Unit 2: Geometric Figures

Transcription

1 Unit 2: Geometric Figures This lesson describes elementary geometric figures as : 1. Circles. 2. Triangles. 3. Quadrilaterals. 4. Polygons.

2 The circle (p. 31) Human brain is very good doing shape recognition. By the end of Kindergarten, kids identify easily the shape circle. In more advanced courses, we want to progress in the study and investigate the relation with other geometry objects. At some point we need a precise definition. Definition: The circle with center C and radius r is the set of points in the plane that are at distance r from point C. An activity to introduce the definition: 1. Divide the class in groups. 2. Ask each kid to draw (using transparent paper) a point C and severay points at a given distance from C (for instance, 6 cm). 3. Put together the drawings of each group (be sure that C coincides).

3 The circle In Primary 4 or Primary 5, students should practice drawing with the compass. Relative position of a circle and a line and relative position of two different circles: disjoint tangents one point in common secants two points in common

4 Triangles (p. 36) As with the rest of geometric figures, we can consider two elementary levels: in Kindergarten and the firsts years of Primary school, triangles are recongnized by their shape. at some point a precise definition is convenient.

5 Levels in Geometry learning (van Hiele) Level 0: visual. Kids notice the general shape and the relation with other objetcs in the environment. Level 1: analytic. Kids understand the properties, but not the relation between them and with the definition. Do not understand definition properties. Make inductive reasoing, but no deductive reasoning. Leveal 2: abstract. Understad definition and properties, but no formal deductive reasoning. Definition of rectangle squares are also rectangles. Level 3: deductive. Correct reasoning, proofs. Level 4: rigorous (abstract geometry).

6 Triangles (p. 36) You can find several definitions (for instance, some of them include the interior, and others do not). It is important to recognize and to be able to deal with a good definition. Definition A region of the plane enclosed by 3 non-collinear points and the line segments joining them. Points are called vertices. Segments are called the side of the triangle (arista).

7 Angles of a triangle Property: The sum of the angles of every triangle is 180. This is a basic geometry fact, and it is important to give arguments that show why it is always true. a b c a + b + c = 180 Let us consider two different arguments: 1. The first one, using materials and easier to understand for young kids:

8 Angles of a triangle 2. The second one, abstract. It is already a proof. Suitable for Primary school? a c b a + b + c = 180 a b An intermediate proposal:

9 Ángulo exterior Angle e in the figure is called exterior angle. Can you find a relation between the measure of e and the angles of the triangle? c e a b

10 Problems 72 Find the measure of angle a? a 39 If we know that AC is the bisector of BAD, what is the measure of BAC. (Points E, A and B are collinear). E A? D C B

11 Triangles and symmetry Symmetry is an easy (intuitive) concept, that will be useful in the study of geometric figures. A line of symmetry divides an object into two equal parts. Somme suggestions in order to understand the concept: construct symmetric objects with paper and scissors. symmetry is everywhere.

12 Symmetry Typical textbook activity: Be careful: lines of symmetry are not necessarily vertical or horizontal. There are figures with more than a line of symmetry. Can you find out a figure with two lines of symmetry? Can you find out a figure with four lines of symmetry? Can you find out a figure with an infinite number of lines of symmetry?

13 A formal definition P r P The point symmetric of P with respect to line r is the point P on the perpendicular line to r through P so that the distance from P to line r is the same distance as P to r.

14 Types of triangles (p. 43) Classified by their angles acute triangle all angles are less than 90 o right triangle one angle is a right angle Cassified by their edges obtuse tringle one angle is greater than 90 o scalene no two equal edges (diferent lengths) isosceles at least two edges have equal length equilateral all edges have equal length

15 Isosceles and equilateral triangles They are defined according to their edge lengths. An isosceles triangle has (at least) a line of symmetry. The angles defined by the base and the the other two sides are equal (the base angles are equal) (los ángulos sobre la base son iguales). The converse is also true: a triangle with two equal angles is an isosceles triangle. An equilateral triangle has three lines of symmetry, thus all three angles are equal. The converse is also true: triangles that have three equal angles are equilateral triangles.

16 Exercises In the figure, AB = AC = BC = CD. Find the value of BDC. A B C? D A Find the value of BCA in the figure. B 40 C