Similar terms 7. Educational and methodological material on algebra (grade 6) on the topic: Similar terms

Let an expression be given that is the product of a number and letters. The number in this expression is called coefficient. For example:

in the expression the coefficient is the number 2;

in the expression - the number 1;

in the expression this is the number -1;

in the expression, the coefficient is the product of the numbers 2 and 3, that is, the number 6.

Petya had 3 candies and 5 apricots. Mom gave Petya 2 more candies and 4 apricots (see Fig. 1). How many sweets and apricots does Petya have in total?

Rice. 1. Illustration for the problem

Solution

Let us write the problem condition in the following form:

1) There were 3 candies and 5 apricots:

2) Mom gave 2 candies and 4 apricots:

3) That is, Petya’s total:

4) Add candies with candies, apricots with apricots:

Consequently, the total became 5 candies and 9 apricots.

Answer: 5 candies and 9 apricots.

In Problem 1, in the fourth step, we dealt with the reduction of similar terms.

Terms that have the same letter part are called similar terms. Similar terms can differ only in their numerical coefficients.

To add (reduce) similar terms, you need to add their coefficients and multiply the result by the common letter part.

By adding similar terms we simplify the expression.

They are similar terms because they have the same letter part. Therefore, to reduce them, it is necessary to add up all their coefficients - these are 5, 3 and -1 and multiply by the common letter part - this is a.

2)

This expression contains similar terms. The common letter part is xy, and the coefficients are 2, 1 and -3. Let's look at these similar terms:

3)

In this expression, similar terms are and let's list them:

4)

Let's simplify this expression. To do this, we find similar terms. In this expression there are two pairs of similar terms - these are and , and .

Let's simplify this expression. To do this, let’s open the brackets using the distribution law:

There are similar terms in the expression - these are and , let's give them:

In this lesson, we became acquainted with the concept of coefficient, learned which terms are called similar, and formulated a rule for bringing similar terms, and we also solved several examples in which we used this rule.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. M.: Gymnasium, 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. M.: Education, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. M.: ZSh MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. M.: Education, Mathematics Teacher Library, 1989.

Homework

1. Internet portal Youtube.com ( ).
2. Internet portal For6cl.uznateshe.ru ().
3. Internet portal Festival.1september.ru ().
4. Internet portal Cleverstudents.ru ().

Instructions

Before bringing similar terms in a polynomial, it often becomes necessary to perform intermediate actions: open all brackets, raise and bring the terms themselves into standard form. That is, write them down as a product of a numerical factor and variables. For example, the expression 3xy(–1.5)y², reduced to standard form, will look like this: –4.5xy³.

Open all brackets. Omit the parentheses in expressions like A+B+C. If there is a plus sign in front, then all terms are retained. If there is a minus sign in front of the brackets, then change the signs of all terms to the opposite. For example, (x³–2x)–(11x²–5ax)=x³–2x–11x²+5ax.

If you need to multiply a polynomial by a polynomial, multiply all the terms together and add the resulting monomials. When raising the polynomial A+B to a power, use abbreviated multiplication. For example, (2ax–3y)(4y+5a)=2ax∙4y–3y∙4y+2ax∙5a–3y∙5a.

Reduce monomials to standard form. To do this, group numbers and powers with bases. Next, multiply them together. Raise the monomial to a power if necessary. For example, 2ax∙5a–3y∙5a+(2xa)³=10a²x–15ay+8a³x³.

Find the terms in the expression that have the same letter part. Highlight them with special underlining for clarity: one straight line, one wavy line, two simple lines, etc.

Add up the coefficients of similar terms. Multiply the resulting number by the letter expression. Similar terms are given. For example, x²–2x–3x+6+x²+6x–5x–30–2x²+14x–26=x²+x²–2x²–2x–3x+6x–5x+14x+6–30–26=10x–50.

Sources:

• Monomial and polynomial
• Wash plz: write down: a) the sum where the first term is

Even the most complex equation ceases to look intimidating if you reduce it to a form that you have already encountered. The simplest way, which helps out in any situation, is to reduce polynomials to standard form. This is a starting point from which you can move forward towards a solution.

You will need

• paper
• colored pens

Instructions

Remember the standard form so you know what you should get as a result. Even the order of writing is significant: the members with the largest should come first. In addition, it is customary to first write down the unknowns, indicated by the letters at the beginning of the alphabet.

Write down the original polynomial and start searching for similar terms. These are the members of the equation given to you, the same letter part and/or digital part. For greater clarity, highlight the pairs found. Please note that similarity does not mean identity - the main thing is that one member of the pair contains the second. So, there will be terms xy, xy2z and xyz - they have a common part in the form of the product of x and y. The same goes for sedate ones.

Label different similar members differently. To do this, it is better to emphasize with single, double and triple lines, use color and other line shapes.

Having found all similar members, start combining them. To do this, remove similar terms from the found ones out of brackets. Remember that in standard form a polynomial has no such terms.

Check to see if you have any duplicate elements in your entry. In some cases, you may have similar members again. Repeat the operation combining them.

Make sure that the second condition required for writing a polynomial in standard form is met: each of its participants must be depicted as a monomial in standard form: in the first place is a numerical factor, in the second place is a variable or variables, following in the order already indicated. In this case, it has a letter sequence specified by the alphabet. Decreasing degrees are taken into account secondarily. Thus, the standard form of a monomial is the notation 7xy2, while y27x, x7y2, y2x7, 7y2x, xy27 are not required.

Video on the topic

Zodiac signs are the main element of astrology. These are 12 sectors (according to the number of months in a year), into which the zodiac zone is divided, according to the astrological tradition of Europe. Each of them has a name, depending on the zodiac constellation located in this area. There is a version according to which the names of the signs are based on ancient Greek myths.

Instructions

Aries is a ram with golden wool. The name of this sign is associated with the myth of the Golden Fleece. People born under the sign of Aries are seemingly meek, like this animal, but at a decisive moment they are capable of bold actions.

Taurus is a kind and at the same time violent animal. The origin of the name of this sign is associated with the legend of Jupiter and Europa. The loving god fell in love with a beautiful girl, and to win her he turned into a beautiful snow-white bull. Europe began to caress the animal and climbed onto its back. And the insidious Jupiter took her to the island of Crete.

The twins are the personification of the myth of the brotherly love of Pollux and Castor, who were ready to die for each other. According to legend, during the battle Castor was wounded and died in the arms of his brother, Pollux was immortal and turned to his father Zeus to allow him to die along with his brother.

A giant crayfish dug its claws into Hercules' leg during his battle with Hydra. He crushed the cancer and continued the battle with the snake, but Juno (it was on her orders that the cancer attacked Hercules) was grateful to him and placed the image of the cancer alongside other heroes.

The Nemean lion is a terrible and formidable animal that for a long time attacked people in the name of preserving the peace of power. Hercules defeated him. From the point of view of mythology, a lion is an attribute of power. People born under this sign have a sense of pride and great self-esteem.

Virgo is mentioned in the ancient Greek myth of the creation of the world. Legend has it that Pandora (the first woman) brought to earth a box that she was forbidden to open, but she could not resist the temptation and opened the lid. All misfortunes, hardships, grief and human vices scattered from the box. After this, the Gods left the earth, the goddess of innocence and purity Astraea (Virgo) was the last to fly away, and the constellation was named after her.

The name of the zodiac sign Libra is associated with the myth of the goddess of justice Themis, who had a daughter, Dika. The girl weighed the actions of people, and her scales became the symbol of the sign.

Scorpio, according to one legend, stung Orion, who tried to rape the goddess Diana. After the death of Orion, Jupiter placed him among the stars.

Sagittarius is a centaur. According to ancient Greek myths, it is half horse, half man. In the myth of the centaur Chiron, the main character knew everything and about everything, taught the gods sports, the art of healing and other knowledge and skills that they should have.

Capricorn is an animal with powerful hooves that is capable of climbing mountain slopes, clinging to ledges. In Ancient Greece, it was associated with Pan (the god of nature), who was half man and half goat.

The sign of Aquarius is named after a young man named Ganymede, who worked as a cupbearer and treated earthly people at holidays and celebrations. The young man had excellent human qualities, was an excellent friend, interlocutor and simply a decent person. For this, Zeus made him the cupbearer of the gods.

The last sign of the zodiac circle is Pisces. The appearance of its name is associated with the myth of Eros and Aphrodite. The goddess was walking with her son along the shore and they were attacked by the monster Typhon. To save them, Jupiter turned Eros and Aphrodite into fish, who jumped into the water and disappeared into the sea.

Bringing fractions to the least denominator otherwise called abbreviation fractions. If your math results in a fraction with large numbers in the numerator and denominator, check to see if it can be reduced.

Simple mathematical operations - addition, subtraction, multiplication, and so on - do not cause much difficulty for students. There is simply nothing to be confused about here. However, it happens that the expression from the problem has a very long alphanumeric notation. This distracts attention, disrupts the train of thought, and most importantly, most often takes a person away from the simplest decision.

It was to simplify mathematical operations that special concepts were invented - for example, similar terms. What is meant by this term, and how can the principle of similarity be used?

Which terms and in what expressions are considered similar?

The expression as such must consist of letter designations or letters and numbers - and of course, it must contain addition, because we are talking about terms. Moreover, in order to talk about similarity, individual terms must have the same letter in their composition.

For example, let's look at the small expression 2a + 3c + 4a. The first and third parts of the expression contain the same letter “a”. Accordingly, by this criterion they are similar terms.

What does this understanding give us in practice?

In order to solve the above expression, you can go in two ways:

• Find the product 2*a, add the product 3*c to it, add the product 4*a to the sum. It's not that difficult - but the longer the expression, the more tedious the calculations become.
• Take advantage of the properties of similar terms and first transform the expression into a simpler and more convenient form in order to find a solution faster.

For any task, it is preferable to choose the second method - it saves time and reduces the possibility of making a mistake.

What does the term “reduction” mean for such terms?

This is a rearrangement of terms so that similar ones are next to each other. From earlier rules we remember that it does not matter in what order the terms of the expression appear when adding - the sum still turns out to be the same.

Thus, our example can be transformed as follows - write it as 2a + 4a + 3c. But that's not all. For simplicity, the numerical coefficients can be put in brackets and added separately - and the letter “a” can be left out of brackets for now.

It will look like this (2 + 4)a + 3c = (6)a + 3c = 6a + 3c. We no longer need to separately calculate the product for each of these terms - we can first add them together, and only then multiply the resulting result.

“Similar terms” - Mathematics textbook, grade 6 (Vilenkin)

Short description:

In this section you will learn what the expression “similar terms” means and how to find them.
You have already learned how to open parentheses, learned the distributive property of multiplication, and know what a numerical-letter expression means (remember, this is an expression like 5a, 6ac). Now let's look at an expression like 8a+8c. Have you noticed that the first term and the second term have the same coefficient - the number 8? In this case, the number 8 can be taken out of brackets and presented as one of the factors of the product, that is, 8 * (a + c). It turns out that 8 is the common factor of the first and second terms.
Now let’s look at this example: 10a+15a-20a. Each of the terms (10a, 15a, -20a) has the same letter part (a), but the coefficients are different (10, 15 and -20). Such terms are called similar (that is, similar to each other). Such an expression can be rewritten in another way, by taking out the literal expression (that is, a) as a factor, and in brackets from each term only a number (coefficient) will remain: a*(10+15-20)=a*5=5a. Thus, we simplified the numerical-letter expression by finding similar terms. That is, similar terms are numerical-letter expressions that have the same letter part. The addition that we performed in the example is called reduction (or addition) of similar terms (that is, their coefficients are summed and the resulting result is multiplied by a letter).

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Slide captions:

Lesson in 6th grade on the topic “Similar terms” 04/06/2018

Lesson objectives: Review the rules for calculating the sum of two numbers. Repeat the coefficients of the terms. Repeat the algorithm for reducing similar terms. Consolidate the acquired knowledge. Develop communication skills.

Oral counting “Adding rational numbers” -22 + 35 -3.7 + 2.8 1.5 + (-6.3) 8.2 + (-8.2) 22 – 27 -13 – 8 19– (- 2) -27 – (-3) -35 + (-9) 13 -0.9 -4.8 0 -5 -21 21 -24 -44

Distributive property of multiplication (a + b) c = ac + sun (a - b) c = ac - sun c (a + b) = ca + ca c (a - b) = ca – ca or OPENING BRACKETS

Open the brackets. 2(x+1); 3(a-2); -2(2x+1); (2a-4b+3)(-3); -(4x-2y+9); -5(-а+2в+3); 5(-2a+4); -(3v-5); -2(-5x-8).

Textbook p. 224 No. 1281 (c, e)

At 5 45. Name the coefficients in these expressions: expression coefficient 2 x - 15 y 18 z - 9 t a -b 2 - 15 18 -9 1 - 1 Name the coefficients of the terms and simplify the expression 3 x – 8 x. Coefficients of terms: 3 and -8. The expression can be simplified: 3 x – 8 x = (3 – 8) x = – 5 x 3 x – 8 x = – 5 x 3 x and – 8 x differ only in similar coefficients

Conclusion: terms with the same letter part are called similar. Similar terms differing only in coefficients

NAME THE COEFFICIENTS OF THE TERMS AND SIMPLIFY THE EXPRESSION: 6 x + 8 x = 6 and 8 14 x 6 x – 8 x = 6 and –8 – 2 x – 6 x – 8 x = – 6 and –8 – 14 x – 6 x + 8 x = – 6 and 8 2 x

NAME THE COEFFICIENTS OF THE TERMS AND SIMPLIFY THE EXPRESSION: x + 3 x = 1 and 3 4 x 5 x – x = 5 and – 1 4 x – x – 7 x = – 1 and – 7 – 8 x – 9 x + x = – 9 and 1 – 8 x

NAME THE COEFFICIENTS OF THE TERMS AND SIMPLIFY THE EXPRESSION: x + x = 1 and 1 2 x x – x = 1 and – 1 0 – x – x = – 1 and – 1 – 2 x – x + x = – 1 and 1 0

Commented completion of tasks. Simplify 1. 3x + 5x; 2. 2x – 4x; 3. – 5у – 3у; 4. – 12a + 2a; 5. V + 15 V; 6. – y – 13u; 7. 8k – k.

Mathematical dictation: “Opening brackets and bringing similar terms.” Simplify the expression: 4 x – 9 x = Check yourself: – 5 x; 1) – 14 y; 2) – 10 a; 3) 1 4 b ; 4) – 19 n; 5) 3 p; 6) – 6 y – 8 y = – 14 a + 4 a = 13 b + b = – n – 18 n = 4 p – p =

Task: give similar terms No. Expression 1) 3t + 4t – 10t = 2) 0.9v - 1.3v + 0.7v = 3) 5t – (3t – 5) + (2t – 5) = 4) 3(v – 5) – (in – 3) = 5) 0.2t – 2/9 – 4t + 2/9 = 6) 1/3(3v – 18) – 2/7(7v – 21) = 7) – 4t + 8t – t = Answer -3 m 0.3b 4m 2b-12 -3.8m -b 3m

Task: bring similar terms 1) 3a + 0.2a – 5.2a + 4a = 2) –4c + 6.7c – 2c +7.3 c = 3) x – 2.45x + 3x + 2.45x = 4 ) –2d + d – 0.2d + 9.2d = 5) 5.6t – 2t – 3.6t + t = 2a 8c 4x 8d m