All the rules that apply to dividing whole numbers also apply to dividing fractions. It is worth remembering that the operation of dividing fractions has no similarity with the operation of addition or subtraction.
In order to divide a fraction by another fraction, you do not need to find a common denominator. To perform the operation of dividing a common fraction, it is necessary to multiply the value being divided by the fraction that is the reciprocal of the divisor.
In other words: during this arithmetic operation, you need to leave the first fraction unchanged, and turn the second one over and multiply both values together.
If the example indicates different types fractions, then in order to divide them, it is necessary to reduce all quantities to one form - to ordinary fractions.
If the example indicates several division and multiplication operations, then they must all be performed in a row, from left to right. This rule Only valid if the example does not contain parentheses.
In order to divide a mixed fraction into another fractional value, you must first convert the mixed fraction into an improper fraction. This can be done as follows: the whole part is multiplied by the denominator of the fraction, and the numerator is added to the resulting number.
After the mixed fraction has been converted to an improper fraction, you can perform the action according to the established rules. If you need to divide a proper fraction by a whole number, then the latter value must also be represented as an improper fraction.
Integers are converted into an incorrect ratio as follows: the number itself is written in the numerator, and the denominator must always be 1, since any value divided by one will be equal to itself.
In order to divide one by a common fraction, you simply need to flip the second fractional value, since any number multiplied by one will be equal to itself.
If, during the process of dividing fractions, it is possible to reduce some quantities, then they must be reduced. However, it is worth remembering that you can reduce the values only after the second number is inverted.
In some examples, dividing ordinary fractions by a number or by another fraction can be written in three-story or even four-story form. In order to make the example normal, you simply need to replace the division line between two fractions with a colon.
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Slide captions:
ORAL WORK What numbers are written on the first line? What fractions are written on the second line? How can you characterize the fractions written on the third line? What are the numbers written on the fourth line called?
Perform actions: What operations with ordinary fractions can you perform?
Examination:
Addition Subtraction Multiplication!
HOMEWORK: No. 626.630(a, b):
ORAL WORK What numbers are called reciprocals? How to write the reciprocal of a fraction? 4.How to write the inverse of a natural number? 5.How to write the inverse of a mixed number?
LET'S SOLVE THE PROBLEM Area of a rectangle. Length of one side. Find the length of the other side.
Solution to the problem Let the second side be x m. The area of the rectangle is found by the formula S = ab We obtain the equation x = Multiply both sides of the equality by the inverse of the number. We get x. = . , apply the commutative law of multiplication, we get. X = . , i.e. we get 1x = . , or x = . , or x = m Now let’s solve separately the equation x = How to find the unknown factor? x = : , we get that: = . = Now let's try to formulate a rule for dividing two fractions. - dividend, - divisor, - reciprocal of the divisor.
Questions: What are the names of the components of the division action? What action was division replaced with? What changed? What hasn't changed? 3/4 and 4/3. What are these numbers called? State the rule for dividing fractions.
Working with the textbook Page 97
Solving on the board No. 596 (b,g,i,m) On our own No. 596 (a,c,f,l,n)
Examination
Questions: 1) How are the examples similar? 2) How are they different? 3) Why were these particular examples chosen? 4) How to divide one fraction by another? 5) How to divide mixed numbers?
Lesson topic:
Division of fractions
Kim Natalya Leonidovna
teacher of mathematics and economics
KSU “Secondary School No. 252 named after G.N. Kovtunov"
Kyzylorda region, Shieli village
Updating knowledge:
What fraction is called rational?
Give examples.
How are fractional expressions multiplied?
How to divide a fractional expression by a fractional expression?
ARCHIMEDES
No, not always funny and narrow The sage, deaf to the affairs of the earth: Already on the roads in Syracuse There were Roman ships. Above the curly mathematician The soldier raised a short knife, And he's on a sandbank I entered the circle into the drawing. Oh, if only death were a dashing guest - I was also lucky to meet Like Archimedes drawing with a cane At the moment of death - a number!
Dmitry Kedrin
Archimedes was obsessed with mathematics.
He forgot about food, completely
took care of himself. Works of Archimedes
applied to almost all areas
mathematicians of that time:
he owns wonderful
geometry research,
arithmetic, algebra. Your best
he considered it an achievement to define
surface and volume of the ball is a problem,
which no one before him could solve.
Archimedes asked to knock out on his
grave a ball inscribed in a cylinder.
Great importance for development
mathematics had a calculated
Archimedes length ratio
circumference to diameter.
287 - 212 BC
Number π
Checking homework:
DIOPHANT
Diophantus - Ancient Greek mathematician from
Alexandria. There is almost nothing about his life
no information. Part preserved
mathematical treatise of Diophantus
"Arithmetic" (6 books out of 13) and excerpts
books about polygonal numbers.
In "Arithmetic", in addition to the presentation
began algebra, many problems are given,
reduced to uncertainty
equations of various degrees, and
Methods for finding solutions to such equations in rational positive numbers are indicated. To denote the unknown and its powers, reciprocal numbers, equality and subtraction, Diophantus used an abbreviated form of words. When multiplying sums and differences of two numbers, I applied the rules of signs. Had an idea about negative numbers.
III century AD
Oral work
- Read the fractions:
- Find the expression that is redundant:
A) ( a+c) 2 ; b)
PYTHAGORAS
Modern historians
suggest that Pythagoras
did not prove the theorem,
but could convey this to the Greeks
knowledge known in Babylon
1000 years before Pythagoras
(according to Babylonian
clay tablets with notes
mathematical equations).
Pythagoras exists, but
weighty arguments
to dispute this, no.
IN modern world Pythagoras
considered a great mathematician
and cosmologist of antiquity.
famous theorem: square
hypotenuse of a rectangular
triangle equals the sum
squares of legs.
570 BC .
Oral work
- For each fraction, find an equal fraction,
using number-letter correspondence
Descartes It took me a while to find what I was looking for
place in life. Nobleman by
origin, graduating from college
in La Flèche, he plunges headlong
into the social life of Paris, then
gives up everything to pursue science.
Descartes gave special attention to mathematics
place in his system, he considered it
principles of truth
a model for other sciences. Main
achievement of Descartes-construction
analytical geometry, in which
geometric problems were translated
into the language of algebra using the method
coordinates He formulated the fundamental theorem of algebra: “the number of roots of an algebraic
equation is equal to its degree", proof
which was received only at the end of the 18th century.
1596-1650
Solution of examples:
Johann Carl Friedrich Gauss
German mathematician, astronomer and physicist.
While still a student he wrote “Arithmetic
research” that determined the development
Number theories up to our time.
At the age of 19 he decided which ones were correct
polygons can be constructed
compass and ruler. I was studying
geodesy and computational astronomy.
created the theory of curved surfaces.
One of the creators of non-Euclidean
geometry.
1777 - 1855
Solution of examples:
Gottfried Wilhelm Leibniz
German mathematician, physicist, philosopher,
founder of the Berlin Academy of Sciences.
Founder of differential
and integral calculus, introduced
Much of the modern symbolism
mathematical analysis. In the works
Leibniz's first ideas
theory of algorithms.
1646 - 1716
SOFIA VASILIEVNA KOVALEVSKAYA
Russian mathematician and mechanic, since 1889
Corresponding member of the St. Petersburg Academy of Sciences.
The first in Russia and Northern Europe
female professor and the first in the world
female mathematics professor.
Kovalevskaya opened the third classical
case of solvability of the rotation problem
rigid body around a fixed point.
Proved the existence of analytical
solutions to the Cauchy problem for systems
differential equations with
partial derivatives, researched
Laplace's ring equilibrium problem
Saturn, received a second approximation.
She also worked in the field of theory
potential, mathematical physics,
celestial mechanics.
1850 - 1891
http://www.kartinki24.ru
http://createpics.ru
21.11.17
Municipal educational institution Ikshinskaya secondary school
Division of fractions
Morozova Ekaterina Sergeevna
Dear Guys!
Today we are conducting the final lesson on the topic "Dividing fractions." In the next lesson you will have to write a test on this topic. Therefore, our task is to repeat and summarize all knowledge on the topic “Division of fractions” in order to prepare for test work. During the lesson we will climb the ladder of success.
Ladder of success
we know how to do it ourselves
all together together
we can do this
let's remember
Let's remember...
Let's remember...
What numbers are written on this line?
Answer : Common fractions
Answer: Correct
and incorrect
Answer: Reducible and irreducible
Answer: Mixed numbers
Answer: Mutually inverse
Convert to improper fraction
(complete the task in notebooks and on the board)
"You can do this" ):
Climbing to the second step "You can do this" , remember all the operations with fractions ( addition, subtraction, multiplication ):
We rise to the third stage "All together together" . Since when dividing ordinary fractions,
Mixed numbers ultimately all come down to multiplying fractions, then remember the rules:
1) How to divide a fraction by a fraction ?
2) But what if you need to divide an ordinary fraction by a natural number or vice versa?
3) How to divide mixed numbers?
Let's solve the equation
Now let's solve the problem
Let's solve the equation
Now let's solve the problem
Dear Guys! We have already overcome half the journey, but there are still many difficulties ahead, so it’s time to relax a little and spend physical education minute . I will read out a certain mathematical statement. You must determine whether it is true or false. If you think that the statement is true, then put your hands on your belt and bend forward, and otherwise, put your hands behind your head and rotate your body to the right and left.
If you think the statement true , That place your hands on your waist and bend forward , and otherwise( false ) – hands behind your head and rotate your body to the right and left .
- Let's move to the fourth step “We can do it ourselves” . Each of you will have to solve two tasks independently
First task - solve the equation . You are invited to choose the level of complexity of the equation yourself and solve one of the proposed ones:
Lesson summary.
Lesson summary.
So we have climbed to the top rung of the ladder of success. Notebooks must be turned in. At the end of the lesson, let's unravel the phrase that I encrypted here
WE ARE GREAT!!!
Homework :
Thank you for your attention!
I wish you success!
Slide 1
Slide 2
Slide 3
Slide 4
Slide 5
Slide 6
The presentation on the topic “Division of ordinary fractions” (grade 6) can be downloaded absolutely free on our website. Project subject: Mathematics. Colorful slides and illustrations will help you engage your classmates or audience. To view the content, use the player, or if you want to download the report, click on the corresponding text under the player. The presentation contains 6 slide(s).
Presentation slides
Slide 1
Math lesson in 6th grade
Dividing ordinary fractions (lesson 4)
⅔ ⅞ ⅕ : ⅗ ₌
Slide 2
Leonardo of Pisa introduced the word “fraction” in 1202 and was the first to use the modern notation of fractions. The names “numerator” and “denominator” were introduced in the 13th century by Maximus Planud, a Greek monk and learned mathematician
Historical reference
Leonardo of Pisa
Maxim Planud
Slide 3
Give the reciprocals of the given numbers:
Verbal counting
Slide 4
6 + 2: 0.6: 0.7 1 2 3 5 Work in groups 0.4 -
A) Name the numbers of cards whose values are reciprocal numbers. B) Find the meaning of the expression written on the card that the monkey took.
Slide 5
Independent work
It's true, kids, I'm good. It looks like a big bag. In the past years I overtook steamships across the seas. Who am I? Solve examples:
1) 2) 3) 4) Solve the equation: 6) Solve the problem:
The area of the square is
Find his side.
d e l f i n 21 Answer: dolphin
What do you know about dolphins?
Slide 6
Find the rule for placing numbers in sectors and insert the missing numbers
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