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Lesson in grade 6 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 like terms. Consolidate the acquired knowledge. Develop communication skills.
Mental counting “Addition of 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 \u003d ac + sun (a - c) c \u003d ac - sun c (a + c) \u003d ca + ca c (a - c) \u003d ca - ca or BRACKET OPENING
Expand the brackets. 2(x+1); 3(a-2); -2(2x+1); (2a-4c+3)(-3); -(4x-2y+9); -5(-a+2b+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. The coefficients of the terms: 3 and -8. The expression can be simplified: 3 x - 8 x \u003d (3 - 8) x \u003d - 5 x 3 x - 8 x \u003d - 5 x 3 x and - 8 x differ only in similar coefficients
Conclusion: terms that have 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 \u003d 6 and 8 14 x 6 x - 8 x \u003d 6 and -8 - 2 x - 6 x - 8 x \u003d - 6 and -8 - 14 x - 6 x + 8 x \u003d - 6 and 8 2 x
NAME THE COEFFICIENTS OF THE TERMS AND SIMPLIFY THE EXPRESSION: x + 3 x \u003d 1 and 3 4 x 5 x - x \u003d 5 and - 1 4 x - x - 7 x \u003d - 1 and - 7 - 8 x - 9 x + x \u003d - 9 and 1 - 8 x
NAME THE COEFFICIENTS OF THE TERMS AND SIMPLIFY THE EXPRESSION: x + x \u003d 1 and 1 2 x x - x \u003d 1 and - 1 0 - x - x \u003d - 1 and - 1 - 2 x - x + x \u003d - 1 and 1 0
Commented execution of tasks. Simplify 1. 3x + 5x; 2. 2x - 4x; 3. - 5y - 3y; 4. - 12a + 2a; 5. in + 15v; 6. - y - 13y; 7. 8k - k.
Mathematical dictation: "Opening brackets and reducing like terms." Simplify the expression: 4 x - 9 x \u003d Test yourself: - 5 x; 1) – 14 y ; 2) – 10 a ; 3) 1 4 b ; 4) – 19n; 5) 3p; 6) – 6 y – 8 y = – 14 a + 4 a = 13 b + b = – n – 18 n = 4 p – p =
Task: bring like terms No. Expression 1) 3t + 4t - 10t \u003d 2) 0.9v - 1.3v + 0.7v \u003d 3) 5t - (3t - 5) + (2t - 5) \u003d 4) 3 (v - 5) - (in - 3) \u003d 5) 0.2t - 2/9 - 4t + 2/9 \u003d 6) 1/3 (3in - 18) - 2/7 (7in - 21) \u003d 7) - 4t + 8t - t \u003d Answer -3 m 0.3b 4m 2b-12 -3.8m -b 3m
Task: bring like terms 1) 3a + 0.2a - 5.2a + 4a \u003d 2) -4c + 6.7c - 2c + 7.3 c \u003d 3) x - 2.45x + 3x + 2.45x \u003d 4 ) -2d + d - 0.2d + 9.2d = 5) 5.6t - 2t - 3.6t + t = 2a 8c 4x 8d m
“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 brackets, learned the distributive property of multiplication, you know what a numerical-literal expression means (remember, this is an expression like 5a, 6ac). Now let's consider 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 represented as one of the factors of the product, that is, 8 * (a + c). It turns out that 8 is a common factor of the first and second terms.
Now consider 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 a different way, taking out the literal expression (that is, a) as a factor, and only the number (coefficient) will remain in brackets from each term: a * (10 + 15-20) \u003d a * 5 \u003d 5a. Thus, we have simplified the numerical-literal expression by finding similar terms. That is, similar terms are numerical-literal expressions that have the same literal part. The addition that we performed in the example is called the reduction (or addition) of similar terms (that is, their coefficients are summed and the result obtained is multiplied by a letter).
Example 1 Let's open the brackets in the expression - 3 * (a - 2b).
Decision. We multiply - 3 by each of the terms a and - 2b. We get - 3 * (a - 2b) \u003d - 3 * a + (- 3) * (- 2b) \u003d - 3a + 6b.
Example 2 Let's simplify the expression 2m - 7m + 3m.
Decision. In this expression, all terms have a common factor m. Hence, by the distributive property of multiplication, 2m - 7m + Зm = m (2 - 7 + 3). The amount in brackets coefficients all terms. It is equal to -2. Therefore 2m - 7m + 3m = -2m.
In the expression 2 m - 7 m + 3m, all terms have a common letter part and differ from each other only by coefficients. Such terms are called similar.
Terms that have the same letter part are called similar terms.
Similar terms can differ only by coefficients.
To add (or say: bring) like terms, you need to add their coefficients and multiply the result by the common letter part.
Example 3 We present similar terms in the expression 5a + a -2a.
Decision. In this sum, all terms are similar, since they have the same letter part a. Let's add the coefficients: 5 + 1 - 2 = 4. So, 5a + a - 2a = 4a.
What terms are called similar terms? How can similar terms differ from each other? Based on what property of multiplication is the reduction (addition) of like terms performed?
1265. Expand the brackets:
a) (a-b + c) * 8; e) (3m-2k + 1)*(-3);
b) -5*(m - n - k); f) - 2a*(b+2c-3m);
c) a*(b - m + n); g) (-2a + 3b + 5c) * 4m;
d) - a*(6b - 3c + 4); h) - a*(3m + k - n).
1266. Perform actions by applying the distribution property multiplication:
1267. Add like terms:
Expressions like 7x-3x+6x-4x read like this:
- the sum of seven x, minus three x, six x and minus four x
- seven x minus three x plus six x minus four x
1268. Reduce like terms:
1269. Open the brackets and give like terms:
1270. Find the value of the expression:
1271. Decide the equation:
a) 3*(2x + 8)-(5x+2)=0; c) 8*(3-2x)+5*(3x + 5)=9.
b) - 3*(3y + 4)+4*(2y -1)=0;
1272. A kilogram of potatoes costs 20 kopecks, and a kilogram of cabbage costs 14 kopecks. Potatoes were bought 3 kg more than cabbages. They paid 1 for everything. 62 k. How many kilos of potatoes and how many cabbages did they buy?
1273. A tourist walked 3 hours and rode a bicycle for 4 hours. In total, he traveled 62 km. At what speed did he walk if he walked 5 km/h slower on foot than he rode a bicycle?
1274. Calculate orally:
1275. What is the sum of a thousand terms, each of which is equal to -1? What is the product of a thousand factors, each of which is -1?
1276. Find the value of the expression
1-3 + 5-7 + 9-11+ ... + 97-99.
1277. Orally solve the equation:
a) x + 4=0; c) m + m + m = 3m;
b) a+3=a -1; d) (y-3)(y + 1)=0.
1278. Multiply: 
1279. What is the coefficient in each of the expressions:
1280. The distance from Moscow to Nizhny Novgorod is 440 km. What should be the scale of the map so that on it this distance has a length of 8.8 cm?
1285. Solve the problem:
1) The combine operator overfulfilled the plan by 15% and harvested grain on an area of 230 hectares. How many hectares, according to the plan, should the combine harvester harvest?
2) A team of carpenters spent 4.2 m3 of planks to renovate the building. At the same time, she saved 16% of the boards allocated for repair. How many cubic meters of boards were allocated for the renovation of the building?
1286. Find the value of the expression:
1) - 3,4 7,1 - 3,6 6,8 + 9,7 8,6; 2) -4,1 8,34+2,5 7,9-3,9 4,2.
1287. Use the graph to solve the problem: “Marina, Larisa, Zhanna and Katya can play on different instruments (piano, cello, guitar, violin), but each only on one. They also know foreign languages (English, French, German, Spanish), but each only one. Known:
1) the girl who plays the guitar speaks Spanish;
2) Larisa plays neither the violin nor the cello and does not know English;
3) Marina does not play the violin or the cello and does not know either German or English;
4) a girl who speaks German does not play the cello;
5) Jeanne knows French, but does not play the violin. Who plays what instrument and what foreign language does he know?”
1288. Expand the brackets:
a) (x+y-z)*3; d) (2x-y+3)*(-2);
b) 4*(m-n-p); e) (8m-2n+p)*(-1);
c) - 8 * (a - b-c); e) (a + 5- b-c) * m.
1289. Find the value of the expression by applying the distributive property of multiplication:
1290. Give like terms:
1291. Open the brackets and give like terms:
1292. Solve the equation:
1293. Bought one table and 6 chairs for 67 rubles. The chair is cheaper than the table by 18 rubles. How much is a chair and how much is a table?
1294. There are 119 students in three classes. There are 4 more students in the first grade than in the second grade and 3 fewer than in the third grade. How many students are in each class?
1295. Determine the scale of the map if the distance between two points on the ground is 750 m, and on the map 25 mm.
1296. What is the length of the segment shown on the map at a distance of 6.5 km, if the scale of the map is 1:25,000?
1297. On the map, a segment has a length of 12.6 cm. What is the length of this segment on the ground if the map scale is 1: 150,000?
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
Mathematics for grade 6 free download, lesson plans, getting ready for school online
Is an . In this article, we will define like terms, figure out what is called the reduction of like terms, consider the rules by which this action is performed, and give examples of reducing like terms with a detailed description of the solution.
Page navigation.
Definition and examples of similar terms.
A conversation about such terms arises after getting acquainted with literal expressions, when it becomes necessary to carry out transformations with them. According to the textbooks of mathematics N. Ya. Vilenkin definition of like terms is given in the 6th grade, and it has the following wording:
Definition.
Similar terms are terms that have the same letter part.
It is worth considering this definition carefully. First, we are talking about terms, and, as you know, terms are constituent elements of sums. This means that such terms can only be present in expressions that are sums. Secondly, in the voiced definition of such terms there is an unfamiliar concept of “literal part”. What is meant by the letter part? When this definition is given in the sixth grade, the letter part refers to one letter (variable) or the product of several letters. Thirdly, the question remains: “What are these terms with a letter part”? These are terms that are the product of a certain number, the so-called numerical coefficient, and the letter part.
Now you can bring examples of similar terms. Consider the sum of two terms 3·a and 2·a of the form 3·a+2·a . The terms in this sum have the same letter part, which is represented by the letter a , therefore, by definition, these terms are similar. The numerical coefficients of these similar terms are the numbers 3 and 2 .
Another example: total 5 x y 3 z+12 x y 3 z+1 the terms 5·x·y 3 ·z and 12·x·y 3 ·z with the same literal part x·y 3 ·z are similar. Note that y 3 is present in the literal part, its presence does not violate the definition of the literal part given above, since it is, in fact, the product of y·y·y .
Separately, we note that the numerical coefficients 1 and −1 for such terms are often not written explicitly. For example, in the sum 3 z 5 +z 5 −z 5 all three terms 3 z 5 , z 5 and −z 5 are similar, they have the same letter part z 5 and coefficients 3 , 1 and −1 respectively, of which 1 and −1 are not clearly visible.
Proceeding from this, in the sum 5+7 x−4+2 x+y, not only 7 x and 2 x are similar terms, but also the terms without the literal part 5 and −4 .
Later, the concept of the literal part also expands - I begin to consider the literal part not only the product of letters, but an arbitrary literal expression. For example, in the algebra textbook for grade 8 authors Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov, edited by S. A. Telyakovsky, a sum of the form is given, and it is said that its components terms are similar. The common literal part of these similar terms is an expression with a root of the form .
Similarly, similar terms in the expression 4 (x 2 +x−1/x)−0.5 (x 2 +x−1/x)−1 we can consider the terms 4 (x 2 +x−1/x) and −0.5 (x 2 +x−1/x) , since they have the same letter part (x 2 +x−1/x) .
Summarizing all the above information, we can give the following definition of similar terms.
Definition.
Similar terms terms in a literal expression are called that have the same literal part, as well as terms that do not have a literal part, where the literal part is understood to be any literal expression.
Separately, we say that similar terms can be the same (when their numerical coefficients are equal), or they can be different (when their numerical coefficients are different).
In concluding this paragraph, we will discuss one very subtle point. Consider the expression 2 x y+3 y x . Are the terms 2 x y and 3 y x similar? This question can also be formulated as follows: “Are the literal parts x y and y x of the indicated terms the same”? The order of the literal factors in them is different, so that in fact they are not the same, therefore, the terms 2·x·y and 3·y·x in the light of the definition introduced above are not similar.
However, quite often such terms are called similar terms (but for the sake of rigor it is better not to do this). In this case, they are guided by the following: according to the permutation of factors in the product, it does not affect the result, so the original expression 2 x y+3 y x can be rewritten as 2 x y+3 x y , whose terms are similar. That is, when they talk about similar terms 2 x y and 3 y x in the expression 2 x y+3 y x , they mean the terms 2 x y and 3 x y in transformed expression of the form 2 x y+3 x y .
Reduction of similar terms, rule, examples
The transformation of expressions containing similar terms implies the addition of these terms. This action has a special name - reduction of like terms.
The reduction of similar terms is carried out in three stages:
- first, the terms are rearranged so that similar terms are next to each other;
- after that, the literal part of similar terms is taken out of brackets;
- finally, the value of the numerical expression formed in brackets is calculated.
Let's analyze the recorded steps with an example. We present similar terms in the expression 3 x y+1+5 x y . First, we rearrange the terms so that the like terms 3 x y and 5 x y are next to each other: 3 x y+1+5 x y=3 x y+5 x y+1. Secondly, we take out the literal part of the brackets, we get the expression x·y·(3+5)+1 . Thirdly, we calculate the value of the expression that was formed in brackets: x·y·(3+5)+1=x·y·8+1 . Since it is customary to write the numerical coefficient before the letter part, we will transfer it to this place: x·y·8+1=8·x·y+1. This completes the reduction of similar terms.
For convenience, the three steps above are combined into rule for reducing like terms: to bring similar terms, you need to add their coefficients and multiply the result by the letter part (if any).
The solution of the previous example using the rule of reduction of like terms will be shorter. Let's bring him. The coefficients of similar terms 3 x y and 5 x y in the expression 3 x y+1+5 x y are the numbers 3 and 5, their sum is 8, multiplying it by the letter part x y , we get the result of reducing these terms is 8·x·y . It remains not to forget about the term 1 in the original expression, as a result we have 3 x y+1+5 x y=8 x y+1 .
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 expression - number 1;
in an 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 sweets and 5 apricots. Mom gave Petya 2 more sweets and 4 apricots (see Fig. 1). How many sweets and apricots did Petya have in total?
Rice. 1. Illustration for the problem
Decision
Let's write the condition of the problem in the following form:
1) There were 3 sweets and 5 apricots:
2) Mom gave 2 sweets and 4 apricots:
3) That is, Petya has everything:
4) We add sweets with sweets, apricots with apricots:
Therefore, there are 5 sweets and 9 apricots in total.
Answer: 5 sweets 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) like terms, you need to add their coefficients and multiply the result by the common letter part.
By reducing like terms, we simplify the expression.
They are similar terms, since they have the same letter part. Therefore, to reduce them, it is necessary to add all their coefficients - these are 5, 3 and -1 and multiply by the common letter part - this is a.
2)
This expression contains like terms. The common letter part is xy, and the coefficients are 2, 1 and -3. Here are these similar terms:
3)
In this expression, similar terms are and , let's bring them:
4)
Let's simplify this expression. To do this, we find similar terms. There are two pairs of similar terms in this expression - these are and , and .
Let's simplify this expression. To do this, open the brackets using the distribution law:
There are similar terms in the expression - this and , let's give them:
In this lesson, we got acquainted with the concept of a coefficient, learned which terms are called similar, and formulated the rule for reducing similar terms, and we also solved several examples in which we used this rule.
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. M.: Gymnasium, 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. Moscow: Education, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6. M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A guide for students in grade 6 of the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. M .: Education, Mathematics Teacher Library, 1989.
Homework
- Internet portal Youtube.com ( ).
- Internet portal For6cl.uznateshe.ru ().
- Internet portal Festival.1september.ru ().
- Internet portal Cleverstudents.ru ().
