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How to make the right proportion. How to calculate proportion. How to solve a problem using proportion

Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, this can be represented as a rectangle in which one side denotes lettuce, the other side denotes water. The sum of these two sides will denote borscht. The diagonal and area of ​​such a "borscht" rectangle are purely mathematical concepts and are never used in borscht recipes.


How do lettuce and water turn into borscht in terms of mathematics? How can the sum of two segments turn into trigonometry? To understand this, we need linear angle functions.


You won't find anything about linear angle functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work whether we know they exist or not.

Linear angular functions are the laws of addition. See how algebra turns into geometry and geometry turns into trigonometry.

Is it possible to do without linear angular functions? You can, because mathematicians still manage without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves can solve, and never tell us about those problems that they cannot solve. See. If we know the result of the addition and one term, we use subtraction to find the other term. All. We do not know other problems and we are not able to solve them. What to do if we know only the result of the addition and do not know both terms? In this case, the result of addition must be decomposed into two terms using linear angular functions. Further, we ourselves choose what one term can be, and the linear angular functions show what the second term should be in order for the result of the addition to be exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we do very well without decomposing the sum; subtraction is enough for us. But in scientific studies of the laws of nature, the expansion of the sum into terms can be very useful.

Another law of addition that mathematicians don't like to talk about (another trick of theirs) requires the terms to have the same unit of measure. For lettuce, water, and borscht, these may be units of weight, volume, cost, or unit of measure.

The figure shows two levels of difference for math. The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the area of ​​units of measurement, which are shown in square brackets and are indicated by the letter U. This is what physicists do. We can understand the third level - the differences in the scope of the described objects. Different objects can have the same number of the same units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same notation for the units of measurement of different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or in connection with our actions. letter W I will mark the water with the letter S I will mark the salad with the letter B- borsch. Here's what the linear angle functions for borscht would look like.

If we take some part of the water and some part of the salad, together they will turn into one serving of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals will turn out. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we do not understand what, it is not clear why, and we understand very poorly how this relates to reality, because of the three levels of difference, mathematicians operate on only one. It will be more correct to learn how to move from one unit of measurement to another.

And bunnies, and ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar problem for adults. What do you get when you add bunnies and money? There are two possible solutions here.

First option. We determine the market value of the bunnies and add it to the available cash. We got the total value of our wealth in terms of money.

Second option. You can add the number of bunnies to the number of banknotes we have. We will get the amount of movable property in pieces.

As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.

But back to our borscht. Now we can see what will happen for different values ​​of the angle of the linear angle functions.

The angle is zero. We have salad but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. Zero borsch can also be at zero salad (right angle).


For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This is because addition itself is impossible if there is only one term and the second term is missing. You can relate to this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram the definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "behind the point zero" and other nonsense. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses all meaning: how can one consider a number that which is not a number. It's like asking what color to attribute an invisible color to. Adding zero to a number is like painting with paint that doesn't exist. They waved a dry brush and tell everyone that "we have painted." But I digress a little.

The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but little water. As a result, we get a thick borscht.

The angle is forty-five degrees. We have equal amounts of water and lettuce. This is the perfect borscht (may the cooks forgive me, it's just math).

The angle is greater than forty-five degrees but less than ninety degrees. We have a lot of water and little lettuce. Get liquid borscht.

Right angle. We have water. Only memories remain of the lettuce, as we continue to measure the angle from the line that once marked the lettuce. We can't cook borscht. The amount of borscht is zero. In that case, hold on and drink water while it's available)))

Here. Something like this. I can tell other stories here that will be more than appropriate here.

The two friends had their shares in the common business. After the murder of one of them, everything went to the other.

The emergence of mathematics on our planet.

All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to the trigonometry of borscht and consider projections.

Saturday, October 26, 2019

Wednesday, August 7, 2019

Concluding the conversation about , we need to consider an infinite set. Gave in that the concept of "infinity" acts on mathematicians, like a boa constrictor on a rabbit. The quivering horror of infinity deprives mathematicians of common sense. Here is an example:

The original source is located. Alpha denotes a real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take an infinite set of natural numbers as an example, then the considered examples can be represented as follows:

To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an "infinite hotel"? An infinity inn is an inn that always has any number of vacancies, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, on the other hand, are not able to move away from banal everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers exist. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on a shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take a unit from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I have written down the operations in algebraic notation and in set theory notation, listing the elements of the set in detail. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same one is added.

Option two. We have many different infinite sets of natural numbers on the shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If another infinite set is added to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler for measurements. Now imagine that you have added one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, they deprive us of free thinking).

pozg.ru

Sunday, August 4, 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical basis of Babylonian mathematics did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:

The rich theoretical basis of modern mathematics does not have a holistic character and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole cycle of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.

May we have many A consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter A, the subscript with a number will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question, how correctly applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.

As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.

As you can see, units of measurement and common math make set theory a thing of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.

Finally, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, this looks like a slowdown in time until it stops completely at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
I will show the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, and there are without a bow. After that, we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received sets "with a bow" and "red" the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.

The letter "a" with different indices denotes different units of measurement. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to which the set is formed, is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it with “obviousness”, because units of measurement are not included in their “scientific” arsenal.

With the help of units of measurement, it is very easy to break one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Remember!

By three known members of the proportion, you can always find its unknown (fourth) member.

solve proportion means to find all its members. Let's solve the proportion below
(find "x").

To find " x", We use the main property of proportion (the "cross" rule).

Now we are ready to figure out how to solve proportion problems.

Solving problems on proportions

Often proportion tasks closely related to percentages. You can brush up on your knowledge of percentages in the "Interests" section.

Task

50 shots were fired from the bow. 5 arrows flew past the target. Determine the hit percentage.

By tradition, we emphasize important and numerical data in the problem.

Note that we need to determine the percentage of hits, not the percentage of arrows that miss.

Therefore, we first calculate how many arrows hit the target. It will not be difficult to do this.

  • 50 − 5 = 45 (arrows) - hit the target.

Next, to solve the problem, we will make a table where we will enter all the data. Remember that opposite 100% in the table is usually written the total amount of something. Unknown percentages will be denoted by the letter x.


To correctly record the necessary data in the table, remember a simple rule.

The equality of two ratios is called proportion.

a :b =c :d. This is proportion. Read: A so applies to b, How c refers to d. Numbers a And d called extreme members of the proportion, and the numbers b And c – average members of the proportion.

Proportion Example: 1 2 : 3 = 16 : 4 . This is the equality of two ratios: 12:3= 4 and 16:4= 4 . They read: twelve is to three as sixteen is to four. Here 12 and 4 are the extreme members of the proportion, and 3 and 16 are the middle members of the proportion.

Basic property of proportion.

The product of the extreme terms of a proportion is equal to the product of its middle terms.

For proportion a :b =c :d or a/b=c/d the main property is written like this: a d \u003d b c .

For our proportion 12 : 3 = 16 : 4 the main property will be written as follows: 12 4 = 3 16 . It turns out the correct equality: 48 \u003d 48 .

To find the unknown extreme term of the proportion, you need to divide the product of the average terms of the proportion by the known extreme term.

Examples.

1) x: 20 = 2: 5. We have X And 5 are the extreme members of the proportion, and 20 And 2 - medium.

Solution.

x = (20 2):5- you need to multiply the middle terms ( 20 And 2 ) and divide the result by the known extreme term (number 5 );

x=40:5 is the product of the middle terms ( 40 ) divide by the known extreme term ( 5 );

x = 8. We got the desired extreme term of the proportion.

It is more convenient to write down the finding of an unknown member of the proportion using an ordinary fraction. Here is how the example we have considered would then be written:

The desired extreme term of the proportion ( X) will be equal to the product of the middle terms ( 20 And 2 ) divided by the known extreme term ( 5 ).

We reduce the fraction by 5 (divide by 5 X.

More such examples of finding an unknown extreme member of the proportion.

To find the unknown middle term of the proportion, you need to divide the product of the extreme terms of the proportion by the known middle term.

Examples. Find the unknown middle term of the proportion.

5) 9: x = 3: 14. Number 3 is the known average term of the given proportion, numbers 9 And 14 are the extreme terms of the proportion.

Solution.

x \u003d (9 14): 3 - multiply the extreme terms of the proportion and divide the result by the known middle term of the proportion;

x= 136:3;

x=42.

The solution to this example can be written differently:

The required average term of the proportion ( X) will be equal to the product of the extreme terms ( 9 And 14 ) divided by the known middle term ( 3 ).

We reduce the fraction by 3 (divide by 3 and the numerator and denominator of a fraction). Finding the value X.

If you forgot how to reduce ordinary fractions, then repeat the topic: ""

More such examples on finding the unknown average member of the proportion.

Sections: Mathematics

Type of lesson: Lesson of studying and primary consolidation of new knowledge.

Lesson form: Lesson-research.

Lesson Objectives:

  • to activate the cognitive activity of students;
  • introduce students to the concepts: proportion, members of the proportion; correct and incorrect proportions;
  • to introduce students to the basic property of proportion and to form the skill of determining the correct proportion.

Equipment:

The route sheets indicate the points that can be obtained for solving tasks. When scoring, the student takes into account the correctness of his decision, the speed of the decision (self-check and mutual check with the help of the presentation). In the “Additional points” line, points are given for answering additional questions, for helping the teacher organize the testing of other students, and also for “guessing” the topic of the lesson.

The cards are cut and distributed in envelopes to students (one envelope per desk).

3. Cards for a magnetic board (Figure 1, Figure 2, Figure 3)

During the lesson, these cards are posted on a magnetic board.

4. Puzzles (Figure 4, Figure 5, Figure 6, Figure 7).

Rebuses compiled by high school students (except for the “Proportion” rebus - this rebus was taken from a lesson presented at the FPI by teacher Kozak Tatyana Ivanovna, secondary school No. 20 Progress, Amur Region) are located on the board, students are invited to solve them after the lesson.

The technical equipment of the lesson is a computer, a projector for demonstrating a presentation, a screen. Computer presentation in Microsoft PowerPoint (Appendix 4).

I. Organization of the beginning of the lesson

Hello! Please check that you have handouts on your desk, that you have a red and blue pencil, and that you are ready for the lesson.

II. Message topics, goals and objectives of the lesson.

Today at the lesson we continue to study a large section of the mathematics course. We have finished studying the topic (what? - "Attitude"). Now we are starting to explore a new topic in this section. A few examples will help us to understand the topic of the lesson. On the title page of your route sheet, you need to fill in the table by solving the examples orally and then you will know the topic of today's lesson. SLIDE 1

So, the topic of today's lesson Proportion. SLIDE 2

Knowing the topic of the lesson, try to make a lesson plan. What should you learn in class today? What do you want to know? What do you want to learn in class?

We will make a plan, which we will supplement during the lesson. (students name the first two and last two points of the plan, the rest are filled in during the lesson, as new knowledge is “discovered”; the lesson plan is written on the board)

- repetition (questions related to attitude)

Definition of proportion

MEMBERS OF PROPORTION

RIGHT AND INCORRECT PROPORTIONS

MAIN PROPERTIES OF PROPORTION

Application in mathematics

Application in life

We will be able to analyze the last two points in the following lessons, as we study the topic.

III. Updating students' knowledge. Preparation for active educational and cognitive activity at the main stage of the lesson.

Discuss questions related to the topic “Attitude” with a classmate.

Who is ready to ask questions related to the last topic? (blizzard) MP1

- What is an attitude?

How can you write a relationship?

What questions does attitude answer?

How can you write the ratio of two numbers?

What can replace the do sign?

Why do you think we repeated these concepts?

They will help us when learning a new topic.

Take the envelopes and make up the relationship A To b And c To d two ways. (only 4 relations) WORK IN PAIRS.

MP2 You have several relationships in front of you. Find the meaning of these expressions. SLIDE 3

4: 0,5=
=
5: 10 =
=
8: 1 =
2,5: 5 =

Group the relationships according to a certain attribute and make the corresponding equalities.

IV. Assimilation of new knowledge.

4: 0,5 = 8: 1 = 5: 10 = 2,5: 5

On what basis did you group these relationships?

- Their values ​​are equal.

The resulting equalities are called proportions.

Think and define proportion.

HINT - proportion is ... ON SCREEN ( equality)

Equality of …WHAT ( relations)

How many relationships? ( two).

Who is sure of his opinion, write down the definition in the route sheet. MP3

Who is ready to go to the blackboard and draw up a definition of proportion? (Appendix 3)

DEFINITION (on a magnetic board): Proportion is the equality of two ratios.

Let's look at the interpretation of the word proportion in the dictionary of the Russian language Ozhegov S.I. SLIDE 4: “Proportion is a certain ratio of parts to each other, proportionality. In mathematics, the equality of two ratios.

You have formulated the definition of proportion as well as in the dictionary of the Russian language!

Think about what mathematical term the word “proportion” is consonant with? ( interest). How is the term "percentage" translated? ( from a hundred). So, "pro" is translated as "from". What part of the word is left? (“ a portion”). Where did you come across this word? (in cooking) What does it mean? ( size)

The word proportion comes from the Latin word proportio - proportionality. (etymological dictionary). SLIDE 4

Using the definition of proportion, write the proportions using the division sign and the fractional bar. (WORK IN PAIRS, envelopes).

In route sheets, write down the proportion using the letters a, b, c, d. MP4

And now we will find out what the numbers that make up the proportion are called.

Numbers a, b, c, d called members of the proportion

What is the first and last term of the proportion? ( a and c)

And what is usually (in life) called the first and last? (extreme)

So the terms a and b are called ...? (extreme)

Where are the terms c and d? (in the middle)

And what are the names of members c and d? ( medium)

Which members are highlighted in red? ( To early)

color (With rare) members.

middle members

Let's go back to the lesson plan - do you have something to add to it? (extreme and middle members of the proportion)

V. Primary consolidation of knowledge

MP5 Fill in the table:

What conclusion can be drawn? Record the output on the itinerary. ( In proportion, the product of the extreme terms is equal to the product of the middle terms)SLIDE 8

MP6 Here are five equalities. Are they all proportions?

Emphasize proportions.

= ; 7 + 11 = 36: 2; 72: 9 = 16: 2; = 20: 4; 5 40 = 100 2

SLIDE 7 Stand up, who finished.

Is everyone sure that there are three proportions here? Indeed, in the last equality, the product of the extreme terms is not equal to the product of the middle ones. Let's return to the definition of proportion ( Proportion - the equality of two ratios). Is the third equality the equality of the two relations? (is). By definition, is it a proportion? (Yes). Is the product of the extreme terms equal to the product of the middle terms? (No). So it's a proportion...? (wrong). This proportion is called incorrect. So, there are incorrect proportions and ...? (faithful). Formulate the main property of the proportion, using the knowledge gained. (In the right proportion, the product of the extreme terms is equal to the product of the middle ones).

VI. Consolidation of knowledge.

Fill in the table.

Correct proportion Wrong proportion

= = 20: 4

How else can you determine the right proportion or the wrong one? (find the value of the relationship)

In the future, we will talk about the correct proportions.

Let's go back to the lesson plan. What can be added? (correct and incorrect proportions)

MP7 Using the letters B and H, mark the correct and incorrect proportions.

= 1: 0,5 = 4,8: 2,4
7,5: 5 = 2: 3 =
10: 3 = 3 : 1 5:x = 20:4x

VII. Generalization and systematization.

MP8 Using the basic property of proportion, make up the correct proportion from the following numbers: 4, 5, 12, 15. How many correct proportions can you make?

VIII. Control and self-test of knowledge

MP9 Math Dictation

  1. Write down the proportion: The number 18 is related to 4 as 27 is related to 6.
  2. Write down the proportion: The ratio of three to five is equal to the ratio of two to seven.
  3. Write down the average terms of the proportion: 1.5: 2 \u003d 4.5: 6
  4. Write down the extreme members of the proportion: 2/1.9 = 3/2.8
  5. Is the proportion in paragraph 3 correct?
  6. Is the proportion in item 4 correct?
  7. Is the statement true: The root of the equation 20/5 \u003d x / 0.5 number 2
  8. Is the following statement true: Any four natural numbers can be used to form a proportion?

SLIDE 10. Peer review

IX. Summing up the lesson.

Please refer to the lesson plan.

What did you learn in class today? (what is a proportion, what does a proportion consist of, proportions are true and incorrect, the main property of a proportion, ...)

What did you learn in class today? (determine the extreme and middle members of the proportion, find out if the proportion is right or wrong, ...)

What other questions can be asked at the end of the lesson?

-How many correct proportions can be made from this correct proportion?

How can you tell if a proportion is right or wrong?

Let's remember the last task of the mathematical dictation.

Any four natural numbers can be used to form a proportion. The correct answer is YES. You can draw up a proportion, but it will not necessarily be true.

From the phrase " Any four natural numbers can be used to form a proportion. Eliminate one word to make this statement incorrect. (natural). Why? (The number 0 cannot be a member of a proportion). Any four numbers can be used to form a proportion

In this phrase Any four natural numbers can be used to form a proportion. insert one word to make the statement incorrect (true). From any four natural numbers, you can make the correct proportion.

Calculate the number of points you have earned in the lesson and assign a mark.

X. Information about homework and instruction on how to complete it

Mathematics - 6, Vilenkin N.Ya. et al. 6th edition

P.21, Nos. 760, 781, 782, 783 (a)

The ratio of two numbers

Definition 1

The ratio of two numbers is their private.

Example 1

    the ratio of $18$ to $3$ can be written as:

    $18\div 3=\frac(18)(3)=6$.

    the ratio of $5$ to $15$ can be written as:

    $5\div 15=\frac(5)(15)=\frac(1)(3)$.

By using ratio of two numbers can be shown:

  • how many times one number is greater than another;
  • what part one number represents from another.

When drawing up the ratio of two numbers in the denominator of a fraction, write down the number with which the comparison is made.

Most often, such a number follows the words "compared to ..." or the preposition "to ...".

Recall the basic property of a fraction and apply it to a relation:

Remark 1

When multiplying or dividing both terms of the relation by the same number other than zero, we obtain a ratio that is equal to the original one.

Consider an example that illustrates the use of the concept of a ratio of two numbers.

Example 2

The amount of precipitation in the previous month was $195$ mm, and in the current month - $780$ mm. How much has the amount of precipitation in the current month increased compared to the previous month?

Solution.

Compose the ratio of the amount of precipitation in the current month to the amount of precipitation in the previous month:

$\frac(780)(195)=\frac(780\div 5)(195\div 5)=\frac(156\div 3)(39\div 3)=\frac(52)(13)=4 $.

Answer: the amount of precipitation in the current month is $4$ times more than in the previous one.

Example 3

Find how many times the number $1 \frac(1)(2)$ is contained in the number $13 \frac(1)(2)$.

Solution.

$13 \frac(1)(2)\div 1 \frac(1)(2)=\frac(27)(2)\div \frac(3)(2)=\frac(27)(2) \cdot \frac(2)(3)=\frac(27)(3)=9$.

Answer: $9$ times.

The concept of proportion

Definition 2

Proportion is called the equality of two relations:

$a\div b=c\div d$

$\frac(a)(b)=\frac(c)(d)$.

Example 4

$3\div 6=9\div 18$, $5\div 15=9\div 27$, $4\div 2=24\div 12$,

$\frac(8)(2)=\frac(36)(9)$, $\frac(10)(40)=\frac(9)(36)$, $\frac(15)(75)= \frac(1)(5)$.

In the proportion $\frac(a)(b)=\frac(c)(d)$ (or $a:b = c\div d$), the numbers a and d are called extreme members proportions, while the numbers $b$ and $c$ are middle members proportions.

The correct proportion can be converted as follows:

Remark 2

The product of the extreme terms of the correct proportion is equal to the product of the middle terms:

$a \cdot d=b \cdot c$.

This statement is basic property of proportion.

The converse is also true:

Remark 3

If the product of the extreme terms of a proportion is equal to the product of its middle terms, then the proportion is correct.

Remark 4

If the middle terms or extreme terms are rearranged in the correct proportion, then the proportions that will be obtained will also be correct.

Example 5

$6\div 3=18\div 9$, $15\div 5=27\div 9$, $2\div 4=12\div 24$,

$\frac(2)(8)=\frac(9)(36)$, $\frac(40)(10)=\frac(36)(9)$, $\frac(75)(15)= \frac(5)(1)$.

Using this property, it is easy to find an unknown term from a proportion if the other three are known:

$a=\frac(b \cdot c)(d)$; $b=\frac(a \cdot d)(c)$; $c=\frac(a \cdot d)(b)$; $d=\frac(b \cdot c)(a)$.

Example 6

$\frac(6)(a)=\frac(16)(8)$;

$6 \cdot 8=16 \cdot a$;

$16 \cdot a=6 \cdot 8$;

$16 \cdot a=48$;

$a=\frac(48)(16)$;

Example 7

$\frac(a)(21)=\frac(8)(24)$;

$a \cdot 24=21 \cdot 8$;

$a \cdot 24=168$;

$a=\frac(168)(24)$;

$3 gardener - $108 trees;

$x$ gardeners - $252$ tree.

Let's make a proportion:

$\frac(3)(x)=\frac(108)(252)$.

Let's use the rule for finding the unknown term of the proportion:

$b=\frac(a \cdot d)(c)$;

$x=\frac(3 \cdot 252)(108)$;

$x=\frac(252)(36)$;

Answer: It will take $7$ gardeners to prune $252$ trees.

Most often, the properties of proportion are used in practice in mathematical calculations in cases where it is necessary to calculate the value of an unknown member of the proportion, if the values ​​of the other three members are known.