Combinative property of multiplication 5. Combinative property of multiplication lesson notes

Matching property multiplication

Goals: introduce students to the associative property of multiplication; teach to use the associative property of multiplication when analyzing numerical expressions; repeat the properties of addition and the commutative property of multiplication; improve computing skills; develop the ability to analyze and reason.

Subject results:

get acquainted with the associative property of multiplication, form ideas about the possibility of using the studied property to rationalize calculations.

Meta-subject results:

Regulatory: plan your action in accordance with the task, accept and save learning task.

Cognitive: use sign-symbolic means, models and diagrams to solve problems, focus on a variety of ways to solve problems; establish analogies.

Communication: construct speech statements in oral and written form, form own opinion, ask and answer questions, proving the correctness of your opinion.

Personal: develop the ability to self-esteem, promote success in mastering the material.

Lesson type: learning new material.

Equipment: task cards, visual material (tables), presentation.

PROGRESS OF THE LESSON

I . Organizational moment (emotional mood)

The long-awaited call is given

The lesson begins.

Did you all have time to rest?

And now - go ahead, get to work!

Guys, let's wish each other to be attentive, collected, and diligent in class. Let's greet each other with smiles and start the lesson.

II. Update background knowledge+ Goal setting

There is an incomplete record of the topic on the board ______________________ the property of multiplication

Looking at the incomplete recording, think about what we will do in class and what the topic of today's lesson is. (Children's reasoning)

Today we will get acquainted with a new property of multiplication, the name of which we will learn by completing the tasks of mental calculation and tasks included in your sheets - lesson cards, we will learn to use the new property of multiplication when analyzing numerical expressions; Let's repeat the properties of addition and the commutative property of multiplication;; We will develop computational skills, the ability to analyze and reason.

We will work together and creatively, in pairs and independently, to complete tasks and draw conclusions.

In your cards, after each task you will have to evaluate your work. If you completed the task without errors, you will give yourself a +, if you failed, then -

Why do we need this?

Where can we apply the acquired knowledge?

Proverb

To teach mathematics is to sharpen the mind

How do you understand the meaning of this proverb?

“Mathematics must be taught later because it puts the mind in order”

M. Lomonosov

III. Oral counting

1. Game “Truth is a lie.” Children show + or - sign

The sum of numbers 6 and 5 is 12

The difference between the numbers 16 and 6 is 9

9 increased by 5 equals 14

100 is the largest three-digit number

A cube is a three-dimensional figure

A rectangle is a flat figure

The letter C opens on the board

2.Ingenuity task

Add the number of colors of the rainbow to the student's favorite grade.

Add the number of days in a week to the number of months in a year.

The letter 0 opens on the board

3.Logic task

There were 2 birch trees, 4 apple trees, 5 cherries growing in the garden. How much in total fruit trees grew in the garden? The letter H opens on the board

4.What groups can the following figures be divided into?

The letter E opens on the board

The letter T opens on the board

The letter A opens on the board

7. Can we say that the area of ​​these figures is the same?

The letter T opens on the board

8. Work in pairs: Divide the numbers into two groups.

Write down each group in ascending order (Sign of teamwork) e

499 75 345 24 521 86

The letter E opens on the board

9. Independent work

Fill out the card

The letter L opens on the board

10. Choose the right sign(+ or )

Increase by 6

Increase 3 times

The letter b opens on the board

11. ,

2 6 … 6 + 6 + 6

5 6…6 4

8 6 … 6 8

The letter H opens on the board

12. Which numerical expression is redundant? Why?

(2 +7) 0 365 0

(9 2) 1 (94-26) 0

The letter O opens on the board

13.Front work

Fill in the missing numbers:

– What properties of addition and multiplication helped you complete the task? (Commutative and associative properties of addition; commutative property of multiplication.)The letter E opens on the board

The topic opens on the boardConjunctive property of multiplication

Fizminutka

To begin with we With you

To begin with, you and I

We only turn our heads.

(Rotate your head.)

We also rotate the body.

Of course we can do this.

(Turns right and left.)

Finally we reached out

Up and to the sides.

We caved in.

(Stretching up and to the sides.)

III. Posting new material

1. Statement of the educational problem

Can we say that the meanings of the expressions in this column are the same?

(For expressions 1 and 2, the combinatory property of addition is applicable - 2 adjacent terms can be replaced by a sum and the meanings of the expressions will be the same;

3 and 1 expression - applied the commutative property of addition

4 and 2 expression is a commutative property.)

-What properties are applicable for calculating data?

expressions?

(Commutative and associative property)

- Is it possible to say that the meanings of the expressions in this column are the same?

This is the question we have to answer.

Today we will find out Is it possible to use the combining property when multiplying?)

2.Primary assimilation of new knowledge

Do the math in different ways the number of all small squares and write it down as an expression.

1 way:(6*4)*2 = 24*2=48

(There are 6 squares in one rectangle, multiplying 6 by 4, we find out how many squares are in one row. By multiplying the result by 2, we find out how many squares are in two rows).

2 way: 6*(4*2)= 6*8=48

(First, we perform the action in brackets - 4 * 2, that is, we find out how many rectangles there are in two rows. There are 6 squares in one rectangle. Multiplying 6 by the result obtained, we answer the question posed.)

Conclusion: Thus, both expressions indicate how many small squares there are in the picture.

This means: (6*4)*2=6*(4*2) - the associative property of multiplication

Familiarity with the formulation of the associative property of multiplication and its comparison with the formulation of the associative property of addition.

IV. Initial check of understanding

Open your textbook to page 50 and find No. 160

Explain what the numerical equalities under each picture mean?

(4*3)*2= 4*(3*2)

(4 snowflakes were placed in 3 squares and 2 rows were taken, or 4 snowflakes were placed in 3 squares of 2 rows each.)

(6 squares took 5 rows and placed in 2 large squares or 6 squares took 5 rows in two large squares)

Let's read the rule:

Primary consolidationWork at the board

Find number 161 (1 column)

Reading the task: ( Write each expression as a product of three single-digit numbers)

Find number 162 (1 column)

Reading the task : Is it true that the values ​​of the expressions in each column are the same?

We work independently in rows (check at the board), using the combinative property: To multiply the product of two numbers by a third, you can multiply the first number by the product of the second and third numbers.

Summing up the lesson.

Assessment

Let's return to the numerical expressions that we met at the beginning of the lesson. Tell me, is it possible to say that the meanings of the expressions in this column are the same?

What discovery did you make in class today? Where can it be used?

(We got acquainted with the new property of multiplication) To multiply the product of two numbers by a third, you can multiply the first number by the product of the second and third numbers.

Homework: rule p.50, no. 163 *Find proverbs or sayings famous people about mathematics

Grading.

“5” marks are given to those guys who have no minuses in the card.

Anyone with 1-2 minuses gets a “4”

3-5 minuses – “3”

More than 5 minuses – “2”

Reflection

Finish the sentence

Today in class I.....

The most difficult thing for me was…..

Today I realized...

Today I learned...

Decide for yourself

Back Forward

Attention! Slide previews are for informational purposes only and may not represent all of the presentation's features. If you are interested in this work, please download the full version.

Target: learn to simplify an expression containing only multiplication operations.

Tasks(Slide 2):

• Introduce the associative property of multiplication.
• To form an idea of ​​the possibility of using the studied property to rationalize calculations.
• To develop ideas about the possibility of solving “life” problems using the subject “mathematics”.
• Develop intellectual and communicative general educational skills.
• Develop organizational general educational skills, including the ability to independently evaluate the results of one’s actions, control oneself, find and correct one’s own mistakes.

Lesson type: learning new material.

Lesson plan:

1. Organizational moment.
2. Oral counting. Mathematical warm-up.
Penmanship line.
3. Report the topic and objectives of the lesson.
4. Preparation for studying new material.
5. Studying new material.
6. Physical education minute
7. Work on consolidating n. m. Solving the problem.
8. Repetition of the material covered.
9. Lesson summary.
10. Reflection
11. Homework.

Equipment: task cards, visual material (tables), presentation.

PROGRESS OF THE LESSON

I. Organizational moment

The bell rang and stopped.
The lesson begins.
You sat down quietly at your desk
Everyone looked at me.

II. Oral counting

– Let’s count orally:

1) “Funny daisies” (Slides 3-7 multiplication table)

2) Mathematical warm-up. Game “Find the odd one out” (Slide 8)

• 485 45 864 947 670 134 (classification into groups EXTRA 45 - two-digit, 670 - there is no number 4 in the number record).
• 9 45 72 90 54 81 27 22 18 (9 is single digit, 22 is not divisible by 9)

Penmanship line. Write the numbers in your notebook, alternating: 45 22 670 9
– Underline the neatest notation of the number

III. Report the topic and objectives of the lesson.(Slide 9)

– Write down the date and topic of the lesson.
– Read the objectives of our lesson

IV. Preparing to study new material

a) Is the expression correct?

Write on the board:

(23 + 490 + 17) + (13 + 44 + 7) = 23 + 490 + 17 + 13 + 44 + 7

– Name the property of addition used. (Collaborative)
– What opportunity does the combining property provide?

The combinational property makes it possible to write expressions containing only addition, without parentheses.

43 + 17 + (45 + 65 + 91) = 91 + 65 + 45 + 43 + 17

– What properties of addition do we apply in this case?

The combinational property makes it possible to write expressions containing only addition, without parentheses. In this case, calculations can be performed in any order.

– In that case, what is another property of addition called? (Commutative)

– Does this expression cause difficulty? Why? (We don’t know how to multiply a two-digit number by a one-digit number)

V. Study of new material

1) If we perform multiplication in the order in which the expressions are written, difficulties will arise. What will help us overcome these difficulties?

(2 * 6) * 3 = 2 * 3 * 6

2) Work according to the textbook p. 70, No. 305 (Make your guess about the results that the Wolf and the Hare will get. Test yourself by performing the calculations).

3) No. 305. Check whether the values ​​of the expressions are equal. Orally.

Write on the board:

(5 2) 3 and 5 (2 3)
(4 7) 5 and 4 (7 5)

4) Draw a conclusion. Rule.

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.
– Explain the associative property of multiplication.
– Explain the associative property of multiplication with examples

5) Teamwork

On the board: (8 3) 2, (6 3) 3, 2 (4 7)

VI. Fizminutka

1) Game "Mirror". (Slide 10)

My mirror, tell me,
Tell me the whole truth.
Are we smarter than everyone else in the world?
Funniest and funniest of all?
Repeat after me
Funny movements of naughty physical exercises.

2) Physical exercise for the eyes “Keen Eyes”.

– Close your eyes for 7 seconds, look to the right, then left, up, down, then make 6 circles clockwise, 6 circles counterclockwise with your eyes.

VII. Consolidation of what has been learned

1) Work according to the textbook. solution to the problem. (Slide 11)

(p. 71, no. 308) Read the text. Prove that this is a task. (There is a condition, a question)
– Select a condition, a question.
– Name the numerical data. (Three, 6, three liter)
– What do they mean? (Three boxes. 6 cans, each can contains 3 liters of juice)
– What is this task in terms of structure? (Compound problem, because it is impossible to immediately answer the question of the problem or the solution requires composing an expression)
– Type of task? (Composite task for sequential actions))
– Solve the problem without a short note by composing an expression. To do this, use the following card:

Help card

– In a notebook, the solution to the problem can be written as follows: (3 6) 3

– Can we solve the problem in this order?

(3 6) 3 = (3 3) 6 = 9 6 = 54 (l).
3 (3 6) = (3 3) 6 = 9 6 = 54 (l)

Answer: 54 liters of juice in all boxes.

2) Work in pairs (using cards): (Slide 12)

– Place signs without calculating:

(15 * 2) *4 15 * (2 * 4) (–What property?)
(8 * 9) * 6 7 * (9 * 6)
(428 * 2) * 0 1 * (2 * 3)
(3 * 4) * 2 3 + 4 + 2
(2 * 3) * 4 (4 * 2) * 3

Check: (Slide 13)

(15 * 2) * 4 = 15 * (2 * 4)
(8 * 9) * 6 > 7 * (9 * 6)
(428 * 2) * 0 < 1 * (2 * 3)
(3 * 4) * 2 > 3 + 4 + 2
(2 * 3) * 4 = (4 * 2) * 3

3) Independent work (using a textbook)

(p. 71, No. 307 – according to options)

1st century (8 2) 2 = (6 2) 3 = (19 1) 0 =
2nd century (7 3) 3 = (9 2) 4 = (12 9) 0 =

Examination:

1st century (8 2) 2 = 32 (6 2) 3 = 36 (19 1) 0 = 0.
2nd century (7 3) 3 = 63 (9 2) 4 = 72 (12 9) 0 = 0

Properties of multiplication:(Slide 14).

• Commutative property
• Matching property

– Why do you need to know the properties of multiplication? (Slide 15).

• To count quickly
• Choose rational way accounts
• Solve problems

VIII. Repetition of covered material. "Windmills".(Slide 16, 17)

• Increase the numbers 485, 583 and 681 by 38 and write three numerical expressions (option 1)
• Reduce the numbers 583, 545 and 507 by 38 and write three numerical expressions (option 2)

485 + 38 523		583 + 38 621		681 + 38 719
583 – 38 545		545 – 38 507		507 – 38 469

Students complete assignments based on options (two students solve assignments on additional boards).

Peer review.

IX. Lesson summary

– What did you learn in class today?
– What is the meaning of the associative property of multiplication?

X. Reflection

– Who thinks that they understand the meaning of the associative property of multiplication? Who is satisfied with their work in class? Why?
– Who knows what he still needs to work on?
- Guys, if you liked the lesson, if you are satisfied with your work, then put your hands on your elbows and show me your palms. And if you were upset about something, then show me the back of your palm.

XI. Homework information

- Which homework would you like to receive?

Optional:

1. Learn the rule p. 70
2. Come up with and write down an expression in new topic with a solution

(4 lessons, No. 113–135)

Lesson 1 (113–118)

Target– introduce students to the combination of their_

the ability of multiplication.

In the first lesson, it is useful to remember what properties

arithmetic operations are already known to children. For this

exercises during which schoolchildren will

use this or that property. For example, you can

Is it possible to assert that the values ​​of the expressions in a given column_

are the same:

875 + (78 + 284)

(875 + 78) + 284

875 + (284 + 78)

(875 + 284) + 78

It makes sense to offer expressions whose meanings

children cannot calculate, in this case they will be_

need to draw a conclusion based on reasoning.

Comparing, for example, the first and second expressions, they

note their similarities and differences; remember the matcher_

new property of addition (two adjacent terms can be

replace them with the sum), which means that the values ​​are expressed

the marriages will be the same. The third expression is appropriate

compare differently with the first and using the commutative

property of addition, draw a conclusion. Fourth expression

can be compared with the second.

– What properties of addition are applicable for calculations?

change the meanings of these expressions? (Commutative

and associative.)

– What properties does multiplication have?

The guys remember that they know the commutative

property of multiplication. (It is reflected on p. 34 of the textbook

nickname “Try to remember!”)

- Today in class we will meet another one of ours_

multiplication!

On the board is the drawing given intask 113 . Teacher

rats in various ways. Children's proposals discussed_

are given. If difficulties arise, you can contact

to the analysis of the methods proposed by Misha and Masha.

(6 · 4) · 2: there are 6 squares in one rectangle, smart_

By pressing 6 by 4, Masha finds out how many squares contain

rectangles in one row. Multiplying the resulting re_

The result is 2, she finds out how many squares contain

rectangles in two rows, i.e. how many small ones are there?

number of squares in the picture.

Then we discuss Misha’s method: 6 · (4 · 2). First you_

we complete the action in brackets – 4 2, i.e. we find out how many

total of rectangles in two rows. In one rectangle_

nick 6 squares. Multiplying 6 by the result obtained,

We answer the question posed. Thus, both

another expression indicates how many small

squares in the picture.

This means (6 · 4) · 2 = 6 · (4 · 2).

Similar work carried out withtask 114 . Pos_

After this, children get acquainted with the formulation of the associative

properties of multiplication and compare it with the formulation

associative properties of addition.

Targettasks 115–117 - find out if children understand

formulation of the associative property of multiplication.

When executingtasks 116 we recommend using_

get a calculator. This will allow students to repeat well_

measurement of three-digit numbers.

Problem 118It's better to decide in class.

If children find it difficult to decide independently_

research institutetasks 118 , then the teacher can use the technique of

judgments ready-made solutions or explanations of expressions,

written down according to the conditions of this problem. For example:

10 5 8 10 8 5

(8 10) 5 8 (10 5)

(2_column),as well as tasks48, 54, 55 TPO No. 1.

Lesson 2 (119–125)

Target

multiplication in calculations; derive the multiplication rule

number by 10.

Working withtask 119 organized according to

instructions given in the textbook:

a) children use the commutative property of multiplication

tion, rearranging the factors in the product 4 10 = 10 4,

find the value of the product 10 · 4 by adding the tens.

The following entries are made in notebooks:

4 10 = 40;

6 10 = 60, etc.

b) children act in the same way as when performing the task_

nia a). In notebooks write down those equalities that do not exist

in task a): 5 10 = 50; 7 10 = 70; 9 10 = 90;

c) analyze and compare the written equalities,

draw a conclusion (when multiplying a number by 10, you must assign

to the first factor zero and write the resulting number in

result);

d) check the formulated rule using calculations_

tore.

Application of the combinatory property of multiplication and pr_

Multiplying by 10 allows students to multiply

"round" tens on single digit number, using on_

table multiplication skills (90 · 3, 70 · 4, etc.).

For this purpose, they are carried outtasks 120, 121, 123, 124.

When executingtasks 120 children first arranging_

draw brackets in a textbook with a pencil and then comment

your actions. For example: (5 · 7) · 10 = 35 · 10 – produced here

maintaining the first and second factors replaced its values

reading. It is useful to immediately find out what the value of pro_ is

production 35 10; 5 · (7 · 10) = 5 · 70 – here is the product

the second and third factors were replaced by its value.

When calculating the value of the product 5 70 children

can reason like this: let’s use the commutative

property of multiplication - 5 · 70 = 70 · 5. Now 7 dec. Can

repeat 5 times, we get 35 des.; this number is 350.

When explaining some equalities intask 121

schoolchildren first use the commutative their_

multiplication, and then associative. For example:

4 6 10 = 40 6

(4 10) 6 = 40 6

each equality on the left and on the right.

By calculating the values ​​of the expressions written on the left,

the guys turn to the multiplication table and then take away_

calculate the result by 10 times:

(4 6) 10 = 24 10

INtask 123 It's useful to consider different ways

would justify the answer. For example, you can in the second expression

we can replace the product with its value, and we get_

what is the first expression:

4 (7 10) = 4 70

In the third expression you need in this case first

Use the associative property of multiplication:

(4 7) 10 = 4 (7 10) and then replace the product of it

meaning.

But you can do things differently, focusing not on

the first, and the second expression. In this case, the number 70 in per_

In this expression you need to represent it as a product:

4 70 = 4 (7 10)

And in the third expression, use to transform_

calling by combining property:

(4 7) 10 = 4 (7 10)

Organizing a discussion in various ways actions

Vtask 123 , the teacher can focus on dialogue

Misha and Masha, who is brought intask 124 .

where to indicate on the diagram known and unknown values_

ranks. As a result, the diagram looks like:

For computational exercises in class, we recommend

blowingtask 125, and alsotasks 59, 60 from TVET No. 1 .

Lesson 3 (126–132)

Target– learn to use the associative property

multiplication for calculations, improve skills

solve problems.

Task 126performed orally. His goal is perfection

development of computational skills and the ability to apply

the associative property of multiplication. For example, comparing

expressions a) 45 10 and 9 50, students reason: number

45 can be represented as the product of 9 5, and then

replace the product of numbers 5 10 with its value.

Task 128also applies to computing

exercises that require active use

analysis and synthesis, comparison, generalization. Formulating the right

When constructing each row, most children used_

They use the concept of “increase by...”. For example: for row – 6,

12, 18, ... – “every next number increases by 6";

for the series – 4, 8, 12, ... – “each next number is increased_

ends at 4”, etc.

But the following option is also possible: “To get a loan_

the first number in each row is increased

2 times, to get the third number in the series, the first

the number of rows was increased by 3 times, the fourth by 4 times,

fifth - 5 times, etc.

By lining up in rows according to this rule, students actually_

They literally repeat all cases of table multiplication.

reading, students can either draw

scheme, or “revive” the scheme that the teacher prepared in advance

will depict it on the board.

Children will write down the solution to the problem in a notebook on their own.

In case of difficulties in solvingtasks 129 reko_

We recommend using the technique of discussing ready-made solutions_

explanations or explanations of expressions written according to the condition

of this task:

10 3 3 4 10 4 (10 3) 4 10 (3 4)

Problem 133It is also advisable to discuss it in class.

(1) 14 + 7 = 21 (days) 2) 21 2 = 42 (days))

tasks 61, 62 TPO No. 1.

Lesson 4 (134–135)

Target– check the mastery of table skills

knowledge and problem solving skills.

134, 135 .

Targettasks 134 – summarize children’s knowledge about the table

multiplication, which can be represented as a table

Pythagoras. Therefore, after the task is completed_

No, it’s useful to find out:

a) In which cells of the table can the same be inserted?

What numbers and why? (These cells are in the bottom row_

ke and in the right column, which is due to the commutative

property of multiplication.)

b) Is it possible, without performing calculations, to say

how much is the next number greater than the previous one in each

row (column) of the table? (In the top (first) line –

by 1, in the second - by 2, in the third - by 3, etc.) This is conditional_

defined by the definition: “multiplication is the addition of one_

kov terms".

Students should also be reminded that

the entire table contains 81 cells. This corresponds to the number

which should be written in its lower right cell.

To test the knowledge, skills and abilities of students

Shmyreva G.G. Tests. 3rd grade. – Smolensk,

Association XXI century, 2004.

Let's draw a rectangle with sides 5 cm and 3 cm on a piece of checkered paper. Divide it into squares with sides 1 cm (Fig. 143). Let's count the number of cells located in the rectangle. This can be done, for example, like this.

The number of squares with a side of 1 cm is 5 * 3. Each such square consists of four cells. That's why total number cells is equal to (5 * 3) * 4.

The same problem can be solved differently. Each of the five columns of the rectangle consists of three squares with a side of 1 cm. Therefore, one column contains 3 * 4 cells. Therefore, there will be 5 * (3 * 4) cells in total.

Counting cells in Figure 143 illustrates in two ways associative property of multiplication for numbers 5, 3 and 4. We have: (5 * 3) * 4 = 5 * (3 * 4).

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.

(ab)c = a(bc)

From the commutative and combinatory properties of multiplication it follows that when multiplying several numbers, the factors can be swapped and placed in parentheses, thereby determining the order of calculations.

For example, the following equalities are true:

abc = cba,

17 * 2 * 3 * 5 = (17 * 3 ) * (2 * 5 ).

In Figure 144, segment AB divides the rectangle discussed above into a rectangle and a square.

Let's count the number of squares with a side of 1 cm in two ways.

On the one hand, the resulting square contains 3 * 3 of them, and the rectangle contains 3 * 2. In total we get 3 * 3 + 3 * 2 squares. On the other hand, in each of the three lines of this rectangle there are 3 + 2 squares. Then their total number is 3 * (3 + 2).

Equal to 3 * (3 + 2 ) = 3 * 3 + 3 * 2 illustrates distributive property multiplication relative to addition.

To multiply a number by the sum of two numbers, you can multiply this number by each addend and add the resulting products.

In literal form this property is written as follows:

a(b + c) = ab + ac

From the distributive property of multiplication relative to addition it follows that

ab + ac = a(b + c).

This equality allows the formula P = 2 a + 2 b to find the perimeter of a rectangle to be written in the following form:

P = 2 (a + b).

Note that the distribution property is valid for three or more terms. For example:

a(m + n + p + q) = am + an + ap + aq.

The distributive property of multiplication relative to subtraction is also true: if b > c or b = c, then

a(b − c) = ab − ac

Example 1 . Calculate in a convenient way:

1 ) 25 * 867 * 4 ;

2 ) 329 * 75 + 329 * 246 .

1) We use the commutative and then the associative properties of multiplication:

25 * 867 * 4 = 867 * (25 * 4 ) = 867 * 100 = 86 700 .

2) We have:

329 * 754 + 329 * 246 = 329 * (754 + 246 ) = 329 * 1 000 = 329 000 .

Example 2 . Simplify the expression:

1) 4 a * 3 b;

2) 18 m − 13 m.

1) Using the commutative and associative properties of multiplication, we obtain:

4 a * 3 b = (4 * 3 ) * ab = 12 ab.

2) Using the distributive property of multiplication relative to subtraction, we obtain:

18 m − 13 m = m(18 − 13 ) = m * 5 = 5 m.

Example 3 . Write the expression 5 (2 m + 7) so that it does not contain parentheses.

According to the distributive property of multiplication relative to addition, we have:

5 (2 m + 7) = 5 * 2 m + 5 * 7 = 10 m + 35.

This transformation is called opening parentheses.

Example 4 . Calculate the value of the expression 125 * 24 * 283 in a convenient way.

Solution. We have:

125 * 24 * 283 = 125 * 8 * 3 * 283 = (125 * 8 ) * (3 * 283 ) = 1 000 * 849 = 849 000 .

Example 5 . Perform the multiplication: 3 days 18 hours * 6.

Solution. We have:

3 days 18 hours * 6 = 18 days 108 hours = 22 days 12 hours.

When solving the example, the distributive property of multiplication relative to addition was used:

3 days 18 hours * 6 = (3 days + 18 hours) * 6 = 3 days * 6 + 18 hours * 6 = 18 days + 108 hours = 18 days + 96 hours + 12 hours = 18 days + 4 days + 12 hours = 22 days 12 hours.