Organs

Relationships of quantities. Test work “Relationships and proportions” Lesson topic: “Relationships of quantities”

Khartsyzsk secondary school No. 25 “Intelligence” with in-depth study of individual subjects

Nakonechnaya Larisa Petrovna

mathematic teacher

Test work

Mathematics, 6th grade

Subject. Relationships and proportions

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

In accordance with the Basic Curriculum for the 2017-2018 academic year, 4 hours per week are allocated for the study of 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 the necessary information, build a logical chain of reasoning, critically evaluate the answer received, 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 to solve 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

Possession of a basic conceptual apparatus: have an idea of relationships, proportions, direct and inverse proportionality, scale, the formation of ideas about patterns in the 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 assimilation by sixth-graders of 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 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 distance 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. Which Russian proverb talks 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 IN 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;

AT 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
p/p	Character errors	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 ordinary 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 test work / 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

Pavlova Natalya Valerievna

mathematic teacher

Municipal educational institution "Lyceum No. 6" Voskresensk

Class: 6

Lesson topic: "Relationships of quantities"

Lesson type:“discovery” of new knowledge.

Basic goals:

To form the concept of a relationship, the ability to simplify relationships and find relationships between numbers and quantities.

Repeat and consolidate: difference and multiple comparisons of numbers and quantities; joint actions with ordinary and decimal fractions; translation of statements into mathematical language.

During the classes

1) Self-determination for activity (organizational moment).

Hello guys! Today we will continue working with numbers.

May today bring you the joy of communication. Let your intelligence, ingenuity and the knowledge that you have already acquired help you.

2) Updating knowledge and fixing difficulties in activities.

2.1. Oral work.

SLIDE2. We know how to compare numbers and quantities. What comparison signs do we use? ( )

SLIDE3. Use a comparison sign instead of an asterisk:

Conclusion about comparison (not all values can be compared).

2.2 (Work in groups of 2 people).

SLIDE 4.- Solve the problem: “A cobra lives about 40 years, and a crocodile lives about 200 years. How can their life expectancy compare?

SLIDE 5.A) 200-40=160 (years). A crocodile lives 160 years longer than a cobra.

B) 200:40=5 (times). A cobra lives 5 times less than a crocodile.

The lifespan of a cobra is one-fifth that of a crocodile.

(Raise your hands who solved the problem. Listen to the guys who solved it in different ways.)

What “clarification” questions can be asked when solving this problem? (What actions can be used to compare? How to write “what part is the life of a cobra from the life of a crocodile”?)

What comparison methods did you use? (found the difference or quotient).

There are two ways to compare values.

SLIDE 6. The first method is to find their difference and answers the question “How much more (less)?” This comparison is called difference. The second is to find the quotient and answer the question “How many times is more (less)?” This comparison is multiple.

2.3 SLIDE 7. Solve the joke problem “Who is stronger: the elephant or the ant?”

SLIDE 8.“The weight of an ant is approximately 50 milligrams or 0.05 g, and an elephant is 5 tons. At the same time, an ant is capable of lifting a load weighing 0.5 g, and an elephant one and a half tons. So which one is stronger?

(Listen to the solution, direct the course of reasoning. Give instructions: find out how many times heavier the load that an ant can lift than it weighs itself. Do the same with the elephant.)

Solution: If we compare the weight of the load being lifted and its own weight (0.5 / 0.05 = 10 and 1.5 / 5 = 0.3), it turns out that the ant lifts a load 10 times more than it weighs itself, and an elephant - three tenths of its weight. It’s probably not without reason that the three-wheeled cargo scooter “Ant” was named after the hardworking ant.

So, what comparison helped us compare the strengths of an ant and an elephant? (MULTIPLE)

3) Setting a learning task.

What question will we work on today?

(We will consider multiple comparison of values).

This is the purpose of the lesson.

In mathematics, the term “ratio” is often used to describe the result of a multiple comparison of two quantities.

Now formulate the topic of the lesson. (Ratios of quantities).

SLIDE 9.- Well done! Write the topic in your notebooks.

(The teacher writes on the board: Ratios of quantities).

4) Building a project for getting out of a problem. Discovery of “new knowledge” by children.

How to write the ratio of numbers from the cobra and crocodile problem? By what action do we determine “how many times or what part is it”? (if anyone knows, write it down on the board)

SLIDE 10.(Make the quotient of the numbers 200 and 40).

So, the ratio is found by division.

Open page 118 of the textbook and read the section “Speak correctly”

Now read this relation in three ways.

(1-the ratio of the number two hundred to the number forty;

2-relation of numbers two hundred and forty;

3 is the ratio of two hundred to forty).

4.2. –You and I already know what “Definition” is and we can give a definition of a divisor that is a multiple of reciprocal numbers.

SLIDE 11.Let's return to the problem about the cobra and the crocodile. Read the dialogue of the animals on the slide. Now try to come up with a definition of the concept of “relationship”.

Suggestive questions:

By what action do we find a relationship? Result of division?

Can numbers be equal to zero?

What does attitude show?

And now, in the problem about a cobra and a crocodile, let us denote a -age of the crocodile, and for b - the age of the cobra and create a definition for the ratio of numbers a And b.

Expected response from students with leading questions from the teacher:

“The ratio of numbers a and b is called:

1.The quotient of two numbers a and b;

Does it make sense to compare numbers multiple times, at least one of which is equal to zero?

2. numbers are different from zero;

-What information can be obtained from the relationship?

(How many times more, less, what part is one number from another).

3.The ratio shows how many times the first number is greater than the second, or what part the first number is of the second.”

-Try to connect all the conclusions and formulate a definition of the relationship yourself. (After listening to the formulations, invite students to read the definition on page 117 of the textbook).

SLIDE 12. 4.3 - There are 6 white and 12 red roses in the flowerbed. What do relationships show?

a) 6:12
b) 12:6
c) 6:18
d) 18:12

a) The number of white roses is half the number of red roses.
b) The number of red roses is 2 times the number of white roses.
c) What part are white roses of all the flowers in the flowerbed?
d) How many times is the number of all flowers in the flowerbed greater than the number of red roses?

What is the relationship?

Pay attention to cases a), b). What are these numbers called?

(Mutually inverse).

What did you notice during the calculation?

(Relationships can be “simplified”; by writing them as a fraction, you can reduce that fraction.)

It is sometimes convenient to express the ratio as a percentage. How to represent a number in %?

(Multiply by 100%). Express it as a percentage, which is convenient.

5) Primary consolidation in external speech.

– Let’s do exercise No. 722 (b, c, d) in your notebooks. (one student at the board: writes, reads, converts to percentage)

B)12.3:3=4.1=410%

D)9.1:0.07=130=13000%

SLIDE 13.- Complete the task: (in a notebook according to the options and on a closed board - 2 students according to the options on cards) ( see Attachment)

1 option There are 10 boys and 15 girls in the class.
Option 2 The notebook has 12 sheets, 4 of which are written on.

SLIDE 14. Solutions:

1 option

There are 1.5 times more girls in the class than boys; there are 50% more girls.

Boys make up two thirds of the number of girls.

Made up of boys from the class.

Made up of girls from the class.

Option 2

The third part of the notebook is covered with writing.

The notebook has only 3 times as many sheets as there are written pages.

Two-thirds of the notebook is not covered with writing.

There are only one and a half times more pages in the notebook than there are unwritten pages.

6) Independent work with self-test according to the standard on the board. (For those who have everything correct, put 5, for those who did not express it as a percentage - 4, for the rest - find errors and correct them)

Find the ratios, if convenient, express them as percentages:

7) Reflection of activity.(Lesson summary).

What new did we learn in class today?

What else needs to be worked on?

If you wish, submit your notebooks for verification.

Well done!

9) Homework: paragraph 20, resp. To questions, No. 722(a,d,f), 723, 747

Equipment:

1.laptop;

2. multimedia projector;

4.handouts (cards with tasks)

1.Vilenkin N.Ya. Mathematics. 6th grade: educational for general education. Institutions / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Shvartsburd. – 30th ed., erased. – M.: Mnemosyne, 2013.-288 p. : ill.;

2. Chesnokov A.S., Neshkov K.I. Didactic materials on mathematics for 6th grade. M.: Education, 2012.

Application.

Option 2

The notebook has 12 sheets, 4 of which are written on.

Based on this condition, make some relationships (at least two) and explain their meaning. Simplify the resulting relationships if possible; If convenient, express it as a percentage.

1 option

There are 10 boys and 15 girls in the class.

Based on this condition, make some relationships (at least two) and explain their meaning. Simplify the resulting relationships if possible; If convenient, express it as a percentage.

MANDATORY PART

1 We check the ability to find the relationship between numbers and quantities.

2 We check the ability to divide a number in a given ratio.

3 We test the ability to solve equations using the basic property of proportion.

4 We check the ability to solve problems on direct and inverse proportional dependence.

ADDITIONAL PART

5 We test the ability to solve non-standard problems using scale.

6 We test the ability to solve non-standard text problems.

Test No. 2. Mathematics 6th grade

on the topic: “INTEREST” (UMK S.M. Nikolsky)

MANDATORY PART

1 We test the ability to find percentages of a number.

2 We check the ability to find a number by its percentage.

3 We check the ability to find what percentage one number is from another.

4 We test the ability to solve a word problem using percentages.

ADDITIONAL PART

5 We check the ability to build a pie chart.

6 We test the ability to solve non-standard problems.

MANDATORY PART

1 We test the ability to perform arithmetic operations with fractions. 2 We test the ability to find a part of a number and a number from its part.

3 We check the ability to solve equations.

4 We check the ability to perform arithmetic operations with decimal fractions.

5 We check the ability to solve percentage problems.

6 We test the ability to solve word problems in 2-3 steps.

ADDITIONAL PART

7 We check the ability to perform calculations in a rational way..

8 We test the ability to solve problems on the probability of an event.

Score sheet for IQR

students____ 6th grade ____________________________Grade

tasks

final grade

conditional

signs

in points

1. We check the ability to perform arithmetic operations with fractions

1. Calculate

2. We check the ability to find a part of a number and a number from its part

Solve a problem

3. Testing your ability to solve equations

3.Solve the equation

4. We check the ability to perform arithmetic operations with decimal fractions.

4. Calculate

5 Testing your ability to solve percentage problems

Solve a problem

6 We check the ability to solve word problems in 2-3 steps.

text

7. We check the ability to make rational calculations

Calculate most convenient

way

8. We test the ability to solve non-standard problems.

task for

probability of an event

I know and I can

I know uncertainly

I don't know and I can't

Option 2.

Independent work on the topic: "Relationships" 6th grade. Option 2.

1. There are 75 apartments in the multi-storey building. 25 apartments are one-room, 30 apartments are two-room, the rest are four-room. What proportion of all apartments are one-room apartments? What percentage of all apartments are two-room apartments? How many times are there fewer four-room apartments than one-room apartments?

2. By what percentage did the cost of the product increase if before the markup it cost 350 rubles, and after the markup it cost 420 rubles?

3. There are 25 students in the class. Of these, 4 students wrote the biology test as “excellent”, 8 students wrote it as “good”, 12 students wrote it as “satisfactory”, the rest did not cope with the work. What percentage of students failed the work?

Independent work on the topic: “Relationships”, 6th grade.

Option 1.

Independent work on the topic: “Relationships”, 6th grade. Option 1.

3. There are 24 students in the class. Of these, 10 children are fond of athletics, 5 children are fond of basketball, and the rest are fond of swimming. What percentage of students in the class are interested in swimming?

1. The tourists planned to travel in three days

45 km. On the first day they walked 18 km, on the second day 15 km, and the rest of the way on the third day. How many percent of the way did they walk on the first day? How many times more did they walk on the second day than on the third? How much of the journey did you cover on the third day?

2. The cost of goods increased from 40 rubles to 50 rubles. By what percentage did the cost of the product increase?

3. There are 24 students in the class. Of these, 10 children are fond of athletics, 5 children are fond of basketball, the rest are fond of swimming. What percentage of students in the class are interested in swimming?

Independent work

MANDATORY PART

1 Checking the ability to find the relationship between numbers and quantities.

2 Checking ability to divide a number in a given ratio .

3 We test the ability to solve equations using the basic property of proportion .

4 Checking ability to solve problems involving direct and inverse proportionality.

ADDITIONAL PART

5 We test the ability to solve non-standard problems using scale.

6 We test the ability to solve non-standard text problems.

Test No. 2. Mathematics 6th grade

on the topic: “INTEREST” (UMK S.M. Nikolsky)

MANDATORY PART

1 Checking ability to find percentages of a number .

2 Checking the ability to find a number by its percentage.

3 Checking ability to find what percentage one number is from another .

4 Checking ability to solve word problems using percentages.

ADDITIONAL PART

5 Checking ability to create a pie chart .

6 Checking

MANDATORY PART

1 Checking ability to perform arithmetic operations with fractions.

2 Checking the ability to find a part of a number and a number from its part.

3 .

4 Checking

5 Checking ability to solve percentage problems.

6 Checking – 3 actions .

ADDITIONAL PART

7 Checking ability to perform calculations in a rational way. .

8 We test the ability to solve problems based on the probability of an event.

Score sheet for IQR

students____ 6th grade ____________________________

Grade

tasks

final grade

conditional

signs

in points

1. Checking ability to perform arithmetic operations with fractions

1. Calculate

2. Checking the ability to find a part of a number and a number from its part

Solve a problem

3. Testing your ability to solve equations

3.Solve the equation

4. Checking ability to perform arithmetic operations with decimal fractions.

4. Calculate

5 Checking problem solving skills percentages

Solve a problem

6 Checking ability to solve word problems in 2 – 3 actions .

text

7. We check the ability to make rational calculations

Calculate most convenient

way

8.Check ability to solve non-standard problems.

task for

probability of an event

I know and I can

I know uncertainly

I don't know and I can't