P 3 ratio and proportion test. Test work “Relationships and proportions. I Organizational moment

In mathematics attitude is the quotient that is obtained by dividing one number by another. Previously, this term itself was used only in cases where it was necessary to express one quantity in fractions of another, and one that is homogeneous to the first. For example, ratios were used when expressing area in fractions of another area, length in fractions of another length, etc. This problem was solved using division.

Thus, the very meaning of the term “ attitude" was somewhat different from the term " division": the fact is that the second meant the division of a certain named value into any completely abstract abstract number. In modern mathematics the concepts " division" And " attitude» are absolutely identical in meaning and are synonyms. For example, both terms are used with equal success for relationship quantities that are inhomogeneous: mass and volume, distance and time, etc. At the same time, many relationship It is customary to express homogeneous quantities as percentages.

EXAMPLE

The supermarket has four hundred different products. Of these, two hundred were produced in the territory Russian Federation. Determine what it's like attitude of domestic goods to the total number of goods sold in the supermarket?

400 – total number goods

Answer: two hundred divided by four hundred equals zero point five, that is, fifty percent.

200: 400 = 0.5 or 50%

In mathematics, the dividend is usually called antecedent, and the divisor is subsequent member of the relation. In the example above, the previous term was the number two hundred, and the next term was the number four hundred.

Two equal ratios form a proportion

In modern mathematics it is generally accepted that proportion is two equal relationship. For example, if the total number of items of goods sold in one supermarket is four hundred, and two hundred of them were produced in Russia, and the same values ​​for another supermarket are six hundred and three hundred, then ratio the number of Russian goods to the total number sold in both trading enterprises is the same:

1. Two hundred divided by four hundred equals zero point five, that is, fifty percent

200: 400 = 0.5 or 50%

2. Three hundred divided by six hundred equals zero point five, that is, fifty percent

300: 600 = 0.5 or 50%

In this case there is proportion, which can be written as follows:

=

If we formulate this expression as is customary in mathematics, then it is said that two hundred applies to four hundred the same as three hundred applies to six hundred. In this case, two hundred and six hundred are called extreme terms of the proportion, and four hundred and three hundred - middle terms of the proportion.

Product of the average terms of the proportion

According to one of the laws of mathematics, the product of the average terms of any proportions equals the product of its extreme terms. If we return to the examples above, this can be illustrated as follows:

Two hundred times six hundred equals one hundred and twenty thousand;

200 × 600 = 120,000

Three hundred times four hundred equals one hundred and twenty thousand.

300 × 400 = 120,000

It follows from this that any of the extreme members proportions equal to the product its middle members divided by the other extreme member. By the same principle, each of the middle terms proportions equal to its extreme members divided by the other middle member.

If we go back to the example above proportions, That:

Two hundred equals four hundred times three hundred divided by six hundred.

200 =

These properties are widely used in practical mathematical calculations when it is necessary to find the value of an unknown term proportions at known values three members of the rest.

Khartsyzskaya secondary school No. 25 "Intelligence" with in-depth study individual items

Nakonechnaya Larisa Petrovna

math teacher

Test test work

Mathematics, 6th grade

Subject. Relationships and proportions

Textbook: Mathematics. 6th grade: textbook for educational institutions/ CM. Nikolsky, M.K. Potapov, N.I. Reshetnikov, A.V. Shevkin. -M.: Education, 2016.

In accordance with the Basic curriculum for 2017 - 2018 academic year 4 hours a week are allocated for studying mathematics in the 6th grade. 12 hours are provided for studying the topic “Relationships and Proportions”.

Planned results of studying this topic:

Students will learn to use the concepts of ratio, scale, and proportion when solving problems. Give examples of using these concepts in practice. Solve problems involving proportional division (including problems from real practice).

Use knowledge about dependencies (direct and inverse proportionality), between quantities (speed, time, distance; work, productivity, time, etc.), when solving word problems: comprehend the text of the problem, extract necessary information, build a logical chain of reasoning, critically evaluate the answer received, and perform simple practical calculations.

Results of mastering the topic content:

Personal

Formation of communicative competence in education and cooperation with peers;

The ability to accurately and competently express one’s thoughts when solving problems, understanding the meaning of the task, the ability to build an argument;

Creative thinking, initiative, resourcefulness, activity in solving arithmetic problems;

Formation of the ability for emotional perception of mathematical objects, problems, solutions, reasoning.

Metasubject

The ability to independently plan alternative ways to achieve goals, consciously choose the most effective ways solving educational and cognitive problems;

Development of the ability to see a mathematical problem in other disciplines, in the surrounding life;

Understanding the essence of algorithmic instructions and the ability to act in accordance with the proposed algorithm.

Subject

Mastery of basic conceptual apparatus: have an idea of ​​relationships, proportions, direct and inverse proportionality, scale, the formation of ideas about patterns in real world;

The ability to apply learned concepts to solve problems of direct and inverse proportionality, dividing a number in a given ratio.

The proposed test covers the material of the entire studied topic “Ratios and Proportions” and consists of 12 tasks differing in the level of complexity and form of presentation, the content of which corresponds to the current mathematics program for the 6th grade of general education organizations.

The purpose of the work is to check the level of mastery by sixth graders educational material on this topic with subsequent correction of knowledge and skills.

The first 9 tasks are tasks for choosing one correct answer. For each task there are four possible answers, of which only one is correct. The task is considered completed correctly if the student indicates in the answer table only one letter that indicates the correct answer. There is no need to provide any explanation. For each correct answer, the student receives 1 point. Maximum quantity points - 9

The next 3 tasks (10 - 12) involve establishing correspondence between tasks (1 - 4) and their answers (A - D). For each of the four rows, indicated by numbers, you must select one answer, indicated by a letter. For each correct answer, the student receives 1 point. The maximum number of points scored for 10 - 12 tasks is 12. Total 21 points

Table for converting points to marks

points	mark
1 - 5	"1"
6 - 10	"2"
11 - 15	"3"
16 - 19	"4"
20 - 21	"5"

45 minutes are allowed to complete the work.

Test work

1. The ratio of 23 and 70 is:

A) B) C) 47; D) 93.

2. Which of the proposed ratios are equal?

A) 4:7 and 8:28; B) 30:5 and 65:13; B) 2:1 and 6:3; D) 3:9 and 13:39.

3. Which of these equalities are proportions?

A) 40: 8 = 4: 2; B) 6:13 = 7:12; B) 7: 2 = 21: 4; D) 36:9 = 16:4;

4. Find the ratio of 40 minutes to 2 hours

A) 1: 3; B) 20: 1; B) 1: 20; D) 3:1.

5.Which quantities are directly proportional?

A) Area of ​​the square and its side;

B) The number of workers and the time during which they will complete the work;

C) The path traveled by the pedestrian and the time he was on the road;

D) The number of pipes filling the pool and the time it takes to fill the pool.

6. In which of the Russian proverbs we're talking about about inversely proportional quantities?

B) The spool is small, but expensive;

C) The higher the stump, the higher the shadow;

D) What is hello, is the answer.

7. What expressions are suitable for calculating the unknown term of the proportionat : 24 = 3: 7

A) .

8. Given proportion 13:X = 17: at. Which of the following equations is not a proportion?

A)x:y= 13:17; B) x: 13 = y: 17; IN)y: x= 17:13; G)x:y = 17: 13.

9. What is the ratio??

A) 8; B) ; IN) ; G).

10. Establish a correspondence between the relations (1 - 4) and the quantities (A - D) that these relations are.

1. ; A) number;

2. ; B) price;

3. ; B) concentration;

4. ; D) speed;

11. Establish a correspondence between the given equations (1 - 4) and the roots of each of them (A - D)

1. 7: 8 = X: 96; A) 2;

2. ; B) 6

3. T B) 1 ;

4. To : D) 50;

D) 84.

12. Establish a correspondence between problems (1 - 4) and numbers (A - D), which are the answers to these problems.

1. In the book by Elena Molokhovets “A Gift for Young Housewives” there is prune pie recipe. For a pie for 10 people, use a pound of prunes. How many grams of prunes should I use for a pie for 3 people? Consider that 1 pound = 400 g. 2. Three tangerine trees together produced 240 fruits, and the number of fruits on them was in the ratio 1:3:4. How many fruits grew on that tree where the number of fruits was neither the largest nor the smallest? 3. To transport cargo by a machine with a carrying capacity of 6 tons, it is necessary to complete 10 trips. How many trips do you need to make to transport this cargo with a vehicle whose carrying capacity is 2 tons less? 4. The distance between two cities on the map is 7cm. Find the distance in kilometers between cities on the ground if the map scale is 1: 200,000.

A) 90; B) 15; B) 12; D) 120; D) 14.

ANSWERS to tasks 1 - 9.

ANSWERS to tasks 10 - 12

Task 10

Task 11

Task 12

To correct knowledge, you can use the following table, which indicates the nature of possible errors

p/p	Character errors	S.M. Nikolsky Mathematics, 5th grade M.: 2016		S.M. Nikolsky Mathematics, 6th grade M.: 2016
		theory	practice	theory	practice
	You don't know the definition of attitude.			clause 1.1	№ 4, №5
	You don't know the properties of relationships.			clause 1.1	№6, №7, №9
	You do not know how to find the ratio of homogeneous quantities with different units of measurement.			clause 1.1	№10, №11
	You don’t know how to find the ratios of quantities of different names.			clause 1.1	№12 - №16 №18, №19
	Don't know the definition of scale			clause 1.2	№ 21
	You do not know how to find the distance on the ground, knowing the scale and distance on the map.			clause 1.2	№24, №28, №29
	You do not know how to divide a number in a given ratio.			clause 1.3	№36, №37, №39, №40
	You don't know the definition of proportion.			clause 1.4	№46 - №48, №50
	You don't know the basic property of proportion.			clause 1.4	№51, №52
	You don't know how to find the unknown term of a proportion.			clause 1.4	№53 - №55, №57, №58, №60, №61
11.	You don't know the definition of directly proportional quantities.			clause 1.5	№72 - №75
12.	You don't know the definition of inversely proportional quantities.			clause 1.5	№76, №77, №79
13.	You don't know how to multiply fractions.	clause 4.9	№892 - №900
14.	You don't know how to divide common fractions.	clause 4.11	№925, №926, №927
	Don't know how to find a fraction of a number?	clause 4.12	№941, №943, №945

List of used literature

1. Mathematics. 5th grade: textbook for educational institutions / S.M. Nikolsky, M.K. Potapov, N.I. Reshetnikov, A.V. Shevkin. -M.: Education, 2016.

2. Mathematics. 6th grade: textbook for educational institutions / S.M. Nikolsky, M.K. Potapov, N.I. Reshetnikov, A.V. Shevkin

3.Mathematics. Grade 6: Collection of tasks and assignments for thematic assessment / A.G. Merzlyak, V.B. Polonsky, E.M. Rabinovich, M.S. Yakir. - Kharkov “Gymnasium”, 2008

4.Didactic materials in mathematics for grade 5: independent and tests/A.S.Chesnokov, K.I.Neshkov. -M.: Education, 1981.

5. Mathematics 6th grade: independent and test work / A.P. Ershova, V.V. Goloborodko. . - Kharkov “Gymnasium”, 2007

P.1. Similarity of figures

Tests

Calculation and graphic work

Example title page

CHOU VPO Institute of Economics, Management and Law (Kazan)

Faculty of Management and Engineering Business

Department of Higher Mathematics

in discipline " Financial mathematics»

Option 1

Performer: 2 OZO SP gr. 112 at ______________ A.V. Petrov

Checked: Art. teacher ______________ E.A. Kasatkina

Naberezhnye Chelny

If there is no “Search for a solution...” item in the “Service” tab, you must execute the command “Service” / “Add-ons...”, and in the window that opens, select the “Search for a solution” add-on, after which the selected item will appear in the “Service” tab.

If the solution search fails to find a solution, then in the solution search window you need to click the “Options” button and set a higher value for the limit number of iterations (for example, 1000) and/or a lower value relative error(eg 0.001).

Option 1

1. Similar geometric shapes have the same shape.

2. The coefficient of similarity of equal figures is equal to one.

3. The similarity coefficient of segments is equal to the quotient of their lengths.

4. The similarity coefficient of circles is equal to the quotient of the lengths of their radii.

5. The similarity coefficient of squares is equal to the quotient of the lengths of their diameters.

6. If the sides of the square are reduced by 5 times, then the perimeter of the resulting square will decrease by 25 times.

7. If the length of the rectangle is increased by k times, then its area will increase by k 2 times.

8. If the edge of a cube is doubled, then its volume of the new cube will be 4 times greater.

9. Any two squares are similar.

10. If the figures are equal, then their areas are equal.

Option 2

Write down a numerical code made up of the numbers of correct statements.

1. Equal geometric figures have the same shape and the same dimensions.

2. The similarity coefficient is a number that shows how many times one of the similar figures is larger or smaller than the other.

3. Similar triangles have equal corresponding angles.

4. The similarity coefficient of triangles is equal to the quotient of the lengths of their similar sides.

5. The similarity coefficient of circles is equal to the quotient of the lengths of their diameters.

6. If the sides of a rectangle are reduced by k times, then its perimeter will decrease by k once.

7. If the side of the square is increased by k times, then the area of ​​the new square will be k 2 times more.

8. If the edge of a cube is reduced by 3 times, then the volume of the new cube will become 9 times smaller.

9. Any two rectangles are similar.

10. If the areas of the figures are equal, then the figures themselves are equal.

Option 1

1. The quotient of two quantities measured in the same units is called the ratio of these quantities.

2. The ratio of the number 150 to the number 250 is equal to .

3. The equality 2:5= 0.1:0.25 is a proportion.

4. In proportion A:b=c:d numbers b And c are called extreme terms of the proportion.

5. In proportion, the product of the extreme terms is equal to the product of the middle terms.

6. In a proportion, the unknown term is 2.4.

7. If triangles ABC And KLM are similar, then Sun:L.M.=A.C.:MK.

8. If the distance between settlements on the ground is 5 km, and on the map 0.5 cm, then the map scale is 1:100,000.

9. 1% of 55 is equal to 0.55.

10. A number of which 20% is the number 5 is 100.

Option 2

Write down a number made up of the numbers of correct statements.

1. True equality of two ratios is called proportion.

2. The ratio of the numbers 350 to 420 is equal to .

3. The equality 7:10=5:9 is a proportion.

4. In proportion to the number A And d are called extreme members.

5. If c:d=k:m, That cm=kd.

6. In a proportion, the unknown term is 4.5.

7. If triangles ABC And KLM are similar, then AB:KL=A.C.:KM.

8. If on the map the distance between villages is 2 cm, and the map scale is 1:100,000, then the distance on the ground is 2 km.

9. 1% of 2 is equal to 0.2.

10. A number of which 10% is 5 is 50.

Test 13-16 "Ratios and proportions".

The proposed tests are designed to test the knowledge and skills of students in a section of the sixth grade mathematics course"Ratio and Proportion" . Using the presented tests, the assimilation of educational material on the following topics is checked: “Relationships”, “Proportions”, “direct and inverse proportional dependencies", "Scale", "Circumference and area of ​​a circle", "Ball". This selection of tests can be used in the system of class-lesson study of the designated section or at home - with independent or distance learning for the purpose of self-control.

The test has a time limit of ten minutes. At the end of this time period, the test finishes its work and prompts you to go to the results window. For ease of orientation in time, there is a timer at the top right with countdown time. This test program provides convenient navigation between questions, and it is also possible to make changes to a previously selected or recorded answer. The tests are presented in two equivalent versions, each of which contains seven questions formulated in the form of tasks of varying levels of difficulty. The first four questions are worth one point and require you to choose one correct answer from four options. Problems numbered five and six are of an average level of difficulty and are worth two points each. The last, seventh, task corresponds high level difficulties and the right decision the test taker receives three points.

After the test is completed, a results window with the points scored is displayed. You can also view the details of the assessment, and, if necessary, you can return to the test tasks with subsequent analysis of the correct and selected (recorded) answers.

Let's make a short analysis of the proposed tests.

First And second tests test knowledge and skills on the topic "Relationship". While passing the tasks of the first test, the student must be able to write down the ratio of two numbers, determine what part one number is relative to another (how many times one number is greater than the other), find what percentage one number is of another, and write the inverse ratio for a given ratio. The seventh task is of particular interest. Here in the condition it is given what a given number of percentages of the percentages of a number are equal to and you need to find what this number is equal to.

Quests second test although they relate to the same topic as the tasks of the first test, they are no longer based on testing the basic theoretical and practical knowledge and skills on a given topic, but are aimed at applying relationships to solve problems. The first question contains a graphical drawing that shows two segments. The student should determine the ratio of the lengths of these segments. In the second task, two quantities are given in different units of measurement and you need to find their ratio. Task number three asks you to determine the percentage ratio of two given numbers. And in the fourth, according to a given relation (written in the form mixed number) we need to find the inverse relation. The fifth question contains a task in which you need to determine what percentage of one number is from another. In the problem, which is in the sixth task, you need to find what part one number is relative to another. In the seventh question, the problem condition contains the ratio of two numbers and you need to find the ratio more to the sum of the two numbers involved.

Third test intended for monitoring by topic "Proportions" And “Direct and inverse proportional relationships”. To successfully pass the test, the student will need to know the terms of proportion (which terms of the proportion are extreme and which are average), find an unknown proportion term using a given notation of a proportional relationship, and be able to compose proportional relationships (and solve them) to solve problems.

IN fourth test assignments test knowledge and ability to work with proportions, as well as on topics "Circumference and Area of ​​a Circle" And "Scale". In the first two questions, you need to solve the proportion. Next, it is proposed to find the length of a circle of a given radius. Then, using the known radius, you need to calculate the area of ​​the circle. The fifth and sixth tasks are essentially opposite to each other. In the fifth, using a known scale, you should determine what the distance will be on the map (on the ground), if this distance on the ground (on the map) is known. The sixth task, on the contrary, suggests finding the scale of the map using known corresponding distances on the map and terrain. When answering the seventh question you will need logical thinking and attention. You need to determine how many even (multiples of 5) two-digit numbers can be made from four given digits.

Relations in mathematics 2 m were cut from a piece of matter 5 m long. What part of the piece of matter was cut off? 5 m 2 m Solution =0.4=40 0 / 0 The quotient of two numbers is called the ratio of these numbers. What does attitude show? The answer can also be written in the form decimal or as a percentage. 2:5=

What does attitude show? The ratio shows how many times the first number is greater than the second 16 kg 8 kg 16: 8 = 2(r.) or what part the first number is from the second. 4 m 20 m 4: 20 = 0.2 (parts) If two quantities are measured by the same unit of measurement, then the ratio of their values ​​is called the ratio of these quantities. Mass ratio Length ratio To dough

P R O P O R T I O N “Proportion is proportionality. 1) A certain relationship between the parts. Proportionality in nature, art, architecture means maintaining certain relationships between sizes individual parts plants, sculptures, buildings and is an indispensable condition for the correct and beautiful image of the subject. 2) In mathematics: the equality of two relations.” Ozhegov S. I.

PROPORTIONS The ratios 3.6:1.2 and 6.3:2.1 are equal. Therefore, we can write the equality 3.6:1.2=6.3:2.1 or a: b = c:d The middle terms of the proportion The extreme terms of the proportion In the correct proportion, the product of the extreme terms is equal to the product of the middle terms. a * d = b * c How to check if the proportion is correct? To the question

PROPORTIONS Basic property of proportion: If the product of the extreme members is equal to the product of the middle members of the proportion, then the proportion is correct. Check if the proportion is correct? 20:16=5:

EXERCISES Make up, if possible, proportions from the following ratios: a) 20:4 and 60: Make up, if possible, proportions from the four given numbers: a) 100; 80; 4; Check in two ways whether the equality is true: a) 49:14=14: Make up a proportion from the following equalities: a) 40*30=20* Find the unknown term of the proportion: a) x:30=54:40

Test 1. Relationships. 1. Which of these ratios is equal? a)7:2; b) 4:14; c) 7:17.5; d)12:17;7:24:147:17,512:17 2. Find the ratio of 1.2 m to 10 cm. a) 12; b) 12 m; c)0.12; d) another answer. 1212 m 0.12 another answer 3. How does one third of an hour relate to eighteen minutes? a)1:54; b)10:8; c)1:6; d) another answer.1:5410:81:6 another answer. 4. The ratio a:b is 5:3. Find the ratio 3a:10c. a) 1:2; b)2; c) 9:30; d) another answer.1:229:30 another answer.

Test 2. Proportions. 1. Find the product of the middle terms of the proportion: a)9.8; b)0.98; c)80; d) another answer.9,80,9880 another answer. 2. Find the unknown member of the proportion: a)0.05; b)20; c)0.5; d) another answer 0.05200.5 another answer. 3. From the given proportions, choose the correct one: a)82:72=64:78; b)15:8=13:6;82:72=64:7815:8=13:6 c)17:2=34:4; d)22:23=81:82.17:2=34:422:23=81:82

Task 4 The distance on the map from the Earth to the Moon is 38.4 cm. Find the distance between them if the map scale is 1: