Division of Fractions — Steps, Method, Properties, Examples
By dividing fractions we mean the division involves at least one fraction. For example,
Let’s consider some examples to see what it means to divide fractions through visual models.
To divide a fraction by another fraction , multiply the dividend fraction by the reciprocal of the divisor fraction. The reciprocal of a fraction can be found by interchanging its numerator and denominator. For example, 34 is reciprocal of 43 and vice-versa.
There is a quick way to remember the steps to divide fractions: Keep >> Change >> Flip.
Quick Tip: In case, the dividend and/or divisor are mixed numbers, we first convert them into improper fractions and then follow the steps mentioned above.
Recall that the reciprocal of a whole number ‘a’ is the unit fraction 1a because we can write a whole in fraction form with denominator equal to 1, that is, a1.
Let’s understand how to divide when divisor is a whole number through an example:
When we divide one number by another, let’s say, a is the dividend and b is the divisor, then we can write it as ab. Also, we can write ab as the product of a and 1b. That is,
Division is the inverse operation of multiplication. So, dividing by a number is the same as multiplying by its reciprocal.
The properties of division with whole numbers hold true for fractions as well. Let’s check!
Solution: Change 123 to improper fraction and then follow the steps to divide fractions.
Example 3: Max is painting toy cars. He has 214 L of paint. If each car requires 73 L of paint, how many cars can Max paint?
Example 4: Melvin sang a medley of songs for 10 minutes. If one song was 212 minutes long, how many songs did Melvin sing?
1
Divide: $\frac{7}{8} \div \frac{3}{9}$
$\frac{21}{72}$
$\frac{24}{63}$
$\frac{21}{8}$
$\frac{8}{21}$
Correct answer is: $\frac{21}{8}$
$\frac{7}{8}\div\frac{3}{9} = \frac{7}{8}\times\frac{9}{3} = \frac{63}{24} = \frac{21}{8}$
2
Divide: $\frac{4}{8}\div 9$
9
$\frac{4}{89}$
$\frac{72}{4}$
$\frac{1}{18}$
Correct answer is: $\frac{1}{18}$
$\frac{4}{8}\div 9 = \frac{4}{8}\times\frac{1}{9} = \frac{4}{72} = \frac{1}{18}$
3
Divide: $1\frac{1}{7}\div \frac{3}{5}$
$\frac{24}{35}$
$\frac{40}{21}$
$\frac{21}{40}$
$\frac{34}{35}$
Correct answer is: $\frac{40}{21}$
$1\frac{1}{7}\div\frac{3}{5} = \frac{8}{7}\div\frac{3}{5} = \frac{8}{7}\times\frac{5}{3} = \frac{40}{21}$
4
Divide: $\frac{3}{7}\div\frac{3}{7}$
1
$\frac{9}{49}$
$\frac{49}{9}$
Correct answer is: 1
When a fraction is divided by itself, the answer is 1.
What does division of fractions mean?
The division of fractions means dividing a fraction into further equal parts. For example, If you have three-fourth of a pizza left and you divided each slice into 2 parts you would get a total of six slices but this would represent six-eighths of the total pizza.
What is the method of dividing fractions?
The first step for dividing fractions is to reciprocate the second fraction, that is, exchange its numerator and denominator. Next, multiply the first fraction with this reciprocal using standard method of fracton multiplication and convert the answer into simplest form.
How does one divide a fraction by a whole number?
To divide a fraction by a whole number, convert the whole number into a fraction with denominator 1 and then follow the standard steps of dividing fractions i.e., Flip the second fraction and multiply this with the first fraction.
How do you divide fractions by mixed number?
To divide a fraction by a mixed number (or vice versa), convert the mixed number into an improper fraction and then follow the standard steps of dividing fractions.
Multiplying and dividing fractions — Krista King Math
Turning fraction division problems into fraction multiplication problems
When we multiply fractions, we multiply their numerators to find the numerator of the result, and we multiply their denominators to find the denominator of the result.
???\frac34\times\frac17???
???\frac{3\times1}{4\times7}???
???\frac{3}{28}???
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When we divide fractions, we actually turn the division problem into a multiplication problem by turning the divisor (the second fraction) upside down (switching its numerator with its denominator) and changing the division symbol to a multiplication symbol at the same time.
We call this process “multiplying by the reciprocal.” Thereciprocalof a fraction ???a/b??? is the fraction ???b/a??? (where the numerator and denominator are flipped.
???\frac34\div\frac17???
???\frac34\times\frac71???
???\frac{3\times7}{4\times1}???
???\frac{21}{4}???
It’s okay that in this last fraction, the numerator is larger than the denominator. When that’s the case, the fraction is called an “improper” fraction.
Examples of multiplying and dividing fractions
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A simple example of multiplying fractions
Example
Multiply the fractions.
???\frac23\times\frac{4}{11}???
To multiply the fractions, we multiply the numerators and the denominators separately.
???\frac{2\times4}{3\times11}???
???\frac{8}{33}???
When we divide fractions, we actually turn the division problem into a multiplication problem by turning the divisor upside down.
Let’s do an example with division.
Example
Divide the fractions.
???\frac23\div\frac{4}{11}???
To do division with fractions, we turn the second fraction upside down and change the division symbol to a multiplication symbol at the same time.
???\frac23\times\frac{11}{4}???
Then we treat this as a multiplication problem, by multiplying the numerators and the denominators separately.
???\frac{2\times11}{3\times4}???
???\frac{22}{12}???
We always like to give our answer in lowest terms, so we’ll simplify this fraction by canceling a ???2??? from the numerator and denominator.
???\frac{22}{12}=\frac{2\cdot11}{2\cdot6}???
???\frac22\cdot\frac{11}{6}???
???1\cdot\frac{11}{6}???
???\frac{11}{6}???
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Learn mathKrista Kingmath, learn online, online course, online math, fundamentals, fundamentals of math, fractions, fraction arithmetric, fraction operations, operations with fractions, multiplying fractions, multiply fractions, fraction multiplication, dividing fractions, divide fractions, fraction division, reciprocal, changing fraction division into fraction multiplication, prealgebra, pre-algebra
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Problems of the new Federal State Educational Standard in mathematics and how to solve them
The standard of education is often criticized by teachers, for a number of reasons.
Our blogger, author of mathematics textbooks Alexander Shevkin, continues to analyze the new Federal State Educational Standard and point out its weaknesses.
We continue the conversation started in the previous post. Let’s move on to dividing the content of education by years of study, which violated the century-old tradition.
1. First year of study
“Appendix 7. Requirements for the subject results of mastering the academic subject “Mathematics (including algebra, geometry, probability, statistics)”, submitted for intermediate and final certification […]
The subject results of mastering the first year of study of the subject «Mathematics» should reflect the formation of skills.
— operate with concepts (hereinafter — recognize specific examples of general concepts by characteristic features, perform actions in accordance with the definition, rule and simplest properties, specify general concepts with examples): natural number, square and cube of a natural number, divisibility of natural numbers; perform arithmetic operations with natural numbers; apply in calculations the commutative, associative laws (properties) of addition and multiplication, the distributive law (property) of multiplication with respect to addition; compare, round natural numbers; estimate and check the results of calculations».
We read carefully: “Subject results of mastering the first year of teaching a subject”. What are we learning? — First year! Who are we teaching? — Subject! Did the writers read their work? How tired of this verbose to the point of losing the sense of office with attempts to be scientific and without reaction to useful remarks!
Why do the first four arithmetic operations with natural numbers come after the fifth and after divisibility? Why does the comparison of natural numbers (and other numbers too) come after actions with them, if, when subtracting, it is necessary to subtract a smaller one from a larger natural number? Why are the concepts of a numeric expression, the value of a numeric expression assigned to the 6th grade? Where did the procedure go? They belong here. Why is there no general concept of «power of a number with a natural exponent»?
We read the requirements of the Standard, not the recipes of the cookbook, where the order of operations «salt» or «pepper», and the recipes themselves, is not so important (if not, the housewives will correct me).
Once again about the «distributive law (property) of multiplication with respect to addition.» In mathematics, there is one «distributive law» — without additions: a*(b + c) = a*b + a*c. And the «distributive law (property) of multiplication with respect to subtraction», which was introduced into textbooks 50 years ago, is a consequence of the distributive law. It is useful for students to show the conclusion of this consequence. What need is there to clarify «with respect to addition» if there is no «with respect to subtraction»? Where is the requirement to represent natural numbers by points on the coordinate ray?
2. Fractions
“Subject results… should reflect the formation of skills:
— operate with concepts: fractions, parts, fractional numbers, ordinary fraction; proper and improper fraction, mixed number; perform addition and subtraction of fractions with the same denominators, compare numbers;
— operate with concepts: decimal fraction, integer and fractional part of a decimal fraction, percentage; perform addition and subtraction of decimal fractions; round decimals; compare numbers;
— operate with concepts: division with remainder, divisibility, divisor, multiple; use the signs of divisibility by 2, 3, 5, 9 and 10 when solving problems.
Once again I will return to the comparison of numbers. Since for the same positive rational number its different representations in the form of a fraction are possible, then when introducing ordinary fractions, one must say which fractions are considered equal, that is, introduce the main property of the fraction, reducing fractions to a new denominator, introduce a comparison of fractions not only for fractions with the same denominators.
The concept of «mixed number» is mentioned here, in my opinion, «mixed fraction» is better. Let’s compare the two entries: ¾ = 0.75 and 11/4 = 2¾ = 2.75. In each of them there are fractions on the left and on the right, in the second there is a number in the middle. For 50 years of studying according to the textbook of N. Ya. Vilenkin, they got used to it. Think about it: there are natural numbers, we consider the set of natural numbers, there are «mixed numbers», then we do not consider the set of mixed numbers. A mixed fraction is a way of writing improper fractions, convenient for comparison, addition, and subtraction.
The requirement to “perform addition and subtraction of fractions with the same denominators” without adding and subtracting fractions with different denominators, multiplying and dividing fractions is surprising — this is a requirement to form incomplete skills in actions with fractions that will never become complete, since in 6- There is no requirement to perform addition and subtraction of fractions with different denominators, to multiply and divide fractions in the first grade. Since the study of mathematics is not reading the mentioned cookbook, here one cannot snatch out a couple of actions for study just because they are simple.
Decimals appear as new entries, they are not related to fractions. Here are the requirements to form incomplete skills. The requirement to “do addition and subtraction of decimals” without the requirement to multiply and divide decimals, which is not in the 6th grade either, is this an attempt to “make children feel good”? Neither in the 5th grade nor in the 6th grade is a single object of study fully studied.
So maybe you need to study completely ordinary fractions in the fifth grade, and decimal fractions and their relationship with ordinary fractions in the sixth?
I was amazed by the last requirement about division with remainder, divisibility, divisor, multiple, use of divisibility criteria. The authors of the Standard have no idea that this is about natural numbers? Why is it written after fractions? Now the questions are: why is division with a remainder separated from division without a remainder? Doesn’t a student, when dividing 56 by 4 without a remainder by a corner, get a remainder of 1 in intermediate calculations?
Why is “division with remainder” repeated in the 6th grade and there is a “remainder from division”, but there is no remainder in the 5th grade?
Why are prime and composite numbers associated with the concepts of divisors and multiples (5th grade) assigned to the 6th grade? Why are there signs of divisibility, but the properties of divisibility, with the help of which they can be justified, are absent? We are not interested in the possibility of creating conditions for a systematic, more complete, as far as the age characteristics of children allow, presentation of the theoretical component of the mathematics course? What is the deep meaning of such cutting of the studied material?
In school mathematics, it was customary to study mathematical objects systematically, this applies to natural and rational numbers, information about which is given to children in the form of a vinaigrette both in the fifth grade and in the sixth, and even with an underinvestment of components! The authors of the Standard are afraid that with proper training in the minds of children, the knowledge gained will form a clear picture — a panel where each colored glass is in its place, and all together they create a clear picture of the material being studied? Instead of such a panel, do they want to give children colored glasses?
Do the authors of the Standard know why, after first places in primary school, our students slide to the bottom of the third ten in the world rankings in three years of study? I will say: they are taught not from textbooks, but from a cookbook, in which there are many unsystematized recipes.
The authors of the Standard strive to tear out of it some pages with «difficult» recipes! By increasing the lack of system in the study of mathematics in grades 5-6, we only exacerbate the situation.
3. Text problems
“Substantive results… should reflect the formation of skills:
— solve plot problems for all arithmetic operations, interpret the results obtained; solve problems of the following types: to find a part of a number and a number by its part; on the relationship between quantities (price, quantity, cost; speed, time, distance; data from household appliances for metering the consumption of electricity, water, gas.
It’s nice to see the impact of my April remarks, it got better, but the questions remained. On a par with three interrelated values »price, quantity, cost» are «data of household appliances» — this is not a trio of interrelated values, but an indication of the plot of the problem. If the plots are so important in the Standard, then write them separately, just don’t forget about sausages, sausage, milk, shoes and many other items from our practical life, for the sake of connection with which electricity, water and gas are mentioned here.
It is natural to ask why rent, garbage collection, etc. are not mentioned? In the 2015 program, I found the only place for which I praised her compilers. There is a requirement to teach schoolchildren to solve problems in arithmetic ways. This is a large reserve for increasing the accessibility of mathematics and its connection with life — the result t of each action is comprehended and justified in relation to those objects and quantities that are discussed in the condition of the problem. This is available for children in grades 5-6, whose thinking is objective. This is a reserve for the development of thinking and speech of students. Such tasks have already appeared in the OGE and the Unified State Examination.
Since neither in the fifth grade nor in the sixth grade there are no equations and their application to solving word problems, then in arithmetic methods for two years of study it is necessary to put things in order. In the 5th grade, it is necessary to mention tasks for movement, for parts, for finding two numbers by their sum and difference, for fractions (the same for finding a part of a number and a number by its part), for joint work.
The latter require the study of all actions with fractions.
4. Geometry
“Subject results… should reflect the formation of skills:
— recognize the simplest shapes: segment, line, ray, polyline, angle; polygon, triangle, quadrangle, rectangle, square; circle, circle; cube, cuboid, pyramid; give examples of figures and recognize them in the world around them;
— depict the studied figures by hand and with the help of drawing tools; measure lengths, distances, including in practical situations,
— measure the area of a figure on checkered paper; apply when calculating the formula of perimeter, area of a rectangle, square; calculate the volume and surface area of a cube, the volume of a cuboid».
Why is there no system again? We recognize, depict, measure and calculate — it’s understandable, but why measure at two separate points? We study and measure segments, even with two values separated by a comma: “measurement of lengths, distances” are these different values? It’s better to write “measurement of distances (lengths of segments)”.
For a broken line, we do not measure either the length or the distance between its ends. We study the angle, but do not measure it. What’s stopping you? Here you can also give types of angles — why postpone for a year? What is the use of such a fine slicing of objects of study? Here, pie charts are also appropriate — their reading and construction. You have to do something with your hands so that knowledge is deposited in your head.
5. Second year of study — sixth grade
“The subject results of mastering the second year of study of the subject “Mathematics” should reflect the formation of skills:
— operate with concepts: a set, an element of a set, a subset, an intersection, a union of sets; set of integers, set of rational numbers; use a graphical representation of sets to describe real processes and phenomena when solving problems from other academic subjects;
— operate with concepts: statement, true statement, false statement, example and counterexample; solve simple logic problems;
— operate with concepts: division with remainder, remainder of division; use division with remainder when solving problems;
— operate with concepts: prime and composite number; find the decomposition of a composite number into a product of prime numbers.
I stumble over the very first concept of «set» — I cannot define this indefinable concept, as required by the footnote I gave in brackets at the very beginning of the material for the 5th grade. What about undefined concepts? Shall we define them according to the footnote? Should the student think that everything in the world can be defined? Wouldn’t it be useful for students to require that the elements of a set be classified according to some attribute? This mental operation must be taught at every opportunity, the sooner the better, then the authors of the next Standard will better classify the objects of study.
Dividing with a remainder, divisibility, prime and composite numbers is mentioned above, only the turnover “decomposition of a composite number into prime factors” is generally accepted.
6. How to correct GEF?
“Objective results… should reflect the formation of skills:
— operate with concepts: a negative number, an integer, a number module, opposite numbers; compare numbers with different signs, add, subtract, multiply and divide numbers with different signs; represent positive and negative numbers on a coordinate line;
— operate with concepts: numerical expression, value of a numerical expression; find the values of numerical expressions, operate with the concept of a rational number; perform arithmetic operations with ordinary and decimal fractions; apply in calculations the commutative, associative laws (properties) of addition and multiplication, the distributive law (property) of multiplication with respect to addition; find decimal approximations of ordinary fractions; round rational numbers; compare rational numbers; estimate and evaluate the results of calculations with rational numbers.
Considering the poor results of actions with positive and negative numbers, which we constantly hear about in the reports on the OGE and the USE, isn’t it time to think about the reason for such results? I consider it expedient to master the idea of the sign of a number on integers, and then transfer the acquired skill to the set of rational numbers (written as fractions of any sign). Assimilation of the idea of the sign of a number when operating with different entries of modules of numbers (whole numbers, ordinary fractions, decimal fractions) is not productive. Once again about the requirement to «compare numbers with different signs.» It’s easier to “compare”, but why only with different signs? Will we compare -3 and 6, 3 and -7, but will we not compare 4 and 5, -4 and -5? The first requirement can be rewritten like this:
“Subject results… should reflect the formation of skills:
— operate with concepts: a positive integer, a negative integer, the modulus of a number, opposite numbers, integers; compare integers, perform arithmetic operations with them; apply the laws of arithmetic operations to simplify calculations; represent integers as points on a coordinate line.
If actions with ordinary fractions are fully studied in the 5th grade, the idea of the sign of a number is studied in the 6th grade using the example of integers, we can enter all rational numbers. You don’t have to lump everything together. Let’s deal with rational numbers without getting confused by their two forms. The second requirement after the transfer of numerical expressions to the 5th grade can be written as follows:
“Substantive results… should reflect the formation of skills:
— operate with concepts: positive rational number, negative rational number, modulus of number, opposite numbers, rational numbers; compare rational numbers, perform arithmetic operations with them; apply the laws of arithmetic operations to simplify calculations; represent rational numbers as points on a coordinate line.
Decimal fractions should be studied as another representation of rational numbers, first positive, then negative. The requirement for decimal fractions must be written separately:
“Subject results… should reflect the formation of skills:
— operate with concepts: a decimal fraction, as another notation of rational numbers, compare decimal fractions, perform arithmetic operations with them; perform arithmetic operations with ordinary and decimal fractions; apply the laws of arithmetic operations to simplify calculations; write a decimal fraction in the form of an ordinary and an ordinary fraction in the form of a decimal; find decimal approximations of ordinary fractions; operate with the concept of a positive infinite periodic decimal fraction, as a record of a positive rational number.
— operate with concepts: a decimal fraction of any sign, compare decimal fractions of any sign, perform arithmetic operations with them; apply the laws of arithmetic operations to simplify calculations; operate with the concept of an infinite periodic decimal fraction, as a record of a rational number; estimate and evaluate the results of calculations with rational numbers; compose a non-periodic infinite decimal fraction, as an example of a non-rational number — an irrational number, give an example of an irrational number «pi»; mark a point on a coordinate line by its coordinate, mark a point in a rectangular Cartesian coordinate system by its coordinates; give examples of using coordinates on a straight line and on a plane (instrument scales, coordinates of points on geographical maps).
Regarding word problems in the Standard for the 6th grade, there are repetitions with the 5th grade, which must be excluded — I do not comment on them. New in the requirements — only tasks for percentages, ratios, proportions and tasks from the field of personal and family finance management.
The requirements for text problems can be described as follows:
“Subject results … should reflect the formation of skills:
— solve word problems for scale, for dividing a number in a given ratio, for direct and inverse proportionality (for proportions), for percentages: find several percent of a number, find a number by its several percent, how many percent one number is from another, an increase (decrease) in a number by several percent, by what percent one number is more (less) than another, a multiple increase (decrease) in a number by several percent (compound interest) ), problems for mixtures and alloys; interpret the results.»
The requirement for a pie chart should be transferred to the 5th grade and studied along with the measurement of angles and their classification. The arithmetic mean is also transferred to the 5th grade, in the section «Positive Rational Numbers» (ordinary fractions). The requirement about spatial figures can be left unchanged, putting before it “approximately calculate the circumference and area of a circle” — this was always at the end of the 6th grade.
Remove repetitions with the 5th grade about measuring the areas of figures, surface areas and volumes.
In the requirement for recognition in the drawing and in the surrounding world, it is not worth putting so much pressure on checkered paper. Preparation for the OGE and the Unified State Examination is sacred, but not at all the main thing. There is also unlined paper. It would be necessary to require the construction of perpendicular and parallel lines with the help of tools (square and ruler) — children should work with their hands. Ahead is a systematic course of planimetry. The requirements about the coordinate line and the coordinate plane should not be attributed to the geometric material, this is a practical application of the studied rational numbers, they are moved up.
One last thing: the historical material looks very poor. I would expand this requirement by adding, in both years of study, the solution of ancient problems from L.F. Magnitsky’s «Arithmetic», Russian mathematical manuscripts, ancient written sources (Babylon, Egypt, China, India), problems of famous Russian and foreign authors: L.
N. Tolstoy, S. A. Rachinsky, A. P. Kiselev, Ya. I. Perelman, L. Euler, I. Newton, D. Poya and others. Studying mathematics in a cultural-historical context gives an understanding of mathematics as a part of human culture, makes mathematics interesting and attractive to study, gives a better connection between the studied material and life than problems about electric meters.
If the Standard is adopted in the proposed edition, then we can safely reduce the forecast for Russia’s position to the end of the sixth decade in the world educational ranking.
I note that the drafters of the Standard in Mathematics treated the task very negligently, as if they knew that they were preparing the document not for work, but for show — to be. They shrugged off many of the actionable proposals of April 2019, which were easy to accept without violating the fabric of a strangely conceived and poorly executed document. Most likely, they lacked professional traditions that do not arise from the performance of one-time assignments, but are born as a result of many years of collective work, discussions, disputes of specialists who have been compiling programs in mathematics for the whole country for many years in a row, controlling the implementation of their own developments, comparing the results of training in experimental textbooks — there is no one to do this work now.
The Educational Standard that we have in the project is not the fault, but rather the misfortune of its authors, driven into the Procrustean bed of an unsuitable idea, the result of their professional life outside the «forge of personnel» who make programs for the whole country. Decades of educational «reforms» have taken their toll. There is no our laboratory in its former quality. There is no one on duty to do the work that has now been assigned to all the teachers of the country.
There is no one to lead the education of the country out of the desert of educational «reforms». But there are plenty of people who want to make the Standard «on the knee» — which one they will order
Standards and programs are not ordered by competition to those who agree to lower wages, to those who are better known to the customer. It must be a long term job. Then teachers will not be engaged in work that is not typical for them every year, the overload, which has been talked about for many years, will decrease.
The collapse of the NII SiMO of the Academy of Medical Sciences of the USSR, the state’s neglect of pedagogical science have brought us to where we are. Should we now be surprised at the results obtained in education?
Photo: Shutterstock (vm2002)
You are in the «Blogs» section. The opinion of the author may not coincide with the position of the editors.
(sit down) To digest knowledge, one must absorb it with appetite. (A. Franz). In order to assimilate the material, one must always do it in a good mood. Ordinary and decimals We can add, subtract, multiply and divide with decimals. With ordinary — add fractions with the same, different denominators, multiply and divide by a natural number, multiply ordinary and mixed fractions. | Slide To digest knowledge, one must absorb it with appetite. (A. Franz). | |||||||||
2. Actualization of Knowledge — game «Loto» (work in pairs, front work) — 4 min | ||||||||||
(Loto’s instruction is necessary sheet, find the result on the corresponding card and attach it with the reverse side to the task. The result is a cipher. If you cannot complete a task, put a “?” sign on it. (Controls the execution of tasks) (Front work, together with the students fill in the scoreboard on the board, analyze the code) What did you do? Were you able to complete all the tasks? What is the difficulty? Why doesn’t it work? How is this task different from the previous ones? (Brings students to the formulation of a problematic question — how to divide an ordinary fraction into a fraction). Well, today our mental operations will be aimed at finding the answer to this problematic question, and this is what we will devote our lesson to. The topic of our lesson is… Course set, what is the purpose of the lesson? So, each one has outlined the field of activity for himself. Write down in your notebook the goal you have set for today. | (Listen carefully, ask questions, if any). (Perform tasks, the couple who filled out the Lotto card raise their hand after the majority completed the task — frontal check) HOW TO DIVIDE A COMMON FRACTION BY? (As a result of the work, they come to a problematic issue — how to divide an ordinary fraction into a fraction, since they failed to solve the last example, due to the fact that the task of a new unexplored topic) (with the teacher formulate the topic of the lesson, write it down in a notebook) Division of ordinary fractions! Several students state their goal for this lesson. Learn to divide ordinary fractions, derive the rule for dividing ordinary fractions and consolidate. | LOTTO Slide Add | Subtract How to find the area of a given rectangle? What happens to the width of a rectangle if the length and area are simultaneously reduced by 10? What number must be multiplied by to get x? What is the width of the rectangle? Which property of the equation did we use? Let’s solve this equation by finding the unknown factor. Analyze the expression and guess how you can divide a fraction by a fraction? And now let’s look at the textbook — whether we have formulated the rule for dividing ordinary fractions correctly. Speak the rule to each other as you understand it. After speaking in pairs, several students say the rule for everyone How can this rule be written using letters? Write in your notebook Well done!!! Here is our discovery! | Given a rectangle whose length is 4cm and area is 20cm 2 . To find the width of a rectangle, divide the area of the rectangle by the length. (Answers are being heard) The width of the rectangle will not change. Let the width be x cm, then to find the area you need to multiply the width by the length of the rectangle, i.e. 10/4 ( X=5cm Both sides of the equation can be multiplied or divided by the same non-zero number. X == 5, since the root is the same X == To divide ordinary fractions, you need to multiply the dividend by the reciprocal of the divisor. Open textbooks and read the rule. Make sure that the rule is formulated correctly According to the scheme: STUDENT-TEACHER-STUDENT the rule is formulated Say the rule in pairs 4 people tell the rule to the class. (recording) | Exploration slide 4cm 4cm cm ? S = 20 cm 2 How to find the width of a rectangle? What is the width of this rectangle? 20 cm 2 : 4cm = 5 cm DM 2 Slide “Research Result” Private ordinary fractions — this is a fraction whose numerator is equal to the work of the first fraction and the signalman of the second shot, and the denominator is the product of the denominator of the first fraction and the numerator of the second fraction. | |||||
4.1. Consolidation (independent work) — 4 min. | solve examples! (The rest decide on their own, then check the solutions by showing on the board) | Reinforcement tasks 1) 4) 2) 5) 3) 6) | ||||||||
4.2. Consolidation (frontal work) — 2 min | ||||||||||
And now, guys, I suggest you become real experts! Find and analyze the error in these tasks. (directs the activities of students to find errors, identify their nature) | (analyze, find errors and their nature) Arithmetic error. An improper fraction must be converted to a number. Fraction not reduced. | (after determining the error, it is distinguished by animation in the example on the board) Slide «Experts» 1) 2) 3) | ||||||||
5. | ||||||||||
(Cards with tasks are distributed, work time is 5 minutes, then mutual check). Who completed all 5 tasks correctly? Who completed 4 tasks correctly? (Analysis is being carried out: the number of those who completed “5”, “4”, the nature of the mistakes, the degree of assimilation of the material is revealed) them, then exchanged with a neighbor on the desk and perform a mutual check using the board). (Analysis is being carried out: the number of students who completed “5”, “4”, the nature of the mistakes) (raise their hands) Students hand over test papers to the teacher to determine the level of assimilation. | (answers are displayed at the stage of mutual verification) | |||||||||
6. Solution of practical problems (work in groups) — 16 min | ||||||||||
and now we’ll look, in which life situations are met, in which life situations are met. The teacher supervises the creation of groups Creates 2 groups of 4 people, | Searches for information from books. Perform the proposed task, select a speaker from the group to explain the solution to the problem. Ask questions if they arise. Perform tasks in groups. Present their solution to problems. | Slide Tasks and their correct solution are displayed alternately | ||||||||
7. Lesson reflection — 2 min | ||||||||||
What was the purpose of the lesson? Did you achieve the objectives of the lesson? Formulate a rule for dividing ordinary fractions. Has the problem posed at the beginning of the lesson been solved? What difficulties have arisen? Did our epigraph match the lesson? Questions? | (They answer the questions, analyzing their own activities in the lesson) | Slide: «Any well-solved mathematical problem is a mental delight.
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We need to find the width of the rectangle. 
STRICTIONAL WORK (independent work (independent work (independent work work, mutual check) — 5 min 
for dividing ordinary fractions. 
