What is the type of problem "Find m for the equation to have two roots x1 x2 satisfying a given condition" under the 10th-grade Mathematics curriculum? What are the regulations on assessing the learning results for 10th-grade Mathematics in Vietnam?

View Post in Vietnamese: >> Môn toán lớp 10 Tìm m để phương trình có 2 nghiệm x1 x2 thỏa mãn điều kiện cho trước là gì? Đánh giá kết quả giáo dục môn toán lớp 10?

What is the type of problem "Find m for the equation to have two roots x1 x2 satisfying a given condition" under the 10th-grade Mathematics curriculum?

This type of problem is common in the Mathematics curriculum at the upper secondary level, especially at the upper secondary level. It requires you to find the value of the parameter m so that the equation (often a quadratic equation) has two distinct roots x1, x2, and these roots satisfy a certain condition.

Find m for the equation to have 2 roots x1 x2 satisfying a given condition

Common solution steps:

Check the condition for the equation to have 2 distinct roots:

For the quadratic equation ax²+bx+c=0, the condition for the equation to have two distinct roots is:

a≠0

Δ>0 (where Δ=b²−4ac)

Apply Viete's theorem:

If the equation has two distinct roots x1, x2, then:

x1+x2=−b/a

x1.x2=c/a

Express the given condition in terms of x1, x2:

Based on the condition provided in the problem, express this condition in the form of an equation or inequality related to x1 and x2.

Substitute the expressions from Viete's theorem into the condition:

Substitute expressions x1+x2 and x1.x2 with −b/a and c/a respectively into the equation or inequality in step 3.

Solve the equation or inequality obtained:

Solve the equation or inequality to find the values of m that satisfy.

*Conclusion:

List the values of m found and conclude on the values of m for the equation to have two distinct roots satisfying the given condition.

Example:

Problem: Find m for the equation x²−2mx+m²−1=0 to have two distinct roots x1, x2 satisfying x1²+x2²=10.

Solution:

The equation has two distinct roots when Δ>0 ⇔ (−2m)²−4(m²−1)>0 ⇔ 4>0 (always true).

Apply Viete's theorem: x1+x2=2m and x1.x2=m²−1.

Problem condition: x1²+x2²=10.

Substitute Viete into the condition: (x1+x2)²−2x1.x2=10 ⇔ (2m)²−2(m²−1)=10.

Solve the equation: 4m²−2m²+2=10 ⇔ 2m²=8 ⇔ m²=4 ⇔ m=±2.

Conclusion: With m=2 or m=−2, the equation has two distinct roots satisfying the problem's condition.

Common problem types:

Find m so that the equation has two roots of opposite signs.

Find m so that the equation has two roots of the same sign.

Find m so that the equation has two roots such that one root is double the other.

Find m so that the equation has two roots satisfying a given relationship between the roots.

*Note: Information is for reference only./.

Grade 10 Math: Find m for the equation to have 2 roots x1 x2 satisfying given condition. Evaluation of grade 10 math education results?

What are the regulations on assessing the learning results for 10th-grade Mathematics in Vietnam?

Under Section 7 of the General Education Program for Mathematics issued with Circular 32/2018/TT-BGDDT:

- The assessment of the learning results for Mathematics aims to provide accurate, prompt information of value regarding the development of competencies and progress of students based on the requirements needed to be achieved at each grade and educational level;

Adjust teaching and learning activities to ensure the progress of each student and improve the quality of Mathematics education in particular and education quality in general.

- Combine multiple forms of assessment (process assessment, periodic assessment), using various methods (observation, performance recording, oral examinations, objective tests, essay tests, written tests, practical exercises, project-based learning products, real-world tasks,...) at appropriate times.

- Process assessment (or frequent assessment) is organized by the subject teacher, in combination with assessments from teachers of other subjects, the self-evaluation of the students being assessed, and evaluation from other students in the group, the class, or parents.

Process assessment accompanies the learning activities of students, avoiding separation between teaching and assessment processes, ensuring the goal of assessment for progress in students' learning.

- Periodic assessment (or summative assessment) primarily aims to evaluate learning objectives achievement.

Results from periodic and summative assessments are used to certify learning levels and recognize student achievements.

Periodic assessments are organized by educational institutions or through national examinations and assessments.

- Periodic assessment is also used for managing educational activities to ensure quality at educational institutions and for the development of the Mathematics curriculum.

- Assess students' competence through evidence showing their achievements during actions taken by students.

The evaluation process involves basic steps such as: determining the evaluation purpose; identifying necessary evidence; selecting appropriate evaluation methods and tools; collecting evidence; interpreting evidence and providing feedback.

- Focus on choosing methods and tools to evaluate competencies and elements of mathematical competence. To be specific:

+ Assess mathematical thinking and reasoning skills: methods and tools like questions (oral, written), exercises,... that require students to present, compare, analyze, aggregate, and systematize knowledge;

Apply mathematical knowledge to explain and reason.

+ Assess mathematical modeling skills: choose real-life situations triggering mathematical problems.

From this, students are required to identify a mathematical model (including formulas, equations, tables, graphs,...) for the situation presented in the problem;

Solve mathematical problems within the established model; express and evaluate solutions within the real-world context and improve the model if the resolution is unsuitable.

+ Assess mathematical problem-solving skills: use methods like encouraging students to identify situations, highlight, and explain problems to be solved;

Describe and explain the initial information, goals, and expectations of the problem situation being considered; collect, select, organize information and connect it with existing knowledge;

Use questions (may require oral or written answers) demanding students apply knowledge to solve problems, especially real-life issues; observe methods (such as checklists with predetermined criteria), observe learners during problem-solving;

Assess through practical work products (for instance, learning project products); reasonably focus on integrated assessment tasks.

+ Assess mathematical communication skills: use methods like requiring students to listen, understand, read, record (summarize), analyze, select, extract the basic, focal mathematical information from spoken or written texts;

Use mathematical language combined with common language in presenting, expressing, questioning, discussing, and debating mathematical content, ideas, and solutions in interaction with others.

+ Assess competence in using mathematical tools and resources: use methods requiring students to recognize the names, functions, specifications, preservation methods, advantages, and limitations of tools and resources for learning mathematics;

Explain how to use (appropriately) tools and resources for learning mathematics to perform learning tasks or to express mathematical reasoning and proofs.

When the teacher plans lessons, it is necessary to establish criteria and evaluation methods to ensure that at the end of each lesson, students achieve the basic requirements based on the established criteria before proceeding to further learning activities.

What are the general objectives when studying 10th-grade Mathematics in Vietnam?

Under Section 3 of the General Education Program for Math, issued with Circular 32/2018/TT-BGDDT the general objectives when studying 10th-grade Mathematics in Vietnam are as follows:

- The Mathematics curriculum assists students to achieve the following main objectives:

+ Develop and enhance mathematical competence comprising the following core elements: mathematical thinking and reasoning competence; mathematical modeling competence; mathematical problem-solving competence; mathematical communication competence; and competence in using mathematical tools and resources.

+ Contribute to developing core qualities and general competencies in students aligned with the subjects' levels and educational stages as regulated in the overall program.

+ Acquire basic, essential common high school Mathematics knowledge and skills; develop the ability to solve integrated, interdisciplinary problems between Mathematics and other subjects like Physics, Chemistry, Biology, Geography, Information Technology, Technology, History, Arts,...; create opportunities for students to experience and apply Mathematics in reality.

+ Have a relatively comprehensive understanding of the usefulness of mathematics in various related professions to facilitate career orientation, as well as have sufficient minimal competence to explore issues related to mathematics throughout life.

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