Topicm6aa259f558402cdb_1528449000663_0Topic

The areaareaarea of the triangletriangletriangle

Levelm6aa259f558402cdb_1528449084556_0Level

Second

Core curriculumm6aa259f558402cdb_1528449076687_0Core curriculum

XI. Calculations in geometry. The student:

2) calculates the areaareaarea of: the triangle, the square, the rectanglerectanglerectangle, the rhombus, the parallelogram and the trapezium presented in the drawing and in practical situations, including data requiring a conversion of units and in situations when the dimensions are not typical, for example the area of the triangletriangletriangle with a side of 1 km and the altitudealtitudealtitude of 1 mm;

3) uses the units of the areaareaarea: mmIndeks górny 2², cmIndeks górny 2², dmIndeks górny 2², mIndeks górny 2², kmIndeks górny 2², are, hectare (without converting the units during in the calculation).

Timingm6aa259f558402cdb_1528449068082_0Timing

45 minutes

General objectivem6aa259f558402cdb_1528449523725_0General objective

Matching a mathematical model to a simple situation and using it in various contexts.

Specific objectivesm6aa259f558402cdb_1528449552113_0Specific objectives

1. Justifying the formula for the areaareaarea of the triangle.

2. Applying the formulaformulaformula for the area of the triangletriangletriangle.

3. Communicating in English; developing mathematical and basic scientific, technical and digital competences; developing learning skills.

Learning outcomesm6aa259f558402cdb_1528450430307_0Learning outcomes

The student:

- justifies the formulaformulaformula for the areaareaarea of the triangle,

- calculates the area of the triangletriangletriangle.

Methodsm6aa259f558402cdb_1528449534267_0Methods

1. Practical exercises.

2. Situational analysis.

Forms of workm6aa259f558402cdb_1528449514617_0Forms of work

1. Individual work.

2. Class work.

Lesson stages

Introductionm6aa259f558402cdb_1528450127855_0Introduction

Before the lesson the student cuts out one paper rectanglerectanglerectangle with the dimensions of 3 cm and 6 cm and the other one with the dimensions of 5 cm and 5 cm.

The teacher introduces the topic of the lesson: learning the formula for calculating the areaareaarea of the triangletriangletriangle.

Revision of the definition of the altitudealtitudealtitude of the triangle and the formulaformulaformula for the area of the rectangle.

The segment connecting the vertex of the triangletriangletriangle with its opposite side (or the extended one) at the right angle is called the altitudealtitudealtitude of the triangle.

The areaareaarea of the rectanglerectanglerectangle equals the productproductproduct of its adjacent sides.

Procedurem6aa259f558402cdb_1528446435040_0Procedure

Task

The students cut the rectangles they have prepared along one of the diagonals and answer the following questions:

How large is the areaareaarea of each rectanglerectanglerectangle?

What figures did you get after cutting the rectangles apart?

What are the sides of right‑angled triangles called?

How large are the areas of the figures you have obtained?

The students should notice that:
- after cutting the rectangle along one of its diagonals we get two identical right‑angled triangles,
- the area of each of these triangles equals half of the area of the rectangle,
and come up with the conclusion:
The area of the right‑angled triangle equals half the product of the length of its legs.m6aa259f558402cdb_1527752263647_0- after cutting the rectangle along one of its diagonals we get two identical right‑angled triangles,
- the area of each of these triangles equals half of the area of the rectangle,
and come up with the conclusion:
The area of the right‑angled triangle equals half the product of the length of its legs.

[Illustration 1]

Task

Using the presented formulaformulaformula the students calculate the areaareaarea of the right‑angled triangletriangletriangle on their own, knowing the length of its legs.

Calculate the area of the right‑angled triangleright‑angled triangleright‑angled triangle whose legs are 9 cm and 12 cm.

Then, the students become familiar with the formula for calculating the areaareaarea of any triangle.

Task

Students work individually using their computers. They are going to observe how the formulaformulaformula for the area of the triangletriangletriangle can be illustrated.

[Geogebra applet]

After completing the task the students discuss their observations and together they draw the following conclusion:

The area of the triangle equals half the product of its base and the altitude drawn to this base. The side of the triangle which the altitude is drawn to it is called the base of the triangle.m6aa259f558402cdb_1527752256679_0The area of the triangle equals half the product of its base and the altitude drawn to this base. The side of the triangle which the altitude is drawn to it is called the base of the triangle.

[Ilustracja 2]

Task 4

Using the written formulaformulaformula the students calculate the areaareaarea of the triangle on their own, knowing the length of the bases and the altitudealtitudealtitude.

Calculate the area of the triangletriangletriangle on the basis of the following data:

a) the triangle base is 11 cm and the altitude drawn to it is 6 cm,

b) one of the altitudes of the triangle is 24 dm and the side that is perpendicularperpendicularperpendicular to it is 3 times shorter.

An extra task

Calculate the altitudealtitudealtitude of the triangletriangletriangle drawn to the base of 15 cm knowing that the areaareaarea of this triangle is 52,5 cm.

Lesson summarym6aa259f558402cdb_1528450119332_0Lesson summary

Students do the exercises summarizing the class.

Then, together they sum up the classes, drawing the conclusions to memorize:

- the areaareaarea of the triangletriangletriangle equals half the productproductproduct of its base and the altitudealtitudealtitude drawn to this base,

- the area of the right‑angled triangleright‑angled triangleright‑angled triangle equals half the productproductproduct of the length of its legs.

Selected words and expressions used in the lesson plan

altitudealtitudealtitude

areaareaarea

diagonaldiagonaldiagonal

formulaformulaformula

leglegleg

perpendicularperpendicularperpendicular

productproductproduct

rectanglerectanglerectangle

right‑angled triangleright‑angled triangleright‑angled triangle

triangletriangletriangle