Mathematics is the queen of sciences. Her greatness is boundless, and her power is great. All other sciences rely on mathematical results. Be it physics, chemistry, biology, and even philology.
Just like a house is made of bricks, every task has small subtasks. And having learned to solve small problems, you can learn to solve more complex problems.
Today we will analyze how to find fractions. The concept of a fraction originated in Ancient Greece, after the Greeks introduced the concept of length, equivalent to whole numbers. Next, a concept was needed that expressed a part of the length, for example, half, one third of the length. This is how the concept of a fraction appeared.
A bunch of rational numbers Q is the set of numbers represented as m/n, where m,n are integers. The number m/n is called common fraction, where m is the numerator and n is the denominator, n≠0.
If n=〖10〗^k, k=1,2,.. , then such a fraction is called a decimal fraction and is written as 0,0..0m, and the number of zeros after the decimal point is equal to k-1.
A number is called composite if it has other divisors besides 1 and itself.
Basic operations
We will move from simple to complex, showing with examples how certain operations are performed.
How to reduce a fraction
To do this, you need to decompose the numerator and denominator into prime factors, if they are composite. And then, if these prime factors are the same, then remove them.
In case of absence prime factors, the fraction is called irreducible. For example, 85/65=(17*5)/(13*5)=17/13
How to find a fraction of a number
Let the number be some length. And the fraction is essentially a part of this length, so to find the integer part, you need to multiply the fraction by the number. For example, 2/3 of 27=27*2/3=27/3*2=18
How to find a fraction from a fraction
In fact, this is a simple multiplication process, to find a fraction from a fraction, you just need to multiply 2 fractions. For example, 2/3 and 13/17: 2/3*13/17=26/51
Division of fractions
When dividing fractions a / b, c / d, the divisor c / d can be represented as d / c and multiply, and then reduce. For example, 27/17?9/34=27/17*34/9=2*3=6.
It must also be remembered that when deciding difficult examples need to come up with a solution algorithm. You may have to change division to multiplication with a change of fraction, it is possible to perform multiplication and division by the same number. Such fairly simple instructions will help in solving the examples.
Let's take a classic word problem as an example. 2/3 was stolen from a warehouse with 150 tons of fuel oil. The stolen parts were divided into parts in the ratio of 5/17 and 12/17, the last one was taken for processing. The fuel oil remaining in the warehouse was taken for processing. How much fuel oil was processed?
150*2/3*12/17+150*(1-2/3)=150*41/51
Problems for fractions - the basis of school arithmetic. They are not difficult in nature, but require perseverance and attentiveness to perform. If these conditions are met, the result will not be long in coming.
In the process of solving problems 149–156, it is necessary to bring students to an understanding of the rule for finding a part of a number:
To find the part of a number expressed as a fraction, you can divide this number by the denominator of the fraction and multiply the result by its numerator.
Of course, students can formulate this rule only for specific situations: in order to find 3 / 4 number 24, you can divide this number by the denominator fractions 4 and multiply the result by the numerator 3.
149 . a) 12 birds were sitting on a branch; 2/3 of their number flew away. How many birds have flown?
b) There are 32 students in the class; 3/4 of all students went skiing. How many students skied?
150 . a) Cyclists traveled 48 in two days km. On the first day they traveled 2/3 of the way. How many kilometers did they drive on the second day?
b) Someone, having 350 rubles, spent 5/7 of his money. How much money does he have left?
c) There are 24 pages in the notebook. The girl filled out all the pages of the notebook on the 5th/8th. How many unwritten pages are left?
151 . Old problem. Bought a chest of drawers for 36 R., I then had to sell it for 7/12 of the price. How many rubles did I lose in this sale?
152 . Autotourists traveled 360 in three days km; on the first day they traveled 2/5, and on the second day they traveled 3/8 of the entire journey. How many kilometers did the autotourists drive on the third day?
153 . 1) There are 24 girls and several boys in the drama club. The number of boys is 3/8 of the number of girls. How many students are in the drama club?
2) There are 45 commemorative ruble coins in the collection. The number of 3 and 5 ruble coins is 2/9 of the number of ruble coins. How many commemorative coins of 1, 3 and 5 rubles are in the collection?
Students must solve tasks 154–156 by first finding the indicated part of the value, and then increasing or decreasing this value by the found part. Another solution will be shown later.
154 . 1) Reduce 90 rubles by 1/10 of this amount.
2) Increase 80 rubles by 2/5 of this amount.
155 . Last month the price of the item was 90 R. Now it has gone down by 3/10 of that amount. What is the price of the item now?
156 . Last month the salary was 400 R. Now it has increased by 2/5 of that amount. What is the salary now?
In the process of solving problems 157-158 and the following problems, you need to bring students to understand and correct application rules for finding a number by its part:
To find a number by its part, expressed as a fraction, you can divide this part by the numerator of the fraction and multiply the result by its denominator.
The formulation of this rule is complicated because of the need
somehow call the number that we have named «
part »
. The authors of textbooks also have to circumvent this difficulty. So in the textbook I.V. Baranova and Z.G. Borchug's rule is formulated only for specific cases: to find a number, 3 / 5 which is 90 km, it is necessary to divide 90 km by the numerator of the fraction 3 and multiply the result by the denominator of the fraction 5.
This is how students can use it. True, when speaking of number, it is better not to use names, since number and magnitude are not the same thing. Later in the same textbook on p. 226 worded general rule, in which the term we use « part » corresponding turnover « the number corresponding to it » , which is hardly easier.
157 . a) 120 R. make up 3/4 of the amount of money available. What is this amount?
b) Determine the length of the segment, 3/5 of which are equal to 15 cm.
158 . a) My son is 10 years old. His age is 2/7 of his father's age. How old is father?
b) Daughter 12 years old. Her age is 2/5 of the mother's age. How old is the mother?
For the purchase of vegetables, the hostess spent 6 R., which amounted to 1/6 of the money she had. Then she bought 2 kg apples 7 R. per kilogram. How much money does she have left after these purchases?
160 . Father bought his son a suit for 24 R., on which he spent 1/3 of his money. After that, he bought several books and had 39 left. R. How much did the books cost?
161 . The son is 8 years old, his age is 2/9 of his father's age. And the age of the father is 3/5 of the age of the grandfather. How old is grandpa?
162 .* From the papyrus of Ahmes (Egypt, c. 2000 BC).
A shepherd comes with 70 bulls. He is asked:
How many do you bring out of your numerous flock?
The shepherd answers:
I bring two-thirds of a third of the cattle. Count!
How many bulls are in the herd?
1) Lesson topic:
"Finding a part of a number and a number by its part"
The purpose of the lesson : the formation of students' ability to solve problems for finding a part of a number and a number in its part.
Practicing the computational skills of students.
To instill in students a sense of responsibility for the assigned task.
Equipment: computer
DURING THE CLASSES
I. ORGANIZING TIME
Checking the readiness of students for work.
II. ORAL WORK
Teacher. We started to explore new big topic"Ordinary fractions".
What number is called a fraction?
· Give an example of a fraction, name its numerator and denominator.
What does the denominator of a fraction show?
What does the numerator of a fraction show?
· Formulate the main property of a fraction.
What is fraction reduction?
Pay attention to the screen. Some tasks will be shown on slides.
Exercise 1 . Reduce the following fractions.
4 9 7 8 4 3 10 6 2 11 4 10
6 " 15 " 14 " 14 " 9 " 9 " 50 " 9 " 4 " 44 " 8 " 15 "
5 4
What is the last fraction called?
What fraction is called irreducible?
Task 2 . Solve the following problems.
1. Vintik and Shpuntik assembled a new car in 15 days. What part of the car did they collect in one day?
2. Dunno decided to do 10 good deeds in a day. But, unfortunately, he could only 1 - part of what he planned. How many good
actions performed Dunno in a day?
3. Znayka read in a day 1 part of the book. How many days will it take Znaika to read
the whole interesting book?
III.STUDYING A NEW TOPIC
Teacher. Pay attention to the screen. The epigraph to this lesson will be the words
D. Poya: "The ability to solve problems is a practical art, like swimming or skiing, or playing the piano: you can learn this only by imitating selected samples and constantly training." In this lesson, we will engage in practical art - learn to find a part of a number and a number by its part. Before starting to study new topic, let's repeat the spelling of some mathematical terms.
Exercise 1 . Write in notebooks following words and phrases in a column one under the other (one student writes on the board):
NUMERATOR
PART OF A NUMBER
Now check the spelling of the words on the blackboard with the spelling in front of you on the screen. Correct the errors if necessary.
When studying a new topic, we must establish a connection between these concepts. During oral work, you solved problems about Dunno and his friends.
Who came up with these wonderful characters?
[N. Nosov.]
N. Nosov wrote another interesting book, which is called "Vitya Maleev at school and at home." Let's solve the problem that the main character solved.
I ask your attention to the screen. Let's try to verbally solve the problem
Task . A boy and a girl were gathering nuts in the forest. The boy collected twice as many nuts as the girl. How many nuts did the boy and the girl collect separately if they collected 120 nuts together?
What fraction of the nuts did the girl collect? What fraction of the nuts did the boy collect?
Task 2. Solve the following problems.
1. The girl collected 1 all nuts. How many nuts did the girl collect, if only
collected 120 nuts?
2. The boy collected 2 all nuts. How many nuts did the boy collect, if only
collected 120 nuts?
Solving these problems, we were looking for a part of the number. Figure out how to find the part of a number.
Conclusion (students do). To find the part of a number, you need to divide the number by the denominator of the fraction and multiply by the numerator .
Teacher. Having formulated this rule, we have connected four mathematical terms
NUMERATOR
PART OF A NUMBER
Task 3. Solve problems to find a part of a number.
1. Mom bought 6 kilograms of sweets. Vitya immediately ate 2 all the sweets and him
got sick. After how many sweets did Vitya have a stomach ache?
2. There were 40 chickens in the chicken coop. For a week the fox dragged 3 all chickens. How many chickens
dragged by a fox?
Task 4. Solve the following "inverse" problems.
1. The girl collected 40 nuts, which is 1 all nuts. How many nuts
was collected?
2. The boy collected 80 nuts, which is 2 all harvested nuts.
How many nuts were collected?
Figure out how to find a number from its part.
Conclusion ( students do). To find a number by its part, you need to divide the part of the number by the numerator of the fraction and multiply by the denominator.
Teacher . Having formulated this rule, we again connected four mathematical terms:
NUMERATOR
PART OF A NUMBER
This record will serve as a support in solving problems of finding a part of a number and a number by its part.
Task 5 . Solve problems to find a number by its part.
1. Alice fell into a fairy tale well and in the first minute of the prolemeters. What is the depth of the well if in the first minute Alice flew 3 the whole distance?
2. The stepmother gave Cinderella a lot of work before the ball. To fulfill 3 this
work, Cinderella took 6 hours. How long does it take Cinderella to complete all the work?
III. INDEPENDENT WORK
No. 000(a, b), 785(a, b), 783.
At the end of the work, the correctness of the solution of problems is checked, the progress of the solution and the answers are discussed.
IV. LESSON SUMMARY
Teacher. What did you learn in class today?
How to find the part of a number from its fraction?
How to find a number by its part?
Solve the following problem orally.
There was a detachment of soldiers: ten rows of seven soldiers in a row.
8 they were mustachioed. How many mustachioed soldiers were there? How many were there
4 of them were nosed. How many nosy soldiers were there? How many were there
snub-nosed soldiers?
V. HOME TASK: Come up with, write down and solve two problems on the topic.
2) Theme of the lesson: Vieta's theorem.
Educational objectives of the lesson:
1. Repeat the formulas for the roots of incomplete quadratic equations.
2. To form students' ability to apply Vieta's theorem when solving quadratic equations.
Educational objectives of the lesson:
1. Contribute to the development of students' desires and needs, the facts being studied.
2. Cultivate independence and creativity.
Developing objectives of the lesson:
1. To develop and improve the ability to apply the knowledge that students have in a changed situation.
2. To promote the development of the ability to draw conclusions and generalizations.
Lesson teaching method:
1. Conversation.
2. Mini-dialogue.
3. Independent work.
During the classes:
1. Organizational moment.
2. Verbal check homework No. 000 (c, e), 544 (b), 546 (c).
3. Repetition of the material covered.
(Two students work with a table at the blackboard.) Task: fill in the empty spaces in the table.
(The rest of the class solves the crossword puzzle using theoretical knowledge)
Task: if you enter the correct words, then in the highlighted line you will get the name of the French mathematician
1. Quadratic equation with
first coefficient
equal to 1. (reduced)
2. Radical expression
in the root formula. (discriminant)
3. One of a kind
quadratic equation. (incomplete)
4. a , b in a quadratic equation.
(coefficients)
The highlighted line will contain the surname of the French mathematician Vieta.
Historical reference (a student's report on the life and work of the mathematician Francois Vieta).
Purpose: Today in the lesson we will explore the relationship between the coefficients and the roots of a quadratic equation.
In dealing with quadratic equations, you have probably already noticed that information about their roots is hidden in the coefficients. Something “hidden” has already been revealed to us.
What determines the presence or absence of the roots of a quadratic equation? (from discriminant)
What is the discriminant of a quadratic equation? (from coefficients a, b, c)
Depending on which coefficients of the quadratic equation, it is possible to determine the roots of incomplete quadratic equations. (we check the filling of the table by students)
How else are the roots and coefficients of a quadratic equation related? To reveal these connections, it will probably be useful to observe the coefficients and roots of various quadratic equations. (The student from each row solves the task on the board, and the rest complete the task in the notebook.)
Exercise. Solve the equation.
x2- x- 6=0
4(3x + 3) =2(1 - x2)
2x2 + 12x + 10 = 0
x2 + 6 x + 5 = 0
x2 - 6 x + 8 = 0
Additionally
(x - 1)(x + 2) + 3x = 10
x2 + x - 2 + 3x - 10 = 0
x2 + 4 x- 12 = 0
What are quadratic equations called after algebraic transformations? (given)
When looking for patterns, researchers often record their observations in tables that help discover those patterns.
Exercise. Fill in the gaps in the table
The equation
x1
x2
x1 + x2
x1 x2
x2 – x – 6 = 0
x2 + 6 x + 5 = 0
x2 – 6 x + 8 = 0
x2 + 4 x –12 = 0
Did this table help you in discovering new relationships between the roots and coefficients of quadratic equations. Express a hypothesis, statement (students draw conclusions). Compare the hypothesis you formulated with the theorem written in the textbook on page 121.
Theorem: The sum of the roots of the given quadratic equation is equal to the second coefficient, taken from opposite sign, and the product of the roots is equal to the free term. (Read the proof yourself)
The theorem is called the Vieta theorem, after the famous French mathematician Francois Vieta ().
He proved his famous theorem in 1591.
Exercise. Using Vieta's theorem, fill in the gaps in the formulas.
The equation
The sum of the roots
Root product
x2 – 5 x – 6 = 0
x2 – 3 x + = 0
x2 + x + 1 = 0
x2 + x + = 0
Vieta's theorem can be used to check the found roots of a quadratic equation. Consider tasks from homework № 000.
v) y2 = 4 y + 96 e) x2 – 20 x = 20 x + 100
y2 – 4 y – 96 = 0 x2 – 40 x – 100 = 0
y1
= – 8
y2
= 12
According to Vieta's theorem:
We check:
Is Vieta's theorem applicable to a quadratic equation in general view? (Yes, if we replace this equation with its equivalent given equation.)
ax2 + bx + c = 0
; if x1 and x2 are the roots of this equation, then by the Vieta theorem:
Formulate a general statement for a quadratic equation.
Theorem: If the roots of the quadratic equation ax2 + bx+ c=0 exist, then the sum of the roots is , and the product of the roots is .
Rightfully worthy to be sung in verse
On the properties of roots, Vieta's theorem.
Which is better, say the constancy of this:
You multiply the roots - and the fraction is ready.
In the numerator c , in the denominator a ,
And the sum of the roots is also a fraction
Even with a minus fraction, what a trouble
In the numerator b , in the denominator a .
Task number 000. Find the sum and product of the roots of a quadratic equation.
The equation
The sum of the roots
Root product
a) x2 – 37 x + 27 = 0
b) y2 + 41y - 371 = 0
v) x2 – 210 x = 0
G) y2 – 19 = 0
e) 2 x2 – 9 x – 10 = 0
e) 5 x2 + 12 x + 7 = 0
g) – z2 + z = 0
h) 3 x2 – 10 = 0
Orally: Without solving this equation, determine which numbers are the roots of the equation.
x2
– 5
x + 4 = 0
-1 and -4
x2 + 5 x + 4 = 0 -1 and 4
x2 – 3 x – 4 = 0 1 and 4
x2 + 3 x – 4 = 0 1 and -4
In some cases, the roots of the equation can be found by selection. The selection of roots greatly facilitates if the dependencies between the roots and the coefficients of the equation are known. The formulas expressing these dependences are reflected in the Vieta theorem.
Formulate a statement converse theorem Vieta.
Theorem. If real numbers x1 and x2 are such that x1 + x2 = – p and x1 x2 = q, then these numbers are the roots of the quadratic equation x2 + px + q = 0.
But more often this theorem is used to find the roots by the selection method.
Students solve task number 000 using this theorem.
Lesson summary:
1. What theorems did you learn in class today?
2. In what situations can the Vieta theorem and its inverse theorem be applicable.
Homework: p. 23 No. 000, 577, 58
3) Algebra lesson (press conference)
Topic:
Abbreviated multiplication formulas
(Repetition and generalization of the material covered)
Target:
during didactic game create conditions for the manifestation of personal functions of students.
Tasks:
1. systematize and generalize knowledge on the topic "Formulas of abbreviated multiplication";
2. continue the formation of cognitive activity;
3. search for your alternative;
4. Expressing your choice of problem solution
During the classes
Introduction.
Teacher: Today your class is a research institute. You - students - employees of this institute. Correspondents from various publications came to the lesson, who want to get answers to their questions. The success of the press conference depends on each employee of the institute. Warm up.
Teacher: To acquaint our guests with how our institute works on the study and application of formulas, I propose to solve the following problem:
There are four boxes and cards with algebraic expressions. Establish a matching principle between cards and boxes, and sort the cards into boxes.
(a±b) (a2±2ab+b2)
a3±3a2b+3ab2±b3
1) (-a-b)2
2) -(a+b)2
3) (b+a)2
4) a2-b2
5) a2+b2
6) (b-a)2
7) (b+a)3
8) (-b+a)3
9) -(a-b)3
10) a3+b3
11) a3-b3
12)-(a3-b3)
Interviews with "correspondents" of magazines. Correspondent for "Quantum" magazine.
- You know many abbreviated multiplication formulas. Explain why they are needed and in what cases you use them. The editors of our magazine received a letter from a 7th grade student Yury Groshev. He earnestly asks for help to factor out the polynomial a3+a2b-ab2-b3 different ways.
(Solution of the problem with the help of an idea).
Three students come to the board and complete this task in different ways; The class is asked to choose their favorite solution.
- Solve the equation: 16x2-(4x-5)2=15 two ways. (Suggest your own ways of solving the equation).
- The interplanetary station, launched to study the planet Mars, took photographs of its surface, visited it, took a soil sample and returned to Earth. Along with the samples, scientists found a piece of hard alloy with mysterious designations. The magazine has placed these designations on its pages, and readers want to know what they mean. Please help the editors answer their question. (5+ )= + +81 472-372=(47- ) (+37) (-3) (+3)=а2- 612=3600+ +292+2 71 29=( + )2= 2
- The criminals stole a large amount of money from the bank. They were caught, but it was not possible to establish the stolen amount. The criminals categorically refuse to name it, claiming that they wrote down this number as a degree and encrypted not only the base, but also its indicator. The experts managed to find out the base of the degree - 597. But to answer the question, what degree was asked. They can not. The criminals then wrote down the equations:
(2y+1)2-4y2=5
4y2+4y+1-4y2=5
4y=5-1
4y=4
y=4/4
y=1
(x-5)2-x2+8=3
x2-10x+25-x+8=3
-10x+33=3
-10x=-30
x=-30:(-10)
x=3
- What formulas were used to solve equations?
And besides, the expression (a-1) (a2+1) (a+1)-(a2-1)2-2 (a2-3)+1 which needs to be simplified. Now, using the alphabet as a cipher, you can read the exponent.
- Find the exponent and raise to it convenient way number 597
5972=(600-3)2=+9=356409
- The editorial office of the newspaper received a letter from Sasha Petrov with a request to publish it. Sasha thinks: in order to square an "integer and a half", you need to multiply this integer by a neighboring, larger number, and add 1/4 to the result.
For example, (71/2)=561/4; (81/2)=721/4.
Fast and simple.
But the editorial board of the newspaper believes that it is necessary to consult with experts. Do you think it is possible to prove this statement?
(two students are invited to the board to prove this statement in different ways).
- I'm compiling content for the "Raisins" page. Dear employees of the research institute, tell me how best to complete the following task: compare which is greater: 361 or 35 37?
Teacher. Our press conference has come to an end. Correspondents of newspapers and magazines, having received answers to questions of interest to readers, will issue them in the form of notes and publish them on the pages of their publications.
You, dear employees, are instructed by the Scientific Council to derive formulas:
(a+b)4 and (a+b+c)2 Thanks to all participants of the game. And in conclusion, I would like to know what impression the game made on you, what difficulties did you experience in the game today? (reflection)
4) Lesson topic: Pythagorean Theorem
Target: Show the historical origins of the theorem.
To teach students to apply the acquired knowledge to solving applied problems.
Learn to perceive the material in a holistic system of various subjects.
Raise cognitive interest in the study of geometry.
During the classes:
1. Organizational moment.
2. Checking homework.
3. oral solution tasks. (slide 2)
1.Find the area of a square with a side
3 cm; 1.2mm; 5\7 m; but see
2. Find the area of a rectangular
triangle with legs 3 cm and 4 cm;
2.2 m and 5 cm; a cm and in cm.
4. Updating the basic knowledge of students.
A special place in geometry, special role plays a right triangle, the ratio between sides and angles in right triangle. For several lessons, we have studied this material with you, and today our goal is to summarize the knowledge gained by studying the Pythagorean theorem. We will approach the issue of generalization from many sides: as historians, lyricists, theorists and practitioners.
5. Explanation of new material.
Biography of Pythagoras (Show 3 slides).
Pythagoras was born around 570 BC. e. on the island of Samos. Pythagoras' father was Mnesarchus, a gem carver. The name of Pythagoras' mother is not known. According to many ancient testimonies, the born boy was fabulously handsome, and soon showed his outstanding abilities. Among the teachers of young Pythagoras, the names of the elder Hermodamant and Pherekides of Syros are mentioned (although there is no firm certainty that Germodamant and Pherekides were the first teachers of Pythagoras).
From the history of the creation of the theorem (4 slide).
Pythagoras did a lot for the development of science, but he began his journey not at all as a scientist, but as a winner of the Olympic Games in fisticuffs!
One of the most remarkable statements is the Pythagorean theorem. с2= a2+b2
As Pythagoras thought, there is no information. Perhaps he drew with a twig in the sand, because the Pythagoreans often walked and did science on walks. According to legend, he sacrificed 100 bulls to the gods as a token of gratitude. And the legends say that when something new is discovered, all the cattle on earth tremble with fear.
Perhaps Pythagoras gathered all the mathematicians and spoke about his discovery. One of the clay tablets tells about this. It has only tasks, but no conclusions. But in Indian manuscripts, a drawing and the word "theorem", which comes from Greek word"theorio" - I consider
Pythagorean theorem (5 slide)
I. Dyrchenko
If we are given a triangle
And, moreover, with a right angle,
That is the square of the hypotenuse
We can always easily find:
We build the legs in a square,
We find the sum of the degrees -
And in such a simple way
We will come to the result.
Pythagorean theorem (6 slide)
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs
There are more than 100 proofs of the famous Pythagorean theorem, which still haunts the minds of scientists.
Let's look at some of them.
Proof of the Pythagorean theorem (slide 7)
Let T- right triangle with legs a, b and hypotenuse With. Let's prove that c2=a2+b2 Construct a square Q with side a + c. Square Q with a party a+b made up of a square R with a party With and four triangles equal to triangle T. Therefore, for their areas, the equality S(Q)= S(P)+4 S(T) .
Because S(Q)=(a+b) 2 ; S(P)=c2 and
S(T)=1/2(ab), then (a+b)2=c2+4*(1/2)abor
a2+b2 +2 ab= c2 +2 aband c2=a2+b2.
Demonstration 8 slide
The simplest proof of the theorem is obtained in the simplest case of an isosceles right triangle. Probably, the theorem began with him. Indeed, it is enough just to look at the mosaic of isosceles right-angled triangles and make sure that the theorem is true. For example, forÙ ABC: the square built on the hypotenuse AC contains 4 initial triangles, and the squares built on the legs contain two. The theorem has been proven.
Demonstration 9 slide
“Pythagorean pants are equal in all directions. To prove it, you need to shoot and show, - so it is sung in one playful song. These " pants” are shown in the figure, where squares are built on each side of the right-angled triangle ABC to the outside. And the drawing itself appeared in the famous first book of Euclid's treatise "Beginnings" and was put by its author as the basis for the proof of the Pythagorean theorem.
2 Oral work.
We will conduct a mathematical warm-up that will help us remember the definitions (slide 5).
1) The median in an isosceles triangle is….
2) The bisector in an isosceles triangle is….
1) A triangle in which all sides are equal is called ……….?
2) A triangle whose two sides are equal is called ………..?
3) A triangle in which one of the corners is a right one is called ……..? Let's check if you answered the questions correctly (slide 6).
3 Independent work (10 min)
Given triangle ABC is isosceles, triangle BSD is equilateral. The perimeter of the triangle ABC is 40 cm, the perimeter of the VSD is 45 cm. Find AB and BC (slide 7).
Let's check the solution of the problem (slide 8)
1) Since ∆ VSD is equilateral, then VD=VS=SD=45:3=15cm.
2) Since ∆ ABC is isosceles, then AB \u003d AC \u003d (40-15): 2 \u003d 12.5 cm.
Answer: AB=12.5cm, BC=15cm.
4. Mathematical test. (Choose the correct answer) (slide 9)
1) How many heights does the triangle have?
2) In an isosceles triangle, the angles at the base
a) not equal b) equal
3) The angles of an equilateral triangle are equal
a) 60° b) 45°
5.game moment (slide 10)
Game "Thinker" (Who will quickly count the number of triangles in this figure)
How many triangles are shown in the picture? (answer 16)
6. Oral survey. (slide 11)
Problem: In a right triangle ABC, one of the acute angles is 30°. Find other corners.
7.Lesson results.
Homework: No. 44(a), No. 47
So, let us be given some integer a. We need to find, for example, a fifth of this number. You can do this with ordinary fractions:
- Since we need to find the fifth part of the number, we are looking for 1/5 of the number a.
- To find 1/5 of the number a, we must multiply the number a by the part we need to find, that is, perform the action: a * 1/5 = a/5. That is, the fifth part of the number a is a / 5.
- Moreover, if we are looking for a part of an integer, then the result will be less than the original number.
There may be different tasks on finding a part of a whole: if you need to find, for example, a tenth of a number a, then you need a * 1/10 = a/10. If you want to find 1/8 of the number a, then you need a * 1/8 = a/8.
Finding any part of a whole is done by multiplying the given integer by the part you want to find.
Consider specific example for even more memorization of the solution.
How to find the sixth of the number 36
We are given an integer - the number 36. We need to find a sixth of it, otherwise we need to find 1/6 of the number 36. Let's perform the action of multiplying the whole by the part: 36 * 1/6 = 6. So the sixth of the number 36 is the number 6. You can also say the following: the number 36 is exactly six times greater than the number 6, or the number 6 is exactly six times less than the number 36.
To find a part of any number, it should be divided by the size of this very part. The actions in this case will differ depending on the form of recording the fraction;
With a common fraction:
If the numerator of an ordinary fraction is divisible without a remainder by a given size of the part, then it is enough to simply divide the numerator by this given size;
If the numerator cannot be divided into a given part without a remainder, then the denominator must be multiplied by the size of this part; With a mixed fraction: We do the same as with an ordinary fraction, but only first you need to convert mixed fraction into the ordinary. With decimal: The calculation will consist of a single division operation. Decimal can be divided into a given size of part in a column.
So, let us be given some integer a. We need to find half of this number. You can do this with ordinary fractions:
- Let's denote the integer as one, then half of the unit is 1/2. So we need to find 1/2 of the number a.
- To find 1/2 of the number a, we must multiply the number a by the part we need to find, that is, perform the action: a * 1/2 = a/2. That is, half of the number a is a / 2.
- Moreover, if we are looking for a part of an integer, then the result will be less than the original number.
There may be different tasks on finding a part of the whole: if you need to find, for example, a quarter of the number a, then you need a * 1/4 = a/4. If you want to find 1/8 of the number a, then you need a * 1/8 = a/8. Finding any part of a whole is done by multiplying the given integer by the part you want to find.
Consider an example.
How to find the third part of the number 75
We are given an integer - the number 75. We need to find the third part of it, otherwise we need to find 1/3. Let's perform the action of multiplying the whole by the part: 75 * 1/3 = 25. So the third part of the number 75 is the number 25. You can also say this: the number 25 is three times less than the number 75. Or: the number 75 is three times greater than the number 25.