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Algebra tests in 8th grade.
Topic: “Rational fractions.”
Option 1.
Mandatory part
1. Shorten the fraction: .
2. Shorten the fraction: .
3. Follow these steps: .
4. Follow these steps: .
5. Follow these steps: .
6. Follow these steps: .
7 . Follow these steps: .
8. Graph the function.
Additional part
9
10. (3 points). At what values of variables and fraction
doesn't make sense? Give an example of such values.
11. .
12.
Thematic test No. 1 in algebra in 8th grade.
Topic: “Rational fractions.”
Option 2.
Mandatory part
1. Shorten the fraction: .
2. Shorten the fraction: .
3. Follow these steps: .
4. Follow these steps: .
5. Follow these steps: .
6. Follow these steps: .
7 . Follow these steps: .
8. Graph the function.
Additional part
9 .(3 points). Simplify the expression:
10. (3 points). Shorten 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 down the one that is contained
between the numbers 4 and 5.
3. Compare:
a) and ; b) 8 and .
4. Find the meaning of the expression:
5. Find the meaning 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 8th grade.
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 meaning of the expression:
5. Find the meaning 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). Place the numbers , , , 2.5 in
ascending order.
11. (5 points). Take the factor out of the root sign:
12. (5 points). Graph 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 the rectangle is 96 cm². Find the side
rectangle if one of them is 1.5 times larger than 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.
the expression accepts only positive ones
meanings..
12. (5 points). Find three consecutive natural numbers,
the sum of squares is 50.
Thematic test No. 3 in algebra in 8th grade.
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.
There were 48 chairs arranged in identical rows in the hall. Rows
there were 8 more chairs in each row. How many
chairs in every row?
Additional part
9 .(3 points). Solve the equation.
10. (3 points). Are there values at which
are the values of binomials and equal?
11. (5 points). Isolating the square of the binomial, show that
the expression only accepts negative ones
meanings..
12. (5 points). Sum of squares of two consecutive
natural numbers are 91 more 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
equals 30 km, a truck leaves. After 10 minutes after him
a passenger car departed 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 a 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 8th grade
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). Use graphs to find out how many roots there are
the equation . 7. (5 points). The cyclist had to travel from the village to
railway station 24 km. After traveling 10 km, he did
stop for 10 minutes. After this, increase 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 it is true
inequality: .
10. (5 points). Determine at what values the values
11. (5 points). At what values does the equation
has two roots?
Thematic test No. 5 in algebra in 8th grade.
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 inequality.
7 . By measuring the length and width of a rectangular piece of land (in
site boundaries.
Additional part
8 .(3 points). Solve the system of inequalities:
9. (3 points). Find the largest integer that is
solution to inequality
10. (5 points). At what values are the values
functions belong to the interval.
11. (5 points). At what values does the expression
has the meaning?
Option 1.
Mandatory part
1. Calculate.
2. Calculate.
3. Calculate.
4.
5. Follow the steps.
6. Follow the steps.
7. Follow the steps.
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 the numbers:
a) and ; b) and .
Thematic test No. 6 in algebra in 8th grade
Topic: “Degree with an integer exponent”
Option 2.
Mandatory part
1. Calculate.
2. Calculate.
3. Calculate.
4. Express the fraction as a product.
5. Follow the steps.
6. Follow the steps.
7. Follow the steps.
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 these 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 the numbers.
Lesson objectives:
- Testing 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.
- Greeting the teams (2-3 minutes).
- Warm up (5-7 minutes).
- Captains competition (5 minutes).
- Competition of sages.
- Competition “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 our study of the large and complex topic “Square Roots”. Our final lesson will take 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, and you will be able to show your erudition. I wish you to score as many points as possible:
9 points and above is a grade 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 piece of paper (indicate the number of the incorrect task and the correct answer). The competition is assessed on a 5-point system. The team's rating is made up of the ratings of the team members.
| For 1 team: | For 2nd team: |
| 1. | 1.  |
| 2. | 2. |
3.  |
3.  |
4.  |
4. |
| 5. | 5.  |
After 5 minutes, the pieces of paper are collected, the jury gives a grade to each student and a total grade to the entire team. While the jury checks the work, representatives from each team comment on one of the options.
3. Captains competition.
Team captains are invited to the board and asked to find the square root of a number without using a calculator or table. Teams! Be ready to come to the aid of your captains. The competition is assessed on a 5-point system.
The jury evaluates the work of the captains.
4. Competition of sages.
Now you have to participate in the “Sage Contest”. 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:
No. 3 Graph the function: y=
No. 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, solving which you will receive an additional point. You need to score the most points. Guys! You decide which card to choose. Captains! Get assignments.
Examples of cards.
No. 1 (3 points)
1) Calculate: .
2) Enter under the root sign:2.
3) Take out from under the root sign:
4) Simplify the expression:
5. Factorize: c 2 -2.
No. 2 (4 points)
1) Calculate: 2
2) Enter a multiplier under the sign of the root:
3) Remove the factor from under the root sign:
4) Simplify the expression: 
5) Reduce the fraction:
No. 3 (5 points)
1) Calculate:
2) Remove the factor from under the root sign: , where
3) Enter a factor 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, and the answers of the “wise men” are heard.
6. Competition “Homework”
The jury has a lot of work to do, so we are now listening to homework - historical information.
Taking the square root of a positive number.
The need for the operations of exponentiation and root extraction was caused, like the other four arithmetic operations, by practical life. So, along with the problem of calculating the area of a square, the side A which is known, the inverse problem has been encountered since ancient times: what length should the side of a square have in order for its area to be equal to b?
Even 4000 years ago, Babylonian scientists compiled, along with multiplication tables and reciprocal tables, 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 root extraction can be illustrated by the following example, set out 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 can be expressed as follows: to extract the root of a number With, it is decomposed into the sum a+b(b should be small enough compared to A) and calculated using the approximate formula ==a+b/2a.
About the root sign.
Beginning in the 13th century, Italian and other European mathematicians designated the root with the Latin word Radix (root) or abbreviated R. The root symbol currently used is derived from the notation used by German mathematicians in the 15th and 16th centuries. They indicated 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. Thus, in a manuscript written in 1480 in Latin, one such symbol of a dot before the number () meant a square root, two such signs () meant a fourth root, and three such signs meant a cubic root. Probably, from these designations a sign was subsequently formed, close to the modern root symbol, but without the upper line. This sign is found for the first time in the German algebra “Quick and beautiful calculation 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. Math crossword(assessed on a 5-point system)
Infinite non-periodic decimal fraction.
Part of the whole.
The science that studies the properties of numbers.
Infinite decimal fraction.
Product of equal factors.
8. Summing up. Homework assignment.
The team that wins the game is awarded the symbol of that game - the square root.
Each of the students received a grade for the lesson and tested their knowledge, so there are no losers. Thanks for the lesson, guys!