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In mathematics on the topic "Peak Formula". Creative work "Application of the peak formula"

This topic will be of interest to students in grades 10-11 in preparation for the exam. The Peak formula can be used when calculating the area of a figure depicted on checkered paper (this task is proposed in the USE test and measuring materials).

During the classes

"The subject of mathematics is so serious

which is useful not to miss the opportunity

make it a little fun"

(B. Pascal)

Teacher: Are there problems that are unusual and do not look like problems from school textbooks? Yes, these are tasks on checkered paper. Such tasks are in the control and measuring materials of the exam. What is the peculiarity of such problems, what methods and techniques are used to solve problems on checkered paper? In this lesson, we will explore tasks on checkered paper related to finding the area of the depicted figure, and learn how to calculate the area of polygons drawn on a checkered sheet.

Teacher: The object of the study will be tasks on checkered paper.

The subject of our study will be problems for calculating the area of polygons on checkered paper.

And the goal of the study will be the Peak formula.

B - the number of integer points inside the polygon

Г - the number of integer points on the border of the polygon

This is a handy formula that can be used to calculate the area of any polygon without self-intersections with the vertices at the knots of the checkered paper.

Who is Peak? Peak Georg Aleksandrov (1859-1943) - Austrian mathematician. Discovered the formula in 1899.

Teacher: Let us formulate a hypothesis: the area of the figure calculated by the Pick formula is equal to the area of the figure calculated by the geometry formulas.

When solving problems on checkered paper, we need geometric imagination and fairly simple information that we know:

The area of a rectangle is equal to the product of the adjacent sides.

The area of a right triangle is half the product of the sides forming the right angle.

Teacher: Grid nodes are points where grid lines intersect.

The interior nodes of the polygon are blue. The nodes on the borders of the polygon are brown.

We will consider only such polygons, all of whose vertices lie at the nodes of the checkered paper.

Teacher: Let's do some research for a triangle. First, we calculate the area of the triangle using the Peak formula.

IN + G/2 − 1 , where IN G— the number of integer points on the border of the polygon.

B = 34, G = 15,

IN + G/2 − 1 = 34 + 15 :2 − 1 = 40, 5 Answer: 40.5

Teacher: Now we calculate the area of the triangle using the geometry formulas. The area of any triangle drawn on checkered paper can be easily calculated by representing it as the sum or difference of the areas of right-angled triangles and rectangles, the sides of which go along the grid lines passing through the vertices of the drawn triangle. Students do the calculations in their notebooks. Then they check their results with the calculations on the board.

Teacher: Comparing the results of the studies, draw a conclusion. We found that the area of the figure calculated using the Peak formula is equal to the area of the figure calculated using the geometry formulas. So the hypothesis turned out to be correct.

Next, the teacher suggests calculating the area of \u200b\u200b"his" arbitrary polygon using the geometry formulas and the Pick formula and comparing the results. You can “play” with the Peak formula on the site of mathematical studies.

At the end of the article, one of the papers on the topic "Calculation of the area of an arbitrary polygon using the Pick formula" is proposed.

More pExample:

The area of a polygon with integer vertices is IN + G/2 − 1 , where IN is the number of integer points inside the polygon, and G is the number of integer points on the boundary of the polygon.

B = 10, G = 6,

IN + G/2 − 1 = 10 + 6 :2 − 1 = 12 ANSWER: 12

Teacher: I suggest that you solve the following tasks:

Answer: 12

Answer: 13

Answer: 9

Answer: 11.5

Answer: 4

Find the area of a triangle drawn on checkered paper with a cell size of 1 cm × 1 cm (see figure). Give your answer in square centimeters.

View document content
"For a performance"

Introduction

Sooner or later, every correct

mathematical idea finds

(A.N. Krylov)

Many students are faced with tasks to find the area of a triangle, parallelogram, polygon and other geometric shapes from a pattern on checkered paper. Applying the rules and theorems from geometry, the student can get confused or forget, and besides, it takes a lot of time for additional construction, and in the conditions of the exam, every minute is precious. In order not to spend a lot of effort, time and not to remember theorems, axioms, rules in a hurry, there is Pick's theorem, with which you can calculate the area of \u200b\u200ba figure located on checkered paper without problems and wasting time.

Seeing such tasks in control and measuring OGE materials and the Unified State Examination, I decided to definitely investigate the tasks on checkered paper related to finding the area of the depicted figure.

Thus, the topic for the study was determined.

Pick's theorem is relevant for all students taking exams. Therefore, you need to know it in order to quickly and correctly solve problems for finding the area.
Object of study : tasks on checkered paper.

Subject of study : tasks for calculating the area of a polygon on checkered paper, methods and techniques for solving them.

Research methods :

Theoretical: analysis and synthesis.

Empirical: comparison.
Inductive method - drawing conclusions from specific examples.

Experiment.

Purpose of the study: Check Peak's formula for calculating the areas of geometric figures in comparison with geometry formulas.

Who is Peak?

Pick entered the University of Vienna in 1875. The very next year he published his first work on mathematics, he was only seventeen years old. He studied mathematics and physics, graduating in 1879 from the university, having received the opportunity to teach both of these subjects. In 1877, Leo Koenigsberger moved from the Dresden Higher Technical School (Technische Hochschule) to take a chair at the University of Vienna. He became the leader of Pick, and on April 16, 1880 Pick defended his doctoral dissertation “On the class of Abelian integrals”

Pick's formula will allow you to find the area of any polygon on checkered paper with integer vertices with extraordinary ease.

We discussed this task in class. Although the polygon looked simple enough, we had to work hard to calculate its area. We spent 10 minutes to solve this problem. I want to note that not all students in our class coped with this task. And when we were told that there is a formula that allows you to calculate the area in one minute, I was very interested and I decided to study this issue.

First, I decided to find out in what ways my classmates calculated the area, who coped with the task and study the formula. No one in our class knew Pick's formula. We also decided to give this task to students in grades 9 and 11. Here's what we got.

Peak Formula:

And now we wanted to show you an example of how, using the Pick formula, you can find the area of \u200b\u200ba figure on a checkered lattice.

Output: Thus, considering the problems of finding the areas of polygons depicted on checkered paper using the geometry formulas and the Pick formula and comparing the results in the tables, we showed the validity of the Pick formula and came to the conclusion that the figure area calculated using the Pick formula is equal to the area of the figure, calculated by the derived geometry formula.

"Formula Peak"

Completed:

Supervisor: Parkina Natalya Ivanovna,

mathematic teacher

Relevance

Object of study:

Tasks on checkered paper.

Subject of study:

Research methods:

Purpose of the study:

application in one way or another.

(A.N. Krylov)

Counting the number of cells;
Peak formula.

Find the area of a polygon

You can search for it in different ways.

S = 5 ・ 6 - 13=17 (square units)

Here's what we got

Class

Right

Not right

Total

Way

Class

Cell count

Splitting a figure

Total

Formula

Pika

Pick's theorem or Peak Formula

Let be IN −

G − S − its area.

S \u003d B + G / 2 - 1

Example.

B = 13 (red dots),

G=6 (blue dots), that's why

S = 13 + 6/2 - 1 = 15 square units.

Proof

Denote:

sides,

Therefore, the area of the polygon is 1/2 m.

180 0 m .

180 0 (G - n).

n – 2) .

The total sum of the angles of all triangles is

360 0 V +180 0 (G– n) + 180 0 (n –2).

So 180 0 m\u003d 360 0 V + 180 0 (G– n) + 180 0 (n – 2),

180 0 m\u003d 360 0 V + 180 0 G - 180 0 n + 180 0 n– 180 0 2,

180 0 m\u003d 360 0 V + 180 0 G– 360 0, 1/2 m \u003d V + G / 2 - 1,

G=4(points on nodes)

B=0(points inside the figure)

Answer: 1cm 2

1 cell = 1 cm
D = 15 (indicated in red)
B = 34 (indicated in blue)

D = 14 (indicated in red)
B = 43 (indicated in blue)

Solving USE tasks

Peak formula-

formula to calculate

area

polygons,

useful in problem solving

Unified State Examination and OGE

Task of the Unified State Examination - 2015

Solution.

By Pick's formula:

S \u003d G: 2 + V - 1

G = 7 , B = 5

S = 7:2 + 5 - 1 =

= 7.5 (cm²)

Answer: 7.5 cm².

USE tasks - 2015

G = 7 B = 2

S= 7:2 + 2 - 1 = 4 ,5

G = 4 B = 0

S= 4: 2 + 0 - 1 = 1

A task. Find the area S

Answer: ≈ 1.11.

A task . ABC .

A task. ABCD

A task. Find the area S

Answer: ≈3.5.

Example #1

G = 14

S= 14:2 + 43–1 =

= 49

Example #2

G = 11

S = 11:2 + 5 – 1= = 9,5

Example #3

S = 15:2 + 22 – 1=

Example #4

S = 8:2 +16 – 1 =

Example #5

G = 10

S= 10:2 + 30 –1 =

Find the area of a quadrangle drawn on checkered paper with a cell size of 1 cm x 1 cm (see figure). Give your answer in square centimeters.

According to Peak's formula S \u003d B + ½G-1 H=4, D=14, S=4+½ 14-1=10

According to Peak's formula S \u003d B + ½G-1 H=36, D=21

According to Peak's formula S \u003d B + ½G-1 H=36, D=21 S \u003d 36 + ½ 21 -1 \u003d 36 + 10.5-1 \u003d 45.5

According to Peak's formula S \u003d B + ½G-1 H=6, D=18, S=6+½ 18-1=14

G = 16 B = 4 S = G : 2 + IN - 1 S = 16 : 2 + 4 – 1 = 11

A task.

Main conclusion:

Conclusion

View presentation content
"Formula Peak2"

Research mathematics

Application of Pick's formula to calculate the area of polygons with vertices at the nodes of the cell

Completed: Vasyakin Mikhail, 10th grade student

Supervisor: Parkina Natalya Ivanovna,

mathematic teacher

Relevance work is that Pick's formula for calculating the area of polygons in school course mathematics (geometry) is not considered. The study of this topic expands the intellectual horizons of students, and its application simplifies finding the area of a geometric figure depicted on checkered paper (grid). The USE test and measurement materials contain tasks of this type, and they can be solved using the Peak formula.

Object of study:

Tasks on checkered paper.

Subject of study:

Problems for calculating the area of a polygon on checkered paper.

Research methods:

Comparison, modeling, generalization, analogy, study of literature and Internet resources, analysis and classification of information.

Purpose of the study:

Justify the rationality of using the Pick formula when solving problems of finding the area of figures depicted on checkered paper.

Study the literature on the topic;
Consider different ways of calculating the areas of polygons;
Show practical use these ways;
Find out the advantages and disadvantages of each method;
Systematize and deepen the knowledge I have accumulated;
Improve the quality of knowledge and skills;
Create an electronic presentation of the work to present the collected material to classmates.

Sooner or later every correct mathematical idea finds

application in one way or another.

(A.N. Krylov)

Georg Alexander Pick was an Austrian mathematician born into a Jewish family.

The circle of mathematical interests of Peak was extremely wide. He wrote works in the field of mathematical analysis, differential geometry, in the theory of differential equations, etc., more than 50 topics in total.

Pick's theorem, discovered by him in 1899, was widely known for calculating the area of a polygon. In Germany, this theorem is included in school textbooks.

Counting the number of cells;
Application of planimetry formulas;
Breaking a figure into simpler figures;
Building a figure to a rectangle;
Peak formula.

Find the area of a polygon

You can search for it in different ways.

Methods used to calculate the area of a given figure

Method 1: Counting the number of cells (approximate for this figure).

Method 2: Try to cut the polygon into fairly simple shapes (Fig. 2), find their areas and add them up.

Method 3: Calculate the area of the figure (Fig. 3), which complements the polygon to a rectangle, and subtract this area from the area of the rectangle. The augmented figure (unlike the original polygon) is easily divided into rectangles and right triangles so that its area is calculated effortlessly.

S = 2+1+0.5 + 3+ 2 + 1 + 2 +1.5=13 (square units)

Therefore, the area of the original polygon is

S = 5 ・ 6 - 13=17 (square units)

Here's what we got

Class

Right

Not right

Total

Way

Cell count

Class

Splitting a figure

Total

Build the shape to a rectangle

Formula

Pika

Try to find the area of the figure

There is a simple and convenient way to do this.

Pick's theorem or Peak Formula

Let be IN − the number of grid nodes inside the polygon,

G − the number of nodes on its boundary, S − its area.

Then Pick's formula is valid: S \u003d B + G / 2 - 1

Example.

For the polygon in the figure B = 13 (red dots),

G=6 (blue dots), so

S = 13 + 6/2 - 1 = 15 square units.

Proof

Consider a polygon whose vertices are at the nodes of an integer lattice, that is, they have integer coordinates.

Denote:

n is the number of sides of the polygon,

m- the number of triangles with vertices at the nodes

lattices that do not contain nodes either inside or on

sides,

B is the number of nodes inside the polygon,

Г is the number of nodes on the sides, including the vertices.

The areas of all these triangles are the same and equal

Therefore, the area of the polygon is 1/2m.

The total sum of the angles of all triangles is 180 0 m .

Now let's find this sum in a different way.

The sum of angles with a vertex at any internal node is 360 0 .

Then the sum of angles with vertices at all internal nodes is 360 0 V.

The total sum of the angles at nodes on the sides but not at the vertices is

180 0 (G - n).

The sum of the angles at the vertices of the polygon is 180 0 ( n – 2) .

The total sum of the angles of all triangles is

360 0 V +180 0 (G– n) + 180 0 (n –2).

So 180 0 m\u003d 360 0 V + 180 0 (G– n) + 180 0 (n – 2),

180 0 m\u003d 360 0 V + 180 0 G - 180 0 n + 180 0 n– 180 0 2,

180 0 m\u003d 360 0 V + 180 0 G– 360 0, 1/2m \u003d V + G/2 - 1,

whence we obtain the expression for the area S of the polygon:

S \u003d B + G / 2 - 1, known as the Pick formula.

G=4(points on nodes)

B=0(points inside the figure)

Answer: 1cm 2

1 cell = 1 cm
D = 15 (indicated in red)
B = 34 (indicated in blue)

D = 14 (indicated in red)
B = 43 (indicated in blue)

Solving USE tasks

Peak formula-

formula to calculate

area

polygons,

useful in problem solving

Unified State Examination and OGE

Task of the Unified State Examination - 2015

Find the area of quadrilateral ABCD

Solution.

By Pick's formula:

S \u003d G: 2 + V - 1

G = 7 , B = 5

S = 7:2 + 5 - 1 =

= 7.5 (cm²)

Answer: 7.5 cm².

USE tasks - 2015

G = 7 B = 2

S = 7:2 + 2 - 1 = 4.5

G = 4 B = 0

S = 4: 2 + 0 - 1 = 1

Now, knowing the new formula, we can easily find the area of this quadrilateral.

Since B \u003d 5; G \u003d 14, then 5 + 14: 2-1 \u003d 11 (cm squared)

The area of this quadrilateral is 11 cm squared.

Using the same formula, we can find the area of a triangle.

Since V=14, D=10, then 14+10:2-1=18 (cm squared)

Area given triangle equals 18 cm squared.

If W=9, D=12, then: 9+12:2-1=14 (cm squared)

The area of this quadrilateral is 14 cm squared.

A task. Find the area S sectors, considering the sides of square cells equal to 1. Specify in your answer.

Solution: G \u003d 5, V \u003d 2, S \u003d B + G / 2 - 1 \u003d 2 + 5/2 - 1 \u003d 3.5.

Answer: ≈ 1.11.

A task . Find the area of a triangle ABC .

Solution: D = 7, V = 5, S = V + D / 2 - 1 = 5 + 7/2 - 1 = 7.5.

A task. Find the area of a quadrilateral ABCD, considering the sides of square cells equal to 1.

Solution: D= 4, V= 5, S = V + D/2 - 1= 5 + 4/2 - 1= 6

A task. Find the area S rings, considering the sides of square cells equal to 1. Indicate in your answer

Solution: G = 8, V = 8, S = V + G / 2 - 1 = 8 + 8/2 - 1 = 11,

Answer: ≈3.5.

Example #1

G = 14

S= 14:2 + 43–1 =

Example #2

G = 11

S = 11:2 + 5 – 1= = 9,5

Example #3

S = 15:2 + 22 – 1=

Example #4

S = 8:2 +16 – 1=

Example #5

G = 10

S = 10:2 + 30 –1=

Find the area of a quadrangle drawn on checkered paper with a cell size of 1 cm x 1 cm (see figure). Give your answer in square centimeters.

According to Peak's formula S \u003d B + ½G-1 H=4, D=14, S=4+½ 14-1=10

According to Peak's formula S \u003d B + ½G-1 H=36, D=21

S \u003d 36 + ½ 21 -1 \u003d 36 + 10.5-1 \u003d 45.5

According to Peak's formula S \u003d B + ½G-1 H=6, D=18, S=6+½ 18-1=14

G = 16 B = 4 S= G : 2 + IN - 1 S= 16 : 2 + 4 – 1 = 11

A task. Find the area of a rectangular parallelepiped, considering the sides of square cells equal to 1.

full surface according to the Pick formula is impossible!

Main conclusion:

Pick's formula has a number of advantages over other methods for calculating the areas of polygons on checkered paper:

1. To calculate the area of a polygon, you need to know only one formula: S \u003d B + G / 2 - 1

2. The Peak formula is very easy to remember.

3.Pick's formula is very convenient and easy to use.

4. The polygon whose area needs to be calculated can be any, even the most bizarre shape.

Conclusion

When performing the work, problems were solved to find the area of polygons depicted on checkered paper in two ways: geometrically and using the Pick formula.

After analyzing the methods of solving problems, we can draw the following conclusions:

1) Pick's formula gives a quick and easy solution to problems of finding the area of a figure whose vertices lie at the nodes of the lattice, that is, finding the areas of polygons.

2) Using the Peak formula to find the area circular sector or rings is impractical, as it gives an approximate result.

3) Pick's formula is not used to solve problems in space.

4) Peak's formula makes it easier and faster to find the area of polygons. But it also has its drawbacks:

The drawing must be very clear (for counting knots);
The formula applies only if the polygon is depicted on checkered paper;

Methods for calculating the areas of polygons, including using the Peak formula, allow the successful study of geometry in high school. This work can be useful for students in preparation for the final certification.

Peak Formula

Sazhina Valeria Andreevna, 9th grade student of MAOU "Secondary School No. 11", Ust-Ilimsk, Irkutsk Region

Supervisor: Gubar Oksana Mikhailovna, higher mathematics teacher qualification category MAOU "Secondary School No. 11", Ust-Ilimsk, Irkutsk Region

2016

Introduction

When studying the topic of geometry "Areas of polygons", I decided to find out: is there a way to find areas that is different from those that we studied in the lessons?

This way is the Peak formula. L. V. Gorina in “Materials for self-education of students” described this formula as follows: “Familiarization with the Pick formula is especially important on the eve passing the exam and GIA. With this formula, you can easily solve big class The problems offered in the exams are problems for finding the area of a polygon depicted on checkered paper. Pick's little formula will replace a whole set of formulas needed to solve such problems. Peak's formula will work "one for all ..."!

In the USE materials, I came across tasks with practical content for finding the area land plots. I decided to check whether this formula is applicable to finding the area of the school, microdistricts of the city, region. And also is it rational to use it to solve problems.

Object of study: Peak's formula.

Subject of study: rationality of the application of the Pick formula in solving problems.

Purpose of the work: to substantiate the rationality of using the Pick formula when solving problems of finding the area of figures depicted on checkered paper.

Research methods: modeling, comparison, generalization, analogies, study of literary and Internet resources, analysis and classification of information.

Select the necessary literature, analyze and systematize the information received;

Consider various methods and techniques for solving problems on checkered paper;

Check experimentally the rationality of using the Peak formula;

Consider the application of this formula.

Hypothesis: if you apply the Peak formula to find the areas of a polygon, then you can find the area of the territory, and solving problems on checkered paper will be more rational.

Main part

Theoretical part

Checkered paper (more precisely, its knots), on which we often prefer to draw and draw, is one of the most important examples of a dotted lattice on a plane. Already this simple lattice served as a starting point for K. Gauss to compare the area of a circle with the number of points with integer coordinates located inside it. The fact that some simple geometric statements about figures in the plane have deep consequences in arithmetic studies was clearly noticed by G. Minkowski in 1896, when he first used geometric methods to consider number-theoretic problems.

Let's draw some polygon on checkered paper (Appendix 1, Figure 1). Let's try to calculate its area now. How to do it? Probably the easiest way is to break it into right-angled triangles and a trapezoid, the areas of which are already easy to calculate and add up the results.

The method used is simple, but very cumbersome, and besides, it is not suitable for all polygons. So the next polygon cannot be divided into right triangles, as we did in the previous case (Appendix 2, Figure 2). You can, for example, try to supplement it to the “good” one we need, that is, to the one whose area we can calculate in the described way, then subtract the areas of the added parts from the resulting number.

However, it turns out that there is a very simple formula that allows you to calculate the areas of such polygons with vertices at the nodes of a square grid.

This formula was discovered by the Austrian mathematician Peak Georg Alexandrov (1859 - 1943) in 1899. In addition to this formula, Georg Pick discovered the Pick, Pick-Julia, Pick-Nevalin theorems, proved the Schwarz-Pick inequality.

This formula went unnoticed for some time after Pick published it, but in 1949 the Polish mathematician Hugo Steinhaus included the theorem in his famous Mathematical Kaleidoscope. Since that time Pick's theorem has become widely known. In Germany, the Pick formula is included in school textbooks.

It is a classical result of combinatorial geometry and the geometry of numbers.

Proof of Pick's formula

Let ABCD be a rectangle with vertices at the nodes and sides going along the grid lines (Appendix 3, Figure 3).

Let us denote by B - the number of nodes lying inside the rectangle, and by G - the number of nodes on its border. Shift the grid half a cell to the right and half a cell

down. Then the territory of the rectangle can be "distributed" between the nodes as follows: each of the B nodes "controls" the whole cell of the shifted grid, and each of the G nodes - 4 boundary non-corner nodes - half of the cell, and each of the corner points - a quarter of the cell. Therefore, the area of the rectangle S is

S = B + + 4 = B + - 1 .

So, for rectangles with vertices at the nodes and sides going along the grid lines, we have established the formula S = B + - 1 . This is the Peak formula.

It turns out that this formula is true not only for rectangles, but also for arbitrary polygons with vertices at the grid nodes.

Practical part

Finding the area of \u200b\u200bfigures by the geometric method and by the Pick formula

I decided to make sure that Pick's formula is correct for all the examples considered.

It turns out that if a polygon can be cut into triangles with vertices at the nodes of the grid, then Pick's formula is true for it.

I reviewed some tasks on checkered paper with cells measuring 1 cm1 cm and carried out comparative analysis for solving problems (Table No. 1).

Table No. 1 Problem solving different ways.

Picture

According to the geometry formula

According to Pick's formula

Task #1

S=S etc -(2S 1 +2S 2 )

S etc =4*5=20 cm 2

S 1 =(2*1)/2=1 cm 2

S 2 =(2*4)/2=4 cm 2

S=20-(2*1+2*4)=10 cm 2

Answer :10 cm ².

H = 8, D = 6

S\u003d 8 + 6/2 - 1 \u003d 10 (cm²)

Answer: 10 cm².

Task #2

a=2, h=4

S=a*h=2*4=8 cm 2

Answer : 8 cm ².

H = 6, D = 6

S\u003d 6 + 6/2 - 1 \u003d 8 (cm²)

Answer: 8 cm².

Task #3

S=S sq. -(S 1 +2S 2 )

S sq. =4 2 =16 cm 2

S 1 \u003d (3 * 3) / 2 \u003d 4.5 cm 2

S 2=(1*4)/2=2cm 2

S\u003d 16- (4.5 + 2 * 2) \u003d 7.5 cm 2

H = 6, D = 5

S\u003d 6 + 5/2 - 1 \u003d 7.5 (cm²)

Answer: 7.5 cm².

Task #4

S=S etc -(S 1 +S 2+ S 3 )

S etc =4 * 3=12 cm 2

S 1 =(3*1)/2=1,5 cm 2

S 2 =(1*2)/2=1 cm 2

S 3 =(1+3)*1/2=2 cm 2

S=12-(1.5+1+2)=7.5 cm 2

H = 5, D = 7

S\u003d 5 + 7/2 - 1 \u003d 7.5 (cm²)

Answer: 7.5 cm².

Task # 5.

S=S etc -(S 1 +S 2+ S 3 )

S etc =6 * 5=30 cm 2

S 1 =(2*5)/2=5 cm 2

S 2 =(1*6)/2=3 cm 2

S 3 =(4*4)/2=8 cm 2

S=30-(5+3+8)=14 cm 2

Answer: 14 cm²

H = 12, D = 6

S\u003d 12 + 6/2 - 1 \u003d 14 (cm²)

Answer: 14 cm²

A task №6.

S tr \u003d (4 + 9) / 2 * 3 \u003d 19.5 cm 2

Answer: 19.5 cm 2

H = 12, D = 17

S\u003d 12 + 17/2 - 1 \u003d 19.5 (cm²)

Answer: 19.5 cm 2

A task №7. Find the area woodland(in m²), depicted on a plan with a square grid 1 × 1 (cm) on a scale of 1 cm - 200 m

S=S 1 +S 2+ S 3

S 1 =(800*200)/2=80000 m 2

S 2 =(200*600)/2=60000 m 2

S 3 =(800+600)/2*400=

280000 m 2

S= 80000+60000+240000=

420000m2

Answer: 420,000 m²

V = 8, D = 7. S\u003d 8 + 7/2 - 1 \u003d 10.5 (cm²)

1 cm² - 200² m²; S= 40,000 10.5 = 420,000 (m²)

Answer: 420,000 m²

Task #8 . Find the area of the field (in m²) shown on the plan with a square grid of 1 × 1 (cm) to scale

1 cm - 200 m.

S= S sq -2( S tr + S ladder)

S sq \u003d 800 * 800 \u003d 640000 m 2

S tr \u003d (200 * 600) / 2 \u003d 60000m 2

S ladder =(200+800)/2*200=

100000m2

S=640000-2(60000+10000)=

320000 m2

Answer: 320,000 m²

Solution. Let's find Sthe area of a quadrangle drawn on checkered paper using Pick's formula:S= B + - 1

V = 7, D = 4. S\u003d 7 + 4/2 - 1 \u003d 8 (cm²)

1 cm² - 200² m²; S= 40000 8 = 320,000 (m²)

Answer: 320,000 m²

Task #9 . Find the areaS sectors, considering the sides of square cells equal to 1. In your answer, indicate .

A sector is one-fourth of a circle and therefore its area is one-fourth that of a circle. The area of a circle is πR 2 , where R is the radius of the circle. In our caseR =√5 and hence the areaS sector is 5π/4. WhereS/π=1.25.

Answer. 1.25.

D= 5, V= 2, S\u003d V + G / 2 - 1 \u003d 2 + 5/2 - 1 \u003d 3.5, ≈ 1,11

Answer. 1.11.

Task number 10. Find the area S rings, considering the sides of square cells equal to 1. Indicate in your answer .

The area of the ring is equal to the difference between the areas of the outer and inner circles. RadiusR the outer circle is

2 , radius r the inner circle is 2. Therefore, the area of the ring is 4and hence. Answer:4.

D= 8, V= 8, S\u003d V + G / 2 - 1 \u003d 8 + 8/2 - 1 \u003d 11, ≈ 3,5

Answer: 3.5

Conclusions: The considered tasks are similar to the task from the variants of control and measuring USE materials in mathematics (tasks No. 5,6),.

From the solutions of the problems considered, I saw that some of them, for example, problems No. 2.6, are easier to solve using geometric formulas, since the height and base can be determined from the drawing. But in most tasks, it is required to split the figure into simpler ones (task No. 7) or complete it to a rectangle (tasks No. 1,4,5), a square (tasks No. 3,8).

From solving problems #9 and #10, I saw that applying the Pick formula to shapes that are not polygons gives an approximate result.

In order to check the rationality of using the Peak formula, I conducted a study on the subject of the time spent (Appendix 4, table No. 2).

Conclusion: from the table and diagram (Appendix 4, diagram 1) it can be seen that when solving problems using the Peak formula, much less time is spent.

Finding the surface area of spatial forms

Let's check the applicability of this formula to spatial forms (Appendix 5, Figure 4).

Find the total surface area of a rectangular parallelepiped, considering the sides of the square cells to be 1.

This is a flaw in the formula.

Applying the Pick Formula to Finding the Area of a Territory

Solving problems with practical content (tasks No. 7,8; table No. 1), I decided to apply this method to find the area of the territory of our school, microdistricts of the city of Ust-Ilimsk, Irkutsk region.

Acquainted with the "Draft Boundaries land plot MAOUSOSH No. 11 of Ust-Ilimsk "(Appendix 6), I found the area of the territory of our school and compared it with the area according to the project of the boundaries of the land plot (Appendix 9, table 3).

Having examined the map of the right-bank part of Ust-Ilimsk (Appendix 7), I calculated the areas of microdistricts and compared them with the data from the Master Plan of Ust-Ilimsk, Irkutsk Region. The results are presented in the table (Appendix 9, table 4).

Having examined the map of the Irkutsk region (Appendix 7), I found the area of the territory and compared it with the data from Wikipedia. The results were presented in the table (Appendix 9, table 5).

After analyzing the results, I came to the conclusion: using the Peak formula, these areas can be found much easier, but the results are approximate.

Of the studies carried out, the most exact value I got when finding the area of the school (Appendix 10, diagram 2). A greater discrepancy in the results was obtained when finding the area of the Irkutsk region (Appendix 10, diagram 3). It's related to that. That not all area boundaries are sides of polygons, and vertices are not anchor points.

Conclusion

As a result of my work, I expanded my knowledge of solving problems on checkered paper, determined for myself the classification of the problems under study.

When performing the work, problems were solved to find the area of polygons depicted on checkered paper in two ways: geometrically and using the Pick formula.

An analysis of solutions and an experiment to determine the elapsed time showed that the application of the formula makes it possible to solve problems for finding the area of a polygon more rationally. This allows you to save time on the exam in mathematics.

Finding the area various figures, depicted on checkered paper, led to the conclusion that the use of the Pick formula to calculate the area of a circular sector and a ring is inappropriate, since it gives an approximate result, and that the Pick formula is not used to solve problems in space.

Also in the work, the areas of various territories were found using the Peak formula. We can conclude that it is possible to use the formula to find the area of various territories, but the results are approximate.

My hypothesis was confirmed.

I came to the conclusion that the topic that interested me is quite multifaceted, the tasks on checkered paper are diverse, the methods and techniques for solving them are also diverse. Therefore, I decided to continue working in this direction.

Literature

Volkov S.D.. Project of the boundaries of the land plot, 2008, p. 16.

Gorina L.V., Mathematics. Everything for the teacher, M: Nauka, 2013, No. 3, p. 28.

Prokopyeva V.P., Petrov A.G., General plan city of Ust-Ilimsk, Irkutsk region, Gosstroy of Russia, 2004. p. 65.

Riss E. A., Zharkovskaya N. M., Geometry of checkered paper. Peak formula. - Moscow, 2009, No. 17, p. 24-25.

Smirnova I. M.,. Smirnov V.A., Geometry on checkered paper. - Moscow, Chistye Prudy, 2009, p. 120.

I. M. Smirnova, V. A. Smirnov, Geometric problems with practical content. – Moscow, Chistye Prudy, 2010, p. 150

Tasks open bank assignments in mathematics FIPI, 2015.

Map of the city of Ust-Ilimsk.

Map of the Irkutsk region.

Wikipedia.

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

Leaders:

Mogutova Tatyana Mikhailovna
Deryushkina Oksana Valerievna

Project motto:

“If you want to learn how to swim, then boldly enter the water.
and if you want to learn how to solve problems, then solve them.”
D. Poya.

The choice of the theme of the project is not accidental. Ways to find the area of a polygon drawn on "cells" is a very interesting topic.

We know different ways performing such tasks: addition method, subtraction method, etc.

We were very interested in this topic, we studied a lot of literature and, to our great joy, found another way, a way not known from the school curriculum, but a wonderful way! Calculating the area using a formula developed by the Austrian mathematician Georg Pick.

We decided to study the Peak formula, with the help of which it is very easy to complete tasks on finding the area!

Purpose of the study

1. Learning the Pick formula.

2. Expansion of knowledge about the variety of tasks on checkered paper, about techniques and methods for solving these problems.

Tasks:

1. Select material for research, choose the main, interesting, understandable information

2. Analyze and systematize the information received

3. Create an electronic presentation of the work to present the collected material to classmates

4. Draw conclusions based on the results of the work.

5. Choose the most interesting, illustrative examples.

Research methods:

1. Simulation

2. Construction

3. Analysis and classification of information

4. Comparison, generalization

5. Study of literary and Internet resources

Georg Pick is an Austrian mathematician. Pick entered the University of Vienna in 1875. He published his first work at the age of 17. The range of his mathematical interests was extremely wide. 67 of his works are devoted to many branches of mathematics, such as: linear algebra, integral calculus, geometry, functional analysis, potential theory.

Wide famous Theorem appeared in a collection of Pick's works in 1899.

The theorem attracted quite a lot of attention and began to be admired for its simplicity and elegance.

The Peak formula, a formula for calculating the area of a polygon drawn on paper in a cage, is useful in solving USE and OGE tasks. That is why we are very interested in it.

Pick's formula is a classic result of combinatorial geometry and the geometry of numbers.

By Pick's theorem, the area of a polygon is:

D: 2 + V - 1

Г is the number of lattice nodes on the boundary of the polygon

B is the number of lattice nodes inside the polygon.

First of all, we set the task: to study what lattice nodes are and how to correctly calculate their number. It turned out to be very simple. Let's give some examples.

Let an arbitrary triangle be given. The nodes on the border are depicted orange, the nodes inside are shown in blue. Finding nodes and counting them is very easy.

In this case, D = 15, V = 35

Example #2 There are 18 nodes on the border, i.e. D = 18, knots inside 20, V = 20.

And one more example. Given an arbitrary polygon. We count the nodes on the border. There are 14 of them. There are 43 nodes inside the polygon. D = 14, V = 43.

We have completed the first task!

The second stage of our work: the calculation of the areas of polygons.

Let's look at a few examples.

Example #1.

G \u003d 14, V \u003d 43, S \u003d + 43 - 1 \u003d 49

Example #2.

G \u003d 11, V \u003d 5, S \u003d + 5 - 1 \u003d 9.5

Example #3.

G \u003d 15, V \u003d 22, S \u003d + 22 - 1 \u003d 28.5

Example number 4.

G \u003d 8, V \u003d 16, S \u003d + 16 - 1 \u003d 19

Example #5

G \u003d 10, V \u003d 30, S \u003d + 30 - 1 \u003d 34

We spent only 1-2 minutes on consideration of five examples. Calculating the area using the Pick formula is not only fast, but also very easy!

But we have a very serious question:

Can Pick's theorem be trusted?

Do you get the same results when calculating areas in different ways?

We find the areas of polygons using the Pick formula and in the usual way, using geometry formulas and methods of completion or partitioning. Here are the results we got:

Example #1.

Calculate the area of the polygon using the Peak formula:

Let's count the number of nodes on the border and inside. D = 3, V = 6.

Calculate the area: S \u003d 6 + - 1 \u003d 6.5

Let's build a polygon to a rectangle. The area of the rectangle is: 3 * 5 = 15, S? == 3,S? ==3, S==2.5

S=15-3-3-2.5=6.5

The result is the same.

Example #2.

D = 4, V = 9, S = 9 + - 1 = 10

Let's build it to a rectangle.

The area of the rectangle is: 5 * 4 = 20, S 1 = 2 * 1 = 2, S 2 = = 3,

S==2, S==1.5, S==2.5

The area of the rectangle is

S = 20 - 2 - 3 - 2 - 1.5 - 2.5 = 10

We got the same results again.

Let's consider one more example.

Example #3

Calculate the area using the Peak formula.

D = 5, V = 6, S = 6 + - 1 = 7.5

Calculate the area using the extension method.

The area of the rectangle is 5 4 = 20

S 1 = 2 * 1 = 2, S 2 = = 1, S 3 = 2 * 1 = 2, S 4 == 1, S 5 == 1, S 6 == 2.5

S = 20 - 2 -1 - 2 - 1 - 1 - 2.5 - 3 = 7.5

The result is the same.

In the presentation we looked at three examples, but in fact we looked at a lot of very different examples. The result was always the same: Calculating the area using Pick's formula and other methods gives the same result.

Conclusion: Pick's formula can be trusted! It gives accurate results.

We are satisfied!

And one more question arose before us: what method of calculation is the most rational, the most convenient for use?

To answer this question, it is enough to use all the previous work. But let's look at three more examples that finally allow us to get an answer to our question.

Example #2

Example #3

Using the Pick formula, it is easy to calculate the area of a polygon, even the most bizarre shape. Consider an example:

The conclusion is unequivocal: the most rational way to calculate the area of a polygon drawn on squared paper: Peak's formula!

We invite each of you to calculate the area of a polygon using the Peak formula:

Calculate the number of nodes on the boundary. They are shown in yellow.

Calculate the number of nodes inside, red color.

Substitute in the formula, name the result. You calculated the area in one minute.

So, the Pick formula has a number of advantages over other methods for calculating the areas of polygons on checkered paper:

To calculate the area of a polygon, you need to know only one formula:

S \u003d D: 2 + V - 1.

The Pick formula is very easy to remember.

The Peak formula is very convenient and easy to use.

The polygon whose area needs to be calculated can be any, even the most bizarre shape.

Using the Peak formula, it is easy to complete the task of the exam and the exam.

Here are some examples of calculating the area from USE options – 2015.

We decided to teach students in grades 9-11 of our school to use the Peak formula. We held the “Formula Peak” festival.

All students got acquainted with the presentation with great interest, learned how to use the Pick formula.

30 minutes practical work students completed a large number of assignments. Each student received a "Peak Formula" memo.

We helped them prepare for the exam and the exam!

After a month of work, we conducted a survey of students in grades 9-11.

The following questions were asked:

Question #1:

Is Pick's formula a rational way to calculate the area of a polygon?

“Yes” - 100% of students.

Question #2:

Do you use Pick's formula?

“Yes” – 100% of students

Our work has not been in vain! We are satisfied!

We have posted the presentation of our project on the Internet. Many views and downloads of our work.

We designed the album "Peak Formula". They were constantly, especially at first, used by students of our school.

Results of the project:

In the process of working on the project, we studied reference, popular science literature on the research topic.

We studied Pick's theorem, learned to find the areas of figures depicted on paper in a cage simply and rationally.
We expanded our knowledge of solving problems on checkered paper, determined for ourselves the classification of the problems under study, and became convinced of their diversity.
We held the “Peak Formula” festival for students 9–11, taught them how to find the area using this formula. Picked up a lot of interesting examples.
We created an electronic presentation to help our peers.
We designed the album “Formula Peak”, which is constantly used by students of the school.

He invites you to complete two tasks so that you are convinced of the rationality of our work.

Thanks for attention!

“Problem solving is a practical art like

swimming, skiing or playing the piano;

you can learn it only by imitating good

samples and constantly practicing "

(D. Poya).

Austrian mathematician,

was born into a Jewish family.

Mother of Joseph Schleisinger

Father Adolf Josef Pick.

George Peak

10.08.1859 - 13.07.1942

Curriculum vitae

Georg Alexander Pick

was gifted child, taught by his father, who headed a private institute. At 16, Georg graduated from high school and entered the University of Vienna. At the age of 20 he received the right to teach physics and mathematics. On April 16, 1880, under the guidance of Leo Koenigsberger, Pick defended his doctoral dissertation "On the class of Abelian integrals". At the German University in Prague in 1888, Pick received a position as an extraordinary professor of mathematics, then in 1892 he became an ordinary professor. In 1900-1901 he served as dean of the Faculty of Philosophy. The Peak matrix, Nevanlinna Peak interpolation, Schwarz Peak's lemma are associated with his name. On July 13, 1942, Pick was deported to the Theresienstadt camp set up by the Nazis in northern Bohemia, where he died two weeks later at the age of 82.