Topicm8094bd980d5aeda4_1528449000663_0Topic

Pythagorean theoremPythagorean theoremPythagorean theorem

Levelm8094bd980d5aeda4_1528449084556_0Level

Third

Core curriculumm8094bd980d5aeda4_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

2) identifies acute, right and obtuse triangles if sidesideside lengths are given (in particular, the student applies the converse of Pythagorean theoremPythagorean theoremPythagorean theorem).

Timingm8094bd980d5aeda4_1528449068082_0Timing

45 minutes

General objectivem8094bd980d5aeda4_1528449523725_0General objective

Selecting and creating mathematical models to solve practical and theoretical problems.

Specific objectivesm8094bd980d5aeda4_1528449552113_0Specific objectives

1. Applying Pythagorean theorem in geometric calculations.

2. Applying the converse of Pythagorean theoremPythagorean theoremPythagorean theorem to identifying different types of triangles.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm8094bd980d5aeda4_1528450430307_0Learning outcomes

The student:

- identifies the type of a triangletriangletriangle using the converse of Pythagorean theorem,

- applies Pythagorean theoremPythagorean theoremPythagorean theorem to solve geometric problems.

Methodsm8094bd980d5aeda4_1528449534267_0Methods

1. Flipped classroom.

2. Instructional conversation.

Forms of workm8094bd980d5aeda4_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm8094bd980d5aeda4_1528450127855_0Introduction

Students revise information about Pythagorean theorem at home.

Task before the lesson
Refresh your knowledge about Pythagorean theoremPythagorean theoremPythagorean theorem. Find out more information about the various proofs of this theorem on the Internet. Write down selected examples of applications of Pythagorean’s theorem and the converse of Pythagorean theorem.

Students present examples in class. The teacher collects the proposals and creates a sample list.

The Pythagorean theoremPythagorean theoremPythagorean theorem can be used to:

1) calculate line segmentline segmentline segment lengths in polygons,

2) calculate the areaareaarea of geometric figures,

3) check if the triangle is acute, right or obtuse,

4) identify the right angle in the triangletriangletriangle,

5) drawing a line segmentline segmentline segment of the length equal to $\sqrt{n}$, where n is a natural number greater than 1.

Procedurem8094bd980d5aeda4_1528446435040_0Procedure

Students revise the Pythagorean theoremPythagorean theoremPythagorean theorem and the converse of Pythagorean theorem.

Pythagorean theorem.

- In the right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and c (as shown in the picture below).m8094bd980d5aeda4_1527752263647_0- In the right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and c (as shown in the picture below).

[Illustration 1]

Converse of Pythagorean theoremPythagorean theoremPythagorean theorem.

- If the square of one side of a triangle is equal to the sum of the squares of the other two sides, the triangle is right. Students recall how to identify the type of the triangle by using the converse of Pythagorean theorem.m8094bd980d5aeda4_1527712094602_0- If the square of one side of a triangle is equal to the sum of the squares of the other two sides, the triangle is right. Students recall how to identify the type of the triangle by using the converse of Pythagorean theorem.

Students work individually or in pairs, using computers. They get to know how to construct line segments of irrational length.

Task
Open the Geogebra applet - Spiral of Theodorus and observe step by step the construction of line segments whose lengths are equal to $\sqrt{n}$ , where n is a natural number greater than 1. This figure is called the spiral of Theodorus. In your notebook, construct a line segmentline segmentline segment with length $\sqrt{\mathrm{50}}$ .

[Geogebra applet]

Task
Using Pythagorean theoremPythagorean theoremPythagorean theorem calculate:

- a diagonaldiagonaldiagonal of the squaresquaresquare with sidesideside a,

- a diagonal of the rectangular with sides a and b,

- the altitudealtitudealtitude of the equilateral triangletriangletriangle with sidesideside a,

- the space diagonaldiagonaldiagonal of the cube with edge a.

Students revise how to identify the type of the triangle by using the converse of Pythagorean theorem.

Let c be chosen to be the longest of the three sides and a + b > c (otherwise there is no triangle according to the triangle inequality). The following statements apply:
- if and only if aIndeks górny 2² + bIndeks górny 2² = cIndeks górny 2², the triangle is right,
- if and only if aIndeks górny 2² + bIndeks górny 2² > cIndeks górny 2², the triangle is acute,
- if and only if aIndeks górny 2² + bIndeks górny 2² < cIndeks górny 2², the triangle is obtuse.m8094bd980d5aeda4_1527752256679_0Let c be chosen to be the longest of the three sides and a + b > c (otherwise there is no triangle according to the triangle inequality). The following statements apply:
- if and only if aIndeks górny 2² + bIndeks górny 2² = cIndeks górny 2², the triangle is right,
- if and only if aIndeks górny 2² + bIndeks górny 2² > cIndeks górny 2², the triangle is acute,
- if and only if aIndeks górny 2² + bIndeks górny 2² < cIndeks górny 2², the triangle is obtuse.

Students solve problems using the information they have learned.

Task
Identify the type of the triangletriangletriangle with the following sides:

a) 6, 8, 10,

b) 2, 7, 7,

c) 4, 5, 7,

d) $n,\frac{n^2-1}{2},\frac{n^2+1}{2},n>1$,

e) 4, 4, 9.

An extra task: 
Points AIndeks dolny 1₁, BIndeks dolny 1₁, CIndeks dolny 1₁, DIndeks dolny 1₁ are midpoints of the sides of the ABCD squaresquaresquare.

Points AIndeks dolny 2₂, BIndeks dolny 2₂, CIndeks dolny 2₂, DIndeks dolny 2₂ are midpoints of the sides of the AIndeks dolny 1₁BIndeks dolny 1₁CIndeks dolny 1₁DIndeks dolny 1₁.

Prove that the areaareaarea of the AIndeks dolny 2₂BIndeks dolny 2₂CIndeks dolny 2₂DIndeks dolny 2₂ square is four times smaller than the area of the ABCD squaresquaresquare.

[Illustration 2]

Lesson summarym8094bd980d5aeda4_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they summarize the class.

- Both the Pythagorean theorem and the converse theorem are true. Pythagorean theoremPythagorean theoremPythagorean theorem provides many measurement relations which are useful in solving geometric problems.

Selected words and expressions used in the lesson plan

altitudealtitudealtitude

areaareaarea

diagonaldiagonaldiagonal

hypotenusehypotenusehypotenuse

line segmentline segmentline segment

Pythagorean theoremPythagorean theoremPythagorean theorem

right triangleright triangleright triangle

sidesideside

squaresquaresquare

triangletriangletriangle