Topicm8baa43e4d3c555a0_1528449000663_0Topic

The areas of polygons

Levelm8baa43e4d3c555a0_1528449084556_0Level

Third

Core curriculumm8baa43e4d3c555a0_1528449076687_0Core curriculum

I. Planimetrics. The student:
4) applies the properties of angles and diagonals in rectangles, parallelograms, rhombuses and trapezoids.

Timingm8baa43e4d3c555a0_1528449068082_0Timing

45 minutes

General objectivem8baa43e4d3c555a0_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesm8baa43e4d3c555a0_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Consolidation of  formulae for the area of a rectangle, squaresquaresquare, parallelogramparallelogramparallelogram, trapezoidtrapezoidtrapezoid, deltoiddeltoiddeltoid.

3. Applying the formulae for the area of quadrangles in tasks.

Learning outcomesm8baa43e4d3c555a0_1528450430307_0Learning outcomes

The student:

- consolidates the formulae for the area of a rectanglerectanglerectangle, squaresquaresquare, parallelogramparallelogramparallelogram, trapezoidtrapezoidtrapezoid, deltoiddeltoiddeltoid,

- applies the formulae for the area of quadrangles in tasks.

Methodsm8baa43e4d3c555a0_1528449534267_0Methods

1. Incomplete senteces.

2. Mind maps.

3. Situational analysis.

Forms of workm8baa43e4d3c555a0_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm8baa43e4d3c555a0_1528450127855_0Introduction

The students use the incomplete sentences technique to recollect concepts and properties connected with quadrangles.

Task 1

Complete the sentences.

A trapezoidtrapezoidtrapezoid is …

A quadrangle whose one pair of sides is parallel is called…

A parallelogramparallelogramparallelogram is a quadrangle which has…

A rhombus is a parallelogramparallelogramparallelogram whose all sides are…

A parallelogramparallelogramparallelogram whose all angles are right is called…

A squaresquaresquare is a rectanglerectanglerectangle whose all sides are…

A deltoiddeltoiddeltoid is a quadrangle which has a pair of sides…

Every squaresquaresquare is a …, but not every … is a squaresquaresquare.

Procedurem8baa43e4d3c555a0_1528446435040_0Procedure

The teacher informs the students that the aim of the class is consolidation of formulae for the areas of quadrangles and using them in tasks.

The students work in groups making mind maps, which include already introduced formulae for calculating the surface areas of quadrangles.

Group 1 – the areas of a rectanglerectanglerectangle and a squaresquaresquare.

Group 2 – the areas of a parallelogramparallelogramparallelogram and a rhombus.

Group 3 – the area of a trapezoidtrapezoidtrapezoid.

Having finished, the representatives of the groups present their posters.

The following information should appear in the posters.

Poster 1
- The area of a rectanglerectanglerectangle equals the product of the lengths of its sides.

- The area of a square can be calculated in two ways: the length of its side squared, or half of the length of its diagonals squared.m8baa43e4d3c555a0_1527752256679_0The area of a square can be calculated in two ways: the length of its side squared, or half of the length of its diagonals squared.

Poster 2
- The area of a parallelogramparallelogramparallelogram equals the product of the length of its side divided by the length of the altitude perpendicular to this side.

- The area of rhombus can be calculated in two ways: the product of the length of its side divided by its altitude, or half of the product of the lengths of its diagonals.

Poster 3
- The area of a trapezoidtrapezoidtrapezoid equals half of the product of its bases divided by the altitude of the trapezoidtrapezoidtrapezoid. The teacher verifies the students’ information and explains any doubts. The students use the information to solve the tasks.

Task 2
The lengths of the sides of a rectanglerectanglerectangle are expressed with natural numbers and its area equals 20 cmIndeks górny 2². How many such rectangles are there? What are the lengths of their sides?

Answer: there are 3 such rectangles. Their sides are 1 cm and 20 cm, 2 cm and 10 cm, 4 cm and 5 cm.

Task 3
One of the sides of a parallelogramparallelogramparallelogram, whose area equals 28 cmIndeks górny 2², has the length of 3,5 cm. Calculate the altitude of this parallelogramparallelogramparallelogram perpendicular to this side.

Answer: h=8 cm

Working in groups, the students consider if there are other formulae for calculating the areas of quadrangles. In order to do it, they analyse the material shown in the INTERACTIVE PRESENTATION. They formulate hypotheses and conclusions.

INTERACTIVE PRESENTATION - formulae for calculating the areas of quadrangles

The conclusion

The areas of quadrangles can be calculated using the trigonometric function of the acute angles between the sides of the quadrangle( in case of a parallelogram, a rhombus, a deltoid) or between the diagonals (in case of a parallelogram, any quadrangle).m8baa43e4d3c555a0_1527752263647_0The areas of quadrangles can be calculated using the trigonometric function of the acute angles between the sides of the quadrangle( in case of a parallelogram, a rhombus, a deltoid) or between the diagonals (in case of a parallelogram, any quadrangle).

The students use the information to solve the tasks individually.

Task 4
Calculate the area of the rectanglerectanglerectangle whose 12 cm long diagonals intersect at an angle of 30Indeks górny o^o.

Answer: 36 cmIndeks górny 2².

Task 5
Calculate the area of the rhombus whose side is 10 dm long and the sum of the diagonals equals 28 dm.

Answer: 96 dmIndeks górny 2²

Task 6
The area of a trapezoidtrapezoidtrapezoid equals 100 cmIndeks górny 2² and its altitude is 10 cm. Calculate the lengths of the trapezoid’s bases, knowing that one of them is three times longer than the other.

Answer: 5 cm and 15 cm

Task 7
The diagonals of a convex quadrangle have the lengths of 16 cm and 18 cm. The area of this quadrangle equals 72 cmIndeks górny 2². Calculate the measure of the angle of the diagonal intersection.

Answer: 30Indeks górny o^o

Having solved all the tasks, the students present their results. The teacher assesses their work and explains any doubts.

An extra task:
Four corners were cut off a square‑shaped wooden board, whose side length is 4 dm. In this way a regular octagon was formed. Is the sum of the areas of the waste (i.e. the cut off corners) larger than 10% of the surface area of the whole wooden board?

Answer: Yes. It is $3-2\sqrt{2}\approx 0,17$ of the board’s surface.

Lesson summarym8baa43e4d3c555a0_1528450119332_0Lesson summary

The students do the consolidation tasks. They cooperate to formulate the conclusion to memorize.

- The areas of quadrangles can be calculated using the trigonometric function of the acute angles between the sides of the quadrangle( in case of a parallelogramparallelogramparallelogram, a rhombus, a deltoiddeltoiddeltoid) or between the diagonals (in case of a parallelogramparallelogramparallelogram, any quadrangleany quadrangleany quadrangle).

Selected words and expressions used in the lesson plan

altitude of the rhombusaltitude of the rhombusaltitude of the rhombus

angle between the diagonals of the quadrangleangle between the diagonals of the quadrangleangle between the diagonals of the quadrangle

any quadrangleany quadrangleany quadrangle

area of the quadranglearea of the quadranglearea of the quadrangle

deltoiddeltoiddeltoid

diagonal of the quadranglediagonal of the quadranglediagonal of the quadrangle

parallelogramparallelogramparallelogram

rectanglerectanglerectangle

squaresquaresquare

trapezoidtrapezoidtrapezoid