Question 11.Multiplication multi-digit numbers. Theoretical material discussed in this topic.
This topic introduces the following computational techniques using algorithms:
Multiplying by single digit number.
Multiplication by place numbers.
Multiplication by two-digit and three-digit numbers.
At each of these stages, the techniques of multiplication and then division are studied first. Other approaches to studying this topic are possible.
At the preparatory stage, repetition, generalization and systematization of the studied material are carried out. At the familiarization stage, we first consider oral computational techniques for multiplying a digit number by a single-digit number of the form: 60003; 4002; 4 hundred. 2= 8 cells = 800
The theoretical basis is the specific meaning of multiplication.
Students are then introduced to the need to introduce written multiplication techniques. For this purpose, the technique of multiplying by a single-digit number with transition through ten or hundred is introduced.
Based on the multiplication algorithm from the mathematics course, a multiplication algorithm is compiled and introduced in elementary school. However, we begin written multiplication with units of the lowest category, and oral multiplication with units of the highest category.
Students’ reasoning could be as follows: “I write down the factors in a column, one below the other. Let's draw a line and put a multiplication sign on the left. I write the second factor under the units.
I start multiplying with units of the lowest rank: I multiply 7 units by 2 = 14 units, that’s 1 ten 4 units, I write 4 units under the units, and I remember 1 ten so that I can then add them to the tens.”
The explanation algorithm can be written in the following sequence:
I multiply the units:
Multiplying tens:
I multiply hundreds:
I read the answer:
First a detailed explanation is given, then a brief one. When the algorithm is mastered, the name of the units of each digit can be omitted.
Children need to be taught:
write down factors correctly;
introduce the multiplication sign;
when multiplying, name each digit;
talk through intermediate results
Complication of techniques occurs in the following order:
the number of bits of the first multiplier increases;
3253; 62855, etc.;
The first factor contains zeros in the middle or at the end; knowledge of the bit composition of the number is necessary;
7056; 60078; 7060005 ….;
Various combinations of these cases.
For example:72500
Explanation: we sign the second multiplier under the first digit of the first multiplier, other than zero. 725 cells 8=4350 cells. Or 435000.
Perform multiplication without paying attention to the zeros written at the end of 1 multiplier and add as many zeros to the resulting product as there are at the end of the first multiplier. The solution moves from a detailed explanation to a brief one, when the name of the bit units and the transformations performed is omitted.
Then we introduce techniques for multiplying a single-digit number by multi-digit ones:
86734 – theoretical basis – commutative property of multiplication.
Multiplication by place numbers.
At the preparatory stage, the following theoretical material is considered:
multiplication by a single digit number;
multiplication and addition tables;
multiplication by 10, 100, 1000.
replacing place numbers with the product of a single-digit number and 10, 100, 1000 (600=6100)
5. property of multiplying a number by a product (combinative law of multiplication):
8 (42)=88=64
8 (42)=(84) 2=64
The combinatory law of multiplication reads like this: in order to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third numbers. The wording of the law may be different: two or more factors in a product can be replaced by their product, but this will not change the value of the arithmetic expression.
(ab) c = a (bc)
The property is the theoretical basis for introducing techniques.
At the familiarization stage, oral cases of the form are introduced first:
1630=16 (310)=(163) 10=480
70060=700 (610)=(7006) 10=42000
Students' reasoning: to multiply 7 hundreds by 60, you need to multiply 7 hundreds by 6, and then multiply the resulting number by 10, there will be 42 hundreds or 42000 units.
Theoretical basis – combinational law multiplication or multiplication of a number by a product.
Then writing techniques are introduced.
For example:
We write the second factor so that the zeros are to the right of the ones of the first factor. We multiply the number 375 by 4 and multiply the resulting result by 10. In the product we write down as many zeros as there were in the second factor.
Next we consider the case of multiplication when both factors end in zeros.
Explanation: multiply 72 hundreds by 6, we get 432 hundreds or 43200 and multiply by 10.
Questions like:
How many zeros are in 1 factor?
How many zeros are in the 2 factor?
How many zeros are there in the product?
Conclusion: to multiply 2 numbers with zeros at the end, you need to multiply them, ignoring the zeros, and then add as many zeros to the resulting product as are written at the end of both factors together.
The theoretical basis is the property of multiplying a number by a product.
Multiplying by two-digit and three-digit numbers.
First, on this basis, a case of the form is introduced:
3013=30 (10+3)=300+90=390
Written techniques for multiplying by double figures is introduced using the example of the form: 7836
The line entry is shown:
7836=78 (30+6)=7830+786=2808
The conclusion is that it is difficult to calculate the result verbally. The products 7830 and 786 are written in columns, the calculation results are called incomplete products; adding them up, we get the product of the numbers 78 and 36.
Then the two columns are combined into one. Another option for introducing multiplication into a column is also possible.
Compare 2 examples.
How to continue the multiplication in the second example?
A multiplication algorithm is introduced.
1. Multiply by units (78 multiplied by 6), we get 1 incomplete product.
2. Multiply by tens (78 multiplied by 30), we get 2 incomplete products.
3. Reading the answer. Adding up the incomplete products, we get the answer.
You don’t have to write a zero at the end of the second incomplete product, since adding the number of units of the first incomplete product with zero we get the number of units of the first incomplete product. When multiplying by the number of tens, we begin to sign the second incomplete product under the tens of the first incomplete product.
The theoretical basis is the properties of multiplying a number by a sum.
Multiplication by a three-digit number is introduced on the basis of multiplication by a two-digit number. You can use this technique: to the numbers 78 and 36 we add a figure indicating the number of hundreds, for example 4 and 5, we get the example 478536.
How to get the third incomplete product?
We multiply 483 by 3, by the number of hundreds and the result by multiplying by 100, sign the 3rd incomplete product under hundreds.
Then special cases of multiplication are included: multiplication of numbers that have zeros at the end or in the middle. The multiplication algorithm remains the same, although there are some peculiarities.
For example:
To multiply 560 by 74, you need 56 decimals. multiply by 74, we get tens, we replace them with ones, adding zero to the right.
In this case, we immediately move from multiplying by units to multiplying by hundreds. We multiply 748 by 300, we get 2244 hundreds or 224400.
The total will not contain units of any category, in in this example there are no tens and we move from multiplying by units to multiplying by hundreds; We sign the second incomplete work under hundreds.
That. All cases of written multiplication are considered sequentially, according to the degree of complexity.
Multiplying by two and three digit numbers.
Action learning is divided into two stages:
multiplication and division by two-digit numbers (the algorithm is mastered, all concepts are formed);
multiplication and division by three-digit numbers (transfer of acquired concepts and skills to more complex material).
An analysis of the multiplication performance shows that the basic principles are the same as when multiplying by a single-digit number: the bitwise nature of the multiplication and the use of a multiplication table in each digit.
However, there are some peculiarities.
For example: 70 4=280 700 4=2800
Find the result using known methods. The similarities and differences of these equalities and the difference in digit units are determined. Then they examine the source of the noticed pattern, realize the main way of performing the action - representing the multiplier not as a product of any numbers, but as the product of a single-digit number by one with zeros. This implies the need to know about the pattern associated with multiplying any number by a digit unit.
Let us highlight the main stages in studying multiplication by a two-digit number: the associative law of multiplication; multiplication by one with zeros based on the use of the associative law of multiplication; multiplication by round tens based on the use of the same law, the distributive law of multiplication relative to addition; multiplying by a two-digit number with all significant figures. It is necessary to establish logical connections between individual stages and between new material and what has been studied.
The algorithm for multiplying by a single-digit number is the basis for mastering the algorithm for multiplying by two-digit and three-digit numbers.
Multiplication by two-digit and three-digit numbers is considered based on properties of multiplying a number by a sum.
A good place to start is by verbally multiplying a two-digit number by a two-digit number. To familiarize yourself with the technique, easier cases are selected, for example:
16 12 = 16 (10 + 2) = 16 10 + 16 2 = 160 + 32 = 192
Then you need to offer a more difficult case, for example:
87 64 = 87 (60 + 4) = 87 60 + 87 4
Children become convinced that it is difficult to solve such an example orally. The teacher suggests doing the calculations in writing:
To multiply 87 by 64, you must first multiply 87 by 4, then multiply 87 by 60 and add the resulting numbers.
Multiply 87 by 4: four times seven - 28; 8 we write down, 2 we remember;
four times eight is 32, yes 2, we get 34, write 34.
We got 348.
Now multiply 87 by 60.
To do this, you need to multiply 87 by 6 and multiply the resulting number by 10, i.e. assign a zero to it on the right, write zero in place of ones.
7 multiplied by 6 is 42, 2 is written in the place of tens, 4 is remembered.
8 multiplied by 6 is 48, yes 4 is 52, we write 52.
We get 5220.
Let's add the numbers 348 and 5220.
Product 5568.
Here 87 and 64 - multipliers,
348 - first incomplete work,
5220 - second incomplete product,
5568 - final result or work numbers 87 and 64.
It is useful that when explaining a computational technique, students first indicate all the basic operations in a certain sequence. This helps to understand the place and meaning of each operation. A detailed explanation is given only to those operations that are new to students; familiar operations are performed independently, while brief explanations are given.
After solving several examples (134 46, 268 37, 451 32), the teacher draws students’ attention to the peculiarity of the second incomplete product: it always ends in zero, therefore, when adding incomplete products, there will always be as many units as there are in the first incomplete product, which means zero You don’t have to write, but start writing down the second incomplete work under tens.
An explanation of multiplication by a three-digit number is also provided.
At the first stages of studying multiplication by a two-digit and especially a three-digit number, along with solving examples, it is useful to include exercises on drawing up a solution plan, which is written in the form of an expression, but the action itself is not performed, for example:
286 374 = 286 4 + 286 70 +286 300
It is also advisable to offer inverse exercises, when according to the solution plan (84 6 + 84 30) you need to create an example (84 36), but in general you can write the following equality: 84 6 + 84 30 = 84 36.
Such exercises focus students' attention on a computational technique and the property that underlies it.
You should pay attention to another group of exercises, the purpose of which is to prevent the mixing of similar computational techniques when multiplying by two-digit numbers. Let's point out some of them.
1) Students are asked to tell a method for solving a pair of examples, compiled in such a way that, against the background of similar ones, the difference in techniques appears more clearly. How to multiply 138 by 14 in writing? (You need to multiply 138 by 4, multiply 138 by 10, add the results obtained: 138 14 = 138 4+ 138 10.)
How to multiply 138 by 40? (You need to multiply 138 by 4 and multiply the resulting result by 10; 138 40 = 138 4 10.)
The opposite exercise to the first. If 376 was multiplied by 4, 376 was multiplied by 10 and the resulting numbers were added, then what number was 376 multiplied by? (376 14) Both the question and the answer can be written like this: 376-4+376-10=376-14. If we multiply 376 by 4 and multiply the resulting result by 10, then what number did we multiply 376 by? (376 40.) Entry: 376 4 10 = 376 40.
Oral and written solution of pairs of examples in one action: 25 12 and 25 20; 194 16 and 194 60, as well as a written solution of pairs of examples in several actions and comparing them. What is greater and by how much: the product 346 7 10 or the sum of the products 346 7 + 346 10?
Solving Examples in different ways, For example:
25 16 = 25 (4 4)=25 4 4
25 16 = 25 (2 8) =25 2 8
25 16 = 25 (10 + 6)
25 16 = 16 25=16 (5 5) = 16 5 5, etc.
5) Solving examples in the most convenient way:
32 2 50 = 32 100 73 6 3 + 73 2 = 73 20
54 80 + 54 20 = 54 100 83 16 + 17 16 = 100 16
The teacher writes on the board only the left side of the given equations, and the students write down the right side.
After the general cases of multiplication by two-digit and three-digit numbers are considered, special cases of multiplication are included: multiplication of numbers in which there are zeros at the end or in the middle of the factors. When studying these cases of multiplication, students are dealing with techniques they are already familiar with, only in new conditions, so they need to be given as much independence as possible.
After multiplying by a two-digit and three-digit number of natural numbers, the multiplication of quantities expressed in units of two names is introduced. In this case, one method is used: a value expressed in units of two items is expressed in units of one item, this value is multiplied by a number and the result is expressed in units of two items.
When studying all cases of multiplication, it is first necessary to achieve an understanding of the computational technique, and then work on developing computational skills. For developing skills, it is of great importance, firstly, timely reduction of explanations for solving examples and corresponding notes, and secondly, a carefully thought-out system of training exercises.
To prevent mistakes, children must be taught to check their solutions. Written multiplication is checked by estimating the result. For this purpose, find the product of numbers of the highest rank of factors and compare it with the result obtained. So, checking the solution to the first of the given examples, we find the product 100-200 = 20,000, but as a result we got only 3288, which means the example was solved incorrectly. You can also check the solutions to examples of multiplication by division.
In connection with the study of multiplying multi-digit numbers, it is necessary to repeat the rules for the order of actions; This is facilitated by exercises: “Write down expressions and find their meanings - to the number 803 add the product of the numbers 254 and 30; increase the product of the numbers 425 and 168 by their difference, etc.”
Methodology for studying a written division algorithm (stage 1).
As already noted, it is advisable to study the division of multi-digit numbers in parallel with multiplication, highlighting the following stages: after multiplication by a single-digit number, division by a single-digit number is introduced, after multiplication by digit numbers, division by digit numbers is given, immediately after studying multiplication division into two-digit and three-digit numbers is studied.
Let's consider each of these stages separately.
- consolidating the ability to multiply by two-digit and three-digit numbers, continuing work on developing computational skills, solving problems involving movement, expressions on the order of actions;
- strengthening the ability to calculate the area and perimeter of a square.
- development of attention, memory, logical thinking, mathematical speech of students.
- instilling interest in the subject, nurturing accuracy, communication skills, and mutual assistance.
Equipment: textbook "Mathematics 4th grade", multimedia projector, PC, screen, signal cards, cards for individual work, tests.
PROGRESS OF THE LESSON
1. Organizational moment.
Very strict science
A very exact science
Interesting science -
This::::::.
Mathematics loves attentive, organized people. Now let’s check who is already ready to do a good job.
2. Updating knowledge.
On your desks are the Pythagorean tables. I will call tabular cases multiplication, and you will paint the square with the correct answer.
7x4, 6x6, 5x8, 5x4, 7x8, 6x5, 5x5, 6x7, 6x8, 7x7, 6x4.
Check: If you have correctly highlighted all the answers, you should receive an “A” on the table for completing this task. Use flashcards to show the results of your work. Slide 1
Knowledge of the multiplication table will be useful to us in today's lesson.
Sit correctly, check if your notebook is in the correct position. Write it down number, great job.
A minute of penmanship.
I will continue the lesson with a riddle.
Listen up, guys.
I thought of a number -
It is ambiguous.
There are exactly as many dozens in it,
How many girls do we have?
Comes to class every day.
There are so many units, children,
How many continents are there in the world?
Well, hundreds of them, including so many,
How many rays are there in the corner?
What is this number?
You name it! Slide 2
This number is 256
What can you say about this number?
(three-digit, even, number of the 1st class. It contains 6 units of the first digit, 5 units of the second digit, 2 units of the third digit, this number has 25 tens, “neighbors” of the number 255 and 257. It can be replaced by the sum of the digit terms).
Write this number down the entire line. (Make sure the correct fit)
Solution of a geometric problem. Slide 3
Create a problem based on the drawing.
The perimeter of the square is 256 cm. Find its area.
Can we immediately answer the question of the task? Why?
Knowing the perimeter of a square, what can we know?
(1 student solves at the board. Check with detailed explanation)
Guys, raise your hands, who has cards for individual work on their desks? You will also solve a geometric problem, but it will be of increased complexity.
256:4= 64 (cm) - side of the square
64x64 = 4096 (cm sq.) - square area (multiplying by column)
Check: using signal cards.
3. Work on the topic of the lesson.
When finding the area of a square, we multiplied by a two-digit number.
Today in the lesson we are strengthening the ability to multiply multi-digit numbers by two-digit and three-digit numbers.
Let's remember the algorithm for multiplying by a two-digit number. Slide 4
1 student solves at the board with a detailed explanation, the children write it down in a notebook.
Now let's repeat the algorithm for multiplying by a three-digit number
Slide 5
986x134 (1 student solves at the board with a detailed explanation).
Students often make mistakes when multiplying when the factors contain 0. Find where the mistake was made? Slide 6
We work according to the textbook: Page. 44 No. 14 (2nd column)
Solve 2 examples at the board with detailed explanations.
4. Independent work of students.
Now you will work independently. During independent work You will demonstrate your skills in multiplying by two-digit and three-digit numbers, adding and subtracting multi-digit numbers, and the procedure for performing arithmetic operations.
Here are mathematical expressions
Let's remember the order in which actions are performed in expressions with and without parentheses.
Solving expressions for the order of actions.
25x (364+242)= 15150 Slide 7
702x69+702x18= 71074
(78213-75209)x207-5x308= 620288
Choose the expression that suits you within my means.
Check: Raise your hands who chose the first expression. Check if you solved your expression correctly. Raise your hands who chose the second expression to solve. Let's check.
Checking the third expression. (Show with cue cards)
5. Fizminutka(dance with musical accompaniment).
6. Solution of the movement problem.
Page 45 No. 22. Read the problem yourself.
What type of task is this? (Movement in opposite directions).
What quantities are known in the problem?
What do you need to find?
What do you need to know to find speed?
How to find unknown time given a given speed and distance?
How to find an unknown distance knowing time and speed?
Choose a drawing that suits the task. Slide 8
Solving the problem at the board with a detailed explanation.
Students use signal cards during testing.
- Who solved the problem in a different way?
Write the solution to the problem on the board. Guys, do you agree with this solution to the problem?
Which solution is more rational? Why?
Educational moment of the lesson - following the rules traffic when riding a bicycle. (Every tenth road accident in the country involves children. Every year Russia loses 1.5 thousand young citizens in accidents. Such cases can be avoided if you know and follow the rules of safe cycling).
7. Exercise for the eyes.
Rest our eyes,
Take a nap under your eyelashes.
Now look into the distance
And then to the desk.
Look left, right,
Up and down.
Now forward .
Let's continue our lesson.
8. Test on the topic studied.
We continue to prepare for the final monitoring.
There are tests on your desk. Sign them. Focus. Start completing the test tasks.
1. What is the value of a number containing 149 units of class I and 37 units of class II?
4. Without performing calculations, determine which product is greater and by how much:
- 45x1254 or 45x1253.
- 45x1254 is 45 more
- 45x1254 is 44 more
- 45x1254 is 1254 more
5. At what speed must a car travel to travel 560 km in 7 hours?
| 1) 60 km/h | 3) 80 km/h |
| 2) 90 km/h | 4) 80 km |
Peer review (swap tests)
KEY. Slide 9
Peer review. Show the results using flashcards.
9. Homework: page 45 No. 21(if desired, create an inverse problem)
Page 44 No. 14 (3rd column) Slide 10
10. Generalization.
That's the end of the lesson.
Let us now summarize.
- What did you do today? (children's answers)
- What did you find a little difficult? (children's answers)
Did you like the lesson?
May the lesson serve you well!
Thanks for your work in class.
