Use a palette to measure the area of ​​the next figure. Measuring areas with palettes

Today you will find out how to find square in a new way.

Answer questions for the teacher (possibly in the video room).

1. Remember what area is?
2. In what units is it measured?
3. Tell me what geometric shapes you know?
4. Areas, what shapes can you find?
5. What formulas can you use for this? Write them down and show them to your teacher. You can show it in the video room or write on .

Can you find the area of ​​a triangle?

Do you know a formula that can help you solve this problem?

Now you don't know this formula. They'll tell you about it in high school.

Still, let's try to solve this problem.

Figure 1 shows a rectangle. Its dimensions and area S = 200 square meters are indicated. units

Suggest options for finding the areas of the triangles shown in Figure 2.

Write down your solutions in your notebook and explain them to the teacher (possibly in the video room).

So, you were able to find the area of ​​a right triangle. The method you proposed is valid only for this type of triangle.

But they are different. Therefore, we need to get acquainted with a new way of finding areas.

Today in the lesson you will learn to find the area of ​​shapes using.

Let's learn how to determine the area using a palette.

Reasoning:

1. One side length of a rectangle is 3 cm.
2. Three sides of a small square fit into three centimeters.
3. Therefore, the side length of the square is 1 cm (3 cm: 3 = 1 cm).
4. The other side length of the rectangle is 5 cm.
5. The 5 sides of a small square fit into five centimeters.
6. Therefore, the side length of the square is 1 cm (5 cm: 5 = 1 cm).
7. We conclude that we really have squares with a side of 1 cm.
8. The area of ​​this small square is 1 cm2.
9. We count how many squares are inside the rectangle. There are 15 of them.
10. Therefore, the area of ​​the rectangle will be equal to: 1 cm 2 · 15 = 15 cm 2.
11. So Kolya has the correct answer.

Get acquainted with the algorithm for finding the area of ​​a figure using a palette.

Watch and repeat, how you can use the palette to find the area of ​​a free-form figure.

Let's look at an example of finding the area of ​​a geometric figure using a palette.

Measuring the area of ​​a figure using a palette

At school, children become familiar with a large number of measuring instruments and devices.
Inna SYCHEVA, teacher at school No. 1936 in Moscow, shows how the area of ​​a figure is calculated using one of these devices - a palette.

Subject."Measuring the area of ​​a figure using a palette."

Goals. Learn to perform approximate calculations of areas; introduce the calculation of area using a palette using an algorithm; repeat length units and area units; develop thinking, attention, memory.

Equipment. Textbook "Mathematics" (4th grade, part 1, author M.I. Moro and others), algorithm table, palettes, individual cards, screen, epidiascope, films with figures.

DURING THE CLASSES

I. Organizational moment

II. Lesson topic message

Teacher. Today in the lesson you will learn how to perform an approximate calculation of area and get acquainted with a device for this.

III. Getting to know new material

U. Look at the figure on the screen.

– How much space does the figure take up? A on surface? In other words, what is its area?

The children's answers are listened to.

– We can answer this question only approximately by indicating the boundaries within which the area of ​​the figure is located A. The area of ​​the figure is more than 6 cells, but less than 16.

On the desk:

The result is written on the board using an approximate equal sign».

– This means that the area of ​​our figure is approximately 11 square units.

We were able to calculate all this thanks to the fact that the figure A was divided into cells. What to do if there are no such cells?

Children. Draw the figure into squares yourself.

U. Correct, but it will take a lot of time. To speed up the work, people came up with a device for determining the area of ​​figures.

The teacher gives the children transparent films, lined with square centimeters, and cards with shapes.

- This is a device in front of you. Open your textbooks to page 49 and read what it is called.

D. To approximately determine the area of ​​a figure, use palette .

Palette - a transparent film divided into equal squares: these can be square decimeters, square centimeters, square millimeters.

U. Look at your palettes. How are they divided?

D. Per square centimeters.

U. In the textbook on page 49, a palette is also superimposed on the colored figures, divided into square centimeters. Read how to find the area of ​​a blue figure.

Children read the text marked with a red line.

– What is the area of ​​this figure?

D. Approximately 31 square centimeters.

U. Let's try to derive a formula that approximately calculates the area.

Children, together with the teacher, derive and write down the formula.

On the desk:

– Find the area of ​​the green and pink shapes.

D. The area of ​​the green figure is approximately 6 + 16: 2 = 14 square centimeters.

– The area of ​​the pink figure is approximately 5 + 16: 2 = 13 square centimeters.

U. Pick up the cards with the figures depicted on them. Using a palette, find their area.

Children complete the task.

– Let’s try to derive an algorithm for finding the area of ​​a figure using a palette.

The teacher writes each step on the board.

On the desk:

IV. Physical education minute

V. Practical work

U. Draw some closed line on a piece of paper and find the area of ​​the figure bounded by this line.

Children complete the task in their notebooks, find the area, and name their answers.

– Draw a circle with a radius of 4 centimeters with a compass, find the area of ​​the resulting circle using a palette.

Children find the square.

VI. Reinforcing the material covered

U. Find task 265 on page 50. We perform the task according to the options: option 1 - the first part of the number, option 2 - the second part.

Children complete the task independently.

– Exchange notebooks and check the work of your neighbors.

Children check.

– Calculate the perimeter and area of ​​the polygon.

On the desk:

– Solve a logic problem. For each figure, explain why it is redundant.

On the desk:

D. First let's remove the figure IN, since among the quadrilaterals there is a triangle. Then we will remove the figure WITH, since there will remain figures with pairwise equal sides. Let's remove the figure D, since the angles in it are not right.

VII. Independent work

U. Complete exercises 267 and 262.

Children do their work and hand in their notebooks.

VIII. Lesson summary

U. What tool did you use to find the approximate area of ​​a figure?

D. Using a palette.

U. What formula did you use?

D. S= A + V: 2.

U. How many of you have learned to approximate the area of ​​a figure?

Children raise their hands.

IX. Homework

The teacher distributes cards with the number 5:

U. At home, calculate the area of ​​the figure and solve problems 261 and 263.

Open lesson plan

Subject: mathematics.

Topic: "Measuring the area of ​​a figure. Palette."

Goals:

· Introduce children to the method of finding the area of ​​shapes of various shapes using a palette.

· Learn to analyze geometric shapes.

· To develop students’ logical thinking, the ability to accurately and reasonably argue, to highlight those aspects of observed phenomena that are necessary for research and comprehension of the problem.

· Improve problem solving skills.

· Cultivate interest in the subject, curiosity, and a friendly attitude towards classmates in joint work.

Lesson objectives: Create conditions for independent search for knowledge.

Equipment:"Measuring the area of ​​a figure using a palette", presentation

Didactic materials for the lesson:

Progress of the training session

1.Organizational moment:

Well, check it out, my friend,
Are you ready to start the lesson?
Is everything in place?
Is everything alright?
Pen, book and notebook?
Is everyone sitting correctly?
Is everyone watching carefully?

Everyone wants to receive
Just a "5" rating?

2. Report the topic of the lesson

Teacher: Guys, we are once again plunging into the world of interesting mathematics. Today we will get acquainted with geometric figures, the area of ​​which is found in a new way. We’ll find out which one in class. We can do this.

3. Preparation of work in notebooks.

Mathematics is the queen of all sciences. It is needed in any science, in any profession, for example, archaeologists. Do you know who archaeologists are? Let's watch the presentation "Who are archaeologists."

7. Work using individual cards.

Finding the area of ​​rectangular and square shapes.

Let's play this profession. You have arrived for an excavation. You need to determine the area of ​​the rectangular or square plot of land on which you will be searching for something. (Everyone has a card with a drawing and data. Children use the formula to find the area, make a drawing and write it down in a notebook.) One student at the blackboard.

S = 5* 9= 45 m2

EXAMINATION. The teacher hands out cards with the correct answer on them, the children read out their answers - Fold back the answers and you will see, archaeologists, who you dug up. Dinosaurs..jpg" width="45" height="59 src=">.jpg" width="49" height="65 src=">

8 . Fizminutka

Are you probably tired?
-Yes!
- And so everyone stood up.
They stretched their necks together
as the dinosaurs hissed, roared.
They hissed and were silent
Like dinosaurs, they jumped.
Let's gallop, gallop
And they disappeared behind a bush.

9. Introducing a new topic.

I found this dinosaur. (A poster is hung on the board.)

- Can we find its area using the formula? Why?

There is a way to find the area of ​​irregular shapes using a palette - a transparent film with square centimeters printed on it.

10. Introduction to the “Palette” presentation

11. Work on the topic of the lesson

Finding the area of ​​a dinosaur using an algorithm. using a large palette. The teacher comments.

Algorithm for calculating area using a palette

1. Apply the palette.

2. Count the quantity

full squares in the figure.

3. Count the number of incomplete squares and divide this number by 2:

4. Add the number of complete squares and the number of incomplete squares divided by 2.

Children write in a notebook.

12. Physical exercise. (Pair dance)

Come on, Dino, let's jump, let's jump. Let's jump.

And we kick our legs, we kick, we kick .

13. Independent work. Find the area of ​​the egg using a palette.

Do you know how small dinosaurs are born? From eggs. Let's continue our archaeological excavations. Whoever finds a dinosaur egg within the square meters marked on the floor has the right to look into it.

(Children find a Kinder egg, inside each there is a palette and the inscription “palette”)

The teacher hands out drawings of an egg and asks them to find its area. Children independently find the area of ​​the egg using a palette.

14. Testing.

Let's test our knowledge. In the test, circle the correct answer.

TEST on the topic “Palette”

1. A rectangle with all sides equal is called

Triangle

2. To find the area of ​​a square or rectangle, you need

Multiply length by width

Find the sum of all sides.

3. Using a palette, the area is found like this:

Add the number of complete squares and the number of incomplete squares divided by 2.

Multiply length by width

4. Area units:

Mm cm m to m

mm2 cm2 m2 to m2

5. Formula for the area of ​​a square or rectangle

15 . RESERVE.

Work in groups.

Group 1 writes the names of objects whose area is convenient to find using the formula

2. The group writes the names of objects whose area is convenient to find with a palette.

16. Lesson summary. Commenting on ratings.

What new did you learn in the lesson?

What did you like?

17. Homework.

Using a palette, find the area of ​​small objects and drawings.

The areas of small areas with curved boundaries can be measured using palettes. A palette for measuring areas is a sheet of transparent material (wax wax, lavsan, plastic, tracing paper), onto which a grid of squares measuring 2x2 mm or a system of equidistant parallel lines is applied.

Having placed a palette with a grid of squares on the plan, count the number of squares that fit in the measured area, evaluating the fractional parts of the squares at the edges of the area by eye. The calculation result is multiplied by the area of ​​one square.

Thus, a square measuring 2 × 2 mm on a plan of scale 1:1000 corresponds to a square 2 × 2 m on the ground, that is, an area equal to 4 m 2. If the counted number of squares is 122.4, then the area of ​​the plot is 122.4 · 4 m 2 = 490 m 2.

The segments of the palette lines, limited by the contour of the area, can be considered as the middle lines of trapezoids enclosed in the drawing between dotted lines. Measuring the lengths of the midlines d 1, d 2, ..., d n, the area of ​​the plot is calculated using formula (5.1):

P = h(d 1 + d 2 + … + d n), (5.1)

Where h- the distance between the lines of the palette (to scale).

Determining the sum of segments d 1 + d 2 + … + d n performed with a measuring compass. Taking a segment into the meter solution d 1, move the meter to the next line, to continue the segment d 2 and increase the solution so that the solution contains the amount d 1 + d 2. Continuing, accumulate the entire sum of distances and determine its value using a scale bar.

The rectangular palette is built in the form of a grid of squares. Determining the area with a rectangular palette is performed according to the method of A.N. Savich (Fig. 5.2).

A. N. Savich’s method is used when measuring on a plan large areas. Part P 0 the area of ​​the site (Fig. 5.2), consisting of whole squares formed by grid lines, does not require measurement - it is equal to the sum of the known areas of the squares. Measure areas P 1, R 2, R 3, R 4, located at the edges of the site and composed

Measuring areas P 1, R 2, R 3, R 4 can be performed by any of the methods described above (by coordinates, by linear-angular measurements).

Areas are calculated similarly R 2, R 3, R 4.

The advantage of Savich’s method is that a significant part of the area (namely - P 0) is determined without measurements, analytically. Reducing the area to be measured and performing controlled measurements increases the accuracy of area determination. In addition, the deformation of the paper is taken into account.

If a significant part of the area is made up of whole squares, and only a small part of it has to be measured, the accuracy of the Savich method is close to the accuracy of analytical methods.

5.2 Methods for determining the area of ​​a plot with curved boundaries