Algebra in 8th grade.
Topic: "Rational fractions".
Option 1.
Mandatory part
1. Reduce the fraction: .
2. Reduce the fraction: .
3. Follow these steps: .
4. Follow these steps: .
5. Follow these steps: .
6. Follow these steps: .
7 . Follow these steps: .
8. Plot the function .
Additional part
9
10. (3 points). At what values of the variables and the fraction
doesn't make sense? Give an example of such values.
11. .
12.
Thematic test No. 1 in algebra in grade 8.
Topic: "Rational fractions".
Option 2.
Mandatory part
1. Reduce the fraction: .
2. Reduce the fraction: .
3. Follow these steps: .
4. Follow these steps: .
5. Follow these steps: .
6. Follow these steps: .
7 . Follow these steps: .
8. Plot the function .
Additional part
9 .(3 points). Simplify the expression:
10. (3 points). Reduce the fraction: .
11. (5 points). Find valid variable values in
12. (5 points). Prove the identity:
Topic: "Square roots".
Option 1.
Mandatory part
1. Calculate:.
2. From the numbers , , write out the one that is concluded
between numbers 4 and 5.
3. Compare:
a) and; b) 8 and .
4. Find the value of the expression:
5. Find the value of the expression:
6.
7 . Simplify the expression: .
8. Simplify the expression: .
Additional part
9 .(3 points). Simplify the expression:
10. (3 points). Prove that .
11. (5 points). Simplify the expression:
12. (5 points). Simplify the expression:
Thematic test No. 2 in algebra in grade 8.
Topic: "Square roots".
Option 2.
Mandatory part
1. Calculate: at =6, =8.
2. Specify two consecutive integers between which
enclosed number .
3. Compare:
a) and; b) 11 and .
4. Find the value of the expression:
5. Find the value of the expression:
6. Enter the multiplier under the root sign: .
7 . Simplify the expression: .
8. Simplify the expression: .
Additional part
9 .(3 points). Eliminate irrationality from the denominator:
10. (3 points). Arrange the numbers , , , 2.5 in
ascending order.
11. (5 points). Take the factor out of the root sign:
12. (5 points). Plot the Function
Option 1.
Mandatory part
1.
2. Determine how many roots the equation has.
3. Solve the equation.
4. Solve the equation.
5. Solve the equation.
6. Solve the equation.
7 . Solve the equation.
8.
The area of a rectangle is 96 cm². Find a side
rectangle if one of them is 1.5 times the other.
Additional part
9
10. (3 points). Find the coefficients and in the equation
If it is known that its roots are equal and
11.
expression accepts only positive
values..
12. (5 points). Find three consecutive natural numbers,
the sum of their squares is 50.
Thematic test No. 3 in algebra in grade 8.
Topic: Quadratic Equations.
Option 2.
Mandatory part
1. Determine how many roots the equation has.
2. Determine how many roots the equation has.
3. Solve the equation.
4. Solve the equation.
5. Solve the equation.
6. Solve the equation.
7 . Solve the equation.
8. Solve the problem using the equation.
48 chairs were arranged in equal rows in the hall. Rows
there were 8 more chairs than there were in each row. How much
chairs in each row?
Additional part
9 .(3 points). Solve the equation.
10. (3 points). Are there values for which
are the values of the binomials and equal?
11. (5 points). By isolating the square of the binomial, show that
expression accepts only negative
values..
12. (5 points). Sum of squares of two consecutive
natural numbers are 91 greater than their product. Find these
Option 1.
Mandatory part
1. Solve the equation.
2. Solve the equation.
3. Solve the equation.
4. Solve the equation.
Additional part
5. (3 points). Solve the equation:
6.
the equation . 7. (5 points). From city A to city B, the distance between them
equal to 30 km, a truck left. 10 minutes after him
a car left and arrived in city B for 5 minutes
before the truck. Find the speed of each car if
it is known that the speed of the truck is 20 km/h less than the speed
passenger car.
8. (5 points). Find the coordinates of the intersection points of the graphs
functions and .
Thematic test No. 4 in algebra in grade 8
Topic: "Fractional rational expressions"
Option 2.
Mandatory part
1. Solve the equation.
2. Solve the equation.
3. Solve the equation.
4. Solve the equation.
Additional part
5. (3 points). Solve the equation:
6. (3 points). Using graphs, find out how many roots has
the equation . 7. (5 points). The cyclist had to pass from the village to
railway station 24 km. After driving 10 km, he did
stop for 10 minutes. After increasing the speed by 2 km / h,
he arrived at the railway station on time. Find
the cyclist's initial speed.
8. (5 points). Find the coordinates of the points of intersection of the graph of the function with the axis and with the axis.
Topic: "Inequalities".
Option 1.
Mandatory part
1. Solve the inequality: .
2. Solve the inequality: .
3. Solve the inequality: .
4. Solve the system of inequalities:
5. Solve the system of inequalities:
6.
sides and (in mm): ,
Additional part
8 .(3 points). Solve the inequality: .
9. (3 points). Prove that for all values is true
inequality: .
10. (5 points). Determine at what values the values
11. (5 points). For what values of the equation
has two roots?
Thematic test No. 5 in algebra in grade 8.
Topic: "Inequalities".
Option 2.
Mandatory part
1. Solve the inequality: .
2. Solve the inequality: .
3. Solve the inequality: .
4. Solve the system of inequalities:
5. Solve the system of inequalities:
6. Solve double non-equality.
7 . By measuring the length and width of a rectangular piece of land (in
plot boundaries.
Additional part
8 .(3 points). Solve the system of inequalities:
9. (3 points). Find the largest integer that is
solving the inequality
10. (5 points). At what values
functions belong to the interval .
11. (5 points). For what values is the expression
has the meaning?
Option 1.
Mandatory part
1. Calculate .
2. Calculate .
3. Calculate .
4.
5. Take action.
6. Take action.
7. Take action.
8. Simplify the expression.
9. Simplify the expression.
10. Write the number 52000 in standard form.
11. Write the number 0.062 in standard form.
12.
Additional part
13. (3 points). Calculate .
14.
15. (5 points). Reduce the fraction.
16. (5 points). Compare numbers:
a) and; b) and .
Thematic test No. 6 in algebra in grade 8
Topic: "Degree with an integer exponent"
Option 2.
Mandatory part
1. Calculate .
2. Calculate .
3. Calculate .
4. Express a fraction as a product.
5. Take action.
6. Take action.
7. Take action.
8. Simplify the expression.
9. Simplify the expression.
10. Write the number 34000 in standard form.
11. Write the number 0.023 in standard form.
12. Follow the steps and write in standard form:
Additional part
13. (3 points). Calculate .
14. (3 points). Simplify the expression.
15. (5 points). Express as a power with base 3
expressions: a) ; b) .
16. (5 points). Compare numbers.
Lesson Objectives:
- Checking theoretical and practical knowledge on the topic.
- Familiarization of students with historical material.
- Activation of students, involving them in various competitions and games.
Lesson plan.
- Team greeting (2-3 minutes).
- Warm-up (5-7 minutes).
- Captains competition (5 minutes).
- Wise Competition.
- Contest “Race for the leader”.(10 minutes)
- Homework. (10 minutes)
- Solving a crossword puzzle. (5 minutes)
- Summing up. (5 minutes)
During the classes
1. Teacher:
Guys! Today we are completing the study of the large and complex topic “Square Roots”. The final lesson will take place in the form of a competition between two teams “Root” and “Radical”. We will test your theoretical and practical knowledge on the topic, get acquainted with historical material, you will be able to show your erudition. I wish you to score as many points as possible:
9 points and above is a mark of “5”;
7-8 points - "4";
5-6 points - "3".
2. Warm up.
Teacher:
There are 5 tasks written on the board with answers for each option. Your task is to check the correctness of the answers, write it down on your sheet (indicate the number of the wrong task and the correct answer). The competition is evaluated on a 5-point system. The team score is made up of the team members' scores.
| For 1 team: | For 2 teams: |
| 1. | 1.  |
| 2. | 2. |
3.  |
3.  |
4.  |
4. |
| 5. | 5.  |
After 5 minutes, the papers are collected, the jury gives a mark to each student and a total mark to the whole team. While the jury checks the entries, representatives from each team comment on one of the options.
3. Competition of captains.
Team captains are invited to the board, they are invited to take the square root of the number without using a calculator or table. Teams! Be ready to come to the aid of your captains. The competition is evaluated on a 5-point system.
The jury evaluates the work of the captains.
4. Competition of wise men.
Now you have to participate in the “Contest of Wise Men”. Two “wise” students are invited from each team, who will have to complete interesting tasks on the board.
#1 Simplify the expression:
#2 Simplify the expression:
#3 Graph the function: y=
#4 Graph the function: y=
5. Competition "Race for the leader".
While the “wise men” solve their tasks, the teams receive a package of cards, each card contains tasks and the number of points that can be received for the correct solution. But on each card there is a task with an asterisk, by solving which you will receive an additional point. You need to get the most points. Guys! You decide which card to choose. Captains! Get assignments.
Card examples.
No. 1 (3 points)
1) Calculate: .
2) Enter under the sign of the root: 2.
3) Take out from under the sign of the root:
4) Simplify the expression:
5. Factor out: c 2 -2.
No. 2 (4 points)
1) Calculate: 2
2) Enter a factor under the root sign:
3) Take out the factor from under the root sign:
4) Simplify the expression: 
5) Reduce the fraction:
No. 3 (5 points)
1) Calculate:
2) Take out the factor from under the root sign: , where
3) Enter a multiplier under the root sign: m, where m>0.
4) Simplify the expression: 
5) Eliminate irrationality in the denominator:
After 10 minutes, the cards are handed over to the jury for verification, the answers of the “wise men” are heard.
6. Competition "Homework"
The jury has a lot of work, so we are now listening to homework - historical references.
Extracting the square root of a positive number.
The need for the operations of exponentiation and extraction of the root was caused, like the other four arithmetic operations, by practical life. So, along with the task of calculating the area of a square, the side a which is known, since ancient times there was an inverse problem: what length should the side of the square have in order for its area to be equal to b?
Even 4000 years ago, Babylonian scientists compiled, along with multiplication tables and tables of reciprocals, tables of squares of numbers and square roots of numbers. At the same time, they were able to find the approximate value of the square root of any integer. The Babylonian method of extracting the root can be illustrated by the following example, set forth in one of the cuneiform tablets found during excavations.
Find the square root of 1700. To solve the problem, this number is decomposed into the sum of two terms: 1700=1600+100=40 2 +100, the first of which is a perfect square. Then it is indicated that =40+100/2*40=41 1 / 4 .
The rule used by the Babylonians may be expressed as follows: to extract the root from the number with, it is decomposed into the sum a+b(b should be small enough compared to a) and calculated by the approximate formula ==a+b/2a.
About the sign of the root.
Beginning in the 13th century, Italian and other European mathematicians designated the root with the Latin word Radix (root), or R for short. They marked the square root with a dot in front of the number or expression. In cursive writing, dots were replaced by dashes, which later turned into a symbol. So, in a manuscript written in 1480 in Latin, one such symbol of a dot in front of a number () meant a square root, two such signs () meant a fourth degree root, and three signs meant a cube root. Probably, from these designations a sign was later formed, close to the modern root symbol, but without the upper line. This sign is found for the first time in the German algebra "Fast and beautiful counting with the help of skillful rules of algebra, usually called Koss", published in 1525 in Strasbourg. It was not until 1637 that Rene Descartes combined the root sign with a horizontal line.
7. Mathematical crossword(assessed on a 5-point system)
Infinite non-periodic decimal.
Part of the whole.
The science that studies the properties of numbers.
Endless decimal.
The product of equal factors.
8. Summing up. Homework assignment.
The team that wins the game is awarded the symbol of this game - the square root.
Each of the students received a grade for the lesson, tested their knowledge, so there are no losers. Thanks for the lesson guys!
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