural numbers (comparisons of numbers, calculations), it is primarily a positional principle of recording natural numbers, on which all standard algorithms of arithmetic actions are constructed over numbers. In addition, when calculations, ideas are used on the procedure for performing actions and relationships between the components of actions.

The following three sections reflect the various semantic aspects of the concept of the number (what the number is for which). Thus, the section "Measurement of values" includes material associated with the concept of magnitude - comparing and measuring values. The main attitude constituting this meaningful area is the relation "Unit - the measured value - the number" expressed by the named number. This relationship can be installed in different ways: directly "laying" units (we note that this method is based on the concept of magnitude), measuring with devices (ruler, scales, etc.) and the calculation according to standard formulas, for example, by the formula of the rectangle area , (Tasks associated with an indirect measurement - calculation according to formulas - can be attributed to the meaningful area "dependence").

The specified section adjoin and issues relating to the actual application aspect of measurement (practical measurement procedures, approximate calculations, representation of the measurement results in the form of diagrams, graphs, tables, etc.). However, at the moment they are not included in the content of the mathematical test, since they were supposed that they would be included in the test in natural science ("surrounding world").

Another aspect of the concept of magnitude is submitted by the "Dependency" section. It covers the content associated with the allocation and description of the mathematical structure of relations between values \u200b\u200b(modeling); Typically, we use text tasks. Here, the emphasis is no longer on obtaining the results of measuring values, but on the analysis of the presentation of these results and their connections (including the analysis of texts), i.e. on a logical aspect. If you expand the test, making it independent of the subject of natural science, then the material associated with the analysis of the data (representation of the measurement results in the form of charts, graphs, tables, etc.) can be referred to here.

The "patterns" section covers the content associated with the construction of numerical and geometric sequences and other structured objects, as well as counting their quantitative characteristics. This section is not enough in Russian primary education, and we believe that the amount of this content should be increased, since it is important in terms of the development of mathematical thinking (first of all - algorithmic and combinatorial) and can serve as a propa defense for the concept of function being studied in the basic school.

Finally, the fifth section, "geometry elements", covers the content associated with the definition of spatial forms (in this test is limited to flat forms) and the relative position of objects. The partition is isolated in a sense by the residual principle, since the concepts of geometric shape and arrangement also operate when measuring geometric values \u200b\u200band when structuring objects.

Dedicated areas, from our point of view, cover the main content of all Russian programs in mathematics for elementary school.

Substitution Matrix (Mathematics / Primary School)

	Means of mathematical action (concepts, presentation)	Mathematical actions
Numbers and calculations	positional principle (multivalued numbers) properties of arithmetic action procedure	comparison of multivalued numbers performing arithmetic actions with multi-valued numbers determining the procedure for action in expression. dipka
Measurement of quantities	the relationship between the number, value and unit the ratio of "whole and parts" the formula of the square of the rectangle	direct measurement of lengths of lines and areas of figures (direct "laying" units, "laying" units with preliminary rearrangement of parts of the object) indirect measurement (measurement with devices, calculation by formulas)
Laws	"Induction step" repeatability (frequency)	detection of patterns in numerical and geometric sequences and other structured objects calculating the number of elements in a structured object
Dependencies	relationship between homogeneous values \u200b\u200b(equality, inequality, multiplicity, difference, "whole and parts") direct proportional relationship between values derived values: speed, labor productivity, etc. relations between units	solving text tasks. description of dependencies between values \u200b\u200bon various mathematical languages \u200b\u200b(representation of dependencies between values \u200b\u200bin drawings, schemes, formulas, etc.) actions with named numbers
Elements of geometry	form and other properties of figures (main types of geometric shapes) spatial relationships between figures symmetry	recognition of geometric figures definition of the mutual location of geometric shapes

Test problems in mathematics

First level (formal)

Section 1. Numbers and calculations

First level indicators are tasks in which directly Apply Standard Action Rules with Numbers:

1) rules for recording numbers;

2) the rules of comparing numbers;

3) algorithms for performing arithmetic action;

4) the rules for the procedure for performing arithmetic action;

5) Communication rules of arithmetic components (find an unknown component of action).

Under the use of rules, it is not referring to the reproduction of their wording, which is characteristic of the traditional interpretation of zunov with distinction knowledge The wording I. skills Apply this knowledge. In this context, we are talking only about the real application of the rule (about the rule as a method of action or the means of organization of action), and without reference to the ability to voice it.

The test does not necessarily cover all algorithms and rules to tasks. It can be limited to checking only the most fundamental (error) options for their use. If the rule is dismembered for applications, it is advisable to check everything. Tasks should not be cumbersome, because in this test the automation of skills is not checked.

Examples of tasks

Direct use of the division algorithm (the most difficult for students). The most fundamental case is presented when it is necessary to consider 0, i.e. Do not miss a discharge.

Direct use of rules defining the procedure. All distractors are answers that are obtained by incorrect procedure. The calculation themselves are minimized, since in this case the calculation algorithms are not checked.

Section 2. Measurement of values

To the first level relate to the tasks assumed separate act.measurements or comparison of values \u200b\u200bin which directly Famous methods are used:

Measurement of geometric values \u200b\u200b(length and area) by directly laying the measurement (units) or comparing the area of \u200b\u200bthe figures by overlay. In the educational process, with the introduction of the concept of magnitude, it is possible to use measurements not only of length and area, but also volume and mass of objects. However, in the test of the task of this kind to submit difficult.

Measurement of values \u200b\u200busing instruments (ruler, scales, clock, etc.). In the test, such tasks can be represented by image of the corresponding measurement situations.

Finding values \u200b\u200bof values \u200b\u200bwith the help of known formulas and rules (for example, a rectangle area formula, a rectangle perimeter formula (square), a rule for calculating the length of the broken line).

The use of formula is understood not only as a direct calculation, but also to find an unknown member (for example, on the formula of a rectangle can not only calculate the area of \u200b\u200bthe rectangle, knowing it, but also to find, for example, a rectangle width along its area and length).

Examples of tasks

Direct placement of units (measure).

The figure represents the situation of the one-act balance of the item and weights on the scales. The result is directly derived from the clearly represented balance conditions.

Task description

The first task checks our skills of computing. This is the easiest task of the entire module and requires only the knowledge of arithmetic. In the first task, arithmetic actions will be the most simple. In the demonstration version of OGE, it is proposed to fold two fractions: ordinary and decimal. Nevertheless, in accordance with the documents on the conduct of OGE, students should be prepared and to fulfill some other uncomplicated tasks. The answer in the first task is an integer or ultimate decimal fraction.

Task Topics: Numbers and Calculations

Primary ball: 1

Quest Quality: ♦ ◊◊

Approximate execution time: 3 min.

Theory of task number 1

So, for successful execution you need to remember:

1. the procedure for conducting arithmetic operations — first, actions are performed in brackets, then the construction of the root or extract, then multiplication and division, and then subtraction and addition.
2. multiplication and division rules in the column
3. rules for calculating ordinary fractions

We remind the rules of operations with ordinary fractions:

Analysis of typical options of task number 1 OGE in mathematics

The first version of the task

Find the value of the expression:

Decision:

The task can be solved by different paths, namely change the sequence of actionbut this solution is recommended for those who i am confident in your capabilities and knows mathematics on excellent. For the rest, we recommend that you follow the steps in the numerator and the denominator, and then split the numerator to the denominator. Numerator Calculate in this example is not necessary, this is the number 9.

Calculate the value of the denominator:

You can produce, then we get:

4,5 2,5 = 11,25

Either translate fraction to simplicity:

4.5 2.5 \u003d 4½ 2 ½ \u003d 9/2 5/2 \u003d 45/4

The last case is preferable, since for a further operation - plugs of the Numerator on the denominator The task is simplified. We divide the numerator to the denominator, multiplying the numerator on an inverted fraction in the denominator:

9 / (45 / 4) = (9 / 1) (4 / 45) = (9 4) / (1 45)

9 and 45 can be reduced by 9:

(9 4) / (1 45) = (1 4)/ (1 5) = 4 / 5 = 8 / 10 = 0,8

We get the answer: 0.8

Summing up, make conclusions:

It is more convenient to immediately move towards fractions of a simple look.Reliable to calculate sequentially in the numerator and denominator.

Second version of the task

Find the value of the expression:

6 (1/3) ² - 17 1/3

Decision:

You can solve the problem directly - calculating the values \u200b\u200bconsistently, it should not be difficult, but the decision will be long and with large calculations. Here it is possible to see that 1/3 is present both in a decrease - 6 (1/3) ² and in subtractable - 17 1/3, so it can be easily taken out of the bracket.

1/3 (6 (1/3) - 17)

After computing in brackets, we get:

1/3 (6 (1/3) - 17) = 1/3 (6 /3 - 17) = 1/3 (2 - 17) = 1/3 (-15)

Now multiply the obtained value is -15 to 1/3:

1/3 (-15) = -5

What conclusions can be made: it is not always worth trying to solve the task "in the forehead", even in OGE.

Third version of the task

Find the value of the expression:

Decision:

Similar to previous tasks, calculate the denominator: To do this, we give a fraction to a common denominator - this is 84. For this, the first fraction is multiplied by 4, and the second to 3, we will get:

1/21 + 1/28 = 4/84 + 3/84

Then we fold:

4/84 + 3/84 = 7/84

So, we got 7/84 in the denominator, now we divide the numerator to the denominator - it's like multiplying 1 to the inverse 7/84 fraction:

Demonstration OGE 2019

Find the value of the expression: ¼ + 0.07

Decision:

To this task, as well as most of the tasks 1 of the algebra module, the approach to the solution is to translate fractions from one species to another. In our case, this is the transition from the ordinary fraction to decimal.

Transfer ¼ from ordinary fraction in decimal. Delim 1 to 4, we get 0.25. Then rewrite the expression using only decimal fractions and calculate:

0,25 + 0,07 = 0,32

Answer: 0.32.

Fourth version of the task

Find the value of the expression:

-0.3 · (-10) 4 + 4 · (-10) 2 -59

Decision:

To obtain the result, it is necessary to consistently perform mathematical actions in accordance with their priority.

-0.3 · (-10) 4 + 4 · (-10) 2 -59 \u003d

We carry out the exercise. We obtain numbers consisting of a unit and following the number of zeros, equal to the degree. In this case, the signs "-" disappear in brackets, as the indicators of the degrees are even. We get:

\u003d -0.3 · 10000 + 4 · 100-59 \u003d

Perform multiplication. To do this, among 0.3, we carry the decimal comma on 4 signs to the right (as in 10,000 four zero), and 2 zero are added to 4, respectively. We get:

= –3000+400–59 =

We perform addition -3000 + 400. Since these are numbers with different signs, then we subtract the smaller module less and before the result we put "-", since the number with a large module is negative. We get:

= –2600–59 =

Since both numbers are negative, then we add them modules and before the result we put "-". We get:

= –(2600+59) = –2659

Answer: -2659

Fifth job option

Find the value of the expression:

-13 · (-9.3) -7.8

Decision:

This task requires a simple ability to perform arithmetic action with decimal fractions.

-13 · (-9.3) -7.8 \u003d

First perform multiplication. I multiply -13 and -9.3 in the column without taking into account the signs "-" before the factors. In the result obtained, separate one - the last - figure of the decimal point:

The workmark will be positive because two negative numbers are multiplied. We get:

This difference can be calculated in the column, but you can and orally. Perform this action in mind: We subtract separately parts and decimal. We get.

Test 1 on the topic "Numbers and calculations"

Option 1

1. Which expressions are the product 0,5 · 0.005 · 0.00005

1) 5 · 10-9 2) 125 · 10-9 3) 5 · 10-5 4) 125 · 10-5

2. Tell the smallest of numbers: 3/5; 0.41; 5/13; 1/2.

1) 3/5; 2) 0.41; 3) 5/13; 4) ½.

3. What about numbers ; ; Not rational

1) ; 2) ; 3) 4) None of these numbers.

4. Match the numerical expressions and the values \u200b\u200btaken by them:

Numerical expressions

A) -0,008: 0.04 b) -0.01 ·· 5 V)

Values

0,002 2) 0,2 3) -0,2 4) -0,002

5. Write in response the numbers of incorrect equalities:

1) (0,9) 2 = 8,1

2) 0.6 · 0.8 \u003d 0.7 2 -1

3) · – 0,1 2 · 100 \u003d 0

4) 0,6 (0,8–0,7)= 0,6

6. From the announcement of the company conducted by training seminars:

"The cost of participation in the seminar is 2000 rubles per person. Groups from organizations are provided with discounts: from 4 to 10 people - 5%; More than 10 people - 8%. " How many rubles must pay the organization, who sent a group of 8 people to the seminar?

7. Express the decimal fraction of 72.5%.

Answer: _______________________

8. What are the integers enclosed between numbers and ?

1) 51, 52, … 89 2) 7, 8, 9, 10 3) 7, 8, 9 4) 8, 9

9. The population of Venezuela is 2.7 · 10 7 Man, and its area is 9 · 105 km 2 . What is the density of the population of Venezuela?

1) 30 2) 3 3) 3,3 4) 0,33

10) Calculate the value of the expression (A + b. ) / (C + b. ) at a \u003d 2.6; b. \u003d - 1.1; C \u003d 1,3

Answer:__________________

Test 2 on the topic "Algebraic expressions"

Option 1

In which case the expression is transformed in identically equal?

1) 3 (x - y) \u003d 3x -y.2 ) (3 + x) (x - 3) \u003d 9 - x 2 3) (X - Y) 2 \u003d X. 2 - Y. 2 4) (x + 3) 2 \u003d X. 2 + 6x + 9

Spread the square three-stakes on the factors x 2 - 4x - 32

(x + 8) (x + 4); 2) (x-8) (x-4); 3) (x-8) (x + 4); 4) (X + 8) (X-4)

1) 2) 3) 4)

Pedestrian passed S. km. Make an expression for calculating a pedestrian speed if it was on the way and minutes (in m / min).

From formula Q. = cm. ( t. 2 – t. 1 ) Express t. 2

Answer:____________

Which expressions does not make sense at x \u003d 1 and x \u003d -2?

Answer:__________________

a. 2 2) a. -4 3) a. 8 4) a. -2

from from a. a. In the garage, the room for washing machines was allocated (in the figure it is shown by hatching). What is Square S. The remaining part of the garage? 1)
2)
3)

Test 3 on the topic "Equations, system of equations"

Option 1

Decide equation 4x 2 - 13x - 12 \u003d 0.

1)0,75; 4 2) -0,75; 4 3) 0,75; -4 4) -0,75; - 4

Roots of which equation are numbers -2; 0; 2?

h. 3 -4x \u003d 0; 2) x (x 2 -4x + 4) \u003d 0; 3) H. 3 -2x \u003d 0; 4) H. 3 -4x + 4 \u003d 0

Callate square equations and their roots.

A) 4.h. 2 + 4 h. - 15 \u003d 0 b) 2h. 2 + 7 \u003d 0 c) 4h. 2 – 9 = 0

1) -2.5; 1.5 2) -1.5; 1.5 3) 1.5; -2.5 4) no roots

1) -9; 2) -6; 3) 36; 4) 2

The distance between the pins on the 12 km river. Boat sailed from one pier to another and returned back, spending on all the way 2 h 30 min. What is the speed of the river flow (in km / h), if its own speed of the boat is 10 km / h?

Select the equation corresponding to the condition of the problem if the letter X is indicated by the flow rate of the river (in km / h).

1) 2) x \u003d

3) 4)

Solve the system of equations

Answer:_____________

Calculate the coordinates of the points of intersection of parabola y \u003d 2x 2 -5 and straight y \u003d 4x-5

(0;2), (-5;3) 2) (-5;0), (2;3) 3) (0;-5), (3;2) 4) (0;-5), (2;3)

The price of goods first increased by 20%, and then decreased by 20%, after which it became 6720 rubles. Find the initial price of the goods.

Answer:______________

How much water should be added to 400 g of an 80% alcohol solution to get a 50% alcohol solution?

1) 200 2) 240 3) 160 4) 400

Decide the equation X. 4 -3x 3 + 4x 2 -12x \u003d 0.

Answer:_____________

Test 4 on the topic "Inequality, inequality systems"

Option 1

On the coordinate direct numbers x, W. and z. . Which of the following differences is negative?

1) x - W. 2) u - H. 3) z. – w. 4) z. – h.

Which of the following inequalities does not follow from inequality k. > m. – n. ?

1) n + k\u003e m 2) n\u003e m - n3) m - n - k\u003e 0 4) n - m + k\u003e 0

How many integers are included in the interval (-2; 4]?

6; 2) 7; 3) 5; 4) 4

Specify the inequality, the solution of which is any number.

1) H. 2 - 16 0 2) x 2 - 16 0 3) x 2 +16 0 4) x 2 +16 0

Solve inequality : 2 y. − 3( y. + 4) ≤ y. +12 .

1) (− ∞;12] 2) [−12;+ ∞) 3) (− ∞;−12] 4) }