In this article we will look at multiplying mixed numbers. First, we will outline the rule for multiplying mixed numbers and consider the application of this rule when solving examples. Next we'll talk about multiplying a mixed number and a natural number. Finally, we will learn how to multiply a mixed number and a common fraction.
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Multiplying mixed numbers.
Multiplying mixed numbers can be reduced to multiplying ordinary fractions. To do this, it is enough to convert mixed numbers to improper fractions.
Let's write it down mixed number multiplication rule:
- First, the mixed numbers being multiplied must be replaced by improper fractions;
- Secondly, you need to use the rule for multiplying fractions by fractions.
Let's look at examples of applying this rule when multiplying a mixed number by a mixed number.
Example.
Perform multiplication of mixed numbers and .
Solution.
First, let's represent the mixed numbers being multiplied as improper fractions: 
And 
. Now we can replace the multiplication of mixed numbers with the multiplication of ordinary fractions: 
. Applying the rule for multiplying fractions, we get 
. The resulting fraction is irreducible (see reducible and irreducible fractions), but it is improper (see proper and improper fractions), therefore, to obtain the final answer, it remains to isolate the whole part from the improper fraction: .
Let's write the entire solution in one line: .
Answer:
.
To strengthen the skills of multiplying mixed numbers, consider solving another example.
Example.
Do the multiplication.
Solution.
Funny numbers and are equal to the fractions 13/5 and 10/9, respectively. Then 
. At this stage, it’s time to remember about reducing a fraction: replace all the numbers in the fraction with their decompositions into prime factors, and perform a reduction of identical factors.
Answer:
Multiplying a mixed number and a natural number
After replacing a mixed number with an improper fraction, multiplying a mixed number and a natural number leads to the multiplication of an ordinary fraction and a natural number.
Example.
Multiply a mixed number and the natural number 45.
Solution.
A mixed number is equal to a fraction, then 
. Let's replace the numbers in the resulting fraction with their decompositions into prime factors, perform a reduction, and then select the whole part: .
Answer:
Multiplication of a mixed number and a natural number is sometimes conveniently carried out using the distributive property of multiplication relative to addition. In this case, the product of a mixed number and a natural number is equal to the sum of the products of the integer part by the given natural number and the fractional part by the given natural number, that is, 
.
Example.
Calculate the product.
Lesson topic: "Multiplication and division of mixed fractions"Goal: to develop in students the ability and skills to apply the rules of multiplication and division of mixed fractions;
development of students’ analytical thinking, formation of students’ ability to highlight the main thing and generalize.
Objectives: repeat the rule for multiplying and dividing ordinary fractions.
Test your ability to apply the rules of multiplication and division of ordinary fractions,
The rule for multiplying a fraction by a natural number and vice versa. Test your ability to convert improper fractions to mixed numbers and vice versa.
Derive a new rule and algorithm for multiplying and dividing mixed numbers.
Practice the new rule by completing tasks.
Subject results: algorithm for multiplying and dividing mixed fractions (memo)
Meta-subject and personal results :
Regulatory UUD: goal setting; plan, getting results
Cognitive UUD: general educational, logical, problem formulation and solution
Communicative UUD: work in pairs
Equipment: mathematics textbook, grade 6
Handout.
Projector.
During the classes:
I. Problem situation and updating of knowledge
1. Survey of children on repetition of the studied material on the topic of multiplication and division of fractions (algorithm for implementation, rule for multiplying a fraction by a natural number).
2. Illustration of examples on the projector. Types of ordinary fractions. How to get a mixed fraction from an improper fraction and vice versa.
3. At the end of the survey, independent work including examples of multiplying and dividing ordinary fractions and containing two examples of multiplying and dividing mixed fractions, where children encounter a problem. The correct answers are displayed on the projector for checking with students.
4. Discussion of the problem. Bring to the topic of the lesson.
II. Collaborative discovery of knowledge.
1/Discussion in pairs is proposed to voice a version of the solution to the problem that has arisen. Write the versions on the school board. How do you know which version is correct?
2/Invite students to refer to the textbook on the relevant topic.
3/ Do some reading, find the paragraph you need and study it to create an algorithm for multiplying and dividing mixed fractions. Control over task completion.
4/Listen to the versions and create a general algorithm from the main one. Display it on a projector and distribute it to students as a reminder.
III.Independent application of knowledge
1/Return to the problem with solving examples from independent work and using the resulting algorithm to solve them. Check in pairs. Display the results on the projector for verification.
2/ Give a task from the textbook. Execution control.
IV. Lesson summary
Start with the problem that arose at the beginning of the lesson, talk about ways to solve it and the result obtained.
Assessing student work.
Homework assignment.
) and denominator by denominator (we get the denominator of the product).
Formula for multiplying fractions:
For example:
Before you begin multiplying numerators and denominators, you need to check whether the fraction can be reduced. If you can reduce the fraction, it will be easier for you to make further calculations.
Dividing a common fraction by a fraction.
Dividing fractions involving natural numbers.
It's not as scary as it seems. As in the case of addition, we convert the integer into a fraction with one in the denominator. For example:
Multiplying mixed fractions.
Rules for multiplying fractions (mixed):
- convert mixed fractions to improper fractions;
- multiplying the numerators and denominators of fractions;
- reduce the fraction;
- If you get an improper fraction, then we convert the improper fraction into a mixed fraction.
Note! To multiply a mixed fraction by another mixed fraction, you first need to convert them to the form of improper fractions, and then multiply according to the rule for multiplying ordinary fractions.
The second way to multiply a fraction by a natural number.
It may be more convenient to use the second method of multiplying a common fraction by a number.
Note! To multiply a fraction by a natural number, you must divide the denominator of the fraction by this number, and leave the numerator unchanged.
From the example given above, it is clear that this option is more convenient to use when the denominator of a fraction is divided without a remainder by a natural number.
Multistory fractions.
In high school, three-story (or more) fractions are often encountered. Example:
To bring such a fraction to its usual form, use division through 2 points:
Note! When dividing fractions, the order of division is very important. Be careful, it's easy to get confused here.
Note, For example:
When dividing one by any fraction, the result will be the same fraction, only inverted:
Practical tips for multiplying and dividing fractions:
1. The most important thing when working with fractional expressions is accuracy and attentiveness. Do all calculations carefully and accurately, concentratedly and clearly. It's better to write a few extra lines in your draft than to get lost in mental calculations.
2. In tasks with different types of fractions, go to the type of ordinary fractions.
3. We reduce all fractions until it is no longer possible to reduce.
4. We transform multi-level fractional expressions into ordinary ones using division through 2 points.
5. Divide a unit by a fraction in your head, simply turning the fraction over.
Then we follow the rule: we multiply the first fraction by the fraction inverse to the second (that is, by an inverted fraction in which the numerator and denominator change places). When multiplying fractions, we multiply the numerator by the numerator, and the denominator by the denominator.
Let's look at examples of dividing mixed numbers.
We begin dividing mixed numbers by converting them into improper fractions. Then we divide the resulting fractions. To do this, multiply the first fraction by the inverted second. 20 and 25 by 5, 3 and 9 by 3. We got the wrong fraction, so we need to.
Convert mixed numbers to improper fractions. Next, according to the rule for dividing fractions, we leave the first number and multiply it by the reciprocal of the second. We reduce 15 and 25 by 5, 8 and 16 by 2. From the resulting improper fraction we select the whole part.
Replace mixed numbers with improper fractions and divide them. To do this, we rewrite the first fraction unchanged and multiply it by the inverted second. We reduce 18 and 36 by 18, 35 and 7 by 7. The result is an improper fraction. We select a whole part from it.