Task 1. A point moves in a straight line from left to right at a speed of 4 dm. per second and per currently passes through point A. Where will the moving point be after 5 seconds?
It is not difficult to figure out that the point will be at 20 dm. to the right of A. Let's write the solution to this problem in relative numbers. To do this, we agree on the following symbols:
1) the speed to the right will be denoted by the sign +, and to the left by the sign –, 2) the distance of the moving point from A to the right will be denoted by the sign + and to the left by the sign –, 3) the period of time after the present moment by the sign + and before the present moment by the sign –. In our problem, the following numbers are given: speed = + 4 dm. per second, time = + 5 seconds and it turned out, as we figured out arithmetically, the number + 20 dm., expressing the distance of the moving point from A after 5 seconds. Based on the meaning of the problem, we see that it relates to multiplication. Therefore, it is convenient to write the solution to the problem:
(+ 4) ∙ (+ 5) = + 20.
Task 2. A point moves in a straight line from left to right at a speed of 4 dm. per second and is currently passing through point A. Where was this point 5 seconds ago?
The answer is clear: the point was to the left of A at a distance of 20 dm.
The solution is convenient, according to the conditions regarding the signs, and, keeping in mind that the meaning of the problem has not changed, write it like this:
(+ 4) ∙ (– 5) = – 20.
Task 3. A point moves in a straight line from right to left at a speed of 4 dm. per second and is currently passing through point A. Where will the moving point be after 5 seconds?
The answer is clear: 20 dm. to the left of A. Therefore, according to the same conditions regarding signs, we can write the solution to this problem as follows:
(– 4) ∙ (+ 5) = – 20.
Task 4. The point moves in a straight line from right to left at a speed of 4 dm. per second and is currently passing through point A. Where was the moving point 5 seconds ago?
The answer is clear: at a distance of 20 dm. to the right of A. Therefore, the solution to this problem should be written as follows:
(– 4) ∙ (– 5) = + 20.
The problems considered indicate how the action of multiplication should be extended to relative numbers. In the problems we have 4 cases of multiplying numbers with all possible combinations of signs:
1) (+ 4) ∙ (+ 5) = + 20;
2) (+ 4) ∙ (– 5) = – 20;
3) (– 4) ∙ (+ 5) = – 20;
4) (– 4) ∙ (– 5) = + 20.
In all four cases, the absolute values of these numbers should be multiplied; the product must have a + sign when the factors identical signs(1st and 4th cases) and sign –, when the factors have different signs(cases 2 and 3).
From here we see that the product does not change from rearranging the multiplicand and the multiplier.
Exercises.
Let's do one example of a calculation that involves addition, subtraction, and multiplication.
In order not to confuse the order of actions, let us pay attention to the formula
Here is written the sum of the products of two pairs of numbers: therefore, you must first multiply the number a by the number b, then multiply the number c by the number d and then add the resulting products. Also in Eq.
You must first multiply the number b by c and then subtract the resulting product from a.
If it were necessary to add the product of numbers a and b with c and multiply the resulting sum by d, then one should write: (ab + c)d (compare with the formula ab + cd).
If we had to multiply the difference between the numbers a and b by c, we would write (a – b)c (compare with the formula a – bc).
Therefore, let us establish in general that if the order of actions is not indicated by parentheses, then we must first perform multiplication, and then add or subtract.
Let's start calculating our expression: let's first perform the additions written inside all the small brackets, we get:
Now we need to do the multiplication inside square brackets and then subtract the resulting product from:
Now let's perform the operations inside the twisted brackets: first multiplication and then subtraction:
Now all that remains is to perform multiplication and subtraction:
16. Product of several factors. Let it be required to find
(–5) ∙ (+4) ∙ (–2) ∙ (–3) ∙ (+7) ∙ (–1) ∙ (+5).
Here you need to multiply the first number by the second, the resulting product by the 3rd, etc. It is not difficult to establish on the basis of the previous one that the absolute values of all numbers must be multiplied among themselves.
If all the factors were positive, then based on the previous one we will find that the product must also have a + sign. If any one factor were negative
e.g., (+2) ∙ (+3) ∙ (+4) ∙ (–1) ∙ (+5) ∙ (+6),
then the product of all the factors preceding it would give a + sign (in our example (+2) ∙ (+3) ∙ (+4) = +24, from multiplying the resulting product by a negative number (in our example +24 multiplied by –1) the new product would have a - sign; multiplying it by the next positive factor (in our example –24 by +5), we again obtain a negative number; since all other factors are assumed to be positive, the sign of the product cannot change any more.
If there were two negative factors, then, reasoning as above, we would find that at first, until we reached the first negative factor, the product would be positive; by multiplying it by the first negative factor, the new product would turn out to be negative, and so would it be. remained until we reach the second negative factor; Then, by multiplying a negative number by a negative, the new product would be positive, which will remain so in the future if the remaining factors are positive.
If there were a third negative factor, then the resulting positive product from multiplying it by this third negative factor would become negative; it would remain so if the other factors were all positive. But if there is a fourth negative factor, then multiplying by it will make the product positive. Reasoning in the same way, we find that in general:
To find out the sign of the product of several factors, you need to look at how many of these factors are negative: if there are none at all, or if there are even number, then the product is positive: if negative multipliers odd number, then the product is negative.
So now we can easily find out that
(–5) ∙ (+4) ∙ (–2) ∙ (–3) ∙ (+7) ∙ (–1) ∙ (+5) = +4200.
(+3) ∙ (–2) ∙ (+7) ∙ (+3) ∙ (–5) ∙ (–1) = –630.
Now it is not difficult to see that the sign of the work, as well as its absolute value, do not depend on the order of the factors.
Convenient when dealing with fractional numbers, find the work immediately:
This is convenient because you don’t have to do useless multiplications, since the previously obtained fractional expression is reduced as much as possible.
In this article we will understand the process multiplying negative numbers. First, we formulate the rule for multiplying negative numbers and justify it. After this, we will move on to solving typical examples.
Page navigation.
We'll announce it right away rule for multiplying negative numbers: To multiply two negative numbers, you need to multiply their absolute values.
Let's write this rule using letters: for any negative real numbers−a and −b (in this case, the numbers a and b are positive), the equality is true (−a)·(−b)=a·b .
Let's prove the rule for multiplying negative numbers, that is, prove the equality (−a)·(−b)=a·b.
In the article multiplying numbers with different signs we have substantiated the validity of the equality a·(−b)=−a·b, similarly it is shown that (−a)·b=−a·b. These results and properties opposite numbers allow us to write the following equalities (−a)·(−b)=−(a·(−b))=−(−(a·b))=a·b. This proves the rule for multiplying negative numbers.
From the above multiplication rule it is clear that the product of two negative numbers is a positive number. Indeed, since the modulus of any number is positive, the product of moduli is also a positive number.
In conclusion of this point, we note that the considered rule can be used to multiply real numbers, rational numbers and integers.
It's time to sort it out examples of multiplying two negative numbers, when solving we will use the rule obtained in the previous paragraph.
Multiply two negative numbers −3 and −5.
The moduli of the numbers being multiplied are 3 and 5, respectively. The product of these numbers is 15 (see multiplication of natural numbers if necessary), so the product of the original numbers is 15.
The entire process of multiplying initial negative numbers is briefly written as follows: (−3)·(−5)= 3·5=15.
Multiplication of negative rational numbers using the analyzed rule can be reduced to multiplication ordinary fractions, multiplication mixed numbers or multiplying decimals.
Calculate the product (−0.125)·(−6) .
According to the rule for multiplying negative numbers, we have (−0.125)·(−6)=0.125·6. All that remains is to finish the calculations, let's do the multiplication decimal on natural number column:
Finally, note that if one or both factors are irrational numbers, given in the form of roots, logarithms, powers, etc., then their product often has to be written as a numerical expression. The value of the resulting expression is calculated only when necessary.
Multiply a negative number by a negative number.
Let us first find the modules of the numbers being multiplied: and 
(see properties of the logarithm). Then, according to the rule of multiplying negative numbers, we have. The resulting product is the answer.
.
You can continue studying the topic by referring to the section multiplying real numbers.
With some stretch, the same explanation is valid for the product 1-5, if we assume that the “sum” is from one single
term is equal to this term. But the product 0 5 or (-3) 5 cannot be explained this way: what does the sum of zero or minus three terms mean?
However, you can rearrange the factors
If we want the product not to change when the factors are rearranged - as was the case for positive numbers - then we must assume that
Now let's move on to the product (-3) (-5). What is it equal to: -15 or +15? Both options have a reason. On the one hand, a minus in one factor already makes the product negative - all the more so it should be negative if both factors are negative. On the other hand, in table. 7 already has two minuses, but only one plus, and “in fairness” (-3)-(-5) should be equal to +15. So which should you prefer?
Of course, you won’t be confused by such talk: from school course Mathematicians You have firmly learned that minus times minus gives a plus. But imagine that your younger brother or sister asks you: why? What is this - a teacher’s whim, an order from higher authorities, or a theorem that can be proven?
Usually the rule for multiplying negative numbers is explained with examples like the one presented in table. 8.
It can be explained differently. Let's write the numbers in a row
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Now let's write the same numbers multiplied by 3:
It is easy to notice that each number is 3 more than the previous one. Now let's write the same numbers in reverse order(starting, for example, with 5 and 15):
Moreover, under the number -5 there was a number -15, so 3 (-5) = -15: plus by minus gives a minus.
Now let's repeat the same procedure, multiplying the numbers 1,2,3,4,5. by -3 (we already know that plus by minus gives minus):
Each next number the bottom row is 3 less than the previous one. Write the numbers in reverse order
Under the number -5 there are 15, so (-3) (-5) = 15.
Perhaps these explanations would satisfy your younger brother or sister. But you have the right to ask how things really are and is it possible to prove that (-3) (-5) = 15?
The answer here is that we can prove that (-3) (-5) must equal 15 if we want the ordinary properties of addition, subtraction and multiplication to remain true for all numbers, including negative ones. The outline of this proof is as follows.
Let us first prove that 3 (-5) = -15. What is -15? This is the opposite number of 15, that is, the number that when added to 15 gives 0. So we need to prove that
(By taking 3 out of the bracket, we used the law of distributivity ab + ac = a(b + c) for - after all, we assume that it remains true for all numbers, including negative ones.) So, (The meticulous reader will ask us why. We honestly admit : we skip the proof of this fact - as well as the general discussion of what zero is.)
Let us now prove that (-3) (-5) = 15. To do this, we write
and multiply both sides of the equality by -5:
Let's open the brackets on the left side:
i.e. (-3) (-5) + (-15) = 0. Thus, the number is the opposite of the number -15, i.e. equal to 15. (There are also gaps in this reasoning: it would be necessary to prove that that there is only one number, the opposite of -15.)
Rules for multiplying negative numbers
Do we understand multiplication correctly?
“A and B were sitting on the pipe. A fell, B disappeared, what’s left on the pipe?
“Your letter I remains.”
(From the film “Youths in the Universe”)
Why does multiplying a number by zero result in zero?
Why does multiplying two negative numbers produce a positive number?
Teachers come up with everything they can to give answers to these two questions.
But no one has the courage to admit that in the formulation of multiplication three semantic errors!
Is it possible to make mistakes in basic arithmetic? After all, mathematics positions itself as an exact science.
School mathematics textbooks do not provide answers to these questions, replacing explanations with a set of rules that need to be memorized. Perhaps this topic is considered difficult to explain in middle school? Let's try to understand these issues.
7 is the multiplicand. 3 is the multiplier. 21-work.
According to the official wording:
- to multiply a number by another number means to add as many multiplicands as the multiplier prescribes.
According to the accepted formulation, the factor 3 tells us that there should be three sevens on the right side of the equality.
7 * 3 = 7 + 7 + 7 = 21
But this formulation of multiplication cannot explain the questions posed above.
Let's correct the wording of multiplication
Usually in mathematics there is a lot that is meant, but it is not talked about or written down.
This refers to the plus sign before the first seven on the right side of the equation. Let's write down this plus.
7 * 3 = + 7 + 7 + 7 = 21
But what is the first seven added to? This means to zero, of course. Let's write down zero.
7 * 3 = 0 + 7 + 7 + 7 = 21
What if we multiply by three minus seven?
— 7 * 3 = 0 + (-7) + (-7) + (-7) = — 21
We write the addition of the multiplicand -7, but in fact we are subtracting from zero multiple times. Let's open the brackets.
— 7 * 3 = 0 — 7 — 7 — 7 = — 21
Now we can give a refined formulation of multiplication.
- Multiplication is the process of repeatedly adding to (or subtracting from zero) the multiplicand (-7) as many times as the multiplier indicates. The multiplier (3) and its sign (+ or -) indicate the number of operations that are added to or subtracted from zero.
Using this clarified and slightly modified formulation of multiplication, the “sign rules” for multiplication when the multiplier is negative are easily explained.
7 * (-3) - there must be three minus signs after zero = 0 - (+7) - (+7) - (+7) = - 21
- 7 * (-3) - again there should be three minus signs after the zero =
0 — (-7) — (-7) — (-7) = 0 + 7 + 7 + 7 = + 21
Multiply by zero
7 * 0 = 0 + . there are no addition-to-zero operations.
If multiplication is an addition to zero, and the multiplier shows the number of operations of addition to zero, then the multiplier zero shows that nothing is added to zero. That's why it remains zero.
So, in the existing formulation of multiplication, we found three semantic errors that block the understanding of the two “sign rules” (when the multiplier is negative) and the multiplication of a number by zero.
- You don't need to add the multiplicand, but add it to zero.
- Multiplication is not only adding to zero, but also subtracting from zero.
- The multiplier and its sign do not show the number of terms, but the number of plus or minus signs when decomposing the multiplication into terms (or subtracted ones).
Having somewhat clarified the formulation, we were able to explain the rules of signs for multiplication and the multiplication of a number by zero without the help of the commutative law of multiplication, without the distributive law, without resorting to analogies with the number line, without equations, without proof from the inverse, etc.
The sign rules for the refined formulation of multiplication are derived very simply.
7 * (+3) = 0 + (-7) + (-7) + (-7) = 0 — 7 — 7 — 7 = -21 (- + = -)
7 * (-3) = 0 — (+7) — (+7) — (+7) = 0 — 7 — 7 — 7 = -21 (+ — = -)
7 * (-3) = 0 — (-7) — (-7) — (-7) = 0 + 7 + 7 + 7 = +21 (- — = +)
The multiplier and its sign (+3 or -3) indicate the number of “+” or “-” signs on the right side of the equation.
The modified formulation of multiplication corresponds to the operation of raising a number to a power.
2^0 = 1 (one is not multiplied or divided by anything, so it remains one)
2^-2 = 1: 2: 2 = 1/4
2^-3 = 1: 2: 2: 2 = 1/8
Mathematicians agree that raising a number to positive degree is the multiple multiplication of one. And raising a number to negative degree is a multiple division of a unit.
The operation of multiplication should be similar to the operation of exponentiation.
2*3 = 0 + 2 + 2 + 2 = 6
2*0 = 0 (nothing is added to zero and nothing is subtracted from zero)
2*-3 = 0 — 2 — 2 — 2 = -6
The modified formulation of multiplication does not change anything in mathematics, but returns the original meaning of the multiplication operation, explains the “rules of signs”, multiplying a number by zero, and reconciles multiplication with exponentiation.
Let's check whether our formulation of multiplication is consistent with the division operation.
15: 5 = 3 (inverse of multiplication 5 * 3 = 15)
The quotient (3) corresponds to the number of operations of addition to zero (+3) during multiplication.
Dividing the number 15 by 5 means finding how many times you need to subtract 5 from 15. This is done sequential subtraction until a zero result is obtained.
To find the result of division, you need to count the number of minus signs. There are three of them.
15: 5 = 3 operations of subtracting five from 15 to get zero.
15 - 5 - 5 - 5 = 0 (division 15:5)
0 + 5 + 5 + 5 = 15 (multiplying 5 * 3)
Division with remainder.
17 — 5 — 5 — 5 — 2 = 0
17: 5 = 3 and 2 remainder
If there is division with a remainder, why not multiplication with an appendage?
2 + 5 * 3 = 0 + 2 + 5 + 5 + 5 = 17
Let's look at the difference in wording on the calculator
Existing formulation of multiplication (three terms).
10 + 10 + 10 = 30
Corrected multiplication formulation (three additions to zero operations).
0 + 10 = = = 30
(Press “equals” three times.)
10 * 3 = 0 + 10 + 10 + 10 = 30
A multiplier of 3 indicates that the multiplicand 10 must be added to zero three times.
Try multiplying (-10) * (-3) by adding the term (-10) minus three times!
(-10) * (-3) = (-10) + (-10) + (-10) = -10 — 10 — 10 = -30 ?
What does the minus sign for three mean? Maybe so?
(-10) * (-3) = (-10) — (-10) — (-10) = — 10 + 10 + 10 = 10?
Ops. It is not possible to decompose the product into the sum (or difference) of terms (-10).
The revised wording does this correctly.
0 — (-10) = = = +30
(-10) * (-3) = 0 — (-10) — (-10) — (-10) = 0 + 10 + 10 + 10 = 30
The multiplier (-3) indicates that the multiplicand (-10) must be subtracted from zero three times.
Sign rules for addition and subtraction
Above we showed a simple way to derive the rules of signs for multiplication by changing the meaning of the wording of multiplication.
But for the conclusion we used the rules of signs for addition and subtraction. They are almost the same as for multiplication. Let's create a visualization of the rules of signs for addition and subtraction, so that even a first-grader can understand it.
What is “minus”, “negative”?
There is nothing negative in nature. No negative temperature, no negative direction, no negative mass, no negative charges. Even sine by its nature can only be positive.
But mathematicians came up with negative numbers. For what? What does "minus" mean?
Minus means opposite direction. Left - right. Top bottom. Clockwise - counterclockwise. Back and forth. Cold - hot. Light heavy. Slow - fast. If you think about it, you can give many other examples where it is convenient to use negative values quantities
In the world we know, infinity starts from zero and goes to plus infinity.
"Minus infinity" in real world does not exist. This is the same mathematical convention as the concept of “minus”.
So, “minus” denotes the opposite direction: movement, rotation, process, multiplication, addition. Let's analyze the different directions when adding and subtracting positive and negative (increasing in the other direction) numbers.
The difficulty in understanding the rules of signs for addition and subtraction is due to the fact that these rules are usually explained on a number line. On the number line, three different components are mixed, from which rules are derived. And because of the mixing, because of the stalling different concepts together, difficulties of understanding are created.
To understand the rules, we need to divide:
- the first term and the sum (they will be on the horizontal axis);
- the second term (it will be on the vertical axis);
- direction of addition and subtraction operations.
This division is clearly shown in the figure. Mentally imagine that the vertical axis can rotate, superimposing on the horizontal axis.
The addition operation is always performed by rotating the vertical axis clockwise (plus sign). The subtraction operation is always performed by rotating the vertical axis counterclockwise (minus sign).
Example. Diagram in lower right corner.
It can be seen that two are nearby standing sign minus (the sign of the subtraction operation and the sign of the number 3) have different meaning. The first minus shows the direction of subtraction. The second minus is the sign of the number on the vertical axis.
Find the first term (-2) on the horizontal axis. Find the second term (-3) on the vertical axis. Mentally rotate vertical axis counterclockwise until (-3) aligns with the number (+1) on the horizontal axis. The number (+1) is the result of addition.
gives the same result as the addition operation in the diagram in the upper right corner.
Therefore, two adjacent minus signs can be replaced with one plus sign.
We are all accustomed to using ready-made rules of arithmetic without thinking about their meaning. Therefore, we often don’t even notice how the rules of signs for addition (subtraction) differ from the rules of signs for multiplication (division). Do they seem the same? Almost. A slight difference can be seen in the following illustration.
Now we have everything we need to derive the sign rules for multiplication. The output sequence is as follows.
- We clearly show how the rules of signs for addition and subtraction are obtained.
- We make semantic changes to the existing formulation of multiplication.
- Based on the modified formulation of multiplication and the rules of signs for addition, we derive the rules of signs for multiplication.
Below are written Sign rules for addition and subtraction,obtained from the visualization. And in red, for comparison, the same rules of signs from the mathematics textbook. The gray plus in parentheses is an invisible plus, which is not written for a positive number.
There are always two signs between the terms: the operation sign and the number sign (we don’t write plus, but we mean it). The rules of signs prescribe the replacement of one pair of characters with another pair without changing the result of addition (subtraction). In fact, there are only two rules.
Rules 1 and 3 (for visualization) - duplicate rules 4 and 2.. Rules 1 and 3 in the school interpretation do not coincide with the visual scheme, therefore, they do not apply to the rules of signs for addition. These are some other rules.
School rule 1. (red) allows you to replace two pluses in a row with one plus. The rule does not apply to the replacement of signs in addition and subtraction.
School rule 3. (red) allows you not to write a plus sign for a positive number after a subtraction operation. The rule does not apply to the replacement of signs in addition and subtraction.
The meaning of the rules of signs for addition is the replacement of one PAIR of characters with another PAIR of characters without changing the result of the addition.
School methodologists mixed two rules in one rule:
— two rules of signs when adding and subtracting positive and negative numbers (replacing one pair of signs with another pair of signs);
- two rules according to which you can not write a plus sign for a positive number.
Two different rules, mixed into one, are similar to the rules of signs in multiplication, where two signs result in a third. They look exactly alike.
Great confusion! The same thing again, for better detangling. Let us highlight the operation signs in red to distinguish them from the number signs.
1. Addition and subtraction. Two rules of signs according to which pairs of signs between terms are interchanged. Operation sign and number sign.
2. Two rules according to which the plus sign for a positive number is allowed not to be written. These are the rules for the entry form. Does not apply to addition. For a positive number, only the sign of the operation is written.
3. Four rules of signs for multiplication. When two signs of factors result in a third sign of the product. The multiplication sign rules contain only number signs.
Now that we have separated the form rules, it should be clear that the sign rules for addition and subtraction are not at all similar to the sign rules for multiplication.
“The rule for multiplying negative numbers and numbers with different signs.” 6th grade
Presentation for the lesson
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Lesson objectives.
Subject:
- formulate a rule for multiplying negative numbers and numbers with different signs,
- teach students how to apply this rule.
Metasubject:
- develop the ability to work in accordance with the proposed algorithm, draw up a plan for your actions,
- develop self-control skills.
Personal:
- develop communication skills,
- form cognitive interest students.
Equipment: computer, screen, multimedia projector, PowerPoint presentation, Handout: table for recording rules, tests.
(Textbook by N.Ya. Vilenkin “Mathematics. 6th grade”, M: “Mnemosyne”, 2013.)
During the classes
I. Organizational moment.
Communicating the topic of the lesson and recording the topic in notebooks by students.
II. Motivation.
Slide number 2. (Lesson goal. Lesson plan).
Today we will continue to study the important arithmetic property– multiplication.
You already know how to multiply natural numbers - verbally and columnarly,
Learned how to multiply decimals and ordinary fractions. Today you will have to formulate the multiplication rule for negative numbers and numbers with different signs. And not only formulate it, but also learn to apply it.
III. Updating knowledge.
Solve the equations: a) x: 1.8 = 0.15; b) y: = . (Student at the blackboard)
Conclusion: to solve such equations you need to be able to multiply different numbers.
2) Checking homework independently. Review rules for multiplying decimals, fractions and mixed numbers. (Slides No. 4 and No. 5).
IV. Formulation of the rule.
Consider task 1 (slide number 6).
Consider task 2 (slide number 7).
In the process of solving problems, we had to multiply numbers with different signs and negative numbers. Let's take a closer look at this multiplication and its results.
By multiplying numbers with different signs, we get a negative number.
Let's look at another example. Find the product (–2) * 3, replacing the multiplication with the sum of identical terms. Similarly, find the product 3 * (–2). (Check - slide No. 8).
Questions:
1) What is the sign of the result when multiplying numbers with different signs?
2) How is the result module obtained? We formulate a rule for multiplying numbers with different signs and write the rule in the left column of the table. (Slide No. 9 and Appendix 1).
Rule for multiplying negative numbers and numbers with different signs.
Let's return to the second problem, in which we multiplied two negative numbers. It is quite difficult to explain such multiplication in another way.
Let's use the explanation that was given back in the 18th century by the great Russian scientist (born in Switzerland), mathematician and mechanic Leonhard Euler. (Leonard Euler left behind not only scientific works, but also wrote a number of textbooks on mathematics intended for students of the academic gymnasium).
So Euler explained the result approximately in the following way. (Slide number 10).
It is clear that –2 · 3 = – 6. Therefore, the product (–2) · (–3) cannot be equal to –6. However, it must be somehow related to the number 6. There remains one possibility: (–2) · (–3) = 6. .
Questions:
1) What is the sign of the product?
2) How was the product modulus obtained?
We formulate the rule for multiplying negative numbers and fill in the right column of the table. (Slide No. 11).
To make it easier to remember the rule of signs when multiplying, you can use its formulation in verse. (Slide No. 12).
Plus by minus, multiplying,
We put a minus without yawning.
Multiply minus by minus
We'll give you a plus in response!
V. Formation of skills.
Let's learn how to apply this rule for calculations. Today in the lesson we will perform calculations only with whole numbers and decimal fractions.
1) Drawing up an action plan.
A scheme for applying the rule is drawn up. Notes are made on the board. Approximate diagram on slide number 13.
2) Carrying out actions according to the scheme.
We solve from textbook No. 1121 (b, c, i, j, p, p). We carry out the solution in accordance with the drawn up diagram. Each example is explained by one of the students. At the same time, the solution is shown on slide No. 14.
3) Work in pairs.
Task on slide number 15.
Students work on options. First, the student from option 1 solves and explains the solution to option 2, the student from option 2 listens carefully, helps and corrects if necessary, and then the students change roles.
Additional task for those pairs who finish work earlier: No. 1125.
Upon completion of work, verification is carried out according to ready-made solution, placed on slide No. 15 (animation is used).
If many people managed to solve No. 1125, then the conclusion is made that the sign of the number changes when multiplied by (?1).
4) Psychological relief.
5) Independent work.
Independent work - text on slide No. 17. After completing the work - self-test using a ready-made solution (slide No. 17 - animation, hyperlink to slide No. 18).
VI. Checking the level of assimilation of the studied material. Reflection.
Students take the test. On the same piece of paper, evaluate your work in class by filling out the table.
Test “Multiplication Rule”. Option 1.
Multiplying negative numbers: rule, examples
In this article we will formulate the rule for multiplying negative numbers and give an explanation for it. The process of multiplying negative numbers will be discussed in detail. The examples show all possible cases.
Multiplying Negative Numbers
Rule for multiplying negative numbers is that in order to multiply two negative numbers, it is necessary to multiply their modules. This rule is written as follows: for any negative numbers – a, – b, this equality is considered true.
Above is the rule for multiplying two negative numbers. Based on it, we prove the expression: (— a) · (— b) = a · b. The article multiplying numbers with different signs says that the equalities a · (- b) = - a · b are valid, as well as (- a) · b = - a · b. This follows from the property of opposite numbers, due to which the equalities will be written as follows:
(— a) · (— b) = — (— a · (— b)) = — (— (a · b)) = a · b .
Here you can clearly see the proof of the rule for multiplying negative numbers. Based on the examples, it is clear that the product of two negative numbers is a positive number. When multiplying moduli of numbers, the result is always a positive number.
This rule is applicable for multiplying real numbers, rational numbers, and integers.
Examples of multiplying negative numbers
Now let's look at examples of multiplying two negative numbers in detail. When calculating, you must use the rule written above.
Multiply the numbers - 3 and - 5.
Solution.
The absolute value of the two numbers being multiplied is equal to the positive numbers 3 and 5. Their product results in 15. It follows that the product given numbers equals 15
Let us briefly write down the multiplication of negative numbers itself:
(– 3) · (– 5) = 3 · 5 = 15
Answer: (- 3) · (- 5) = 15.
When multiplying negative rational numbers, using the discussed rule, you can mobilize to multiply fractions, multiply mixed numbers, multiply decimals.
Calculate the product (— 0 , 125) · (— 6) .
Using the rule for multiplying negative numbers, we obtain that (− 0, 125) · (− 6) = 0, 125 · 6. To obtain the result, you must multiply the decimal fraction by the natural number of columns. It looks like this:
We found that the expression will take the form (− 0, 125) · (− 6) = 0, 125 · 6 = 0, 75.
Answer: (− 0, 125) · (− 6) = 0, 75.
In the case when the multipliers are irrational numbers, then their product can be written in the form numerical expression. The value is calculated only when necessary.
It is necessary to multiply the negative - 2 by the non-negative log 5 1 3 .
Finding the modules of the given numbers:
- 2 = 2 and log 5 1 3 = - log 5 3 = log 5 3 .
Following from the rules for multiplying negative numbers, we get the result - 2 · log 5 1 3 = - 2 · log 5 3 = 2 · log 5 3 . This expression is the answer.
Answer: — 2 · log 5 1 3 = — 2 · log 5 3 = 2 · log 5 3 .
To continue studying the topic, you must repeat the section on multiplying real numbers.
In this article we will formulate the rule for multiplying negative numbers and give an explanation for it. The process of multiplying negative numbers will be discussed in detail. The examples show all possible cases.
Yandex.RTB R-A-339285-1
Multiplying Negative Numbers
Definition 1Rule for multiplying negative numbers is that in order to multiply two negative numbers, it is necessary to multiply their modules. This rule is written as follows: for any negative numbers – a, - b, this equality is considered true.
(- a) · (- b) = a · b.
Above is the rule for multiplying two negative numbers. Based on it, we prove the expression: (- a) · (- b) = a · b. The article multiplying numbers with different signs says that the equalities a · (- b) = - a · b are valid, as is (- a) · b = - a · b. This follows from the property of opposite numbers, due to which the equalities will be written as follows:
(- a) · (- b) = - (- a · (- b)) = - (- (a · b)) = a · b.
Here you can clearly see the proof of the rule for multiplying negative numbers. Based on the examples, it is clear that the product of two negative numbers is a positive number. When multiplying moduli of numbers, the result is always a positive number.
This rule is applicable for multiplying real numbers, rational numbers, and integers.
Now let's look at examples of multiplying two negative numbers in detail. When calculating, you must use the rule written above.
Example 1
Multiply numbers - 3 and - 5.
Solution.
The absolute value of the two numbers being multiplied is equal to the positive numbers 3 and 5. Their product results in 15. It follows that the product of the given numbers is 15
Let us briefly write down the multiplication of negative numbers itself:
(- 3) · (- 5) = 3 · 5 = 15
Answer: (- 3) · (- 5) = 15.
When multiplying negative rational numbers, using the discussed rule, you can mobilize to multiply fractions, multiply mixed numbers, multiply decimals.
Example 2
Calculate the product (- 0 , 125) · (- 6) .
Solution.
Using the rule for multiplying negative numbers, we obtain that (− 0, 125) · (− 6) = 0, 125 · 6. To obtain the result, you must multiply the decimal fraction by the natural number of columns. It looks like this:
We found that the expression will take the form (− 0, 125) · (− 6) = 0, 125 · 6 = 0, 75.
Answer: (− 0, 125) · (− 6) = 0, 75.
In the case when the factors are irrational numbers, then their product can be written as a numerical expression. The value is calculated only when necessary.
Example 3
It is necessary to multiply negative - 2 by non-negative log 5 1 3.
Solution
Finding the modules of the given numbers:
2 = 2 and log 5 1 3 = - log 5 3 = log 5 3 .
Following from the rules for multiplying negative numbers, we get the result - 2 · log 5 1 3 = - 2 · log 5 3 = 2 · log 5 3 . This expression is the answer.
Answer: - 2 · log 5 1 3 = - 2 · log 5 3 = 2 · log 5 3 .
To continue studying the topic, you must repeat the section on multiplying real numbers.
If you notice an error in the text, please highlight it and press Ctrl+Enter
Topic of the open lesson: "Multiplying Negative and Positive Numbers"
Date of: 03/17/2017
Teacher: Kuts V.V.
Class: 6 g
Purpose and objectives of the lesson:
introduce rules for multiplying two negative numbers and numbers with different signs;
promote development mathematical speech, random access memory, voluntary attention, visual and effective thinking;
formation internal processes intellectual, personal, emotional development.
cultivate a culture of behavior during frontal work, individual and group work.
Lesson type: lesson of initial presentation of new knowledge
Forms of training: frontal, work in pairs, work in groups, individual work.
Teaching methods: verbal (conversation, dialogue); visual (working with didactic material); deductive (analysis, application of knowledge, generalization, project activities).
Concepts and terms : modulus of numbers, positive and negative numbers, multiplication.
Planned results training
-be able to multiply numbers with different signs, multiply negative numbers;Apply the rule for multiplying positive and negative numbers when solving exercises, consolidate the rules for multiplying decimals and ordinary fractions.
Regulatory – be able to determine and formulate a goal in a lesson with the help of a teacher; pronounce the sequence of actions in the lesson; work according to a collectively drawn up plan; evaluate the correctness of the action. Plan your action in accordance with the task; make the necessary adjustments to the action after its completion based on its assessment and taking into account the errors made; express your guess.Communication - be able to formulate your thoughts into orally; listen and understand the speech of others; jointly agree on the rules of behavior and communication at school and follow them.
Cognitive - be able to navigate your knowledge system, distinguish new knowledge from already known knowledge with the help of a teacher; gain new knowledge; find answers to questions using a textbook, your life experience and information received in class.
Formation of a responsible attitude to learning based on motivation to learn new things;
Formation of communicative competence in the process of communication and cooperation with peers in educational activities;
Be able to carry out self-assessment based on the criterion of success of educational activities; focus on success in educational activities.
During the classes
Structural elements lesson
Didactic tasks
Designed teacher activity
Designed student activities
Result
1.Organizational moment
Motivation to successful activities
Checking readiness for the lesson.
- Good afternoon guys! Have a seat! Check if you have everything ready for the lesson: notebook and textbook, diary and writing materials.
I'm glad to see you in class today in a good mood.
Look into each other's eyes, smile, and with your eyes wish your friend a good working mood.
I also wish you good work today.
Guys, the motto of today's lesson will be a quote from the French writer Anatole France:
“The only way to learn is to have fun. To digest knowledge, you need to absorb it with appetite.”
Guys, who can tell me what it means to absorb knowledge with appetite?
So today in class we will absorb knowledge from great pleasure, because they will be useful to us in the future.
So let’s quickly open our notebooks and write down the number, great job.
Emotional mood
-With interest, with pleasure.
Ready to start lesson
Positive motivation to study new topic
2. Activation cognitive activity
Prepare them to learn new knowledge and ways of acting.
Organize a frontal survey on the material covered.
Guys, who can tell me what is the most important skill in mathematics? ( Check). Right.
So now I’ll test you how well you can count.
We will now do a mathematical warm-up.
We work as usual, count verbally and write down the answer in writing. I'll give you 1 minute.
5,2-6,7=-1,52,9+0,3=-2,6
9+0,3=9,3
6+7,21=13,21
15,22-3,34=-18,56
Let's check the answers.
We will check the answers, if you agree with the answer, then clap your hands, if you do not agree, then stomp your feet.
Well done boys.
Tell me, what actions did we perform with numbers?
What rule did we use when counting?
Formulate these rules.
Answer questions by solving small examples.
Addition and subtraction.
Adding numbers with different signs, adding numbers with negative signs, and subtracting positive and negative numbers.
Readiness of students for production problematic issue, to find ways to solve the problem.
3. Motivation for setting the topic and goal of the lesson
Encourage students to set the topic and purpose of the lesson.
Organize work in pairs.
Well, it's time to move on to learning new material, but first, let's review the material from previous lessons. A mathematical crossword puzzle will help us with this.
But this crossword is not an ordinary one, it encrypts keyword, which will tell us the topic of today's lesson.
Guys, the crossword puzzle is on your tables, we will work with it in pairs. And since it’s in pairs, then remind me how it’s like in pairs?
We remembered the rule of working in pairs, and now let’s start solving the crossword puzzle, I’ll give you 1.5 minutes. Whoever does everything, put your hands down so I can see.
(Annex 1)
1.What numbers are used for counting?
2. The distance from the origin to any point is called?
3.Numbers that are represented by a fraction are called?
4. What are two numbers that differ from each other only in signs?
5.What numbers lie to the right of zero on the coordinate line?
6.What are the natural numbers, their opposites and zero called?
7.What number is called neutral?
8. Number showing the position of a point on a line?
9. What numbers lie to the left of zero on the coordinate line?
So, time is up. Let's check.
We have solved the entire crossword puzzle and thereby repeated the material from previous lessons. Raise your hand, who made only one mistake and who made two? (So you guys are great).
Well, now let's get back to our crossword puzzle. At the very beginning, I said that it contains an encrypted word that will tell us the topic of the lesson.
So what will be the topic of our lesson?
What are we going to multiply today?
Let's think, for this we remember the types of numbers that we already know.
Let's think about what numbers we already know how to multiply?
What numbers will we learn to multiply today?
Write down the topic of the lesson in your notebook: “Multiplying positive and negative numbers.”
So, guys, we found out what we will talk about today in class.
Tell me, please, the purpose of our lesson, what should each of you learn and what should you try to learn by the end of the lesson?
Guys, in order to achieve this goal, what problems will we have to solve with you?
Absolutely right. These are the two tasks that we will have to solve with you today.
Work in pairs, set the topic and purpose of the lesson.
1.Natural
2.Module
3. Rational
4.Opposite
5.Positive
6. Whole
7.Zero
8.Coordinate
9.Negative
-"Multiplication"
Positive and negative numbers
"Multiplying Positive and Negative Numbers"
The purpose of the lesson:
Learn to multiply positive and negative numbers
First, to learn how to multiply positive and negative numbers, you need to get a rule.
Secondly, once we have the rule, what should we do next? (learn to apply it when solving examples).
4. Learning new knowledge and ways of doing things
Gain new knowledge on the topic.
-Organize work in groups (learning new material)
- Now, in order to achieve our goal, we will proceed to the first task, we will derive a rule for multiplying positive and negative numbers.
And research work will help us with this. And who will tell me why it is called research? - In this work we will research to discover the rules of “Multiplication of positive and negative numbers”.
Your research work will be carried out in groups, we will have 5 research groups in total.
We repeated in our heads how we should work as a group. If someone has forgotten, then the rules are in front of you on the screen.
Your goal research work: While exploring the problems, gradually derive the rule “Multiplying negative and positive numbers” in task No. 2; in task No. 1 you have a total of 4 problems. And to solve these problems, our thermometer will help you, each group has one.
Make all your notes on a piece of paper.
Once the group has a solution to the first problem, you show it on the board.
You are given 5-7 minutes to work.
(Appendix 2 )
Work in groups (fill out the table, conduct research)
Rules for working in groups.Working in groups is very easy
Know how to follow five rules:
first of all: don’t interrupt,
when he talks
friend, there should be silence around;
second: don’t shout loudly,
and give arguments;
and the third rule is simple:
decide what is important to you;
fourthly: it is not enough to know verbally,
must be recorded;
and fifthly: summarize, think,
what could you do.
Mastery
the knowledge and methods of action that are determined by the objectives of the lesson
5. Physical training
Establish the correct assimilation of new material on at this stage, identify misconceptions and correct them
Okay, I put all your answers in a table, now let's look at each line in our table (see presentation)
What conclusions can we draw from examining the table?
1 line. What numbers are we multiplying? What number is the answer?
2nd line. What numbers are we multiplying? What number is the answer?
3rd line. What numbers are we multiplying? What number is the answer?
4th line. What numbers are we multiplying? What number is the answer?
And so you analyzed the examples, and are ready to formulate the rules, for this you had to fill in the blanks in the second task.
How to multiply a negative number by a positive one?
- How to multiply two negative numbers?
Let's take a little rest.
A positive answer means we sit down, a negative answer we stand up.
5*6
2*2
7*(-4)
2*(-3)
8*(-8)
7*(-2)
5*3
4*(-9)
5*(-5)
9*(-8)
15*(-3)
7*(-6)
Multiplying positive numbers, the answer always turns out to be a positive number.
When you multiply a negative number by a positive number, the answer is always a negative number.
When multiplying negative numbers, the answer always results in a positive number.
Multiplying a positive number by a negative number produces a negative number.
To multiply two numbers with different signs, you needmultiply modules of these numbers and put a “-” sign in front of the resulting number.
- To multiply two negative numbers, you needmultiply their modules and put the sign in front of the resulting number «+».
Students perform physical exercise, reinforcing the rules.
Prevents fatigue
7.Primary consolidation of new material
Master the ability to apply acquired knowledge in practice.
Organize frontal and independent work based on the material covered.
Let's fix the rules, and tell each other these same rules as a couple. I'll give you a minute for this.
Tell me, can we now move on to solving the examples? Yes we can.
Open page 192 No. 1121
All together we will make the 1st and 2nd lines a)5*(-6)=30
b)9*(-3)=-27
g)0.7*(-8)=-5.6
h)-0.5*6=-3
n)1.2*(-14)=-16.8
o)-20.5*(-46)=943
three people at the board
You are given 5 minutes to solve the examples.
And we check everything together.
Creative task in pairs. (Appendix 3)
Insert the numbers so that on each floor their product is equal to the number on the roof of the house.
Solve examples using acquired knowledge
Raise your hands if you haven't made any mistakes, well done...
Active actions students to apply knowledge in life.
9. Reflection (lesson summary, assessment of student performance results)
Ensure student reflection, i.e. their assessment of their activities
Organize a lesson summary
Our lesson has come to an end, let's summarize.
Let's remember the topic of our lesson again? What goal did we set? - Did we achieve this goal?
What difficulties did it cause you? this topic?
- Guys, in order to evaluate your work in class, you must draw a smiley face in the circles that are on your tables.
A smiling emoticon means that you understand everything. Green means that you understand, but need to practice, and a sad smiley if you haven’t understood anything at all. (I'll give you half a minute)
Well, guys, are you ready to show how you worked in class today? So, let’s raise it and I’ll also raise a smiley face for you.
I am very pleased with you in class today! I see that everyone understood the material. Guys, you are great!
The lesson is over, thanks for your attention!
Answer questions and evaluate their work
Yes, we have achieved it.
Students’ openness to the transfer and comprehension of their actions, to identifying positive and negative points lesson
10 .Homework information
Provide an understanding of the purpose, content and methods of implementation homework
Provides understanding of the purpose of homework.
Homework:
1.
Learn multiplication rules
2.No. 1121(3 column).
3.Creative task: make a test of 5 questions with answer options.
Write down your homework, trying to comprehend and understand.
Realization of the need to achieve conditions for successful implementation homework by all students, in accordance with the task and the level of development of the students
Back forward
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.
Lesson objectives.
Subject:
- formulate a rule for multiplying negative numbers and numbers with different signs,
- teach students how to apply this rule.
Metasubject:
- develop the ability to work in accordance with the proposed algorithm, draw up a plan for your actions,
- develop self-control skills.
Personal:
- develop communication skills,
- to form the cognitive interest of students.
Equipment: computer, screen, multimedia projector, PowerPoint presentation, handouts: table for recording rules, tests.
(Textbook by N.Ya. Vilenkin “Mathematics. 6th grade”, M: “Mnemosyne”, 2013.)
During the classes
I. Organizational moment.
Communicating the topic of the lesson and recording the topic in notebooks by students.
II. Motivation.
Slide number 2. (Lesson goal. Lesson plan).
Today we will continue to study an important arithmetic property - multiplication.
You already know how to multiply natural numbers - verbally and columnarly,
Learned how to multiply decimals and ordinary fractions. Today you will have to formulate the multiplication rule for negative numbers and numbers with different signs. And not only formulate it, but also learn to apply it.
III. Updating knowledge.
1) Slide number 3.
Solve the equations: a) x: 1.8 = 0.15; b) y: = . (Student at the blackboard)
Conclusion: to solve such equations you need to be able to multiply different numbers.
2) Checking homework independently. Review rules for multiplying decimals, fractions and mixed numbers. (Slides No. 4 and No. 5).
IV. Formulation of the rule.
Consider task 1 (slide number 6).
Consider task 2 (slide number 7).
In the process of solving problems, we had to multiply numbers with different signs and negative numbers. Let's take a closer look at this multiplication and its results.
By multiplying numbers with different signs, we get a negative number.
Let's look at another example. Find the product (–2) * 3, replacing the multiplication with the sum of identical terms. Similarly, find the product 3 * (–2). (Check - slide No. 8).
Questions:
1) What is the sign of the result when multiplying numbers with different signs?
2) How is the result module obtained? We formulate a rule for multiplying numbers with different signs and write the rule in the left column of the table. (Slide No. 9 and Appendix 1).
Rule for multiplying negative numbers and numbers with different signs.
Let's return to the second problem, in which we multiplied two negative numbers. It is quite difficult to explain such multiplication in another way.
Let's use the explanation that was given back in the 18th century by the great Russian scientist (born in Switzerland), mathematician and mechanic Leonhard Euler. (Leonard Euler left behind not only scientific works, but also wrote a number of textbooks on mathematics intended for students of the academic gymnasium).
So Euler explained the result roughly as follows. (Slide number 10).
It is clear that –2 · 3 = – 6. Therefore, the product (–2) · (–3) cannot be equal to –6. However, it must be somehow related to the number 6. There remains one possibility: (–2) · (–3) = 6. .
Questions:
1) What is the sign of the product?
2) How was the product modulus obtained?
We formulate the rule for multiplying negative numbers and fill in the right column of the table. (Slide No. 11).
To make it easier to remember the rule of signs when multiplying, you can use its formulation in verse. (Slide No. 12).
Plus by minus, multiplying,
We put a minus without yawning.
Multiply minus by minus
We'll give you a plus in response!
V. Formation of skills.
Let's learn how to apply this rule for calculations. Today in the lesson we will perform calculations only with whole numbers and decimal fractions.
1) Drawing up an action plan.
A scheme for applying the rule is drawn up. Notes are made on the board. Approximate diagram on slide No. 13.
2) Carrying out actions according to the scheme.
We solve from textbook No. 1121 (b, c, i, j, p, p). We carry out the solution in accordance with the drawn up diagram. Each example is explained by one of the students. At the same time, the solution is shown on slide No. 14.
3) Work in pairs.
Task on slide number 15.
Students work on options. First, the student from option 1 solves and explains the solution to option 2, the student from option 2 listens carefully, helps and corrects if necessary, and then the students change roles.
Additional task for those pairs who finish work earlier: No. 1125.
At the end of the work, verification is carried out using a ready-made solution located on slide No. 15 (animation is used).
If many people managed to solve No. 1125, then the conclusion is made that the sign of the number changes when multiplied by (?1).
4) Psychological relief.
5) Independent work.
Independent work - text on slide No. 17. After completing the work - self-test using a ready-made solution (slide No. 17 - animation, hyperlink to slide No. 18).
VI. Checking the level of assimilation of the studied material. Reflection.
Students take the test. On the same piece of paper, evaluate your work in class by filling out the table.
Test “Multiplication Rule”. Option 1.
1) –13 * 5
A. –75. B. – 65. V. 65. D. 650.
2) –5 * (–33)
A. 165. B. –165. V. 350 G. –265.
3) –18 * (–9)
A. –162. B. 180. C. 162. D. 172.
4) –7 * (–11) * (–1)
A. 77. B. 0. C.–77. G. 72.
Test “Multiplication Rule”. Option 2.
A. 84. B. 74. C. –84. G. 90.
2) –15 * (–6)
A. 80. B. –90. V. 60. D. 90.
A. 115. B. –165. V. 165. G. 0.
4) –6 * (–12) * (–1)
A. 60. B. –72. V. 72. G. 54.
VII. Homework.
Clause 35, rules, No. 1143 (a – h), No. 1145 (c).
Literature.
1) Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. “Mathematics 6. Textbook for educational institutions”, - M: “Mnemosyne”, 2013.
2) Chesnokov A.S., Neshkov K.I. “Didactic materials in mathematics for grade 6”, M: “Prosveshchenie”, 2013.
3) Nikolsky S.M. and others. “Arithmetic 6”: a textbook for educational institutions, M: “Prosveshchenie”, 2010.
4) Ershova A.P., Goloborodko V.V. “Independent and test papers in mathematics for 6th grade.” M: “Ilexa”, 2010.
5) “365 tasks for ingenuity”, compiled by G. Golubkova, M: “AST-PRESS”, 2006.
6) “Great encyclopedia Cyril and Methodius 2010”, 3 CD.