What is scale? Types of scales. Discovery of new knowledge

Math lesson in 6th grade

Lesson topic: “Scale”

Lesson objectives:

Review the concepts of “ratio”, “proportion”, “distance ratio”;

Introduce the concept of “scale”;

to form in students practical skills and skills related to mathematical calculations when solving word problems;

promote the development of the ability to draw conclusions

Develop cognitive interest students

Cultivate love for your native land.

Tasks:

expand knowledge about scale, give it a practical orientation

Equipment: computer, projector, tables, map of the Stavropol Territory, world map, cards with a site plan, with a building plan, with drawings of details; thread, ruler, compass - meter.

Lesson progress:

Opening remarks teachers.

Today we will continue to study the topic “Proportions”. Let's consider their application in one of the areas of life. Let's prove how important practical application has the material for today's lesson.

So, let's start by repeating what we've covered.

Solution of oral exercises.

What is an attitude? ( Quotient of two numbers.)

What does attitude show? (How many times is one number greater than the other, if the larger number is divided by the smaller, or what part one number is of the other, if divided smaller number for more.)

What is the main condition for creating a relationship? (The ratio of quantities is found if they are expressed in the same units of measurement).

So that you don't forget about last fact, I will give an example of solving one problem. (Slide No. 2) Sasha decided to find the ratio of the mass of the mouse to the mass of the elephant. A mouse weighs 50 grams, and an elephant weighs 5 tons. “Let’s make a ratio of 50: 5,” said Sasha. “A mouse is 10 times heavier than an elephant.” Is Sasha right?

Answer: No, that's wrong. After all, Sasha used quantities expressed in different dimensions. You need to do this: 5 T= 5 000 000 G; 50: 5 000 000 = 1: 100 000.

Find the ratio of the numbers 8 and 2; 1 and 10; sizes: 2cm and 4cm; 3cm and 6m; 5cm and 4.5km. (Slide No. 3)

How many times is the number 8.4? more number 4,2? (Slide No. 4)

Learning new material

A geographical map is one of the most important documents of human culture. Large territories, states or parts of the world, are depicted on geographical maps. Every wrinkle on the map, every stroke is the result of enormous long-term work of explorers, brave travelers and researchers. (Slide No. 5)
Look at the map of Russia - our Motherland. To make your work more convenient, open atlases. Find the capital of our homeland - Moscow and main city our region - Stavropol. Let's find the distance from Moscow to Stavropol (using a ruler). What distance did you get? (6.2 cm) Tell me, is this actually the same distance? No. We will calculate the actual distance from Moscow to Stavropol a little later.

Areas on the map earth's surface are shown in reduced size. For example, a distance of 1 kilometer on a map is a segment of 1 centimeter.

How many times is the segment on the map smaller than the distance on the ground? (Slide No. 6)

We found the ratio of the length of a segment on the map to the length of the corresponding distance on the ground; this value is called scale.

The ratio of the length of a segment on the map to the length of the corresponding segment on the ground is called the map scale.

The scale shows how many times the distance on the map is less than the distance on the ground. (Slide No. 7)

Note the scale notation. Scale 1: 100000 is called numerical.

Consolidating a new concept

Task No. 1

X=(3*100000): 1=300000cm=3000m=3km

Answer: 3 km

No. 821 (independently, with subsequent verification)

Attitude 8.5:x write down and say out loud the definition of scale. The two relationships can be equated.

8.5: x = 1: 1000000

X=(8.5*1000000):1= 8500000cm=85000m=85km

Answer: 85km

Task No. 2

The length of the segment on the ground is 20 m. What is the length of this segment on a map made on a scale of 1: 1000?

(independently with subsequent verification)(Slide No. 9)

x: 2000 = 1: 1000,

Answer: 2 cm

Task No. 3 The length of the segment on the ground is 240 m, and the length of the corresponding segment on the map is 2.4 cm. Find the scale of the map. (Slide No. 10)

According to the definition, the scale of this map is equal to the ratio 2.4cm To 240m.

Answer: 1: 10000

Fizminutka(conducted by a physiologist)

We will leave the desks together,
But there is no need to make noise at all.
Stand up straight, legs together,
Turn around in place.
Let's clap our hands a couple of times
And we'll drown a little.
Now let's imagine, kids,
It’s as if our hands are branches.
Let's shake them together
Like the wind blows from the south. The wind blows in our faces and shakes the tree. The tree is getting higher and higher, And the guys are quieter, quieter.
The wind died down. We sighed together.
We need to continue the lesson.
We caught up and sat down quietly
And they looked at the board

Practical task(by groups)

Find on the map of the Stavropol Territory:

1st group – distance from our x. Controversial before district center Abundant; (Answer: 7.3 km)

2nd group - distance from our farm to the regional center - the city of Stavropol ( Answer: 32.85 km)

See how the scale is set on the map of the Stavropol Territory: 1: 365000

1st group

X= 365000*2:1=730000cm=7300m=7.3km

Answer: distance from x. Spornoye to Izobilny is 7.3 km

2nd group

X= 365000*9:1=3285000cm=32850m=32.85km

Answer: distance from x. Spornogo to Stavropol 32.85 km

Now let’s return to the problem of the distance from Moscow to Stavropol. You have measured the distance on the map of Russia from Moscow to Stavropol. Calculate the actual distance from Moscow to Stavropol.

6.2:x = 1:20000000

X=6.2*20000000=124000000cm=1240km

Answer: the distance from Moscow to Stavropol is 1240 km

Look at the map. This is not a simple map, but a topographic one. It gives a complete picture of the nature of the area. This card is taken if you need to build a bridge, conduct railway, lay the gas pipeline route. It is used by people of many professions. On this map the scale is set differently than on previous maps. This linear scale. (Slide No. 11)

In this lesson, we were introduced to only one area of ​​​​use of scale, namely, when depicting areas of land on a map.

In practice, it is necessary to image very large parts (Slide No. 12, 13)(for example, a car, parts of an airplane, a ship) and very small (model of an atom, parts of a clock mechanism, etc.). Therefore, when drawing, images large parts are reduced, and small ones are increased. For this, scale is also used. We will talk about this in more detail in drawing lessons.

Summing up. Reflection.

What new did you learn today?

What questions were answered?

What new questions have arisen?

What types of tasks caused difficulties?

What parts of the lesson made you happy or sad?

I hope today's lesson helped you discover the unknown in the previously known concept of “scale”. About smart person They say: “thinks big.” Let's learn this! (Slide No. 15)

Homework assignment(Slide No. 16)

1)Practical task:

Measure the size of the room you live in. Draw a plan of the room on a scale of 1:50. Show the location of the furniture on the plan

2) paragraph 23, No. 840

Thanks for the lesson. (Slide No. 17)

Without it, it is impossible to construct a single geographical map. What is scale? And what types of scales exist in cartography and geodesy? This will be discussed in this article.

What is scale?

Scale is a German word (masstab), which consists of two parts: mass - “measure, magnitude” and stab - “stick, pole”. Measuring pole - this is how this term can be translated.

What is scale? In general terms this is mathematical quantity, which shows how many times the model (image) is reduced compared to the original. This concept is actively used in mathematics, cartography, modeling, geodesy and design, photography, and programming.

In other words, scale is the ratio of two linear dimensions. In cartography, it shows how many times a segment on a map (or plan) is reduced compared to the actual length of the same segment. When compiling any geographical map, it is impossible to depict objects (forest, village, building, etc.) in real size. Therefore, all values ​​are reduced many times (by 5, 10, 100, 1000 times, and so on). The scale of the map is precisely this value expressed as a number.

Types of scales

Scale is shown on maps and drawings using numbers or graphically. Accordingly, several types are distinguished.

The numerical scale is in the form of a fraction. It is most common in cartography. Many of us have seen this designation at the bottom topographic map or site plan. The numerical scale of the map has the following form (for example): 1:100,000. This means that the real length of the segment on the ground is 100,000 times greater than its length on this map.

A named scale is used when you need to know what the map scale is. It is also quite often indicated on geographical maps. It looks like this: 1 cm - 1 km.

Linear scale is already a graphic type of scale. It is a ruler, which is divided into columns of appropriate sizes. The photo above shows this type scale.

Transverse scale is a more sophisticated version of the graphical view. It is used for the most accurate measurements and can be found on more serious maps.

How to use the map scale correctly? Suppose you need to find out from a specific card real distance between villages A and B. At the same time, you are given the following scale: 1 cm - 0.5 km (or 1:50,000). To do this, you need to take a regular ruler and measure the distance between two points on the map. Then the resulting value (suppose this is a segment 5 centimeters long) should be multiplied by 0.5 km, according to the scale of our map. Thus, we will get the correct answer: the distance between village A and village B is 2.5 kilometers.

Types of maps (by scale)

Scale is one of the criteria for classifying geographic maps. So, according to him, all cards are divided into:

• small-scale (scale up to 1:1,000,000);
• medium-scale (from 1:1,000,000 to 1:200,000);
• large-scale (from 1:200,000 and more).

Of course, on large-scale maps the terrain is more detailed: individual streets or even buildings can be shown here. The larger the scale of the map, the more terrain objects can be depicted on it.

Small-scale geographic maps, as a rule, are used to depict hemispheres and continents, medium-scale - for states and their parts, large-scale - for individual, small areas. Military personnel, local historians, and tourists are very familiar with large-scale maps.

Cartographic generalization

No matter how detailed the map is, it still will not be able to display absolutely all the objects and details that are present in a given area. This is precisely the essence of the concept of “cartographic generalization”.

The word generalis can be translated from Latin language as "generalized". Generalization is the process of selecting those geographical objects, which will be depicted on a specific map. Moreover, this process is objective, expedient and scientifically sound.

To understand what generalization is, it is enough to remember the cards that you probably held in your hands. So, on the map of Eurasia you are unlikely to find the city of Cherepovets. And here it is on the map Vologda region it will definitely be noted.

Cartographic generalization helps to make the map of the highest quality, functional, and easy to read. Of course, it directly depends on the scale.

In conclusion

So what is scale? This value shows how much the image is reduced in comparison with the actual size of the depicted object. This concept has received greatest distribution in cartography and geography. There are several types of scales: numerical, named, linear and transverse.

The concept of cartographic generalization is closely related to the term “scale”. This process allows surveyors to select the most important geographical features and display them on a geographical map.

Scale

Quite often in life we ​​use maps, drawings, floor plans, where all dimensions are significantly smaller than natural ones. Because it is impossible to depict, for example, sections of the earth's surface or the layout of an apartment in life-size on a small sheet of paper. So they came up with the idea of ​​depicting large objects in a reduced form. Thus, each segment on the map is many times smaller than the corresponding segment on the ground. In order to maintain the proportions of all quantities, the concept of “scale” was introduced.

Scale is the ratio of the length of a segment in the image to the length of the corresponding segment on the ground, in other words, to its real length.

Let us determine the scale of the map if the actual actual length of the segment and the length of the segment on the drawing or map are known. Let the length of the segment on the map be 1 cm, but the length of the segment in reality is 3 km.

3 km = 3000 m = 300000 cm.

Then the scale of this punishment will be equal to 1: 300000. They say that the map was made on a scale of one three hundred thousandth.

How can you determine the length of a segment on a map if the scale and length of the segment on the ground are indicated?

Let's solve the problem:

The length of the section on the ground is 6.3 km. Find the corresponding length of the segment on a map made at a scale of 1:100000. Such problems are solved using proportions. Therefore, let us remember what proportion is.

Proportion is the equality of two ratios.

Let's denote the length of the segment (in kilometers) on the map with the letter X and draw up a proportion.

X: 6.3 = 1: 100000. Therefore, x = 6.3 · 1: 100000. X = 0.000063 km.

0.000063 km = 6.3 cm. We find that the length of the segment on the map is 6.3 cm.

Let's consider another problem:

The length of the segment on the map is 2 cm, the scale of the map is 1:1000.

It is necessary to find the length of the segment on the ground. Let us denote the length of the segment on the ground by X and find the ratio of the length of the segment on the map to the length of the segment on the ground, which will be equal to the scale of the map. Those. let's make a proportion: 2: X = 1: 1000. Solve it: X = 2 · 1000: 1. X = 2000 (cm). We find that the length of the segment on the ground is 2000 cm, that is, 20 meters.

Math lesson in 6th grade

Lesson topic: “Scale”

Lesson objectives:

• Review the concepts of “ratio”, “proportion”, “distance ratio”;
• Introduce the concept of “scale”;
• to develop in students practical skills related to mathematical calculations when solving word problems;
• promote the development of the ability to draw conclusions
• Develop students' cognitive interest
• Cultivate love for your native land.

Tasks:

expand knowledge about scale, give it a practical orientation

Equipment: computer, projector, tables, map of the Stavropol Territory, world map, cards with a site plan, with a building plan, with drawings of details; thread, ruler, compass - meter.

Lesson progress:

1. Teacher's opening speech.

Today we will continue to study the topic “Proportions”. Let's consider their application in one of the areas of life. Let's prove what an important practical application the material in today's lesson has.

So, let's start by repeating what we've covered.

1. Solution of oral exercises.

1. What is an attitude? (Quotient of two numbers.)
2. What does attitude show? (How many times is one number greater than the other if the larger number is divided by the smaller, or what part one number is of the other if the smaller number is divided by the larger.)
3. What is the main condition for creating a relationship?(The ratio of quantities is found if they are expressed in the same units of measurement).

So that you do not forget about the last fact, I will give an example of solving one problem.(Slide No. 2) Sasha decided to find the ratio of the mass of the mouse to the mass of the elephant. A mouse weighs 50 grams, and an elephant weighs 5 tons. “Let’s make a ratio of 50: 5,” said Sasha. “A mouse is 10 times heavier than an elephant.” Is Sasha right?

Answer: No, that's wrong. After all, Sasha used quantities expressed in different dimensions. You need to do this: 5 t = 5,000,000 g; 50: 5,000,000 = 1: 100,000.

1. Find the ratio of the numbers 8 and 2; 1 and 10; sizes: 2cm and 4cm; 3cm and 6m; 5cm and 4.5km.(Slide No. 3)
2. How many times is 8.4 greater than 4.2?(Slide No. 4)

1. Learning new material

A geographical map is one of the most important documents of human culture. Large territories, states or parts of the world, are depicted on geographical maps. Every wrinkle on the map, every stroke is the result of enormous long-term work of explorers, brave travelers and researchers.(Slide No. 5)
Look at the map of Russia - our Motherland. To make your work more convenient, open atlases. Find the capital of our homeland - Moscow and the main city of our region - Stavropol. Let's find the distance from Moscow to Stavropol (using a ruler). What distance did you get? (6.2 cm) Tell me, is this actually the same distance? No. We will calculate the actual distance from Moscow to Stavropol a little later.

On the map, areas of the earth's surface are depicted in a reduced form. For example, a distance of 1 kilometer on a map is a segment of 1 centimeter.

How many times is the segment on the map smaller than the distance on the ground?(Slide No. 6)

We found the ratio of the length of a segment on the map to the length of the corresponding distance on the ground; this value is called scale.

The ratio of the length of a segment on the map to the length of the corresponding segment on the ground is called the map scale.

The scale shows how many times the distance on the map is less than the distance on the ground.(Slide No. 7)

Note in notebooks:(Slide No. 8)

Note the scale notation. Scale 1: 100000 is called numerical.

1. Consolidating a new concept

Task No. 1

3: x=1: 100000

X=(3*100000): 1=300000cm=3000m=3km

Answer: 3 km

No. 821 (independently, with subsequent verification)

Ratio 8.5:x write down and say out loud the definition of scale. The two relationships can be equated.

8.5: x = 1: 1000000

X=(8.5*1000000):1= 8500000cm=85000m=85km

Answer: 85 km

Task No. 2

The length of the segment on the ground is 20 m. What is the length of this segment on a map made on a scale of 1: 1000?

(independently with subsequent verification)(Slide No. 9)

x: 2000 = 1: 1000,

Answer: 2 cm

Task No. 3 The length of the segment on the ground is 240 m, and the length of the corresponding segment on the map is 2.4 cm. Find the scale of the map.(Slide No. 10)

According to the definition, the scale of this map is equal to the ratio 2.4cm to 240m.

Answer: 1: 10000

Fizminutka (conducted by a physiologist)

We will leave the desks together,
But there is no need to make noise at all.
Stand up straight, legs together,
Turn around in place.
Let's clap our hands a couple of times
And we'll drown a little.
Now let's imagine, kids,
It’s as if our hands are branches.
Let's shake them together
Like the wind blows from the south. The wind blows in our faces and shakes the tree. The tree is getting higher and higher, And the guys are quieter, quieter.
The wind died down. We sighed together.
We need to continue the lesson.
We caught up and sat down quietly
And they looked at the board

1. Practical task (in groups)

Find on the map of the Stavropol Territory:

1st group – distance from our x. Disputed to the regional center of Izobilny;(Answer: 7.3 km)

2nd group – distance from our farm to the regional center - the city of Stavropol ( Answer: 32.85 km)

See how the scale is set on the map of the Stavropol Territory: 1: 365000

1st group

2:x = 1:365000

X= 365000*2:1=730000cm=7300m=7.3km

Answer: distance from x. Spornoye to Izobilny is 7.3 km

2nd group

9:x = 1:365000

X= 365000*9:1=3285000cm=32850m=32.85km

Answer: distance from x. Spornogo to Stavropol 32.85 km

Now let’s return to the problem of the distance from Moscow to Stavropol. You have measured the distance on the map of Russia from Moscow to Stavropol. Calculate the actual distance from Moscow to Stavropol.

6.2:x = 1:20000000

X=6.2*20000000=124000000cm=1240km

Answer: the distance from Moscow to Stavropol is 1240 km

Look at the map. This is not a simple map, but a topographic one. It gives a complete picture of the nature of the area. This card is taken if you need to build a bridge, lay a railroad, or lay a gas pipeline. It is used by people of many professions. On this map the scale is set differently than on previous maps. This linear scale. (Slide No. 11)

In this lesson, we were introduced to only one area of ​​​​use of scale, namely, when depicting areas of land on a map.

In practice, it is necessary to image very large parts(Slide No. 12, 13) (for example, a car, parts of an airplane, a ship) and very small (model of an atom, parts of a clock mechanism, etc.). Therefore, when drawing, images of large parts are reduced, and small ones are enlarged. For this, scale is also used. We will talk about this in more detail in drawing lessons.

1. Summing up. Reflection.

• What new did you learn today?
• What questions were answered?
• What new questions have arisen?
• What types of tasks caused difficulties?
• What parts of the lesson made you happy or sad?

I hope today's lesson helped you discover the unknown in the previously known concept of “scale”. They say about an intelligent person: “he thinks big.” Let's learn this!(Slide No. 15)

1. Homework assignment (Slide No. 16)

1)Practical task:

Measure the size of the room you live in. Draw a plan of the room on a scale of 1:50. Show the location of the furniture on the plan

2) paragraph 23, No. 840

Thanks for the lesson.(Slide No. 17)

Preview:

To use presentation previews, create a Google account and log in to it: https://accounts.google.com

Slide captions:

“Scale” Lesson topic

Mouse - 50 g Elephant - 5 t 50: 5 = 10 ??? 50: 5,000,000 = 1: 1,000,000 5 t = 5,000,000 g

Solve orally

How many times is the number 8.4 greater than the number 4.2? 2 times

How many times is the segment on the map smaller than the distance on the ground?

The ratio of the length of a segment on the map to the length of the corresponding segment on the ground is called the map scale. The scale shows how many times the distance on the map is less than the distance on the ground.

On the map 1 cm On the ground 1 km = 100000 cm Scale 1: 100000

On the map 3 cm On the ground X cm Scale 1: 100000 Task No. 1 3: x = 1: 100000

On the map 8.5 cm On the ground X cm Scale 1: 1000000 No. 821 8.5: x = 1: 1000000 X = (8.5*1000000) : 1= =8500000cm=85000m=85km

On the map x cm On the ground 20 m = 2000 cm Scale 1: 1000 Task No. 2 x: 2000 = 1: 1000

Problem No. 3 The length of the segment on the ground is 240 m, and the length of the corresponding segment on the map is 2.4 cm. Find the scale of the map. On the map 2.4 cm On the ground 240m = 24000cm Scale (M) ? According to the definition, the scale of this map is equal to the ratio of 2.4 cm to 240 m.

On the map 2 cm On the ground X cm Scale 1: 365000 X. Sporny - Izobilny 2: x = 1: 365000 X = (365000 * 2) : 1= =730000cm==7.3 km

On the map 9 cm On the ground X cm Scale 1: 365000 X. Sporny - Stavropol 9: x = 1: 365000 X = (365000 * 9) : 1= =3285000 cm = 32.85 km

1 2 3 4 5 6 km Novosergievsky district

Scaled-down model of a fire truck High-scaled model of an atom

They say about an intelligent person: “he thinks big.” Let's learn this!

Homework Practical work Measure the size of the room you live in. Draw a plan of the room on a scale of 1:50. Show the location of the furniture on the plan