T lesson type: ONZ (discovery of new knowledge - using the technology of the activity-based teaching method).
Basic goals:
- Deduce methods for dividing a fraction by a natural number;
- Develop the ability to divide a fraction by a natural number;
- Repeat and reinforce division of fractions;
- Train the ability to reduce fractions, analyze and solve problems.
Equipment demonstration material:
1. Tasks for updating knowledge:
Compare expressions:
Reference:
2. Trial (individual) task.
1. Perform division:
2. Perform division without performing the entire chain of calculations: .
Standards:
- When dividing a fraction by a natural number, you can multiply the denominator by that number, but leave the numerator the same.
- If the numerator is divisible by a natural number, then when dividing a fraction by this number, you can divide the numerator by the number and leave the denominator the same.
During the classes
I. Motivation (self-determination) for educational activities.
Purpose of the stage:
- Organize the updating of requirements for the student in terms of educational activities (“must”);
- Organize student activities to establish thematic frameworks (“I can”);
- Create conditions for the student to develop an internal need for inclusion in educational activities (“I want”).
Organization of the educational process at stage I.
Hello! I'm glad to see you all at the math lesson. I hope it's mutual.
Guys, what new knowledge did you acquire in the last lesson? (Divide fractions).
Right. What helps you do division of fractions? (Rule, properties).
Where do we need this knowledge? (In examples, equations, problems).
Well done! You did well on the assignments in the last lesson. Do you want to discover new knowledge yourself today? (Yes).
Then - let's go! And the motto of the lesson will be the statement “You can’t learn mathematics by watching your neighbor do it!”
II. Updating knowledge and fixing individual difficulties in a trial action.
Purpose of the stage:
- Organize the updating of learned methods of action sufficient to build new knowledge. Record these methods verbally (in speech) and symbolically (standard) and generalize them;
- Organize the actualization of mental operations and cognitive processes sufficient to construct new knowledge;
- Motivate for a trial action and its independent implementation and justification;
- Present an individual task for a trial action and analyze it in order to identify new educational content;
- Organize fixation of the educational goal and topic of the lesson;
- Organize the implementation of a trial action and fix the difficulty;
- Organize an analysis of the responses received and record individual difficulties in performing a trial action or justifying it.
Organization of the educational process at stage II.
Frontally, using tablets (individual boards).
1. Compare expressions:
(These expressions are equal)
What interesting things did you notice? (The numerator and denominator of the dividend, the numerator and denominator of the divisor in each expression increased by the same number of times. Thus, the dividends and divisors in the expressions are represented by fractions that are equal to each other).
Find the meaning of the expression and write it down on your tablet. (2)
How can I write this number as a fraction?
How did you perform the division action? (Children pronounce the rule, the teacher posts letter symbols on the board)
2. Calculate and record the results only:
3. Add up the results and write down the answer. (2)
What is the name of the number obtained in task 3? (Natural)
Do you think you can divide a fraction by a natural number? (Yes, we'll try)
Try this.
4. Individual (trial) task.
Perform division: (example a only)
What rule did you use to divide? (According to the rule of dividing fractions by fractions)
Now divide the fraction by a natural number in a simpler way, without performing the entire chain of calculations: (example b). I'll give you 3 seconds for this.
Who couldn't complete the task in 3 seconds?
Who did it? (There are no such)
Why? (We don't know the way)
What did you get? (Difficulty)
What do you think we will do in class? (Divide fractions by natural numbers)
That's right, open your notebooks and write down the topic of the lesson: “Dividing a fraction by a natural number.”
Why does this topic sound new when you already know how to divide fractions? (Need a new way)
Right. Today we will establish a technique that simplifies the division of a fraction by a natural number.
III. Identifying the location and cause of the problem.
Purpose of the stage:
- Organize the restoration of completed operations and record (verbal and symbolic) the place - step, operation - where the difficulty arose;
- Organize the correlation of students’ actions with the method (algorithm) used and fixation in external speech of the cause of the difficulty - that specific knowledge, skills or abilities that are lacking to solve the initial problem of this type.
Organization of the educational process at stage III.
What task did you have to complete? (Divide a fraction by a natural number without going through the entire chain of calculations)
What caused you difficulty? (We couldn’t solve it in a short time using a quick method)
What goal do we set for ourselves in the lesson? (Find a quick way to divide a fraction by a natural number)
What will help you? (Already known rule for dividing fractions)
IV. Building a project for getting out of a problem.
Purpose of the stage:
- Clarification of the project goal;
- Choice of method (clarification);
- Determination of means (algorithm);
- Building a plan to achieve the goal.
Organization of the educational process at stage IV.
Let's return to the test task. You said you divided according to the rule for dividing fractions? (Yes)
To do this, replace the natural number with a fraction? (Yes)
What step (or steps) do you think can be skipped?
(The solution chain is open on the board: 
Analyze and draw a conclusion. (Step 1)
If there is no answer, then we lead you through questions:
Where did the natural divisor go? (Into the denominator)
Has the numerator changed? (No)
So which step can you “omit”? (Step 1)
Action plan:
- Multiply the denominator of a fraction by a natural number.
- We do not change the numerator.
- We get a new fraction.
V. Implementation of the constructed project.
Purpose of the stage:
- Organize communicative interaction in order to implement the constructed project aimed at acquiring the missing knowledge;
- Organize the recording of the constructed method of action in speech and signs (using a standard);
- Organize the solution to the initial problem and record how to overcome the difficulty;
- Organize clarification of the general nature of new knowledge.
Organization of the educational process at stage V.
Now run the test case in a new way quickly.
Now you were able to complete the task quickly? (Yes)
Explain how you did this? (Children talk)
This means that we have gained new knowledge: the rule for dividing a fraction by a natural number.
Well done! Say it in pairs.
Then one student speaks to the class. We fix the rule-algorithm verbally and in the form of a standard on the board.
Now enter the letter designations and write down the formula for our rule.
The student writes on the board, saying the rule: when dividing a fraction by a natural number, you can multiply the denominator by this number, but leave the numerator the same.
(Everyone writes the formula in their notebooks).
Now analyze the chain of solving the test task again, paying special attention to the answer. What did you do? (The numerator of the fraction 15 was divided (reduced) by the number 3)
What is this number? (Natural, divisor)
So how else can you divide a fraction by a natural number? (Check: if the numerator of a fraction is divisible by this natural number, then you can divide the numerator by this number, write the result in the numerator of the new fraction, and leave the denominator the same)
Write this method down as a formula. (The student writes the rule on the board while pronouncing it. Everyone writes the formula in their notebooks.)
Let's return to the first method. You can use it if a:n? (Yes, this is the general way)
And when is it convenient to use the second method? (When the numerator of a fraction is divided by a natural number without a remainder)
VI. Primary consolidation with pronunciation in external speech.
Purpose of the stage:
- Organize children’s assimilation of a new method of action when solving standard problems with their pronunciation in external speech (frontally, in pairs or groups).
Organization of the educational process at stage VI.
Calculate in a new way:
- No. 363 (a; d) - performed at the board, pronouncing the rule.
- No. 363 (e; f) - in pairs with checking according to the sample.
VII. Independent work with self-test according to the standard.
Purpose of the stage:
- Organize students’ independent completion of tasks for a new way of action;
- Organize self-test based on comparison with the standard;
- Based on the results of independent work, organize a reflection on the assimilation of a new method of action.
Organization of the educational process at stage VII.
Calculate in a new way:
- No. 363 (b; c)
Students check against the standard and mark the correctness of execution. The causes of errors are analyzed and errors are corrected.
The teacher asks those students who made mistakes, what is the reason?
At this stage, it is important that each student independently checks their work.
VIII. Inclusion in the knowledge system and repetition.
Purpose of the stage:
- Organize the identification of the boundaries of application of new knowledge;
- Organize repetition of educational content necessary to ensure meaningful continuity.
Organization of the educational process at stage VIII.
Organization of the educational process at stage IX.
1. Dialogue:
Guys, what new knowledge have you discovered today? (Learned how to divide a fraction by a natural number in a simple way)
Formulate a general method. (They say)
In what way and in what cases can you use it? (They say)
What is the advantage of the new method?
Have we achieved our lesson goal? (Yes)
What knowledge did you use to achieve your goal? (They say)
Did everything work out for you?
What were the difficulties?
2. Homework: clause 3.2.4.; No. 365(l, n, o, p); No. 370.
3. Teacher: I’m glad that everyone was active today and managed to find a way out of the difficulty. And most importantly, they were not neighbors when opening a new one and establishing it. Thanks for the lesson, kids!
Ordinary fractional numbers first meet schoolchildren in the 5th grade and accompany them throughout their lives, since in everyday life it is often necessary to consider or use an object not as a whole, but in separate pieces. Start studying this topic - shares. Shares are equal parts, into which this or that object is divided. After all, it is not always possible to express, for example, the length or price of a product as a whole number; parts or fractions of some measure should be taken into account. Formed from the verb “to split” - to divide into parts, and having Arabic roots, the word “fraction” itself arose in the Russian language in the 8th century.
Fractional expressions have long been considered the most difficult branch of mathematics. In the 17th century, when first textbooks on mathematics appeared, they were called “broken numbers,” which was very difficult for people to understand.
The modern form of simple fractional remainders, the parts of which are separated by a horizontal line, was first promoted by Fibonacci - Leonardo of Pisa. His works are dated to 1202. But the purpose of this article is to simply and clearly explain to the reader how mixed fractions with different denominators are multiplied.
Multiplying fractions with different denominators
Initially it is worth determining types of fractions:
- correct;
- incorrect;
- mixed.
Next, you need to remember how fractional numbers with the same denominators are multiplied. The very rule of this process is not difficult to formulate independently: the result of multiplying simple fractions with identical denominators is a fractional expression, the numerator of which is the product of the numerators, and the denominator is the product of the denominators of these fractions. That is, in fact, the new denominator is the square of one of the initially existing ones.
When multiplying simple fractions with different denominators for two or more factors the rule does not change:
a/b * c/d = a*c / b*d.
The only difference is that the formed number under the fractional line will be a product of different numbers and, naturally, it cannot be called the square of one numerical expression.
It is worth considering the multiplication of fractions with different denominators using examples:
- 8/ 9 * 6/ 7 = 8*6 / 9*7 = 48/ 63 = 16/2 1 ;
- 4/ 6 * 3/ 7 = 2/ 3 * 3/7 <> 2*3 / 3*7 = 6/ 21 .
The examples use methods for reducing fractional expressions. You can only reduce numerator numbers with denominator numbers; adjacent factors above or below the fraction line cannot be reduced.
Along with simple fractions, there is the concept of mixed fractions. A mixed number consists of an integer and a fractional part, that is, it is the sum of these numbers:
1 4/ 11 =1 + 4/ 11.
How does multiplication work?
Several examples are provided for consideration.
2 1/ 2 * 7 3/ 5 = 2 + 1/ 2 * 7 + 3/ 5 = 2*7 + 2* 3/ 5 + 1/ 2 * 7 + 1/ 2 * 3/ 5 = 14 + 6/5 + 7/ 2 + 3/ 10 = 14 + 12/ 10 + 35/ 10 + 3/ 10 = 14 + 50/ 10 = 14 + 5=19.
The example uses multiplication of a number by ordinary fractional part, the rule for this action can be written as:
a* b/c = a*b /c.
In fact, such a product is the sum of identical fractional remainders, and the number of terms indicates this natural number. Special case:
4 * 12/ 15 = 12/ 15 + 12/ 15 + 12/ 15 + 12/ 15 = 48/ 15 = 3 1/ 5.
There is another solution to multiplying a number by a fractional remainder. You just need to divide the denominator by this number:
d* e/f = e/f: d.
This technique is useful to use when the denominator is divided by a natural number without a remainder or, as they say, by a whole number.
Convert mixed numbers to improper fractions and obtain the product in the previously described way:
1 2/ 3 * 4 1/ 5 = 5/ 3 * 21/ 5 = 5*21 / 3*5 =7.
This example involves a way of representing a mixed fraction as an improper fraction, and can also be represented as a general formula:
a bc = a*b+ c / c, where the denominator of the new fraction is formed by multiplying the whole part with the denominator and adding it with the numerator of the original fractional remainder, and the denominator remains the same.
This process also works in the opposite direction. To separate the whole part and the fractional remainder, you need to divide the numerator of an improper fraction by its denominator using a “corner”.
Multiplying improper fractions produced in a generally accepted way. When writing under a single fraction line, you need to reduce fractions as necessary in order to reduce numbers using this method and make it easier to calculate the result.
There are many helpers on the Internet to solve even complex mathematical problems in various variations of programs. A sufficient number of such services offer their assistance in calculating the multiplication of fractions with different numbers in the denominators - so-called online calculators for calculating fractions. They are able not only to multiply, but also to perform all other simple arithmetic operations with ordinary fractions and mixed numbers. It’s not difficult to work with; you fill in the appropriate fields on the website page, select the sign of the mathematical operation, and click “calculate.” The program calculates automatically.
The topic of arithmetic operations with fractions is relevant throughout the education of middle and high school students. In high school, they no longer consider the simplest species, but integer fractional expressions, but the knowledge of the rules for transformation and calculations obtained earlier is applied in its original form. Well-mastered basic knowledge gives complete confidence in successfully solving the most complex problems.
In conclusion, it makes sense to quote the words of Lev Nikolaevich Tolstoy, who wrote: “Man is a fraction. It is not in the power of a person to increase his numerator - his merits - but anyone can reduce his denominator - his opinion about himself, and with this decrease come closer to his perfection.
Sooner or later, all children at school begin to learn fractions: their addition, division, multiplication and all the possible operations that can be performed with fractions. In order to provide proper assistance to the child, parents themselves should not forget how to divide integers into fractions, otherwise you will not be able to help him in any way, but will only confuse him. If you need to remember this action, but you just can’t bring all the information in your head into a single rule, then this article will help you: you will learn to divide a number by a fraction and see clear examples.
How to divide a number into a fraction
Write your example down as a rough draft so you can make notes and erasures. Remember that the integer number is written between the cells, right at their intersection, and fractional numbers are written each in its own cell.
- In this method, you need to turn the fraction upside down, that is, write the denominator into the numerator, and the numerator into the denominator.
- The division sign must be changed to multiplication.
- Now all you have to do is perform the multiplication according to the rules you have already learned: the numerator is multiplied by an integer, but you do not touch the denominator.
Of course, as a result of this action you will end up with a very large number in the numerator. You cannot leave a fraction in this state - the teacher simply will not accept this answer. Reduce the fraction by dividing the numerator by the denominator. Write the resulting integer to the left of the fraction in the middle of the cells, and the remainder will be the new numerator. The denominator remains unchanged.
This algorithm is quite simple, even for a child. After completing it five or six times, the child will remember the procedure and will be able to apply it to any fractions.
How to divide a number by a decimal
There are other types of fractions - decimals. The division into them occurs according to a completely different algorithm. If you encounter such an example, then follow the instructions:
- First, convert both numbers to decimals. This is easy to do: your divisor is already represented as a fraction, and you separate the natural number being divided with a comma, getting a decimal fraction. That is, if the dividend was 5, you get the fraction 5.0. You need to separate a number by as many digits as there are after the decimal point and divisor.
- After this, you must make both decimal fractions natural numbers. It may seem a little confusing at first, but it is the fastest way to divide and will take you seconds after a few practice sessions. The fraction 5.0 will become the number 50, the fraction 6.23 will become 623.
- Do the division. If the numbers are large, or the division will occur with a remainder, do it in a column. This way you can clearly see all the actions of this example. You don't need to put a comma on purpose, as it will appear on its own during the long division process.
This type of division initially seems too confusing, since you need to turn the dividend and divisor into a fraction, and then back into natural numbers. But after a short practice, you will immediately begin to see those numbers that you simply need to divide by each other.
Remember that the ability to correctly divide fractions and whole numbers by them can come in handy many times in life, therefore, a child needs to know these rules and simple principles perfectly so that in higher grades they do not become a stumbling block due to which the child cannot solve more complex tasks.
Multiplying and dividing fractions.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
This operation is much nicer than addition-subtraction! Because it's easier. As a reminder, to multiply a fraction by a fraction, you need to multiply the numerators (this will be the numerator of the result) and the denominators (this will be the denominator). That is:
For example:
Everything is extremely simple. And please don't look for a common denominator! There is no need for him here...
To divide a fraction by a fraction, you need to reverse second(this is important!) fraction and multiply them, i.e.:
For example:
If you come across multiplication or division with integers and fractions, it’s okay. As with addition, we make a fraction from a whole number with one in the denominator - and go ahead! For example:
In high school, you often have to deal with three-story (or even four-story!) fractions. For example:
How can I make this fraction look decent? Yes, very simple! Use two-point division:
But don't forget about the order of division! Unlike multiplication, this is very important here! Of course, we will not confuse 4:2 or 2:4. But it’s easy to make a mistake in a three-story fraction. Please note for example:
In the first case (expression on the left):
In the second (expression on the right):
Do you feel the difference? 4 and 1/9!
What determines the order of division? Either with brackets, or (as here) with the length of horizontal lines. Develop your eye. And if there are no brackets or dashes, like:
then divide and multiply in order, from left to right!
And another very simple and important technique. In actions with degrees, it will be so useful to you! Let's divide one by any fraction, for example, by 13/15:
The shot has turned over! And this always happens. When dividing 1 by any fraction, the result is the same fraction, only upside down.
That's it for operations with fractions. The thing is quite simple, but it gives more than enough errors. Take practical advice into account, and there will be fewer of them (mistakes)!
Practical tips:
1. The most important thing when working with fractional expressions is accuracy and attentiveness! These are not general words, not good wishes! This is a dire necessity! Do all calculations on the Unified State Exam as a full-fledged task, focused and clear. It’s better to write two extra lines in your draft than to mess up when doing mental calculations.
2. In examples with different types of fractions, we move on to ordinary fractions.
3. We reduce all fractions until they stop.
4. We reduce multi-level fractional expressions to ordinary ones using division through two points (we follow the order of division!).
5. Divide a unit by a fraction in your head, simply turning the fraction over.
Here are the tasks that you must definitely complete. Answers are given after all tasks. Use the materials on this topic and practical tips. Estimate how many examples you were able to solve correctly. The first time! Without a calculator! And draw the right conclusions...
Remember - the correct answer is received from the second (especially the third) time does not count! Such is the harsh life.
So, solve in exam mode ! This is already preparation for the Unified State Exam, by the way. We solve the example, check it, solve the next one. We decided everything - checked again from first to last. But only Then look at the answers.
Calculate:
Have you decided?
We are looking for answers that match yours. I deliberately wrote them down in disarray, away from temptation, so to speak... Here they are, the answers, written with semicolons.
0; 17/22; 3/4; 2/5; 1; 25.
Now we draw conclusions. If everything worked out, I’m happy for you! Basic calculations with fractions are not your problem! You can do more serious things. If not...
So you have one of two problems. Or both at once.) Lack of knowledge and (or) inattention. But this solvable Problems.
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Last time we learned how to add and subtract fractions (see lesson “Adding and subtracting fractions”). The most difficult part of those actions was bringing fractions to a common denominator.
Now it's time to deal with multiplication and division. The good news is that these operations are even simpler than addition and subtraction. First, let's consider the simplest case, when there are two positive fractions without a separated integer part.
To multiply two fractions, you must multiply their numerators and denominators separately. The first number will be the numerator of the new fraction, and the second will be the denominator.
To divide two fractions, you need to multiply the first fraction by the “inverted” second fraction.
Designation:
From the definition it follows that dividing fractions reduces to multiplication. To “flip” a fraction, just swap the numerator and denominator. Therefore, throughout the lesson we will mainly consider multiplication.
As a result of multiplication, a reducible fraction can arise (and often does arise) - it, of course, must be reduced. If after all the reductions the fraction turns out to be incorrect, the whole part should be highlighted. But what definitely won't happen with multiplication is reduction to a common denominator: no criss-cross methods, greatest factors and least common multiples.
By definition we have:
Multiplying fractions with whole parts and negative fractions
If fractions contain an integer part, they must be converted to improper ones - and only then multiplied according to the schemes outlined above.
If there is a minus in the numerator of a fraction, in the denominator or in front of it, it can be taken out of the multiplication or removed altogether according to the following rules:
- Plus by minus gives minus;
- Two negatives make an affirmative.
Until now, these rules have only been encountered when adding and subtracting negative fractions, when it was necessary to get rid of the whole part. For a work, they can be generalized in order to “burn” several disadvantages at once:
- We cross out the negatives in pairs until they completely disappear. In extreme cases, one minus can survive - the one for which there was no mate;
- If there are no minuses left, the operation is completed - you can start multiplying. If the last minus is not crossed out because there was no pair for it, we take it outside the limits of multiplication. The result is a negative fraction.
Task. Find the meaning of the expression:
We convert all fractions to improper ones, and then take the minuses out of the multiplication. We multiply what is left according to the usual rules. We get:
Let me remind you once again that the minus that appears in front of a fraction with a highlighted whole part refers specifically to the entire fraction, and not just to its whole part (this applies to the last two examples).
Also pay attention to negative numbers: when multiplying, they are enclosed in parentheses. This is done in order to separate the minuses from the multiplication signs and make the entire notation more accurate.
Reducing fractions on the fly
Multiplication is a very labor-intensive operation. The numbers here turn out to be quite large, and to simplify the problem, you can try to reduce the fraction further before multiplication. Indeed, in essence, the numerators and denominators of fractions are ordinary factors, and, therefore, they can be reduced using the basic property of a fraction. Take a look at the examples:
Task. Find the meaning of the expression:
By definition we have:
In all examples, the numbers that have been reduced and what remains of them are marked in red.
Please note: in the first case, the multipliers were reduced completely. In their place there remain units that, generally speaking, need not be written. In the second example, it was not possible to achieve a complete reduction, but the total amount of calculations still decreased.
However, never use this technique when adding and subtracting fractions! Yes, sometimes there are similar numbers that you just want to reduce. Here, look:
You can't do that!
The error occurs because when adding, the numerator of a fraction produces a sum, not a product of numbers. Consequently, it is impossible to apply the basic property of a fraction, since this property deals specifically with the multiplication of numbers.
There are simply no other reasons for reducing fractions, so the correct solution to the previous problem looks like this:
Correct solution:
As you can see, the correct answer turned out to be not so beautiful. In general, be careful.