Solving triangles methodical development in geometry (grade 9) on the topic. Test Solving Right Triangles Test Solving Triangles

Goal: to consolidate the students' knowledge of the theorems of sines and cosines, to teach how to apply these theorems in the course of solving problems.

Equipment:

tables with triangles;
cards with formulas;
calculators;
bradis tables;
test for each student.

DURING THE CLASSES

I. Organization of the class. Checking readiness for the lesson. Communication of the topic and purpose of the lesson.

II. Repetition of the material learned (or warm-up phase)

1. Continue:

The square of the side of a triangle is ... (cosine theorem)

2. Fill in the blanks:

3. Continue:

The sides of a triangle are proportional ... (sine theorem)

4. Fill in the blanks

5. Connect the parts of the phrases that correspond to each other with a line:

The solution to triangles consists of

Finding unknown heights, medians and bisectors at known angles and sides of a triangle;

Finding an unknown perimeter at known angles and sides of a triangle;

Finding unknown sides and angles of a triangle from its known angles and sides.

III. Consolidation of the studied material.

1. Solving problems using ready-made formulas

Determine the formula by which you need to find this unknown element:

formula cards:

2. Solving problems by pulling out one of the cards:

IV. Intermediate control. Test for the whole class by options:

Option 1.

a) The square of any side of a triangle is equal to the sum of the squares of its other two sides;

b) The square of any side of a triangle is equal to the sum of the squares of the other two sides without the double product of these sides by the cosine of the angle between them;

c) The square of any side of the triangle is equal to the sum of the squares of the other two sides, minus the product of these sides by the cosine of the angle between them.

3. The cosine of an angle of 120 ° is equal to ...

d) there is no correct answer.

4. Find the sine 29 ° 30 ". Underline the correct answer:

5. To calculate KMD in a triangle, you need to know ...

a) KM, MD, KD;

b) KM, MD,;

d) there is no correct answer.

6. The sides of the triangle are 5 cm and 4 cm, and the angle between them is 30 °. Find the third side of the triangle.

Option 2

1. Put a “+” sign next to the correct statement:

a) The sides of the triangle are proportional to the sines of the opposite angles;

b) The sides of the triangle are inversely proportional to the sines of the opposite angles;

c) The sides of the triangle are proportional to the sines of the opposite angles.

2. For this triangle, the equality is true ...

3. The sine of an angle of 135 ° is ...

d) there is no correct answer.

4. Find the cosine 67 ° 18 ". Underline the correct answer:

5. In triangle ABC, the length of the side BC and the value of the angle C are known. To calculate AB, you need to know ...

d) there is no correct answer.

6. The sides of the triangle are 5 cm and 3 cm, and the angle between them is 60 °. Find the third side of the triangle.

Teacher of KSU Secondary School No. 30 -Kovalevskaya ON

In the geometry lesson in the 9th grade, with the help of a presentation, various types of problems on the topic "Solving triangles" are considered. When solving problems, special attention is paid to the correct choice of the theorem, which allows solving the problem in the most rational way. To consolidate the material studied, it is proposed to perform a verification test on a computer in Excel.

Subject:

Grade 9 geometry

Date:

02.03.2015

Occupation:

Theme:

Solving triangles

Common goals:

Strengthening and deepening students' knowledge of the theorems of sines and cosines and their application to solving triangles, as well as the relationship between the angles of a triangle and opposite sides.

Learning outcomes:

increased interest in the subject,

improving learning outcomes,

the formation of skills for self and mutual learning;

self and mutual appreciation.

Key ideas:

Modules: "New Approaches in Teaching and Learning", "Teaching Critical Thinking", "Assessment for Learning and Assessment of Learning", "Using ICTs in Teaching and Learning", "Teaching Talented and Gifted Students", "Teaching and Learning in accordance with age characteristics of students "," Management and leadership in teaching. "

Geometry textbook for grade 9

Requisites:

Stickers, paper, markers, handouts, interactive whiteboard

During the classes:

Time

Lesson steps

Teacher actions

Student actions

1 min

Org moment

Greeting. Positive wishes for the lesson.

Responsiveness

1 min

Division into groups - 4 colors and 6 geometric shapes (4 groups)

Allows each student from the package to choose a geometric shape of a certain color. Explains the meanings of the shapes:

Square - group leader

Parallelogram - speaker

Rectangle - secretary

The rest are ideas generators

They are seated in groups by color (blue, yellow, pink and red).

4 minutes

Brainstorming (oral)

The teacher asks questions:

Cosine theorem?

The sine theorem?

A theorem on the sum of the angles of a triangle?

Acute and obtuse angle reduction formulas for sine and cosine?

Student answers:

The square of any side of the triangle is equal to the sum of the squares of the other two sides without twice the product of these sides by the cosine of the angle between them.

Sides of a triangle

proportional to the sines of the opposite angles.

The angles of a triangle add up to 180̊ .

3 min

Brainstorming (written individual work)

According to the drawing given at the presentation, write down the theorem of sines and cosines and, after completion, check the correctness of your writing on the board and evaluate yourself.

Write theorems on their own for this drawing. Upon completion, students check the teacher's answer key on the interactive whiteboard with the key and give themselves scores on the assessment sheets.

2 minutes

Brainstorming (oral)

The teacher asks questions. Types of tasks:

Solving triangles by side and by two corners.

Solving triangles on two sides and the angle between them.

Solving triangles on three sides.

Solving triangles on two sides and an angle lying opposite one of them.

Answers the questions posed.

Student answers:

We apply the theorem on the sum of the angles of a triangle and the cosine theorem.

We apply the triangle sum theorem and the sine theorem.

13 minutes

Mathematical dictation (written individual work)

Find the unknown element of the triangle according to the drawings given on the presentation slides, writing out the theorems of sines and cosines. After completing, check the correctness of your writing on the board and evaluate yourself. The slides in the presentation are switched in time for the first 3 give 2 minutes each, the last 2 for 3 minutes.

Pupils solve problems independently. Upon completion, students check the teacher's answer key on the interactive whiteboard with the key and give themselves scores on the assessment sheets.

1 min

Physiotherapy for the eyes

The teacher observes the students and guides them to calm music

Positive attitude

7 minutes

PISA : Solving a logical problem on a poster (work in groups). Protection of the poster with the speaker's comments from the group.

The teacher reads the problem and offers to solve it geometrically in a group. After asking all the groups for answers, he offers one of them to defend his decision.

Using open-ended and problematic questions to find out how well students understood the problem. (56 trees)

Gathering information - knowledge that they have at the time of the lesson (knowledge and understanding). During work, students can turn to each other for help. Pupils in groups try to find a more complete explanation of the problem.

10 min

The stage of consolidation and control of students' knowledge on this topic:

independent work in groups with the test

The teacher offers to solve the problems independently by performing a test test on a computer in Excel.

1 min

Homework

Students listen carefully and write down their homework.

3 min

Reflection stage. Summarizing.

The teacher asks you to choose one of the 6 thinking hats and try to reflect on the lesson and your knowledge at the end of the lesson. This method is based on the idea of \u200b\u200bparallel thinking. Parallel thinking - This is constructive thinking, in which different points of view and approaches do not collide, but coexist. Why hats? The hat is easy to put on and take off, and the hats also indicate the role.

Assess their knowledge after the lesson. Control, correction, assessment of the partner's actions, the ability to express their thoughts with sufficient completeness and accuracy.

« Trying on»On a hat of certain flowers, students learn to think in a given direction. Changing hats teaches you to see the same object from different positions, resulting in the most complete picture.

Appendix # 1:

Assessment sheet (group number 1)

FI student

Grades for assignments

Overall score

Homework

Frontal poll

Mathematical dictation

Poster protection

test

Additional assessment

Appendix # 2:

Test on the topic: "Solving triangles".

I. Instructions for working with the test:

1. Tasks of the 1st variant of the test are on Sheet 2. The tasks of the 2nd variant of the test are on Sheet 3. To go - click LMB on the tab Sheet2 or Sheet3.

2. After reading the next task, choose the correct answer. Then switch to the Sheet1 tab and enter the number of the correct answer into the answer table of your choice.

3. Repeat step 2 of the instruction until you have completed all the tasks of the test.

4. The test will take 10 minutes to complete. Check the time on a computer clock!

5. Report the test to the teacher. - The score is logged.

II. Answer tables for the test:

Option 1

Option 2

№ tasks

№ answer

№ tasks

№ answer

Number of correct answers:

Rating:

How to enter the number of the selected answer:

1. Click LMB (Left Mouse Button) in the required cell of the "Answer No." column.

2. Enter the number corresponding to the number of the correct answer.

3. Press the Enter key.

Test on the topic "Solving triangles"

Option 1

In tasks 1-4, select the correct answer and enter its number in the table on Sheet 1 by clicking LMB on the Sheet 1 tab in the lower left corner of the screen.

In a triangle ABC AB \u003d BC \u003d 2. If acosB \u003d - 1/8, then speaker sideis equal to:

1) √ 7

2) 7

3) 3

4) 9

In triangle ABC, side AB \u003d 3, side AC \u003d 5. Then the relation (sin B) :( sin C) equals:

1) 5 / 3

2) 3 / 5

3) 4 / 5

4) 5 / 4

In a right-angled triangle ABC, angle C \u003d 45 0 ... If AB \u003d 4, then the hypotenuse BCis equal to:

1) 8

2) 4√ 3

3) 2√ 2

4) 4√ 2

In a triangle ABC AB \u003d 2, BC \u003d 3. If angle A \u003d 36 0, then

1) angle B obtuse

2) angle B straight

3) angle B acute

4) angle type B cannot be set

Auelbekova Gavkhar Umurbekovna

Lyceum at KazGASA

Question 1: Choose the correct wording for the definition of a right triangle:

A triangle with only two sharp corners

Triangle with straight sides

Triangle with all angles right

A triangle with one straight corner and two sharp angles

Question 2: What is the name of the side of a right triangle opposite to a right angle?

Base

Cathetus

Hypotenuse

I find it difficult to answer

Question 3: Continue the wording:

If the acute angle of a right-angled triangle is 30 °, then ...

leg is half the hypotenuse

the hypotenuse is equal to the leg

the leg lying opposite this angle is half the hypotenuse

hypotenuse larger than leg

Question 4:

Which triangle is called the Egyptian? What is equal to

cos 45 °?

Question 5:

In triangle ABC (  C \u003d 90 °)  A \u003d 30 °, BC \u003d 12 cm

Find the length of the hypotenuse AB.

6 cm

12cm

24 cm

Can't define

Question 6: In an isosceles triangle ABC with the base BC, the height AD is drawn.

Find the angles B and C if

the side of the triangle AC \u003d 7 cm, and CD \u003d 3.5 cm

Can't define

Question 7: In a right-angled isosceles triangle, the hypotenuse is 18 cm. Determine the height of the triangle, dropped from the apex of the right angle.

Can't define

You did a good job !

Start solving the next problem .

Review the theory one more time and return to the problem.