Presentation for the lesson "Addition and subtraction of polynomials". Presentation on the topic "Addition and subtraction of polynomials" Addition and subtraction of polynomials presentation

Warm-up "Own game"
Myths and mathematics
Game "Arrow"
Pair work "Don't let me down"
Constructor

Slide 2-Category Selection

This slide is the main game board. You go here to begin the game, and you return here after each Question/Answer slide. This is where the “contestant” selects one of the five categories and a dollar value for the question. The higher the value, the more difficult the question. When you open this slide, the categories appear one at a time, and the dollar values appear at random with an accompanying laser beep. Here's how it works: if the contestant selects the first category for $300, you would click on the $300 text under

Polynomials

monomials

From theory

Properties

degrees

the 1st category(i.e., the 3rd dollar box in column one). As a result, the corresponding Question/Answer slide will automatically appear. Once the question, and then the answer, for that slide have been shown, you will click on the arrow in the bottom right of that slide to return to this main slide. When you return to this slide, the dollar amount for the box you selected will have changed from white to blue to show that that particular question has already been used. Below, you will see how to tailor the game for your particular categories.

Five different categories are used in the game. The category names appear at the top of the columns on this slide and on the five associated Question/ Answer slides (one for each dollar value). Rather than changing all of these separately, you will use the Replace command to change each placeholder category name only once.

1. Under Edit , choose Replace

Type the placeholder name for category 1 as shown in the pop-up at the right. Type in your
Type the placeholder name for category 1 as shown in the pop-up at the right.
Type in your category name (e.g., Mixed Numbers) under Replace with:

The Replace pop-up should now look like the one on the right, only with your category name.

Click the Replace All button to make the changes.

You will then see this pop-up

You will then see this pop-up

Click the OK button. This replaces the six occurrences of the specified placeholder category name with your category name. After this, the top of the slide will look like this:

Notice that in this case, “Mixed Numbers” doesn’t fit on the line. To fix this, simply click on the text right before the “N” and press Backspace followed by Enter. Now it's on two lines:

2. Now, repeat Step 1 for the remaining four category placeholder names:

Slide 3-Question/Answer (Cat1, $100)

This slide is the first Question/Answer slide. It corresponds to Category 1 for $100. Once you have followed the instructions on Slide 2 to replace category name placeholders with your actual categories, the text “Cat1” on this slide will be replaced with your 1st category name.

When you click on Category 1 for $100 on the main slide, this slide opens automatically, with the Question appearing at the top. (Note: On TV Jeopardy, the contestant is actually shown an

Properties of degrees for 10

Perform transformations:

answer and is asked to offer a related question. Since this concept is sometimes difficult to understand and implement, this PowerPoint version shows a question followed by the corresponding answer.)

One way to play the game in class is to set up three teams. For each round, have one person from each team stand up as contestants. Have one pick the category and dollar value; click on that box and then ready the question that appears. Call on the first contestant that raises his or her hand for the answer. If they are correct, their team gets corresponding points or dollars (e.g., 1 point for each $100). If the first contestant misses the question or does not answer quickly enough, his or her team loses the corresponding points. Then, offer the question to the remaining two contestants in order of their raised hands. After the question has been correctly answered, or after all three contestants miss it, or after no contestant wants to try, return to the main slide by clicking on the yellow arrow. The current contestants then sit down, and the game moves to the next round.

Note that this Jeopardy game does not have a Double Jeopardy question.

To tailor this slide, follow these instructions:

You are now ready to put in your questions and answers, but you might want to go ahead and save this file first, using Save As and giving it a new name-one that makes sense for this particular Jeopardy game (eg, Fractions Jeopardy) .

If your Question is short, simply double click on the word “Question” and type in your specific question (e.g., “50% of 150” or “Capitol of France”). If the text you enter will not fit on one line, there's room for two lines at this font size. If you need more room, reduce the font size by triple clicking on the text and using the Font Size selector in the toolbar. In some cases, your question may need a drawn figure or graphic. You can use PowerPoint features to draw the figure you need or to insert graphics. A few examples are shown below.
Double click on the word “Answer” and type in your answer in the same way.
Do the same steps to tailor the remaining Question/Answer slides, remembering to make questions of higher dollar value more difficult. Also remember to save your work.

Example Questions:

Properties of degrees for 20

Perform transformations:

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Calculate:

Properties of degrees for 30

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You can easily customize this template to create your own Jeopardy game. Simply follow the step-by-step instructions that appear on Slides 1-3.

Calculate:

Properties of degrees for 40

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Name the coefficients

monomial:

Monomials for 10

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Determine the degree

monomial:

Monomials for 20

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Monomials for 30

Bring monomial to standard form

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Represent in the form

monomial square:

Monomials for 40

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From theory for 10

Formulate a definition

polynomial

A polynomial is called

sum of monomials

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You can easily customize this template to create your own Jeopardy game. Simply follow the step-by-step instructions that appear on Slides 1-3.

Formulate a definition

monomial

A monomial is a product

numbers, variables

and their degrees

From theory for 20

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What monomials

called similar?

monomials that differ

only from each other

coefficients are called

similar

From theory for 30

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You can easily customize this template to create your own Jeopardy game. Simply follow the step-by-step instructions that appear on Slides 1-3.

What is a ratio?

The numerical factor of the monomial,

written in standard

form called

coefficient

From theory for 40

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Give similar

Polynomials in 10

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Give similar

Polynomials for 20

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What is the degree

polynomial?

Polynomials for 30

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Find the value

expressions

Polynomials for 40

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The appearance of some mythical characters consists

from the head and torso, taken from different creatures.

Decipher their names.

the character

ANSWER

Centaur

Minotaur

Sphinx

Chimera

Output

2a+4c a-3c 3a+c 4a-2c

5x-3y -2x+y 3x-2y x-y

2a+4s a-3s a +7s -10s

5x-3y -2x+y 7x-4y -9x+5y

1 option

6a - 5a = a

Option 2

- 3a + (-5 b) = -8b

3 a

2 a

3 option

- 4c - 6c = -10c

4 option

-12x+ 10 x = - 2 x

90 points and above - score "5"
70 - 89 points - score "4"
50 - 69 points - score "3"
below 50 points - score "2"

"4" - No. 596, No. 606 (a)

"5" - No. 596, No. 606 (a), No. 609 *

Mathematics in translation from ancient Greek means study, knowledge, science. This queen of sciences puts the mind in order, helps to discipline oneself and, having understood its principles, skillfully apply them in life. To everyone who comes into contact with her, she gives clear thinking.

It is time to clearly recall the previously studied concept of "polynomial". The answer is simple: polynomial (or polynomial) is the sum of monomials.

slides 1-2 (Presentation topic "Addition and subtraction of polynomials", example)

Now we need to learn how to perform simple arithmetic operations with polynomials. Let's start with the usual addition.

For example: we have two polynomials: the first is a^3-7a^2-1 and the second is 3a^3-a^2+6

Let's try to put them together. And as we solve this problem, we will formulate the rule for the addition of polynomials.
So, let's begin. We put each individual polynomial in brackets and put a “+” sign between the brackets like this: (a ^ 3-7a ^ 2-1) + (3a ^ 3-a ^ 2 + 6)
Then we open the brackets, and since there is a “+” sign between the brackets, we do not change the signs. It looks like this: (a^3-7a^2-1)+(3a^3-a^2+6)=a^3-7a^2-1+3a^3-a^2+6

Connect: a^3-7a^2-1+3a^3-a^2+6=4a^3-8a^2+5
We got the answer: 4a^3-8a^2+5

slides 3-4 (examples, rules for opening brackets)

We will perform similar actions with another simple function - subtraction. Again, it is proposed to take two polynomials: the first 5b^2 - b + 1 and the second 8b^2 + 3b - 6

Again we put them in brackets and put a minus sign between the brackets: (5b^2 - b + 1) - (8b^2 + 3b - 6)

We open the brackets, changing the signs to the opposite if there was a “minus” before the bracket, and again we give similar terms:
(5b^2 - b + 1) - (8b^2 + 3b - 6) = 5b^2 - b + 1 - 8b^2 - 3b + 6 = - 3b^2 - 4b + 7
Answer: - 3b^2 - 4b + 7

Now we will do the opposite, namely, we will learn how to correctly put brackets after the plus or minus signs.
Take, as an example, the following polynomial 5x - 3y + 1.
Task: correctly put brackets after 5x and signs "+" or "-", considering the following rules:

1. If a plus sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs. Let's use plus.
Then the polynomial will look like this: 5x - 3y + 1 = 5x + (- 3y + 1)

2. If a minus sign is placed before the brackets, then all members enclosed in brackets must be reversed.

The same polynomial only with a minus sign 5x - 3y + 1 = 5x - (3y - 1)

slide 5 (example)

As it turned out - "everything ingenious is simple."

It remains only to conclude: When adding and subtracting polynomials, we use the same principle, so there is no need to distinguish between these operations. Naturally, there is no need to use the two terms "addition of polynomials" and "subtraction of polynomials". Incredibly, two seemingly different opposite functions are actually one concept "algebraic sum of polynomials".

The simplest problems with a polynomial
We did it today, friends.
And the conclusion was bold:
That the brothers "plus" and "minus"
Two sides of the same coin -
Algebraic sum of being.

This once again shows the unity of opposites, such as yes and no, day and night, rest and movement, action and reaction. All this is our one huge concept - life!

Addition and subtraction of polynomials
Algebra lesson
in the 7th grade
Teacher MOSSh No. 29 Khachankova T.V.

Goals and objectives:

Educational:
To test the knowledge, skills and abilities of students on the topic of the sum and difference of polynomials.
Educational:
Raise interest in algebra by applying interesting tasks using various forms of work.
Developing:
To develop the ability of students to work both individually (independently) and collectively (work in pairs and in a group).
Develop the ability to assess your strengths, using tasks of different levels of complexity.

Name the coefficients of monomials Give similar terms:

Answer:

Solve the polynomial addition example:

Answer:

Solve the polynomial subtraction example:

Answer: a

After opening parentheses:

Work on cards Work in pairs Answers of pair tasks Work in groups Mathematical

"Brain Ring"

Question:

In the Middle Ages, people who knew how to produce THIS IS AN ARITHMETIC OPERATION could almost be counted on the fingers. They were respectfully called "masters ...".
They moved from city to city at the invitation of merchants who wanted to put their accounts in order.
In Italy, the saying is still preserved: “This is a difficult task - ...”. This is what they usually say when they face an almost insoluble problem.

What is this action?

Question:

A man who wanted to be both a lawyer and a philosopher, but became a mathematician. He was the first to introduce a rectangular coordinate system.

What is the name of this person?

Question:

This word among jewelers means the proportion of gold in the product, equal to 1/24 and a unit of mass equal to 200 mg.

What is this value?

Question:

Before us is the picture of Bogdanov-Belsky "Oral Account". 11 students find in their minds the meaning of the expression written on the blackboard by teacher Rachinsky. Let's help these students find the answer. Example written on the board:

What answer?

Question:

Name an ancient geometrical instrument, which, according to the Roman poet Ovid (Iv.), was invented in ancient Greece.
Prompt. We often use this tool in algebra and geometry lessons.

What is this geometric tool?

To the question: "How many fish did you catch?", the fisherman replied: "A half of eight, six without heads and nine without a tail."

To the question: "How many fish did you catch?", the fisherman replied: "A half of eight, six without heads and nine without a tail."

How many fish did the fisherman catch?

Question:

How old is the ancient Oak, if the number-lovers reported that it has been standing in this place for exactly 2964 months.

How old is the oak tree?

Question:

This number comes from the Latin word "solus".
Both in Ancient Russia and in Ancient Rome it was associated with the Sun, while among the ancient Greeks this number was not considered a number.

What is this number?

Question:

The player bets $30. When he wins, he returns his bet plus $60. He spends a third of the total amount on a gift for his wife, $ 10 on a taxi and 10% of the remaining amount he gives to the driver for a tip.

How much money does he have left?

Calculation of the points scored for the lesson and filling out the sheets of personal achievements Thanks for attention!

Thanks for attention!

The purpose of the lesson: The formation of positive personality traits in the process of developing a skill on the topic: “Addition and subtraction of polynomials” Lesson objectives: 1. Subject: repeat the rule of addition and subtraction of polynomials, the rules of multiplication, division of degrees, the rules for expanding brackets and simplifying expressions. 2. Meta-subject: to adequately use speech to solve various communicative tasks, to speak and write, to consolidate the skills of partially search cognitive activity: to be aware of the problem, to draw conclusions. 3. Personal: the ability to conduct a dialogue based on equal relations and mutual respect, independently analyze the conditions for achieving the goal.

Orally a) 5av b) 1.5a 0.6c c) (2av) 2 - 1 d) 3c + c e) 7xy e) 6.7 - k Among the expressions, select polynomials Name the letters under which the polynomials of the standard form are written Name the degree of each polynomial a) 5av c) (2av) 2 - 1 d) 3c + c e) 6.7 - k 3411

Polynomials. Addition of polynomials. Present the polynomial in standard form -4ava - 2a 2 in 2 5a 2 0.2a 2 in 3 + 2a 4 in 3 - av 13a - 8c -5a 2 - 5c 2 a 3 -1.4c 2 + 5a 2 5a 2 in - 13c 2 a -4a 2 in - 2a 2 in 2 3a 4 in 3 - av

Arrange the polynomials in powers in the order: 4, 3, 5, 7, 7, 2, 1. and E x A 13a - 8c -5a 2 - 5c 2 a 3 -1.4c 2 + 5a 2 5a 2 c - 13c 2 a 3 a 4 c 3 - av l A ch i l l E What catch phrase is associated with the name of this hero? c -4a 2 c - 2a 2 c 2 5a 2 c - 13c 2 a -5a 2 - 5c 2 a 3 3a 4 c 3 - ab -1.4c 2 + 5a 2 13a - 8c

Achilles' mother, Thetis, dipped the baby into the waters of an underground river, making a person invulnerable. In this dive, she held Achilles by the heel, which remained dry and therefore vulnerable. During the Trojan War, an enemy arrow hit Achilles in the heel, as a result of which he died. The expression "Achilles' heel" in a figurative sense means "a weak, vulnerable spot."

Symbolically depicts the head, and the body Ox man + Lion goat + Man horse + Man lion bird + + body more 3x 2 y - 2x 2 7x 2 - 5x 2 y 7x 2 y 2 - 8x 2 y 3x 2 y - 2x 2 6x 2 y -2x 2 y 2 x 2 y 2 - 3x 2 y - xy 2 7x 2 y 2 - 8x 2 y2xy 2 -6x 2 y 2 5x 2 - 2x 2 y 5x 2 y 2 - 2x 2 y x 2 y 2 - 3x 2 x 2 y 2 - 5x 2 y

Checking homework 25.9 Find the sum and difference of polynomials A) p(a)=2a 5 +7a 4 +7a 3 +2a 2 +a+1 3a + 1 C) p (a) \u003d -2a 5 + a 4 + 9a 3 + a + 1 D) p (a) \u003d -2a 5 -7a 4 -3a 3 + 4a 2 -3a (a, b) A )p(x;y)=57x 3 -30x 2 y+8xy 2 -3y 3 B)p(x;y)=17x 3 +3y (a,b) Solve equation A) 3 B) -1

White - the easiest (2 points) Yellow - medium difficulty (3 points) Red - the most difficult (4 points) White (2a+5)+(3a-7)= (3a-4)+(11+3a)= Yellow (4x 5 +2x+1)+(x 5 +x-2)= (x 11 +x 6 -3)+(2x 11 +3x 6 +1)= Red (4y 4 +2y 2 -13)+ (4y 4 -4y 2 +13)= (18a 3 -3a 2 b-5ab 2 +2b 3)+(8a 3 +3a 2 b-5ab 2 +b 3)= 5a-2 6a+7 5x 5 +3x -1 3x 11 +4x y 4 -2y 2 26a 3 -10ab 2 +3b 3

White - the easiest (2 points) Yellow - medium difficulty (3 points) Red - the most difficult (4 points) White (3a-4)-(-1-5a)= (5a-2)-(3a+4) = Yellow (2y 3 +8y-11)-(3y 3 -6y+3)= (15-7y 2)-(y 3 -y 2 -15)= Red (y 3 -y+7)-(y 3 +5y+11)= (x 5 +x-2)-(4x 5 +2x-1)= 8a-3 2a-6 -y 3 +14y y 3 -6y 2 -6y-4 -3x 5 -x- one

C6C6 C 11 C5C5 C 12 C3C3 C 10 C 13 -4x3x-7x-2x7x-3x k) 6) (s 5) 2 \u003d (o) 3) (s 2) 6 * s \u003d (m) 7) (s 3) 4 \u003d (i) 4) s 7 * s 3 * s \u003d (o) 1) 4x-8x \u003d (e) 4) -9x + 15x + x \u003d (i) 2) -3x + x \u003d (l) 5) -2x + 5x \u003d (c) 3) x 3 - (x 3 + 7x) \u003d (k) 6) 7x 2 - (3x + 7x 2) \u003d (e) C6PS6P C 11 O C5LS5L C 12 And C3HC3H C 10 O C 13 M -4x E 3x B -7x K -2x L 7x I - 3x D

Group work In the Middle Ages, people who knew how to perform THIS ARITHMETICAL OPERATION could be counted almost on the fingers. They were respectfully called "masters ...". They moved from city to city at the invitation of merchants who wanted to put their accounts in order. In Italy, the saying is still preserved: “This is a difficult task - ...”. This is what they usually say when they face an almost insoluble problem.

Algebra Lesson Plan Grade 7

"Addition and subtraction of polynomials"

Type of lesson: lesson learning new material.

Equipment and materials: computer, projector, interactive whiteboard.

Educational: introduce the rule of addition and subtraction of polynomials; teach how to apply the rule when simplifying an expression; to consolidate the skills of partially exploratory cognitive activity: to be aware of the problem, to draw conclusions and generalizations.

Developing: to excite students' interest in educational material and cognitive actions in which the above skills are formed; development of logical thinking, intuition, attention; development of the ability to independently solve educational problems and work with additional literature.

Educational: to instill interest in the subject; the formation of communication skills, the ability to work in a team.

During the classes.

I. Organizational moment

Polynomials are the foundation on which the majestic edifice of algebra rests. Actions with polynomials are widely used in solving various kinds of exercises both in the 7th grade and in the senior grades. Historical information.

The topic “Polynomials” is a very important topic in algebra. Many scientists have worked on this topic. In 1799, the German scientist Gauss proved the fundamental theorem of the algebra of polynomials with complex coefficients, at the end of the 18th century. French mathematician Bezout proved the fundamental polynomial theorem with real coefficients.

II. Updating the basic knowledge of students

Let's check how you learned the material of the last lesson!

III. Learning new material

So, in today's lesson, we have to find out what happens as a result of adding two or more polynomials or subtracting from one other polynomial

a) Sum the polynomials 5x 2 + 2x - 1 and 7x + 4 and convert it to a polynomial of standard form. The teacher decides and explains, with the involvement of students.

b) Compose the difference of the polynomials 5x 2 + 2x - 1 and 7x + 4 and transform it into a polynomial of standard form.

Ask students to draw conclusions.

Adding and subtracting polynomials again results in a polynomial .

Find the rule in the textbook and review the examples on page 109 of the textbook.

In order to perform the inverse problem - to represent a polynomial as a sum or difference of polynomials, you must use the rule:

If a plus sign is placed before the brackets, then the terms that are enclosed in brackets are written with the same signs; if a minus sign is placed in front of the brackets, then the terms enclosed in brackets are written with opposite signs.

For example, 3x 3 -2x 2 -x+4=3x 3 -2x 2 +(-x+4)

3x 3 -2x 2 -x+4=3x 3 -2x 2 -(x-4)

Algorithm for addition and subtraction of polynomials

Expand brackets

Bring Like Members

Two polynomials whose sum is zero are called opposite.

Fill the gaps:

a) (2a -3b) + _____________ \u003d 0

b) (7 a 2 - 12a + 4) - (___________) = 0

c) (__________) + (-4a +3b) = 0

d) (___________) + (-3a 2 -2a +1) = 0

IV. Consolidation of the studied material

1. Find the algebraic sum of polynomials

a) (7x-19y) -(18y -3x) + (6x-16y)

b) (x 3 -2x 2 -x-7) - (-3x -2x 2 + x 3 +5)

2. Solve the equations:

(2x - 1) + (- x + 5) = 2

(43 - 12x) - (- 7x + 33) = -2

(2x - 10) - (3x - 4) = 6.

Solve at the blackboard No. 3.35 (h), No. 3.39 (h)

Fizkultminutka.

V. Primary control of mastering the material

Checking the test results.

VI. Homework

P. 3.5, No. 3.35(n), 3.39(n)

VII. Lesson summary

Review the rules for adding and subtracting polynomials.

Orally solve No. 3.34(1 - 4)

IX. Reflection.

The children are invited to choose a token with a certain color:

Black - boring, not interesting. Blue is not always clear. Green is interesting.

This survey allows you to assess the quality of the lesson and adjust it for further use.

It might be useful to read: